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11 Canonical quantization of classical fields In Sections 5-10 a formalism was developed which allows to describe quantum mechanically interactions of relativistic particles. Starting with the second quantized formulation of quantum mechanics of many-particle systems we have identified in Section 6 the basic properties one-particle states should possess, if the theory is to be Poincaré covariant. States describing many noninteracting particles were constructed as appropriate tensor products of one-particle states; such many-particle states are in one-to one correspondence (also as far as their transformation properties are concerned) with the asymptotic in and out states (introduced in Section 7) of interacting particles. The results of Sections 5 and 6 provided the starting point for for constructing in Sections 8 and 9 interactions of particles and formulating the perturbative expansion satisfying the requirements formulated in Section 7, necessary to obtain relativistically covariant S matrices. The resulting formalism turns out to have many features of quantum field theory but the basic building entities remain particles: their set and transformation properties must be specified from the outset to specify the theory. Here we present another, complementary approach to the formulation of theory of interactions of relativistic particles. It is based on the canonical quantization of relativistic fields. Particles emerge in this formalism as states of quantized fields. The basic entities - at least as far as bosonic degrees of freedom are concerned - are, however, fluctuating quantum fields. The advantage of the field theory approach is twofold: firstly, it provides a well defined prescription for constructing generators of the full Poincaré group (whose structure in terms of the creation and annihilation operators in the approach of Section 6 can only be guessed at) and, secondly, allows for an easy discussion of various internal symmetries (providing through the Noether theorem a recipe for constructing associated conserved charges) and their spontaneous breaking. Moreover, in the case of fields “whose quanta”1 are bosons (integer spin particles) the picture of fluctuating field values at different space-time points (complementary to the description in terms of particles) seems more fundamental; it allows for deeper understanding of global aspects of the theory (such as the role of topologically nontrivial classical field configurations, symmetry breaking etc.) and definitely proves useful in diverse modern applications of quantum field theory (e.g. in cosmology). The picture of fluctuating fields seems however not applicable to quantum fields, “whose quanta” are fermions (half-integer spin particles). While 1 We put the quotation marks bacaus, as will become clear in section 13 the corespondence between the fields entering the Lagrangian and the particle states predicted by a given theory need not be direct, nor one-to-one. 315 it is possible to take as a starting point for such quantum theories the Lagrangian formalism, whose formal aspects are analogous to that of classicla fields, the basic entities in this case are anticommuting generators of abstract Grassmann (or Bieriezin) algebra which have no classical counterpart and can hardly be considered physical. The difference between bosonic and fermionic fields2 becomes particularly clear in the path integral approach to field quantization (see Section 16). While quantizing in this way bosonic fields one integrates functionally (over (that is performs summation of contributions of) all possible classically realisable (and measurable) field configurations, for fermionic fields one uses for this purposes an operation formally defined for Grassmann algebra generators only formal aspects of which resemble the integration. Thus no “classical fermionic fields can classically be measurable and the only physical picture that in quantum field theory can be associated with fermions is that provided in Sections 5 and 6. Quantization of classical fermion fields is only a mathematical trick allowing to easily reproduce the results of the formalism presented there. In particular it cannot be viewed (as is sometimes presented in older textbooks) as “quantization of the wave function satisfying the Dirac equation”. We begin this Section with a brief review of the classical theory of relativistic fields and their symmetries and then proceed to their quantization. As the next step we show, that quantum description of noninteracting fields, whose Lagrangians consist of terms at most bilinear in field variables, leads to the interpretation of the corresponding Hamiltonian eigenstates in terms of (noninteracting) particles. In principle, quantization of classical fields is very similar to the quantization of the system of many coupled harmonic oscillators discussed in Section 5.5: the field value at a given point x in space can be treated as an oscillator coupled to other oscillators in neighbouring points (in particular, the common features of both systems: of quantized coupled anharmonic oscillators and of interacting quantized fields, is the nonconservation of the numbers of phonons and particles, respectively). However, despite these deep similarities, quantization of some classical fields (including the electromagnetic one) has some subtle points which require special treatment and we devote separate subsections to these problems. The formalism allowing to treat fermionic fields on formally equal footing with bosonic ones is given in section 11.8. After quantizing various types of fields we perform transition to the so-called interaction picture which for theories of interacting fields (with more complicated Lagrangians) allows to compute the S-matrix 2 The reason for this sharp difference between bosonic and fermionic fields seems to be that in the classical ~ → 0 limit bosonic quantum fields go over into true, measurable (at least in principle) classical fields of forces (like the electromagnetic forces, and various potential forces) whereas fermionic fields become matter particles described first, at low energies, by the Schrödinger (or Pauli) equation and ultimately, for ~ → 0, by the classical mechanics. 316 elements using the methods developed in Sections 7 and 9. 11.1 Action and the Noether theorem Dynamics of a system of classical fields3 φi is determined by the action I[φ] which is a functional over all differentiable space-time field configurations. Actions considered in connection with quantum field theory are given in terms of Lagrangians L obtained as space integrals of local Lagrangian densities Z t1 Z Z t1 3 I[φ] = dt d x L(φ, ∂µ φ) ≡ dt L . (11.1) t0 V t0 The Lagrangian density L is a function of fields φi , i = 1, . . . , N and their space-time derivatives. In the following we consider Lagrangian densities depending on first order derivatives only. The volume V may be finite or infinite; in the latter case one assumes that all the fields φi vanish sufficiently fast at spatial infinity; in the finite volume V some spatial boundary conditions must be specified. The action I[φ] must be a real and - for relativistic field theories - Poincaré invariant quantity. In general the fields φi transform nontrivially (as some, in general reducible, representation of the SO(1, 3) group or, in the case of fermionic fields to be discussed in section 11.8, of its universal covering SL(2, C)) under changes of the inertial frame and can also transform nontrivially under some internal symmetries. Equations of motion of a system of fields are obtained by requiring that for the true field configurations the action (11.1) is stationary (i.e. it does not change to first order in δφi (x)) with respect to substitutions φi (x) → φi (x) + δφi (x), where the variations δφi (x) are assumed to vanish for t = t1 and t = t2 as well as for |x| → ∞ (if the volume V in (11.1) is finite, δφi (x) are assumed to vanish at the boundaries). Concisely this requirement is written as δI = 0. For Lagrangian densities which depend only on fields and their first derivatives the variation δI of the action due to any variations δφi (x) of fields (not necessarily subject to some specific boundary conditions) is4 Z ∂L ∂L 4 δI = d x δφi + δ(∂ν φi ) (11.2) ∂φi ∂(∂ν φi ) 3 The formalism presented in this section carries over unmodified to fermionic fields discussed in section 11.8 provided all derivatives of the Lagrangian are treated as right derivatives and the variations stand to the right of these derivatives. 4 Generalization to Lagrangian densities depending on higher field derivatives is straightforward but r equires assuming appropriate boundary conditions also for derivatives of field variations. However the Hamilton’s formalism presented here, which is important for formulating transition to the quantum theory (without appealing to functional integrals which will be introduced in Section 16), is adapted to Lagrangian densities depending on first derivatives only. It can be extended to theories whose Lagrangians depend on higher derivatives using the formalism of constraints presented in section 11.6 317 (summation over the index i is understood). In the following we will consider variations φi (x) → φi (x) + δφi (x) for which δ(∂ν φi ) = ∂ν (δφi ). The variation δI can be then rewritten as Z ∂L ∂L ∂L 4 δφi + ∂ν − ∂ν δφi . (11.3) δI = d x ∂φi ∂(∂ν φi ) ∂(∂ν φi ) Due to the boundary conditions imposed on the variations δφi considered in determination of the true field configurations, the second, surface term in (11.3) vanishes and the condition δI = 0 leads to the Euler-Lagrange equations of motions: ∂ν ∂L(φ, ∂φ) ∂L(φ, ∂φ) = 0. − ∂(∂ν φi ) ∂φi (11.4) It is important to remember that Lagrangian densities which differ from each other by a total derivative of an arbitrary function of fields5 L′ = L + ∂µ X µ (φ) , (11.5) give the same classical equations of motion (due to vanishing of the field variations on the boundary). Canonical quantization of a system of fields requires6 going over to the Hamilton’s formalism. To this end one defines the momenta Πi (t, x) canonically conjugated to the field variables φi (t, x) by Πi (t, x) = ∂L δL ≡ , ∂(∂0 φi (t, x)) δ φ̇i (t, x) and forms the Hamiltonian density X H(x) = Πi (x) φ̇i (x) − L(x) , (11.6) (11.7) i and the Hamiltonian H= Z d3 x H(t, x) . (11.8) We have assumed here that the momenta Πi and the field variables φi are not subject to any constraints and that the relations (11.6) can be inverted to give φ̇i (x) = φ̇i (Π(x), φ(x)) , 5 (11.9) For Lagrangian densities depending only on fields and their first field derivatives X µ can depend also on field derivatives but only in such a way that its variation vanishes for vanishing variations δφi of fields without imposing any conditions on variations of field derivatives. 6 In the alternative approach based on path integrals (see Section 16) fields are quantized using directly the action I[φ]. 318 where the dependence on φ means possibly also a dependence on ∂i φ(x). As we will see, in physically interesting cases this assumption is not always true and we will have to consider appropriate modifications of the canonical formalism. If no such problems arise, the equations of motions in the Hamilton’s formalism read φ̇i (t, x) = {φi (t, x), H}PB , Π̇i (t, x) = {Πi (t, x), H}PB , (11.10) where the Poisson bracket (PB) of any two functionals F [φ, Π] and G[φ, Π] is defined as XZ δF [φ, Π] δG[φ, Π] 3 F [φ, Π], G[φ, Π] ≡ dx δφi (t, x) δΠi (t, x) PB i δG[φ, Π] δF [φ, Π] , (11.11) − δφi (t, x) δΠi (t, x) so that {φi (t, x), Πj (t, y)}PB = δij δ (3) (x − y) , {φi (t, x), φj (t, y)}PB = {Πi (t, x), Πj (t, y)}PB = 0 . (11.12) If the canonical variables φi and Πi are not subject to any constraints, the canonical equations of motion (11.10) are exactly equivalent to the original Euler-Lagrange equations (11.4). Formulation of the field dynamics in terms of the action I allows to easily identify symmetries of the field equations of motion and, via the Noether theorem, of the corresponding conserved quantities. We begin with the case of symmetries which do not affect space-time coordinates (these can be ordinary global symmetries or, in the case of theories involving fermionic fields, rigid supersymmetries). We consider first a change of the field variables φi (x) → φ′i (x) = φi (x) + δ0 φi (x) . (11.13) L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) . (11.14) In the new field variables φ′i (x) the dynamics is described by a new Lagrangian density L′ depending on the fields φ′i , which can be chosen so that7 7 This is just as in classical mechanics in which one is allowed to use any set of dynamical variables, qi (t) or qi′ (t) = qi′ (q(t), t), to describe a given system. The equations of motion in the new variables follow then from the new Lagrangian L′ d ∂L′ (q ′ , q̇ ′ , t) ∂L′ (q ′ , q̇ ′ , t) = . dt ∂ q̇i′ ∂qi′ Because Lagrangian has physically a well defined interpretation (e.g. L = T − V in nonrelativistic mechanics), the new Lagrangian L′ (q ′ , q̇ ′ , t) is obtained just by inverting the relations qi′ = qi′ (q(t), t) and inserting qi (t) = qi (q ′ (t), t) into the original Lagrangian: L′ (q ′ (q), q̇ ′ (q), t) = L(q(q ′ (t), t), q̇(q ′ (t), t), t). 319 In this case ′ I −I ≡ Z 4 ′ ′ ′ d x L (φ , ∂µ φ ) − Z d4 x L(φ, ∂µ φ) ≡ 0 , (11.15) i.e. I ′ = I. Consequently, if I is stationary for a configuration φi (x) of the original fields, that is, if φi (x) satisfy the equations (11.4), then I ′ is stationary for φ′i (x) related to φi (x), by (11.13), i.e. the configuration φ′i (x) of the primed fields satisfy the equations ∂ν ∂L′ (φ′ , ∂φ′ ) ∂L′ (φ′ , ∂φ′ ) = 0, − ∂(∂ν φ′i ) ∂φ′i (11.16) which are in general of different form than the equations (11.4) because L′ (· , ·) is in general a different function of its arguments than L(· , ·). The choice of L′ satisfying the condition (11.14) is not the only possibility: any L′ such that L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) + ∂µ X µ (φ) , is equally good. In this case, instead of (11.15), one has Z ′ I = I + d4 x ∂µ X µ (φ) , (11.17) (11.18) but still, if I is stationary for a configuration φi (x), then I ′ is such for φ′i (x) R 4 related to φi (x) by (11.13) because δ d x ∂µ X µ (φ) = 0 for variations δφi vanishing for t1 and t2 and for |x| → ∞. It is important to realize, that we are not yet speaking of a symmetry of the field equations of motion, but rather about the well known fact that in the Lagrangian formalism a given system can be described by arbitrarily chosen dynamical variables. A change of variables8 φi → φ′i (φi ) is a symmetry of the equations of motion if L′ (· , ·) leading to (11.17) is such that L′ (· , ·) = L(· , ·) + ∂ µ Xµ′ (·) , (11.19) or, using the freedom to redefine L′ by subtracting from it ∂ µ Xµ′ , if for L′ (·, ·) leading to (11.17) one can just take L(· , ·), because then the equations 8 Mathematically speaking,the fields φi (x), i = 1, . . . , N define mappings from the space-time into some N -dimensional “target” space T (N ) , called by physicists the internal space. Therefore φi should be viewed as coordinates of a point in T (N ) onto which the space-time point characterized by the coordinates xµ is mapped. A change of variables φi → φ′i can be due to changing the coordinate system of T (N ), in which case we have to do with a passive transformation, or due to considering a transformed system (active transformation) - cf. Section 4.1. 320 of motion (11.4) satisfied by the fields configuration φi (x) for which I is stationary have the same form as the equations of motion (11.16) satisfied by the configuration φ′i (x) for which I ′ is stationary. In other words, φi (x) and φ′i (x) are both solutions of the same Euler-Lagrange equations. Thus, the condition that the change φi → φ′i (φi ) is a symmetry reads L(φ′ , ∂φ′ ) = L(φ, ∂φ) + ∂µ X µ (φ) . (11.20) Considering now an infinitesimal symmetry transformation (in the sense specified above) φi → φi + δ0 φi , we write L(φ′ , ∂φ′ ) = L(φ + δ0 φ, ∂φ + δ0 ∂φ) = L(φ, ∂φ) + δL(φ, ∂φ) , (11.21) with δL of first order in δ0 φi . Using then (11.3) as well as the condition (11.20) combined with (11.18) we can write the equality9 Z 0 ≡ d4 x [L(φ′, ∂φ′ ) − L(φ, ∂φ) − ∂µ X µ (φ)] (11.22) Z ∂L ∂L ∂L 4 µ = dx δ0 φi + ∂µ − ∂µ δ0 φi − δX (φ) + . . . . ∂φi ∂(∂µ φi ) ∂(∂µ φi ) The first bracket in the above formula vanishes for the fields φi satisfying the equations of motion (11.4). Thus, for such a fields configuration φi (x) the quantity jµ = ∂L δ0 φi − δXµ (φ) , ∂(∂ µ φi ) (11.23) is conserved, i.e. satisfies the equation ∂ µ jµ = 0 , (11.24) because for a symmetry transformation (11.22) is identically zero, and no special requirements on the values at t = t1 and t = t2 of the field changes δ0 φi (x) due to the symmetry transformations are imposed (of course, δ0 φi (x) must vanish at spatial infinity or, in the finite volume, should not change the boundary conditions satisfied by the fields).10 If the infinitesimal changes δ0 φi are parametrized by n independent infinitesimal constant parameters θa 9 This reasoning can easily be generalized to Lagrangian densities depending on higher field derivatives. 10 For this reason the second term on the right hand side of (11.18) is nonvanishing - it receives contributions from the t = t1 and t = t2 hypersufaces. Notice that the situation is quite different in the case of theories formulated in the Euclidean space with coordinates x̄µ : because in this case fields and, therefore, also the symmetry changes δ0 φi are bound to vanish for |x̄| → ∞ in all directions, from (11.22) one cannot infer the existence of conserved currents and, hence, there are no conserved charges (11.27). 321 (in theories with fields taking values in Grassmann algebras the parameters θa can be anticommuting Grassmann variables), δ0 φi = θa δ0a φi , (11.25) there are n conserved Noether currents jµa = ∂L δ a φi − Xµa (φ) , ∂(∂ µ φi ) 0 a = 1, . . . , n, (11.26) (for (11.25) δX µ must also be of first order in the transformation parameters, i.e. δXµ (φ) = θa Xµa (φ)). The Noether charges Qa given by Z a Q = d3 x j0a (t, x) , a = 1, . . . , n, (11.27) are then time independent (conserved) quantities provided the fields fall off sufficiently rapidly at spatial infinity (or, in a finite volume, satisfy the appropriate boundary conditions). In most cases the transformations (11.25) are linear11 in the fields φi θa δ0a φi = −iθa Tija φj , (11.28) (summations over a and j are understood) with T a being a set of Hermitian matrices - a matrix representation of the generators of a symmetry group G - forming a basis of a representation of the dimension n Lie algebra of G (see Section 4). The matrix generators T a satisfy the commutation rule a T , T b = iT c fc ab , (11.29) with some structure constants fc ab and will be assumed to be normalized by the condition tr(T a T b ) = 21 δ ab . The transformations (11.13) are then infinitesimal forms of finite field transformations. For such internal symmetries12 Xµa ≡ 0 and it is then easy to see that in the canonical Hamilton’s formalism the Noether charges Qa generate via Poisson brackets the infinitesimal symmetry transformations (11.28) of field variables. Indeed, the time component of the current is then simply j0a (t, x) = −iΠi (t, x) Tija φj (t, x) , 11 (11.30) However, symmetries realized on fields nonlinearly also play an important role, mainly in so-called effective quantum field theories, e.g. in the chiral Lagrangian description of low energy strong interactions of mesons. 12 More generally, whenever X µ (φ) cannot be removed by a suitable redefinition of L by a total four-divergence, we have to do with a space-time symmetry (for example, X µ (φ) cannot be removed for supersymmetric transformations); otherwise we speak about internal symmetries. 322 and using (11.12) one gets θa {Qa , φi (t, x)}PB = iθa Tija φj (t, x) , (11.31) so that φ′i (x) ≈ φi (x) − θa {Qa , φi (x)}PB . In order to discuss space-time transformations (like translations, rotations or Lorentz boosts) the formalism has to be generalized. An infinitesimal such a transformation xµ → x′µ x′µ = xµ + δxµ (x) = xµ + θa δ a xµ (x) , (11.32) should be accompanied by a corresponding transformation φi (x′ ) → φi (x′ ) of the fields:13 φ′i (x′ ) = φi (x) + δφi (x) = φi (x) + θa δ a φi (x) . (11.33) For example, for an infinitesimal transformation xµ → x′µ with x′µ = xµ + ω µν xν − ǫµ , (11.34) corresponding to a change of the reference frame, the fields φi (x) transform under some regular (in general reducible) matrix representation (J µν )ij of the Lorentz group14 φ′i (x′ ) = φi (x) − i ωµν (J µν )ij φj (x) . 2 (11.35) One can also consider transformations xµ → x′µ which correspond e.g. to infinitesimal conformal transformations etc. The system of fields can be described in the new space-time coordinates x in terms of the new field variables φ′ (x′ ) if one uses such a new Lagrangian density L′ , that for any two fields configurations φi (x) and φ′i (x′ ) related to each other by (11.32) and (11.33) Z Z ′ 4 ′ ′ ′ ′ ′ ′ ′ d x L (φ (x ), ∂µ φ (x )) − d4 x L(φ(x), ∂µ φ(x)) = 0 , (11.36) I −I = ′ Ω′ Ω where ∂µ′ denotes the derivative with respect to x′µ and Ω′ is the image of Ω under the change of variables (11.32). (11.36) is a sufficient condition for the new fields φ′ (x′ ) obtained via (11.33) from the solutions φ(x) of the 13 This means that the change of the space-time coordinate system entails a related coordinate change in the internal (target) space. 14 λν µ The simplest nontrivial is the vector representation with Jvec given by (B.3) κ acting on vector fields φi ≡ V κ , but higher tensor or spinorial representations can also be considered. 323 equations of motion (11.4) following from L to be solutions of the equations of motion following from L′ . However as previously, the same conclusion for φ′i (x′ ) follows if L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) is chosen such that Z ′ I = I + d4 x ∂µ X µ (φ) . (11.37) Ω for some four-vector function X µ (φ). Again, one speaks about a space-time symmetry, if (by an appropriate choice of the factor Xµ′ (φ′ )) for the new Lagrangian density L′ (·, ·) leading to (11.37) one can take the original L(·, ·): L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) = L(φ′(x′ ), ∂µ′ φ′ (x′ )) , (11.38) because then the equations of motion satisfied by φ′i (x′ ) (in the space-time coordinates x′µ ) are the same as the equations of motion (in xµ ) satisfied by φi (x). Taking into account (11.38), the condition (11.37) can be written in the form d4 x′ L(φ′ (x′ ), ∂µ′ φ′ (x′ )) = d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ(x))] , (11.39) (d4 x′ 6= d4 x, i.e. det(∂x′ /∂x) 6= 1 for a general transformation (11.32)). In a relativistic field theory, the transformations (11.34) accompanied by (11.35) should be symmetries in the above sense. For such transformations det(∂x′ /∂x) = 1 and the condition (11.39) takes the form similar to (11.20), except for different space-time coordinates on both sides. Conserved quantities corresponding to symmetry transformations can be found using the condition (11.37). To this end it is technically convenient to split the total change of the field φi (x) as follows δφi (x) ≡ φ′i (x′ ) − φi (x) = φ′i (x′ ) − φ′i (x) + φ′i (x) − φi (x) ≡ δxµ ∂µ φi (x) + δ0 φi (x) , (to the first order in the transformation parameters θa the fields φ′i have been replaced by φi in the term with δxµ ) and to work with the functional changes of fields δ0 φi (x) = φ′i (x) − φi (x) for which δ0 (∂µ φi (x)) = ∂µ (δ0 φi (x)) . (11.40) Thus, to the first order in the parameters θa , we write the transformed fields and their derivatives as15 φ′i (x′ ) = φi (x) + δ0 φi (x) + δxλ ∂λ φi (x) + O(θ2 ) , 15 From (11.34) it follows that ∂x′λ /∂xκ = δκλ + ∂(δxλ )/∂xκ , and ∂xλ /∂x′µ = δµλ − ∂(δxλ )/∂xµ . Thus, ∂µ′ = ∂µ − (∂µ (δxλ ))∂λ and the terms with derivatives of δxλ cancel out in (11.41). 324 ∂µ′ φ′i (x′ ) = ∂µ φi (x) + ∂µ (δ0 φi (x)) + δxλ ∂λ ∂µ φi (x) + O(θ2 ) , ∂ν′ ∂µ′ φ′i (x′ ) λ (11.41) 2 = ∂ν ∂µ φi (x) + ∂ν ∂µ (δ0 φi (x)) + δx ∂λ ∂ν ∂µ φi (x) + O(θ ) , etc. This allows to represent L(φ′ (x′ ), ∂µ′ φ′ (x′ )) in the identity (in which x and x′ are connected by (11.32)) Z Z 4 ′ ′ ′ ′ ′ ′ 0≡ d x L(φ (x ), ∂µ φ (x )) − d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ)] , (11.42) Ω′ Ω 16 following from (11.37) in the form L(φ′ (x′ ), ∂µ′ φ′ (x′ )) = L(φ(x), ∂µ φ(x)) + δL , (11.43) ∂L ∂L δ0 φi + ∂µ δ0 φi + δxµ ∂µ L + O(θ2 ) , ∂φi ∂(∂µ φi ) (11.44) ∂L ∂L . + ∂µ ∂λ φi (x) δx ∂µ L ≡ δx ∂µ φi (x) ∂φi ∂(∂λ φi ) (11.45) in which δL = where µ µ The Lagrangian density change (11.44) can also be rewritten as ∂L ∂L ∂L µ δL = δx ∂µ L + δ0 φi + ∂µ − ∂µ δ0 φi . (11.46) ∂φi ∂(∂µ φi ) ∂(∂µ φi ) We then make in the first integral is (11.42) the change of the integration variables d4 x′ → d4 x (upon which Ω′ → Ω) using the Jacobian17 ′ ∂xµ ∂δxµ (x) ∂δxµ ν = det δµ + ≈ 1 + tr = 1 + ∂µ δxµ , (11.47) det ∂xν ∂xν ∂xν so that finally to first order the integrand of the first term of (11.42) becomes d4 x′ L(φ′(x′ ), ∂µ′ φ′ (x′ )) = d4 x′ [L(φ(x), ∂µ φ(x)) + δL] = d4 x L(φ(x), ∂µ φ(x)) + d4 x [δL + L ∂µ δxµ ] . (11.48) For field configurations φi (x) satisfying the equations of motion (11.4) the identity (11.42) then gives: Z ∂L 4 µ µ µ d x L ∂µ δx + δx ∂µ L + ∂µ δ0 φi − ∂µ δX (φ) + O(θ2 ) ∂(∂ φ ) µ i Ω Z ∂L µ 4 µ δ0 φi − δX (φ) + O(θ2 ) = 0 . (11.49) = d x ∂µ L δx + ∂(∂ φ ) µ i Ω 16 We again assume, that L depends only on fields and their first derivatives. Conserved Noether charges corresponding to more complicated Lagrangian densities can be derived using similar methods. 17 We use the relation det(1 + A) = exp{tr ln(1 + A)} ≈ 1 + tr(A). 325 Taking into account the arbitrariness of Ω and reexpressing δ0 φi in terms of δφi ≡ φ′i (x′ ) − φi (x) = δ0 φi + δxµ ∂µ φi we see that conserved is the quantity ∂L ∂L µ µ J = g ρL− ∂ρ φi δxρ + δφi − δX µ (φ) . (11.50) ∂(∂µ φi ) ∂(∂µ φi ) That is, ∂µ J µ (x) = 0 for fields satisfying the equation (11.4). We now consider the conserved currents associated with the Poincaré transformations of the form (11.34). For translations (ωµν = 0 and ǫµ 6= 0) φ′i (x′ ) = φi (x), so that δφi = 0. The quantity given by (if X µ (φ) = 0, which is usually the case) µν Tcan = ∂L ∂ νφi − g µν L , ∂(∂µ φi ) (11.51) called the canonical energy-momentum tensor is then conserved: µν ∂µ Tcan = 0. The four constants of motion (i.e. time independent quantities) Z µ 0µ P = d3 x Tcan (t, x) , (11.52) (11.53) play the role of the total energy P 0 of the system of fields and of its total νµ momentum vector P i . It can be shown that if Tcan is conserved (and only µ then), P given by (11.53) transforms as a true four-vector when the reference frame is changed. In the case of Lagrangian densities which are of the general form 1 L(φi , ∂µ φi ) = ∂µ φi ∂ µ φi − V (φ) , 2 (11.54) where V (φ) is called the field potential, going over to the Hamilton’s formalism, one finds 1 1 00 Tcan = Πi Πi + ∇φi ·∇φi + V (φ) ≡ H , 2 2 0k k Tcan = Πi ∂ φi . (11.55) In this case it is easy to see that the transformations of the field φi (x) corresponding to translations in time and space are generated by Poisson brackets: {P µ , φi (x)}PB = − ∂ µ φi (x) , so that φ′i (x) = φi (x) − {P µ , φi (x)}PB ǫµ . 326 (11.56) The canonical energy-momentum tensor (11.51) is not always symmetric µν in its indices µν. However, if instead of Tcan one uses another tensor µν T µν (x) = Tcan (x) + H µν (x) , (11.57) where H µν (x) is an arbitrary conserved tensor, ∂µ H µν (x) = 0, such that Z d3 x H 0ν (x) = 0 , (11.58) the associated conserved charges P µ obtained from the new tensor T µν are (for fields vanishing sufficiently fast at spatial infinity) the same as the ones µν obtained from Tcan . This is so if one takes H µν (x) = ∂ρ H ρµν (x) , (11.59) where H ρµν (x) is antisymmetric in its ρµ indices. Using this freedom one can µν µν νµ replace Tcan by a symmetric tensor Tsymm = Tsymm . For fields transforming nontrivially under the Lorentz group of particular interest is the Belinfante symmetric tensor obtained with ∂L 1 ρµν (−iJ µν )ij φj H = 2 ∂(∂ρ φi ) ∂L ∂L ρν ρµ − (−iJ )ij φj − (−iJ )ij φj . (11.60) ∂(∂µ φi ) ∂(∂ν φi ) It is the Belinfante symmetric energy-momentum tensor which appears as the right hand side of the Einstein’s equations of General Relativity. Conserved currents associated with the Lorentz transformations (ǫµ = 0, 6= 0 in (11.34)) are derived in a similar way. Using δxµ = ω µν xν and δφi = − 2i ωµν (J µν )ij φj (x) following from (11.35) we get (again for X µ = 0) ω µν µνκ µκ µν Mcan (x) = xν Tcan − xκ Tcan + ∂L (−iJ νκ )ij φj . ∂(∂µ φi ) The conserved (i.e. time independent) charges Z νκ 0νκ J = d3 x Mcan (t, x) , (11.61) (11.62) µνκ are antisymmetric J νκ = J κν . Again, it can be shown that if Mcan is conνκ served, J transform as a true four-dimensional tensor. The spatial components J ij play the if (11.62) role of the total angular momentum of the considered system of fields. For Lagrangian densities of the form (11.54) it 327 is straightforward to check that through the Poisson brackets the tensor J νκ generates Lorentz transformations of fields φi (x). i h (11.63) {J µν , φi (x)}PB = − (xµ ∂ ν − xν ∂ µ )δij + (−iJ µν )ij φj (x) , so that φ′i (x) = φi (x) − 21 ωµν {J µν , φi (x)}PB . It can also be easily shown, that the tensor (11.61) differs by a total four-divergence from the tensor µκ µν M µνκ = xν Tsymm − xκ Tsymm , (11.64) µν in which Tsymm is the Belinfante symmetric energy-momentum tensor obtained from (11.60), which gives therefore the same conserved Noether charges J µν (for field configurations satisfying the equations of motion (11.4) and vanishing at spatial infinity). 11.2 Canonical quantization of a real scalar field In this subsection we describe in details quantization of the simplest example of a relativistic field - of the real scalar field ϕ(x). No special difficulty arises in this case. More complicated cases of vector (in particular of the electromagnetic field) and spinor fields will be discussed in subsequent subsections. We consider first quantization of the noninteracting field ϕ(x) and then outline modifications introduced by interactions. Finally, the results are generalized to the case of many fields. The simplest dynamics of a real field is given (in units in which c = ~ = 1) by the Poincaré invariant Lagrangian density 1 1 L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 . 2 2 (11.65) By a rescaling of the field variable ϕ the first term quadratic in first derivatives, called the kinetic term, can always be brought into the canonical form as above. In the absence in L of higher powers of ϕ (a feature which makes this theory solvable both classically and quantum mechanically) the negative sign of the second term ensures the relativistic relation E 2 (k) = k2 + M 2 and boundedness from below of the Hamiltonian spectrum. To stress the similarity to the system of coupled oscillators we write the corresponding Lagrangian in the more general, apriori nonlocal, form Z Z Z 1 1 3 2 3 d x ϕ̇ (t, x) − d x d3 y ϕ(t, x)K(x, y)ϕ(t, y) , (11.66) L= 2 2 with a symmetric kernel K(x, y) = K(y, x). Formally, the value of ϕ at every point x can be treated as an independent canonical variable Qx (t) = ϕ(t, x). 328 The Euler-Lagrange equation of motion of the system described by (11.66) in this formulation read δL d δL − = 0, dt δ ϕ̇(t, x) δϕ(t, x) (11.67) (where the derivatives are in the functional sense). Applying (11.67) to (11.66) gives Z d2 ϕ(t, x) + d3 y K(x, y)ϕ(t, y) = 0 . (11.68) dt2 For the local Lagrangian density (11.65) the kernel K(x, y) takes (restoring c and ~ for decoration) the form K(x − y) = −c2 ∇2(x) δ (3) (x − y) + M 2 c4 (3) δ (x − y) , ~2 (11.69) so that18 1 L= 2 Z M 2 c4 2 2 2 2 d x ϕ̇ (t, x) − c (∇ϕ(t, x)) − ϕ (t, x) . ~2 3 (11.70) The Euler-Lagrange equation (11.67) is in this case equivalent to the field equation (11.4) and reads 2 M 2 c4 ∂ 2 2 ϕ(t, x) = 0 . (11.71) −c ∇ + ∂t2 ~2 To quantize the system we have first to set up the canonical (Hamilton’s) formalism. We define the momenta Px (t) ≡ Π(t, x) conjugated to the canonical variables Qx (t) = ϕ(t, x): Π(t, x) = δL = ϕ̇(t, x) , δ ϕ̇(t, x) and construct the Hamiltonian Z H = d3 x Π(t, x)ϕ̇(t, x) − L , (11.72) (11.73) in which ϕ̇(t, x) has to be expressed in terms of Π(t, x) and ϕ(t, x): Z Z Z 1 1 3 2 3 d x Π (t, x) + d x d3 y ϕ(t, x)K(x − y) ϕ(t, y) . (11.74) H= 2 2 18 R In units [M ], [T ] and [L] (mass, time and length) the action I = dtL has dimension of ~, that is, [M ][L]2 [T ]−1 . It follows that the field ϕ has dimension [M ]1/2 [L]−1/2 . Furthermore, since Π = ϕ̇, it has dimension [M ]1/2 [L]−1/2 [T ]−1 and the Hamiltonian (11.74) has the right dimension [M ][L]2 [T ]−2 of energy. See also Appendix F. 329 It is easy to check, that the classical Hamilton’s equations δH d ϕ(t, x) = {ϕ(t, x), H}PB = , dt δΠ(t, x) d δH Π(t, x) = {Π(t, x), H}PB = − , dt δϕ(t, x) (11.75) are fully equivalent to the Euler-Lagrange equation (11.4), i.e. to (11.71). Quantization in the Schrödinger picture of a classical system means promoting its canonical variables Qi and Pj taken at one particular instant, usually t = 0, to time independent operators Q̂i and P̂j (acting in some as yet unspecified Hilbert space) satisfying the commutation rules19 h i Q̂i , P̂j = i~ {Qi , Pj }PB = i~ δij (11.76) (and [Q̂i , Q̂j ] = [P̂i , P̂j ] = 0). Following this prescription, in the case of the real scalar field we impose h i ϕ̂(x), Π̂(y) = i~δ (3) (x − y) , h i [ϕ̂(x), ϕ̂(y)] = Π̂(x), Π̂(y) = 0 . (11.77) Upon quantization the Hamiltonian (11.74) also becomes an operator. In principle, in full analogy with ordinary quantum mechanics in position space, in which the system is described by a wave function ψ(Qi , t) (the probability amplitude of finding the system in the classical state characterized by the values Qi of its canonical variables) on which Q̂i ’s act by multiplication by Qi and P̂j ’s act as −i~∂/∂Qj , one can introduce the “wave functional” Ψ[ϕ, t] ≡ hϕ(x)|Ψ(t)i which represents the probability amplitude that the field takes on the configuration ϕ(x). On Ψ[ϕ, t] the operators ϕ̂(x) and Π̂(x) act as ϕ̂(x)Ψ[ϕ, t] = ϕ(x)Ψ[ϕ, t] , Π̂(x)Ψ[ϕ, t] = −i~ δΨ[ϕ, t] . δϕ(x) (11.78) 19 A stronger requirement, that for any pair of classical observables F (Q, P ) and G(Q, P ) such that {F (Q, P ), G(Q, P )}PB = A(Q, P ) in the quantum theory one has h i F (Q̂, P̂ ), G(Q̂, P̂ ) = i~A(Q̂, P̂ ) , cannot in general be imposed because of problems with ordering of operators; it can hold only for observables F (Q, P ) and G(Q, P ) which are at most linear in the canonical variables Q and P . 330 While this approach offers some useful insight (we will return to it in the section devoted to path integrals), it does not lead immediately to the interpretation of field states in terms of particles. Such an interpretation emerges naturally if we find a representation of the ϕ(x) and Π(x) operators (from now on we omit “hats”) in terms of some other operators, so that the Hamiltonian becomes similar to the one of uncoupled harmonic oscillators. To this end we define first new operators (Fourier transforms) ϕ̃(k) and Π̃(k) by Z Z 3 −ik·x ϕ̃(k) = d x ϕ(x) e , Π̃(k) = d3 x Π(x) e−ik·x . (11.79) Since upon quantization real classical field ϕ and real Π go over into Hermitian operators ϕ† (x) = ϕ(x) and Π† (x) = Π(x), their Fourier transforms satisfy ϕ̃† (k) = ϕ̃(−k) , Π̃† (k) = Π̃(−k) . From the commutation relations (11.77) it then follows that h i ′ ϕ̃(k), Π̃(k ) = i~(2π)3 δ (3) (k + k′ ) , h i [ϕ̃(k), ϕ̃(k′ )] = Π̃(k), Π̃(k′ ) = 0 . (11.80) (11.81) Below we consider only translationally invariant kernels K(x, y) = K(x−y). It is then convenient to introduce the quantity ω 2 (k) by Z Z 3 dk 2 2 3 −ik·x ω (k) eik·x . (11.82) ω (k) = d x K(x) e , K(x) = (2π)3 For a real kernel K(x) one has ω 2 (−k) = ω 2∗ (k), and because K(x) = K(−x), ω 2(−k) = ω 2(k). Hence, ω 2 (k) is real. We assume it is positive (this restricts possible forms of the kernel K(x)). For the local K(x) of the form (11.69) corresponding to the Klein-Gordon Lagrangian density (11.65) ω 2 (k) = c2 |k|2 + M 2 c4 . ~2 (11.83) Expressing the Hamiltonian (11.74) in terms of ϕ̃(k), Π̃(k) and ω 2 (k) we obtain Z 3 h i 1 dk 2 H= Π̃(k)Π̃(−k) + ω (k) ϕ̃(k)ϕ̃(−k) . (11.84) 2 (2π)3 By analogy with the harmonic oscillator case we define now the operators r ω(k) i ϕ̃(k) + Π̃(k) , a(k) = 2~ ω(k) r ω(k) i † a (k) = ϕ̃(−k) − Π̃(−k) , (11.85) 2~ ω(k) 331 whose commutation rules follow directly from (11.81): a(k), a† (k′ ) = (2π)3 δ (3) (k − k′ ) , [a(k), a(k′ )] = a† (k), a† (k′ ) = 0 . In terms of a(k) and a† (k) one finds s ~ a(k) + a† (−k) , ϕ̃(k) = 2ω(k) 1 Π̃(k) = i r (11.86) ~ω(k) a(k) − a† (−k) , 2 and inserting these expressions in (11.84) we get Z 3 † 1 d k † H= ~ω(k) a (k)a(k) + a(k)a (k) 2 (2π)3 Z 3 dk 1 † 3 (3) = E(k) a (k)a(k) + (2π) δ (0) , (2π)3 2 (11.87) where E(k) = ~ω(k). The delta function (2π)3 δ (3) (0) in the second term can be interpreted as the volume factor. The term Z 3 Z 3 d k dk 1 1 3 (3) E(k) (2π) δ (0) → V × E(k) , (11.88) 3 2 (2π) 2 (2π)3 can be therefore identified with the contribution to the field energy contained in the volume V of zero point oscillations of infinitely many field modes (numbered by the wave vectors k). This infinite constant contribution can be discarded as long as we do not couple the quantized field to gravity and as long as we do not compare energies of vacuum states of quantized fields subject to different boundary conditions (see Section 11.3). Expressed in terms of the operators a† (k) and a(k), the Schrödinger picture (time independent) field operators ϕ(x) and Π(x) take the form Z 3 s dk ~ ik·x † −ik·x ϕ(x) = a(k) e + a (k) e , (11.89) (2π)3 2ω(k) 1 Π(x) = i Z d3 k (2π)3 r ~ω(k) a(k) eik·x − a† (k) e−ik·x , 2 (11.90) One can then easily check, that with (11.86) the basic commutation rules (11.77) are satisfied. The commutation rules (11.86) of a(k) and a† (k) differ from the rules obtained for their counterparts in the case of the harmonic oscillators in that 332 a(k) and a† (k′ ) commute to the delta function instead of the Kronecker delta. Therefore the standard algebraic arguments cannot be directly used to determine the spectrum of the Hamiltonian (11.87). There are two alternative methods of dealing with this difficulty. Firstly, one can take an arbitrary complete countable set of normalizable functions fα (k) such that Z 3 X dk ∗ f (k)f (k) = δ , fα (k)fα∗ (k′ ) = (2π)3 δ (3) (k − k′ ) , β αβ α 3 (2π) α and define the new operators20 Z 3 d k ∗ aα = f (k) a(k) , (2π)3 α a†α = Z d3 k fα (k) a† (k) , (2π)3 (11.91) such that i h † aα , aβ = δαβ . It is then possible to use the same argument as for ordinary harmonic oscillators that there must exist a vector |0, . . . , 0, . . .i ≡ |Ω0 i annihilated by all aα ’s and to construct the states (a† )nα (a†1 )n1 (a†2 )n2 √ . . . √α . . . |Ω0 i , |n1 , n2 , . . . , nα , . . .i = √ n1 ! n2 ! nα ! (11.92) which span the whole Hilbert space and represent quantum excitations of the field ϕ, which can be given a particle interpretation: the state |n1 , n2 , . . .i is interpreted as the state in which n1 particles are in the first one-particle state characterized by the momentum space wave function f1 (k), n2 particles are in the second one-particle state characterized byf2 (k), etc. Particles can occupy infinitely many one-particle states (in the case of N coupled oscillators there were only N types of phonons corresponding to N possible 1-phonon states). However, except for the state |Ω0 i, the states (11.92) are not eigenstates of the Hamiltonian (although, for the appropriately chosen functions fα (k) they can be made arbitrarily close to the true eigenstates). The second alternative is to enclose the classical system in a box of finite volume V = L3 and to impose periodic boundary conditions on the allowed field configurations ϕ(x). The wave vectors k form then a discrete set: k = (2π/L)n, where n = (nx , ny , nz ) are vectors whose components are integers and the integrals over d3 k are replaced by the sums over discrete k, that is, 20 Similarly as ϕ(x) and Π(x), a† (k) and a(k) are said to be operator-valued distributions; this wise terminology does not shed however much light on physical problems. 333 over (nx , ny , nz ). Periodic boundary conditions21 for the operators ϕ(x) and Π(x) are ensured by the expansions 1 X 1 X ϕ̃k eik·x , Π(x) = √ Π̃k eik·x , ϕ(x) = √ V k V k Z Z 1 1 3 −ik·x d x ϕ(x) e , Π̃k = √ d3 x Π(x) e−ik·x . (11.93) ϕ̃k = √ V V Imposing the relations h i ϕ̃k , Π̃k′ = i~ δ−k,k′ (11.94) etc. ensures that the commutation rules (11.77) are satisfied. The Hamiltonian (11.74) takes then the form X 1 X 1 † 2 H= ~ωk ak ak + Π̃k Π̃−k + ωk ϕ̃k ϕ̃−k = . (11.95) 2 k 2 k The fields can again be written in terms of the creation and annihilation operators s X ~ ak eik·x + a†k e−ik·x , ϕ(x) = 2V ω(k) k r 1 X ~ω(k) Π(x) = (11.96) ak eik·x − a†k e−ik·x , i k 2V satisfying the rules i h ak , a†k′ = δk,k′ , h i [ak , ak′ ] = a†k , a†k′ = 0 . (11.97) One can therefore use the standard reasoning to show that there is a state |Ω0 i annihilated by all ak ’s and that the Hilbert space is spanned by the vectors22 1 |nk1 , nk2 , . . .i ≡ p |k1 , . . . , k1 , k2 , . . . , k2 , . . .i nk1 ! nk2 ! . . . nk nk 1 2 a†k1 a†k2 p = p . . . |Ω0 i , (11.98) nk1 ! nk2 ! 21 The necessary orthogonality and completeness relations read Z X ′ d3 x eix·(k−k ) = V δk,k′ , eik·(x−y) = V δ (3) (x − y) . V k 22 Since the Hamiltonian (11.95) is a sum of Hamiltonians of independent harmonic oscillators (one per each k), one can, using the analogy with the nonrelativistic quantum p̂2 + 21 mω 2 q 2 and the wave function of the ground state mechanics, in which for H = 2m 334 which are interpreted as representing nk1 bosons of momenta k1 , nk2 bosons of momenta k2 , etc. The interpretation of the H eigenstates in terms of noninteracting particles (bosons) given above23 is supported by the statistical properties of the quantized field: To show this let us consider the quantum field enclosed in the box of volume V = L3 in contact with a heat bath of temperature T , and compute the statistical sum Zstat = e−βF , where24 β = 1/kB T and F is the Helmholtz free energy of the Gibbs Canonical Ensemble. This reduces to −βH Zstat = Tr e = ∞ ∞ X X nk1 =0 nk2 =0 = Y k " . . . hnk1 , nk2 , . . . |e−βH |nk1 , nk2 , . . .i e−~ωk /2kB T ∞ X e−~ωk /kB nk =0 T nk # . (11.99) The geometric series can easily be summed and one finds F (T, V ) = −kB T ln Zstat = X 1X ~ωk + kB T ln 1 − e−~ωk /kB T , (11.100) 2 k k or, going over to the continuous normalization, Z 3 d k 1 −~ωk /kB T ~ωk + kB T ln 1 − e , (11.101) F (T, V ) = V (2π)3 2 p with ωk = c2 k2 + c4 M 2 /~2 . Apart from the first term which represents the contribution of the zero point oscillations (which could have been subtracted from the beginning) the temperature dependent part of F is precisely the free energy of a system of nointeracting relativistic bosons (which do not carry any conserved number and whose total number - similarly to the total number of photons - is therefore determined only by the condition of thermal is ψ0 (q) = (mω/π~)1/4 exp −mωq 2 /2~ , represent the vacuum functional of the ground state of the field in the form ! Y ωk 1/4 1 X exp − hϕk |Ω0 i ≡ ΨΩ0 [ϕ] = ωk ϕk ϕ∗k . π~ 2~ k k The operators ϕk act in this representation as multiplication whereas the operators Πk act as derivatives. Wave functionals of particle states can in principle be obtained by acting with a†k in this representation on the vacuum ΨΩ0 [ϕ]. The analogy with ordinary quantum mechanics has to be exploited cautiously, however, because the variables ϕk and Πk are complex here. 23 Notice that having quantized the field in a finite box we cannot appeal to the Poincaré transformation properties of the states. 24 kB = 8.6 × 10−5 eV/K is the Boltzmann constant. 335 equilibrium). Thus, one of the arguments that allow us to interpret states of quantized fields as particles is essentially identical to the statistical argument used by Einstein in 1905 to argue that electromagnetic radiation behaves as a collection of noninteracting particles (photons). Calculation of the partition function of an interacting quantum field is most easily formulated in the path integral approach. Still, presenting that approach in Section 16 we will need the result (11.101) to fix an additive constant in F which in the functional approach is difficult to compute. Fields quantized in the infinite space can be treated similarly. We assume that there is a ground state |Ω0 i which has zero energy (after subtracting the infinite contribution of infinitely many zero mode oscillations) and the other states spanning the Hilbert space are obtained by acting on it with the creation operators a† (k) |ki = a† (k)|Ω0 i , |k1 , k2 i = a† (k1 ) a† (k2 )|Ω0 i , |k1 , k2 , k3 i = a† (k1 ) a† (k2 ) a† (k3 )|Ω0 i , ... (11.102) where all momenta k, k1 , k2 , etc. are arbitrary. (These states are not properly normalized when two or more momenta coincide; this has to be taken care of in the completeness relation as it was done in (5.14)). Although nonnormalizable (if we assume that the state |Ω0 i has the norm equal to 1) these are true eigenstates of the Hamiltonian H (11.87). Their particle interpretation follows now from their Poincaré transfomation properties which we discuss at the end of this subsection. If the Lagrangian defining the theory is not quadratic in the field ϕ but instead of (11.65) takes the more general form (we set c = 1 but for decoration keep ~ in some formulae below) 1 1 1 L = ∂µ ϕ∂ µ ϕ − V (ϕ) = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) , 2 2 2 (11.103) where Hint (ϕ), and therefore also the whole field potential V (ϕ), is usually a polynomial of fields,25 e.g. Hint (ϕ) = (λ/4!)ϕ4 , the quantization in the Schrödinger picture proceeds in the same way as described above: the commutation rules (11.77) of ϕ and Π remain unchanged. The Hamiltonian takes then the form Z H = H0 + Vint = H0 + d3 x Hint (ϕ) , (11.104) 25 We will see that in the presence of Hint (ϕ) the sign of the term quadratic in ϕ can also be positive (negative M 2 ). 336 H0 = Z 1 2 2 1 2 1 2 d x Π + (∇ϕ) + M ϕ . 2 2 2 3 The form (11.104) of H remains valid for Hint (ϕ), or even V (ϕ) explicitly depending on time (e.g. if the quantized field ϕ interacts with some external agents). As in the case of the free field, one can expand the Schrödinger picture operators ϕ(x) and Π(x) as in (11.89) and (11.90) into the operators a(k) and a† (k) with the same commutation rules as previously but the Hamiltonian (11.104) will now contain terms (a(k))4 , (a† (k))4 , etc. and the states (11.102) will not be its eigenstates.26 Still, if the coefficient M 2 does not depend on time, they are eigenstates of the free part H0 of the Hamiltonian (11.104) and will play an important role in the formulation of the S-matrix approach to the scattering theory. This will be discussed in Section 11.9. In older formulations one used to define the quantum theory by taking for the Hamiltonian operator not the expression (11.104), but its normal odered counterpart Ĥ = : H(11.104) :. Normal ordered form : OS : of a Schrödinger picture operator OS is the form, in which all annihilation operators are placed to the right with respect to creation operators; for example (−) (−) (−) (+) : ϕi (x)ϕj (x) : = ϕi (x)ϕj (x) + ϕi (x)ϕj (x) (−) (+) (+) (+) +ϕj (x)ϕi (x) + ϕi (x)ϕj (x) , for bosonic field operators (i.e. ones satisfying commutation relations) and (−) (+) : ψα (x)ψβ (x) : = ψα(−) (x)ψβ (x) + ψα(−) (x)ψβ (x) (−) (+) −ψβ (x)ψα(+) (x) + ψα(+) (x)ψβ (x) , for fermionic operators (i.e. ones satisfying anticommutation relations -see section 11.8), where ϕ(−) , (ϕ(+) ) is the part of (11.89) with the creation (annihilation) operator. Operators ordered in this way have all (except for the unit operator) zero expectation value in the Fock state |Ω0 i annihilated (by definition) by all the annihilation operators. This prescription for Ĥ removes some of the infinities encountered in practical calculations, in particular it removes the addtive infinite part (11.88) in the expectation value of the free field Hamiltonian (11.87) in the state |Ω0 i. Normal ordering, ubiquitus on older approaches to quantum field theory, has been now largely abandoned. First of all, it does not remove all divergences which must be renormalized anyway (see Section 14) and there is no point to invoke two different prescription for removing divergences having a common ultraviolet origin. Furthermore, normal ordering of operators defined in terms of the creation and annihilation operators diagonalizing H0 (i.e. normal ordering with respect 26 Moreover, if Hint depends explicitly on time, the Hamiltonian H does not, strictly speaking, possess time-independent eigenstates. 337 to the Fock state |Ω0 i) does not imply that hΩ− | : OS : |Ω+ i = 0 - matrix elements can have nonperturbative contributions. Finally, and probably most importantly, in the modern functional approach (see Section 16) quantum field theory is viewed as a formalism allowing to take into account real quantum fluctuations of fields; from this point of view every quantum field theory is defined with Fourier momenta bounded by an ultraviolet cutoff Λ which should have a real physical meaning. All contributions to amplitudes, expectation values, etc. one computes using the quantum field theory formalism are then equally physical and should not be subtracted using an arbitrarily defined prescription. The divegences (arising for Λ → ∞) disappear however if the computed quantities are expressed in terms of some other measurable quantities; subtracting divergences is then merely done for computational convenience and has no fundamental meaning; hence defining the Hamiltonian as a normal ordered operator becomes then a completely useless and obsolete prescription.27 We have quantized the classical real scalar field ϕ in the Schrödinger picture in which the operators ϕ(x) and Π(x) satisfying the rules (11.77) do not depend on time. Transition to the Heisenberg picture, in which the states of the system do not vary with time whereas the operators do, is usually convenient, particularly in relativistic field theories. In general (see Section 1), this is achieved by choosing the moment t0 at which the two pictures coincide and defining Heisenberg picture operators OH (t, x) corresponding to Schrödinger picture operators OS (t, x) = O(t, ϕ(x), Π(x)) by the formula OH (t, x) = U † (t, t0 )OS (t0 , x)U(t, t0 ) = O(t, ϕH (t, x), ΠH (t, x)) , (11.105) in which i U(t, t0 ) = T exp − ~ Z t t0 ′ ′ dt H(t ) , (11.106) is the Schrödinger picture evolution operator. By construction the operators OH (t, x) satisfy then the Heisenberg equation (see Sec. 1.1) 1 ∂O d QH (t, x) = [OH (t, x), HH (t)] + dt i~ ∂t H 1 † ∂O ≡ , (11.107) U (t, t0 ) [OS (t0 , x), H(t0 )] U(t, t0 ) + i~ ∂t H in which HH (t) ≡ U † (t, t0 )H(t0 )U(t, t0 ). (Most of the Schrödinger picture operators OS do not depend on time and the second term in (11.107) is 27 Normal ordering is therefore only relevant for the Wick theorem (13.12) used to set the perturbative expansion. 338 absent; one notable exception is the boost generators given by space integrals (11.62) of M 00i given by (11.61)). It is easy to check that the Heisenberg picture operators ϕH (t, x) and ΠH (t, x), corresponding to the canonical variables and obtained from the prescription (11.105), satisfy the canonical equal time commutation rules [ϕH (t, x), ΠH (t, y)] = iδ (3) (x − y) , [ϕH (t, x), ϕH (t, y)] = [ΠH (t, x), ΠH (t, y)] = 0 , (11.108) for arbitrary time t. These rules together with the Hamiltonian H (following from the action I) are in fact the basic relations defining the quantum version of the classical field theory model. Furthermore, for H of the form (11.104) (even if V (ϕ) in the Lagrangian density (11.103) depends explicitly on time, i.e. for a system interacting with external agents) the Heisenberg equations (11.107) satisfied by ϕH and ΠH , d ϕH (t, x) = dt d ΠH (t, x) = dt 1 [ϕH (t, x), HH (t)] , i~ 1 [ΠH (t, x), HH (t)] , i~ (11.109) ′ yield ϕ̇H = ΠH and Π̇H = c2 ∇2 ϕH − M 2 ϕH − Hint (ϕH ). As a result, the Heisenberg operator ϕH satisfies the “classical” field equation of motion 2 ∂ 2 ′ 2 (ϕH (t, x)) , (11.110) − ∇ + M ϕH (t, x) = −Hint 2 ∂t while the operator ΠH satifies the equation 2 ∂ 2 2 ′′ − ∇ + M + Hint (ϕH (t, x)) ΠH (t, x) = 0 . ∂t2 (11.111) Together with the equal time commutation rules (11.108) these two equations can also constitute the complete specification of the theory. Of course, for closed systems, which the system of fields usually are, H is independent of time. The evolution operator U(t, t0 ) (11.106) reduces then to the ordinary exponent, HH (t) = H, so that in the considered theory (11.103) it takes the form Z 1 2 1 3 2 H = d x ΠH (t, x) + (∇ϕH ) (t, x) + V (ϕH (t, x)) , (11.112) 2 2 formally the same as in the classical theory, and choosing t0 = 0 as it is customary, one has ϕH (t, x) = eiHt/~ ϕ(x) e−iHt/~ , ΠH (t, x) = eiHt/~ Π(x) e−iHt/~ . 339 (11.113) For the free field whose Hamiltonian H (11.74) can be represented in the form (11.87), the equations (11.109) can be easily solved. ϕH (t, x) and ΠH (t, x) are then given by the expressions (11.89) and (11.90) but with a(k) and a† (k) replaced by aH (t, k) and a†H (t, k) satisfying the equations 1 d aH (t, k) = [aH (t, k), H] , dt i~ d † 1 † aH (t, k) = [a (t, k), H] , dt i~ H (11.114) with the initial conditions aH (0, k) = a(k), a†H (0, k) = a† (k). One then finds [aH (t, k), H] = ~ω(k)aH (t, k) and [a†H (t, k), H] = −~ω(k)a†H (t, k). The equations (11.114) can be then easily integrated28 to give aH (t, k) = e−iω(k)t a(k) , a†H (t, k) = eiω(k)t a† (k) , (11.115) so that ϕH (t, x) = Z 1 ΠH (t, x) = i Z d3 k (2π)3 s d3 k (2π)3 ~ a(k) e−iω(k)t+ik·x + a† (k) eiω(k)t−ik·x , (11.116) 2ω(k) r ~ω(k) a(k) e−iω(k)t+ik·x − a† (k) eiω(k)t−ik·x ,(11.117) 2 that is, the exponents in the Heisenberg operators depend on the Lorentz invariant products xµ kµ , where xµ = (ct, x) and k µ = (ω(k)/c, k). Both field operators, ϕH and ΠH , satisfy in this case the free Klein-Gordon equation. It is also good to remark that in certain cases (the most notable being the one of the free electromagnetic field quantized using the approach of Gupta and Bleuler discussed in section 11.11 with the gauge fixing parameter ξ 6= 1) even if the Hamiltonian is quadratic in field variables and conjugated momenta their expansions analogous to (11.89), (11.90) are not easy to find and do not render the Hamiltonian diagonal; the time dependendent operators (11.113) are then not simply given by replacing in (11.89), (11.90) e±ik·x by e∓ikx as in (11.116), (11.117). Still, the explicit form of the time dependent operators and their momenta can be found by directly solving (though not so easily) the canonical equations (11.109). 28 Notice, that if the Lagrangian contains terms with ϕ in powers higher than the second, the Heisenberg equations resulting from (11.114) are more complicated and their solution cannot be found in a closed form. Hence, the Heisenberg picture operators ϕH (x) and ΠH (x) corresponding to the interacting field cannot be written in the forms (11.116) and (11.117); only ϕH (0, x) ≡ ϕ(0, x) and ΠH (0, x) ≡ Π(0, x) (recall, we have chosen to equate the Schrödinger and Heisenberg pictures at t0 = 0) can be written as in (11.89) and (11.90). 340 At this point, in order to simplify the notation, we set also ~ = 1 and drop the subscript H on Heisenberg picture operators. In addition, it is convenient to change thep normalization of the thepcreation and annihilation operators a(k) → a(k)/ 2E(k), a† (k) → a† (k)/ 2E(k), where E(k) = ~ωk , so that their commutator becomes a(k), a† (k′ ) = 2E(k)(2π)3 δ (3) (k − k′ ) ≡ δΓ (k − k′ ), (11.118) and the (Heisenberg picture) free field operators (11.116), (11.117) take the simple form Z ϕ(x) = dΓk a(k) e−ik·x + a† (k) eik·x , (11.119) 1 Π(x) = i Z dΓk Ek a(k) e−ik·x − a† (k) eik·x , (11.120) where the measure dΓk ≡ d3 k/(2π)3 2E(k) is separately Lorentz invariant. In theories of interacting fields (when in the full Hamiltonian there are terms with powers fields higher than the second) the time dependent operators termed the interaction picture operators obtained by solving the equations (11.109) but with time independent H0 replacing the full HH (t) and hence given in closed form by (11.113) again with H0 instead of H, play an important role in formulating the perturbation expansion for computing S-matrix elements. This will be discussed in section 11.9. We can now discuss the Poincaré transformations. Using the rules given in Section 11.1 one finds the canonical energy-momentum tensor corresponding to the Lagrangian density (11.103) 1 2 2 µν µ ν µν 1 λ Tcan = ∂ ϕ∂ ϕ − g ∂ ϕ∂λ ϕ − M ϕ − Hint (ϕ) . (11.121) 2 2 It is in this case automatically symmetric.29 In the quantum theory ∂ 0 ϕ must of course be expressed in terms of the canonical variables ϕ and Π. One then has (we drop the subscript H) 1 2 Π (t, x) + ∇ϕ(t, x) ·∇ϕ(t, x) + M 2 ϕ2 (t, x) + Hint (ϕ(t, x)) , 2 0i Tcan (t, x) = Π(t, x) ∂ i ϕ(t, x) . (11.122) 00 Tcan (t, x) = 29 µν Still, as it turns out, even in this simple case Tcan must be modified by adding to it µν ρµν a tensor H = ∂ρ H as in (11.57), in order to ensure finiteness of matrix elements of the energy-momentum tensor (treated as an operator) in the theory of interacting fields. 341 00 P 0 given by the integral of Tcan over d3 x is just the Hamiltonian (11.104) written in terms of the Heisenberg field operators whereas the momentum operator is given by Z Z i 3 0i P = d x Tcan (t, x) = d3 x Π(t, x)∂ i ϕ(t, x) . (11.123) µν From Tcan the generators of the Lorentz transformations are obtained according to the formulae (11.61) and (11.62): Z µν 0ν 0µ J = d3 x xµ Tcan − xν Tcan . (11.124) Using the canonical commutation rules (11.77) these generators can be shown to satisfy the commutation rules of the Poincaré algebra (6.19) or (6.21). Moreover, P µ and J µν generate transformations of the field operator ϕ(x) related to changes of the reference frame µ µ eiaµ P ϕ(x) e−iaµ P = ϕ(x + a) , i µν i e 2 ωµν J ϕ(x) e− 2 ωµν J µν = ϕ(Λ−1 (ω) ·x) , (11.125) or, in the infinitesimal form, i [P µ , ϕ(x)] = ∂ µ ϕ(x) , i [J µν , ϕ(x)] = (xµ ∂ ν − xν ∂ µ )ϕ(x) . (11.126) It is important to stress that the operators P i , H, J i and K i satisfy the commutation rules of the Poincaré algebra and generate the field transformations as in (11.125) only by virtue of the canonical commutation relations (11.108), independently of the form of Hint (ϕ) in (11.103), i.e. also in the quantum theory of the interacting field ϕ. The Poincaré symmetry generators P i , H, J i and K i are by construction independent of time. This can be checked by using the Heisenberg equation (11.107). As such they can be computed for any time t. A particularly convenient (for checking their commutation relations) is the choice of t = 0, for which the Heisenberg picture operators ϕH and ΠH can be expanded into the creation and annihilation operators. For V (ϕ) = 21 M 2 ϕ2 , i.e. in the quantum theory of the free field (with Hint = 0), H takes in terms of the creation and annihilation operators the form (11.87) and the generators P i and J ij and K i = J 0i are given by Z i P = dΓk k i a† (k)a(k) , Z ∂ ij † i ∂ a(k) , (11.127) − kj J = i dΓk a (k) k ∂kj ∂ki Z ∂ K i = i dΓk a† (k) Ek a(k) . ∂ki 342 In this case, the states (11.102) are the eigenstates of H and P with the eigenvalues (Ek , ~k), (Ek1√+ Ek2 , ~k1 + ~k2 ), etc., where Ek is given by the relativistic formula Ek = ~2 k2 c2 + M 2 c4 = ~ω(k). One can also check that the one particle states |ki transform properly under the Lorentz transformations, U(Λ)|ki = U(Λ) a† (k) U −1 (Λ) U(Λ)|Ω0 i = a† (kΛ )|Ω0 i , (11.128) i.e. that they transform in the way appropriate for states of a spin 0 particle (see Section 6.2) provided the vacuum state |Ω0 i is assumed to be Lorentz invariant U(Λ)|Ω0 i = |Ω0 i. In the interacting field case, if ϕH (0, x) and ΠH (0, x) are expanded into creation and annihilation operators, the operators P i and J ij take usually30 the same form as in (11.127), whereas the boost generators K i are modified by the interaction, as argued in Section 7.3. Quantization of a single real scalar field described above can immediately be generalized to the case of many real scalar fields ϕi (x) where i = 1, . . . , N, whose classical dynamics is governed by the Lagrangian density (11.54). Furthermore, a system of N classical complex fields can always be represented as a system of 2N real fields and quantized using the same prescriptions. For several fields ϕi the Lagrangian density may be invariant (or invariant up to a total divergence - see the formula (11.20)) under a group of continuous transformations which in the infinitesimal form can be written as ϕi → ϕ′i = ϕi + θa δ a ϕi . In this case, upon replacing classical fields ϕi and their derivatives by the corresponding field operators (as ∂ 0 ϕi one has of course to use the expressions ϕ̇i = ϕ̇i (Π, ϕ) given by the canonical formalism), the classical expressions for Noether currents jµa become current operators and the quantities Z a Q̂ = d3 x ja0 (t, x) , (11.129) become the symmetry generators acting in the Hilbert space.31 By using the canonical commutation rules (11.108) one can show that independently of the precise form of L (provided it is symmetric) the time-like components of the Noether currents satisfy the commutation relations 0 (11.130) ja (t, y), jb0 (t, x) = jc0 (t, x) ifc ab δ (3) (x − y) , 30 Exception are theories quantized around a nontrivial classical background ϕcl (x) in which the field operator takes the form ϕH (t, x) = ϕcl (x) 1̂ + χH (t, x). 31 Of course, the operators Q̂a in (11.129) are well defined if their matrix elements R between physical states: hΦ|Q̂a |Ψi = d3 x hΦ|ja0 (t, x)|Ψi are well defined; it may then happen that the right hand side is not integrable due to the presence in the H spectrum of massless particles which mediate long range forces. This is so whenever a classical continuous symmetry of the Lagrangian is spontaneously broken by the vacuum state (see Section 22). 343 which ensure that the Noether charges (11.129) satisfy the symmetry algebra relations [Q̂a , Q̂b ] = Q̂c ifc ab . (11.131) Moreover, by virtue of the canonical commutation rules (11.77) the Noether charges Q̂a generate symmetry transformations of the field operators iθa [Q̂a , ϕi (x)] = θa δ a ϕi (x) . (11.132) In contrast, space-like components of the Noether currents usually can satisfy the commutation rules analogous to (11.130) only in the theory of noninteracting fields. For general interactions Hint (ϕ) it is only possible to infer (using the Lorentz covariance) that (x) ik ja0 (t, y), jbi (t, x) = jci (t, x) ifc ab δ (3) (x − y) + Sab (t, x) ∂k δ (3) (x − y) , ik where Sab (t, x) are the so-called Schwinger terms. Even in theories (like electrodynamics of spin 1/2 particles) in which the canonical (anti)commutation relations formally imply the absence of Schwinger terms,32 it is possible to prove, using arguments based on general principles of quantum mechanics, that they cannot vanish. The contradiction with the reasoning based on the canonical (anti)commutation relations stems from the fact that Noether currents are composite operators (products of elementary field operators taken at the same space-time point) which are in general singular objects - their matrix elements require some regularization; Schwinger terms can therefore depend on the adopted regularization prescription, and a regularization can, a priori, also induce nontrivial Schwinger terms in the commutators (11.130) of the time components of the currents. If a given symmetry can be recovered after regularization in the renormalized theory, Schwinger terms in the commutators like (11.130) are absent (although are generally nonvanishing in the commutators involving spatial components of the Noether currents). In fact, anomalies discussed in Section 23 can be understood as manifestation of nontrivial Schwinger terms induced by the necessity of regularization in the commutators like (11.130) of the time components of the currents. Finally, if the Hamiltonian commutes with the Noether charges Q̂a and the symmetry is not spontaneously broken by the vacuum state, i.e. if Q̂a |Ωi = 0 for all a (see Section 22), the Hamiltonian eigenstates (the in and out states) form multiplets of the symmetry algebra. Usually also the H0 part quadratic in field operators of the full Hamiltonian H commutes 32 In theories in which symmetries are realized on scalar fields, the presence of nontrivial (regularization independent) Schwinger terms is usually revealed already by using the canonical commutation relations. 344 with Q̂a ’s separately, and the H0 eigenstates, i.e. particles created and annihilated by the Hermitian free-field (interaction picture) operators ϕi (x) also form multiplets of the symmetry algebra. As all members of the symmetry multiplets have the same mass, one then forms linear combinations of the particle states (both in, out and the free-particle ones) which diagonalize the Noether charges Q̂a forming the Cartan subalgebra (see Section 4) of the full symmetry algebra. It is then convenient to form also the appropriate (complex in general) linear combinations of the free-field operators ϕi (x) creating and destroying one-particle eigenstates of the the Cartan subalgebra generators.33 In the case of the symmetry multiplets transforming as complex representations, such that each particle (except for essentially neutral ones) finds its antiparticle within the same multiplet, this corresponds precisely to forming (non-Hermitian) field operators creating a particle and annihilating its antiparticle as described in section 8.2. Existence of conserved charges has also important consequences for statistical properties of the system of quantum fields. If the Hamiltonian commutes with Noether charges Q̂a , the quantum numbers corresponding to the Cartan subalgebra generators can have simultaneously definite values (because all these generators commute with each other) which are constant of motion. This would have to be taken into account in the Gibbs Canonical Ensemble statistical sum Zstat (T, V ), making it essentially intractable analytically (similarly as in the case of a system consisting of a fixed number N of particles) even for noninteracting fields. One then passes to the Grand Canonical Ensemble introducing the chemical potentials µa for each Cartan subalgebra generator. Choosing in the Hilbert space the basis in which the Cartan subalgebra generators are diagonal facilitates then computation of the statistical sum Ξ(T, V, µa ). For example, the Lagrangian 2 1X L= ∂µ ϕl ∂ µ ϕl − M 2 ϕ2l , 2 l=1 (11.133) of two noninteracting scalar fields ϕ1 and ϕ2 has the O(2) ≃ U(1) symmetry whose conserved Noether current is jµ = ϕ2 ∂µ ϕ1 −ϕ1 ∂µ ϕ2 . After quantization it is the operators a† (k) and ac† (k) formed as appropriate linear combinations of a†l (k), l = 1, 2 which, acting on the vacuum |Ωi, generate the states 33 The corresponding Heisenberg picture operators obtained from the free-field ones as in (11.113) (or, more generally, as in (11.105)) have then “diagonal” matrix elements between the one-particle states of the full Hamiltonian H and the vacuum: hΩ|φiH |p, σ, ji ∝ δ ij with the same proportionality constant for all φiH and all states |p, σ, ji belonging to the same symmetry multiplet. 345 |nk , . . . , nck , . . .i of the Hilbert space basis in which the Hamiltonian X c ~ωk a†k ak + ac† a + 1 , (11.134) H= k k k where ωk = p c2 k2 + c4 M 2 /~2 , and the Noether charge X † c Q̂ = (ak ak − ac† k ak ) , (11.135) k are simultaneously diagonal. Obtaining the potential Ω(T, V, µ) = −kB T ln Ξ by computing the statistical sum Ξ = e−β Ω = Tr e−β(H−µQ̂) , is then straightforward: Z 3 dk Ω(T, V, µ) = V ~ωk + kB T ln 1 − e−(~ωk −µ)/kB T 3 (2π) +kB T ln 1 − e−(~ωk +µ)/kB T . (11.136) (11.137) We have passed already to the continuum (thermodynamical) limit as in (11.101). The value of the chemical potential µ determines the mean (in the sense of the ensemble) total charge Q of the field in the box V . Convergence of the summations leading to (11.137) imposes the constraint |µ| < M =mink ωk , where M is the mass of the particles. Of course, for µ = 0 the system is neutral (Q = 0 on average), and |µ| grows as the (mean) charge |Q| of the system grows. Eventually, if |Q| becomes so large that |µ| → M, the occupancy of the zero momentum (k = 0) state becomes macroscopical (it has to be extracted before the transition to the continuous normalization) and the usual Bose-Einstein condensation occurs. 11.3 Another view on fields quantization In the preceding Section we have quantized field(s) not subject to any special boundary conditions, i.e. defined in the empty space. We have used the Schrödinger picture to determine the Hamiltonian eigenstates (in the case of free fields) and performed the transition to the Heisenberg picture only at the later stage. For greater generality we have allowed for Hamiltonians explicitly dependent on time, but such cases, even if the Hamiltonian is at most quadratic in field variables, are difficult to analyze using the formalism presented in Section 11.2 because of technical difficulties in constructing the evolution operator (11.106). 346 Here we present another view on fields quantization which makes clear the central role played in this procedure by the complete set of solutions to the classical equations of motion (11.4) derived from the underlying Lagrangian. The advantages offered by this view are twofold. Firstly, it allows to take into account various types of boundary conditions imposed on fields, usually intended to model complicated interactions of fields of interest with external agents. Secondly, it allows to quantize fields directly in the Heisenberg picture even for systems whose Hamiltonians depend on time explicitly, without solving34 the complicated operator equations (11.106), (11.107). Thus, in this case, this view facilitates studies of the time evolution of quantum systems. One should stress however, that the approach discussed here, similarly as the one presented in the preceding Section, applies evidently only to systems described by Lagrangians depending on first derivatives of field variables only, or more precisely, to systems which classically admit the Hamiltonian description. To begin with, we quantize once again, this time directly in the Heisenberg picture, the free real field ϕ in the empty space whose classical dynamics is governed by the Lagrangian density (11.65). As no special boundary conditions are imposed, the most general solution to the classical Klein-Gordon equation (∂µ ∂ µ + M 2 )ϕ(x) = 0 following from (11.65) reads Z 4 dk ϕ(x) = A(k) e−ik·x 2πδ(k 2 − M 2 ) , (11.138) (2π)4 where (because ϕ is a real field) A(k) = A∗ (−k). Using then the formula δ(k 2 − M 2 ) = 1 1 δ(k 0 − E(k)) + δ(k 0 + E(k)) , 2E(k) 2E(k) (11.139) we get ϕ(x) = Z d3 k A(k 0 = E, k) e−iEt+ik·x + A(k 0 = −E, k) e+iEt+ik·x . 3 (2π) 2E(k) In the second term we change the integration variable k → −k and introduce the new coefficients a(k) ≡ A(E, k) and a∗ (k) ≡ A(−E, −k) so that the most general solution to the classical Klein-Gordon equation takes the form Z ϕ(x) = dΓk a(k) e−ik·x + a∗ (k) eik·x . (11.140) 34 Of course, to apply this method one has to find the complete set of solutions to the classical equations of motion, which is usually not possible in a closed form for Lagrangian densities not quadratic in field variables. The only possible quantizaton method is then the one presented in the preceding Secion. 347 The canonical momentum Π(x) is obtained by using the classical definition (11.72) as ϕ̇(x). One then quantizes the system by promoting the arbitrary coefficients a(k) and a∗ (k) (which classically are determined by some initial data ϕ(t0 , x) = q(x), Π(t0 , x) = p(x) given at some t0 , or more generally on some space-like hypersurface), to operators and imposing on them the commutation rules ensuring that the Heisenberg picture field operators ϕ(t, x) and Π(t, x) satisfy the canonical equal time commutation rules (11.108) [ϕ(t, x), Π(t, y)] = iδ (3) (x − y) , [ϕ(t, x), ϕ(t, y)] = [Π(t, x), Π(t, y)] = 0 . (11.141) It is easy to see that this requires a(k) and a∗ (k) to be promoted to annihilation and creation operators, respectively with the commutation rule (11.118). Thus, we directly recover the Heisenberg field operator ϕH (t, x) given by (11.116). The same procedure, with appropriate modifications can be applied to quantize the field ϕ with periodic boundary conditions. Strictly speaking, it is just the approach adopted here that lends justification to the quantization of the field ϕ in a box of volume V performed in the Schrödinger picture in the preceding Section: as a matter of facts, we have there written down ϕ(x) and Π(x) = ϕ̇(x) as most general functions of x consistent with the Klein-Gordon equation and the periodic boundary conditions and only then imposed the appropriate commutation relation on the coefficients of these functions so that to satisfy the canonical commutation rules (11.77). The equivalence of the quantization method in the Heisenberg picture described here and the one based on the Schrödinger picture, the evolution operator (11.106) and the equations (11.107), follows immediately if one notices that the Heisenberg picture operators ϕH (t, x) and ΠH (t, x) obtained from the prescription (11.105) satisfy the canonical equal-time commutation relations (11.141) as well as the “classical” field equations (11.110) and (11.111), respectively; in the method described above, field operators having the same properties are constructed directly, using the complete set of classical equation of motions.35 In order demonstrate the working of this approach in nontrivial circumstances, let us consider a real scalar field ϕ between two large parallel plates (perpendicular to the z axis) of area L2 each, separated by the distance d. Let us impose as the boundary conditions that the field ϕ vanishes on the plates and satisfies periodicity conditions (with the period L) in the remaining directions. Furthermore, for simplicity let us assume that the Lagrangian 35 This suggests that it should be possible to construct the evolution operator U (t, t0 ) defined in (11.106) using the complete set of solutions of the classical equations of motion. 348 density of the field ϕ is given by (11.65) with M = 0. Classically, ϕ satisfies the Klein-Gordon equation ∂µ ∂ µ ϕ(t, x) = 0 with the boundary conditions ϕ(t, x, y, z = 0) = ϕ(t, x, y, z = d) = 0 , ϕ(t, x + L, y, z) = ϕ(t, x, y + L, z) = ϕ(t, x, y, z) . The most general form of the field ϕ therefore is ∞ ∞ ∞ π 2π X X X ϕnml (t) sin l z ei L (nx+my) , ϕ(t, x) = d n=−∞ m=−∞ l=1 (11.142) (11.143) with the coefficients ϕnml (t) satisfying the equation d2 2 ϕnml (t) + ωnml ϕnml (t) = 0 , dt2 (11.144) where 2 2 ωnml 2 π = l + (m2 + n2 ) c2 d 2π L 2 . The general solution for ϕnml (t) can be written as r ~ ϕnml (t) = Anml e−iωnml t + Bnml e+iωnml t 2 ωnml L d (11.145) (11.146) (the prefactor has been chosen for further convenience). Reality of the field ϕ(x) imposes the relation Bnml = A∗−n,−m,l . Thus, ∞ ∞ ∞ r π X X X ~ 2π sin l z Anml e−iωnml t ei L (nx+my) ϕ(t, x) = 2 ωnml L d d n=−∞ m=−∞ l=1 2π +A∗nml eiωnml t e−i L (nx+my) , (11.147) after changing in the second term −n → n, −m → m. It is now easy to check that if the coefficients Anml and A∗nml are promoted to annihilation and creation operators, respectively, with the commutation rules i h † (11.148) Anml , An′ m′ l′ = δnn′ δmm′ δll′ , the field operators ϕ(t, x) and Π(t, x) = ϕ̇(t, x) satisfy the right canonical equal time commutation rules36 h i ϕ̂(t, x), Π̂(t, y) = i~δ (3) (x − y) , h i [ϕ̂(t, x), ϕ̂(t, y)] = Π̂(t, x), Π̂(t, y) = 0 . (11.149) 36 The necessary formulae are Z d π Z d π d π π ′ dz cos l′ z cos l z = δl′ l , dz sin l z sin l z = d d d d 2 0 0 349 It is then straightforward to express the Hamiltonian Z Z L Z L 1 d dz dx dy ϕ̇2 + c2 (∇ϕ)2 , H= 2 0 0 0 in terms of the operators Anml and A†nml : H= ∞ X ∞ ∞ X X n=−∞ m=−∞ l=1 1 † ~ωnml Anml Anml + . 2 (11.150) so that the energy of the vacuum fluctuation of the field ϕ(t, x) for L ≫ d can be written as r Z ∞ X π2 ~c L2 2 2 2 k + l Ebc ≡ hΩbc |H|Ωbc i = d k , (11.151) k k 2 (2π)2 d2 l=1 where |Ωbc i is the ground state of the field quantized with the boundary conditions (11.142), annihilated by all Anml , and where we have replaced the summations over n and m by the two dimensional integral (L/2π)2 dkx dky . One can now compare the energy contained in the box L × L × d of the field ϕ(t, x) subject to the boundary conditions (11.142) with the energy Eper contained in the same volume L2 d of the field ϕ quantized in the box L × L × L with the periodic boundary conditions (with period L) in all the three directions. We are therefore interested in the difference ∆E ≡ Ebc − Eper , in which Z q ~c L2 d L2 d 3 2 d k . (11.152) k2k + k⊥ Eper ≡ 3 × hΩper |H|Ωper i = L 2 (2π)3 Writing k⊥ formally as l(π/d) and integrating over dl instead of dk⊥ , we get ) (∞ r Z Z ∞ r 2 2 X π π ~c L2 k2k + l2 2 − ∆E = d2 kk dl k2k + l2 2 . (11.153) 2 2 (2π) d d 0 l=1 This energy difference should be finite. To jsutify this claim we can appeal to the physics of the problem. Nontrivial boundary conditions as (11.142) ∞ X l=1 Z 0 L 2π 2π ′ π π d sin l z sin l z ′ = δ(z − z ′ ) , d d 2 dx ei L nx ei L n x = L δn′ n , ∞ X n=−∞ 350 2π ′ ei L n(x−x ) = L δ(x − x′ ) . considered here, arise usually when fields are quantized in a non-empty space. The complete Hamiltonian of the system should then, strictly speaking, include terms describing interactions of the considered fields with external agents. As these are usually complicated, they are modeled by imposing appropriate boundary conditions on fields themselves (from the quantum theory point of view, this means imposing some restrictions on the Hilbert space of field states). Of course, by itself the example of the scalar field is not very physical. One can however interpret it as a simplified model of the electromagnetic field quantized between two conducting plates. Classically in this case, the component of the electric field E tangent to the conductor surface vanishes at this surface, at least for electrostatic fields, i.e. for fields with small frequencies ω. Microscopically this is of course due to complicated interactions between the field and electrons in a metal out of which the plates are made. But one may expect that field modes whose frequencies are much higher than any frequencies that can be associated with electrons in matter, almost do not feel the presence of the plates. Thus, the microscopic physics of the interaction between the field and the metal provides an effective ultraviolet cutoff on the expression (11.153): high frequency modes of the field ϕ (11.147) should effectively approach modes of the field quantized without the presence of the plates. The attractive force between the two plates, which will be foud below, is mostly due to the difference in the low frequency modes. Despite the argument in favour of the finiteness of ∆E given above, the practical implementation of the cancellation in the actual calculation requires some care. To facilitate the calculation one traditionally regularizes both infinite sums introducing a cutoff function f (ω) which model the intuitive picture discussed above. We assume f (ω) is such that f (0) = 1, all its derivatives f (n) (0) = 0 for n ≥ 1, and vanishing for ω → ∞. Thus, we rewrite the expression (11.153) as ) (∞ Z ∞ Z X ∆E ~c ∞ ω(k, l) f (ω(k, l)) − dl ω(k, l) f (ω(k, l)) , dk k = L2 4π 0 0 l=1 p with ω(k, l) = k 2 + l2 π 2 /d2 . Introducing the variable u = k 2 (d2 /π 2 ) brings this into the form (∞ Z π √ X√ ∆E π 2 ~c ∞ 2f 2 u + l u + l = du L2 8d3 0 d l=1 Z ∞ √ π √ 2 2 u+l − dl u + l f d 0 Z ∞ π 2 ~c ≡ F (1) + F (2) + . . . − dl F (l) , (11.154) 8d3 0 351 where F (l) = Z 0 Z l π √ π √ 2 2 2 u + l = −2 du u + l f ξ . (11.155) dξ ξ f d d +∞ ∞ One can then use the Euler-Mclaurin formula Z ∞ ∞ X 1 1 1 F (0) + F (n) − dn F (n) = − B2 F ′ (0) − B4 F ′′′ (0) + . . . (11.156) 2 2! 4! 0 n=1 in which Bi are the Bernoulli numbers: B2 = 1/6, B4 = −1/30, etc., to obtain the final result: since we have assumed the f (n) (0) = 0 for n ≥ 1 it is easy to see that F ′ (0), F ′′ (0) and all derivatives F (n) (0) with n > 3 vanish. The only nonzero contribution to (11.154) comes from F (0) and F ′′′ (0) = −4: ∆E ~cπ 2 1 1 = − F (0) + B4 . (11.157) L2 8d3 2 3! It is also easy to see (by appropriately rescaling the integration variable), that F (0) ∝ d3 , so that the force per area between the plates, due to the fluctuations of the quantized field, ∂ ∆E 1 π 2 ~c π 2 ~c F = − = B = − (11.158) 4 L2 ∂d L2 2 8d4 480d4 is attractive37 and does not depend on the details of the cutoff imposed in order to regularize the expression (11.153). This is the celebrated Casimir effect which has been experimentally confirmed and measured in 1958. Being initially only a curiosity, it plays now an important role in planning nanoscale devices. As another application of this approach to field quantization we consider a real scalar field coupled linearly to an external source. (This example is analogous to the one considered in Section 1.2). The Lagrangian density is 1 1 L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 + ϕJ , 2 2 (11.159) where J(t, x) is a given function. We assume also that J vanishes for t → ±∞ (or, better, it is just zero for |t| > T ), because then the states of the quantized 37 Of course, for M 2 = 0 proportionality of F/L2 to ~c/d3 follows from the dimensional analysis (at least for L → ∞, in which limit the result should be independent of L). The computation is needed only to establish the dimensionless proportionality constant and the sign. Notice, that from the thermodynamical point of view the negative sign means that the quantum vacuum behaves as a gas exerting negative pressure on the walls of the recipient! 352 field can have particle interpretation in the far past and in the far future. The classical field equation following from the Lagrangian (11.159) is ∂µ ∂ µ + M 2 ϕ(x) = J(x) . (11.160) The solutions can be written as the sum of two terms: Z ϕ(x) = ϕ0 (x) + i d4 y D(x − y)J(y) , (11.161) where ϕ0 (x) satisfies the homogeneous equation (∂µ ∂ µ + M 2 )ϕ0 (x) = 0. The Green’s function D(x − y) is defined by ∂µ ∂ µ + M 2 D(x − y) = −iδ (4) (x − y) . (11.162) and can formally be written as the Fourier integral Z 4 i dk e−ik·(x−y) . D(x − y) = 4 2 2 (2π) k − M (11.163) Different prescriptions of going around the singularities at k 0 = ±E(k), √ 2 where E(k) ≡ k + M 2 , specify different Green’s functions (satisfying different boundary conditions) of the Klein-Gordon equation. In particular, writing the denominator k 2 − M 2 as [k 0 − E(k) + i0][k 0 + E(k) + i0] gives the retarded Green’s function Dret (x − y) vanishing for x0 − y 0 < 0, while writing it with two −i0 factors gives the advanced function Dadv (x − y) vanishing for x0 − y 0 > 0. We can thus write down two different forms of the complete solution to the classical equation of motion (11.160) Z ϕ(x) = ϕin (x) + i d4 y Dret (x − y)J(y) , Z ϕ(x) = ϕout (x) + i d4 y Dadv (x − y)J(y) . (11.164) Classically, ϕin (x) is the initial field, before the source J started to act, while ϕout (x) is the final field. Of course, both ϕin (x) and ϕout (x) can be written in the form (11.140) with some coefficients ain (k), a∗in (k) and aout (k), a∗out (k). Specifying ain (k) and a∗in (k) or aout (k) and a∗out (k) determines uniquely ϕ(x) for all x. Quantization consists of promoting ϕ(x) to the operator ϕH (x) satisfying with the conjugated momentum ΠH (x) = ϕ̇H (x) the canonical commutation relations (11.141). This can be done in two ways, by imposing the commutation rules on the in and out creation and annihilation operators. In this way the same operators ϕH (x) and ΠH (x) are represented in two alternative 353 ways given in (11.164). In both representations ϕH (x) and ΠH (x) satisfy the fundamental commutation rules (11.141) and the two representations are unitarily equivalent.38 The operators ain (k) and a†in (k) and the operators aout (k) and a†out (k) define two different bases in the same Hilbert space of the field states. These bases have obvious physical meaning: the Hamiltonian Z 1 2 1 1 2 2 2 3 H = d x ϕ̇H + (∇ϕH ) + M ϕH − ϕH J (11.165) 2 2 2 in the first (the in) representation (11.164) of the field ϕH (t, x) takes in the far past the form Z H = dΓk E(k) a†in (k)ain (k) + const , (11.166) because for x0 → −∞ the retarder Green’s function Dret (x − y) vanishes and ϕH (t, x) → ϕin (t, x). Similarly, the Hamiltonian takes the form (11.166) with a†out (k)aout (k) replacing a†in (k)ain (k) in the far future. The states |Ω+ i, annihilated by all ain (k), a†in (k)|Ω+ i, etc., are therefore the eigenstates of H(−∞) having the particle interpretation in the far past39 whereas the analogous states |Ω− i, a†out (k)|Ω− i etc., have the similar interpretation in the far future. It is straightforward to relate the in and out operators to each other. From (11.164) we get Z ϕout (x) = ϕin (x) + i d4 y [Dret (x − y) − Dadv (x − y)] J(y) . (11.167) Explicitly the difference of the Green’s functions is d3 k (2π)3 ∞ 0 0 0 dk 0 e−ik (x −y ) ik·(x−y) Dret (x − y) − Dadv (x − y) = e 2E(k) −∞ 2π i i i i . − − + k 0 − E(k) + i0 k 0 + E(k) + i0 k 0 − E(k) − i0 k 0 + E(k) − i0 Z Z Upon using the formula 38 1 1 =P ∓ iπδ(x − a) , x − a ± i0 x−a (11.168) In the similar problem of the interaction of the electromagnetic field the two representations might turn out not to be unitarily equivalent, which means that the states created by a†in (k) from |Ω+ i have all zero scalar products with the states created by a†out (k) from |Ω− i. Physically this corresponds to creation by the external source of an infinite number of arbitrarily soft photons. 39 Since H depends on time it cannot have global, time independent eigenstates. 354 it can be brought into the form Dret (x − y) − Dadv (x − y) = Thus, from (11.167) we get Z dΓk e−ik·(x−y) − eik·(x−y) . aout (k) = ain (k) + iJ(E(k), k) , a†out (k) = a†in (k) − iJ ∗ (E(k), k) , (11.169) (11.170) where J(k) = J(k 0 , k) = J ∗ (−k 0 , −k) is defined by Z 4 d k −ik·y e J(k) . J(y) = (2π)4 (11.171) One can now introduce the unitary operator S, S −1 = S † , such that (compare (1.32) and (7.22)) S|α− i = |α+ i . (11.172) It must therefore have the following properties: S a†out (k) S † = a†in (k) , S aout (k) S † = ain (k) , (11.173) S † a†in (k) S = a†out (k) , S † ain (k) S = aout (k) . (11.174) and It is not difficult to construct the operator S explicitly Z Z iδ 4 iδ′ 4 S = e exp i d x ϕin (x)J(x) = e exp i d x ϕout (x)J(x) , (11.175) where δ is an arbitrary phase factor and δ ′ can be easily related to δ with the help of (11.167). Using the operator identity 1 eA+B = aA eB e− 2 [A, B] 1 = eB eA e 2 [A, B] , (11.176) valid for [A, [A, B]] = [B, [A, B]] = 0, the operators S and S † can be written in the forms which are more convenient for computation of their matrix elements (we omit the phase factor δ): S = ei R R R dΓk a†in (k) J(k) i dΓk ain (k) J ∗ (k) − 12 dΓk |J(k)|2 S † = e−i e R e R dΓk ain (k) J ∗ (k) −i dΓk a†in (k) J(k) e 355 1 e2 R , dΓk |J(k)|2 (11.177) . (11.178) Similar formulae in terms of the out operators can also be written down. The formulae (11.174) can be now checked with the help of the operator expansion eA B e−A = B + [A, B] + 1 [A, [A, B]] + . . . 2! (11.179) it is easy to check that indeed S † a†in (k) S = a†in (k) − iJ ∗ (k) , (11.180) etc. Various transition probabilities can now be computed. For example, the vacuum persistence probability is P (0 → 0) = |hΩ− |Ω+ i|2 = |hΩ+ |S|Ω+ i|2 = e− R dΓk |J(k)|2 . (11.181) Similarly, the probability of creating n quanta of the field ϕ by the source is given by Z Z 1 P (0 → n) = dΓk1 . . . dΓkn |h(k1 , . . . kn )+ |S|Ω+ i|2 n! 1 (11.182) = (n)n e−n n! (the Rfactor 1/n! is inserted because of identity of the field quanta), where n = dΓk |J(k)|2 , i.e. it is given by the Poisson distribution, whereas the mean total energy emitted by the source can be computed as Z 3 d k 1 † |J(k)|2 . (11.183) Erad = hΩ+ |H(aout aout )|Ω+ i = 2 (2π)3 where H(a†out aout ) stands for the Hamiltonian taken at t = +∞ which when expressed in terms of the out operators has the free field form. The field radiated by the source if the initial state of the field at t = −∞ was |Ψ+ i can in turn be identified with the matrix element lim hΨ+ |ϕH (t, x) − ϕH (−t, x)|Ψ+ i = lim hΨ+ |ϕout (t, x) − ϕin (−t, x)|Ψ+ i . t→∞ t→∞ For |Ψ+ i = |Ω+ i, using (11.167), this gives (as hΩ+ |ϕin(x)|Ω+ i = 0) Z lim hΩ+ |ϕout (x)|Ω+ i = 0lim i d4 y [Dret (x − y) − Dadv (x − y)] J(y) . 0 x →∞ x →∞ Finally we apply this approach to quantization to the case of a scalar field in a curved space-time i.e. in a nontrivial gravitational background. We will 356 consider only the simplest case of the Friedman-Robertson-Walker (FRW) metric in which fixed x0 sections are flat spaces. Although very simple, this example has important application in modern cosmology. We start with the simplest scalar field action in the general space-time40 Z 1 µν 4 √ g ∂µ ϕ∂ν ϕ − V (ϕ) , (11.184) I[ϕ] = d x −g 2 where g = det(gµν ) is the determinant of the metric tensor. The equation of motion obtained by varying the field ϕ (with the usual requirements that the variation vanishes at large distances and at the initial and final time hypersurfaces) reads √ 1 √ ∂µ −gg µν ∂ν ϕ + V ′ (ϕ) = 0 . −g (11.185) We now specify to V (ϕ) = 12 M 2 ϕ2 and to the spatially flat FRW metric with the line element ds2 ≡ gµν dxµ dxν = dt2 − a2 (t)(dx)2 = a2 (η) dη 2 − (dx)2 , (11.186) where we have introduced the conformal time coordinate η defined by Z t dt . (11.187) η(t) = t0 a(t) Expressed through the conformal time variable, the equation of motion (11.185) takes the form a′ ϕ′′ + 2 ϕ′ − ∇2 ϕ + M 2 a2 ϕ = 0 . a (11.188) By introducing the rescaled field χ(η, x) ≡ a(η)ϕ(η, x) (11.189) the equation (11.188) can be simplified to χ′′ − ∇2 χ + m2eff (η) χ = 0 , (11.190) i.e. to the Klein-Gordon equation but with an η dependent effective mass term m2eff (η) ≡ M 2 a2 (η) − 40 a′′ (η) . a(η) (11.191) The most general Raction√quadratic in the real scalar field ϕ and its first derivatives includes also the term d4 x −g 21 ξRϕ2 , where R is the Ricci scalar. For ξ = 61 the action I[ϕ] is conformally invariant. We set ξ = 0 for simplicity. 357 The equation of motion (11.190) follows of course from the effective action Z Z 1 (11.192) Ieff = dη d3 x (χ′ )2 − (∇χ)2 − m2eff (η) χ2 , 2 which serves to define the momentum Π canonically conjugated to χ: Π = χ′ . One can then write down the (conformal) time-dependent Hamiltonian Z 1 H= d3 x Π2 + (∇χ)2 + m2eff (η) χ2 . (11.193) 2 To quantize the field χ(η, x) following the procedure adopted in this Section, we must first write down the most general solution to the classical equation of motion (11.190). The solution can be written in the form41 Z 3 dk χ(η, x) = χ(η, k) eik·x , (11.194) (2π)3 where for each k the function χ(η, k) satisfies independently the equation χ′′ (η, k) + ω 2 (η, k)χ′′ (η, k) = 0 , (11.195) ω 2 (η, k) ≡ k2 + m2eff (η) . (11.196) with Since the equation (11.195) is a linear (in χ) second order ordinary differential equation with the real coefficient ω 2 , it has two linearly independent solutions. The important property of the equation (11.195) is that for any two its solutions, χ1 and χ2 , their combination W (χ1 , χ2 ) ≡ χ′1 χ2 − χ1 χ′2 , (11.197) called the Wrońskian, is independent of η: d W (χ1 , χ2 ) = 0 . dη (11.198) It is also clear that W (χ1 , χ2 ) = 0, if χ1 and χ2 are linearly dependent. Furthermore, from any two real linearly independent solutions χ1 and χ2 , of (11.195) a complex solution of the equation (11.195) v(η, k) = χ1 (η, k) + iχ2 (η, k) , 41 (11.199) We exploit the fact that the metric is spatially flat. For more general metrics with nonzero spatial curvature, but still with g0i = gi0 we would have to do first the harmonic analysis on spatial sections of the space-time. 358 can be formed. For such complex solutions Im (v ′ v ∗ ) = −W (χ1 , χ2 ) . (11.200) With the help of the complex functions v(η, k) the most general solution of the equation of motion (11.190), such that χ(η, x) is a real function can be written in the form Z 3 d k ik·x e [av (k) v ∗ (η, k) + a∗v (−k) v(η, −k)] . (11.201) χ(η, x) = (2π)3 Quantization of the field χ(η, x) directly in the Heisenberg picture consists now of promoting χ(η, x) and Π(η, x) = χ′ (η, x), and hence the coefficients av (k) and a∗v (k), to operators Z 3 dk ∗ ik·x † −ik·x a (k) v (η, k) e + a (k) v(η, k) e , (11.202) χ(η, x) = v v (2π)3 Z 3 d k ′∗ ik·x † ′ −ik·x Π(η, x) = a (k) v (η, k) e + a (k) v (η, k) e , v v (2π)3 satisfying the canonical equal-(conformal)time commutation relation [χ(η, x), Π(η, y)] = iδ (3) (x − y) . (11.203) It is then easy to check, that this is satisfied for av (k) and a†v (k) such that (11.204) av (k), a†v (k′ ) = (2π)3 δ (3) (k − k′ ) , and provided the complex solutions are normalized so that v ∗ (η, k) v ′ (η, k) − v(η, −k) v ′∗ (η, −k) = i . (11.205) The normalization (11.205) is consistent with the (conformal)time evolution of the functions v(η, k), provided they satisfy the relation42 v(η, k) = v(η, −k); this follows from the relation (11.200) and the property (11.198). Complex solutions of (11.195) normalized in this way are called mode functions. It is however important to remember that the normalization condition (11.205) by no means specifies the mode functions uniquely: different choices of the basis of the real solutions χ1 and χ2 of (11.195) lead to different (conformal)time dependence of the functions v(η, k); by a trivial rescaling (by a complex factor) any such function can be adjusted to satisfy (11.205). For 42 Since ω 2 defined by (11.195) depends only on the length |k| of k and not on its direction, its solutions v(η, k) can also be chosen to be isotropic, i.e. v(η, k) = v(η, |k|). In specific situations it might prove more convenient to chose the functions v(η, k) nonisotropic but they always must satisfy the equality v(η, k) = v(η, −k). 359 example, in the standard quantization of the real scalar field ϕ in the flat Minkowski space-time (in which η = t) one takes χ1 (t, k) = (2ω)−1/2 cos(ωt), χ2 (t, k) = (2ω)−1/2 sin(ωt), so that v ∗ = (2ω)−1/2 exp(−iωt), but one could also take v1 = A1 cos(ωt) + B1 sin(ωt) and v2 = A2 cos(ωt) + B2 sin(ωt) with real coefficients A1,2 and B1,2 ; however in the flat Minkowski space-time only the first choice makes the Hamiltonian diagonal (see below). The operators av (k), a†v (k) formally resemble the ordinary creation and annihilation operators. In particular, one can define the “vacuum” state |0(v) i annihilated by all av (k) associated with the given set of complex solutions v(η, k) and the Hilbert space of states of the quantum field χ(η, x) is spanned by the vectors of the form a†v (k)|0(v) i, a†v (k2 )a†v (k1 )|0(v) i, etc. similar to the vectors (11.102). However, while owing to our choice (11.186) of the gravitational background these states transform under spatial rotations as do the ordinary particles, they do not have all the necessary properties to be justly called particles. Similarly, as we will see, neither the state |0(v) i can be termed the vacuum. To see this in more detail, let’s choose another set of linearly independent complex solutions u(η, k) and u∗ (η, k) to the equation (11.195). It must of course be possible to express v(η, k) as a linear combination43 of u(η, k) and u∗ (η, k): v ∗ (η, k) = α(k) u∗ (η, k) + β(k) u(η, k) , (11.206) with the coefficients α(k) and β(k) independent of η and such that α(−k) = α(k), β(−k) = β(k) (as follows from the condition v(η, k) = v(η, −k)). The normalization condition (11.205) satisfied by both sets of complex solutions imposes the constraint |α(k)|2 − |β(k)|2 = 1 (11.207) on the Bogolyubov coefficients. By comparing the field operator χ(η, x) written in terms of the new mode functions u(η, k) and u∗ (η, k) and the new operators au (k) and a†u (k) Z 3 dk ∗ ik·x † −ik·x χ(η, x) = a (k) u (η, k) e + a (k) u(η, k) e , (11.208) u u (2π)3 with the same field operator given in (11.202) in terms of the operators a(k) and a† (k) and substituting in the latter form (11.206) one finds that au (k) = α(k) av (k) + β ∗ (k) a†v (−k) , a†u (k) = α∗ (k) a†v (k) + β(k) av (−k) , 43 (11.209) Due to the choice (11.186) of the metric v(η, k) depends only on u(η, k) and u∗ (η, k) with the same k; in spatially non-flat gravitational backgrounds v(η) would in general be linear combinations of all functions (with all k) of the alternative complete set of solutions. 360 where we have taken into account that α(−k) = α(k) and β(−k) = β(k). The relations (11.209) are called the Bogolyubov transformations and the coefficients α(k) and β(k) are the Bogolyubov coefficients. The relations (11.209) show that the state |0(v) i annihilated by all operators av (k) is not annihilated by the operators au (k), which all annihilate the state |0(u) i and vice versa. In particular, since h0(v) |a†u (k)au (k)|0(v) i = |β(k)|2 (2π)3 δ (3) (0) , (11.210) the quantity |β(k)|2 can be given the meaning of the density of the “uquanta” associated with the solutions u in the “vacuum” associated with the v modes (the factor (2π)3 δ (3) (0) in (11.210) should be interpreted as the volume factor). It is also possible to show that if the field χ(η, x) is quantized in the finie spatial volume V = L3 , so that k = (2π/L)n, these two “vacuum” states are related by44 Y 1 β ∗ (k) † † p exp − |0(u) i = av (k)av (−k) |0(v) i . (11.211) 2α(k) |α(k)| k This shows, that with respect to |0(v) i the state |0(u) i is the so-called squeezed state. Expressing the Hamiltonian (11.193) in terms of the operators av (k) and by inserting the expansions (11.202) one finds Z 3 d k † 1 av (k)av (k) + av (k)a†v (k) E(η, k) H= 3 2 (2π) +av (k)av (−k)F ∗ (η, k) + a†v (k)a†v (−k)F (η, k) , a†v (k) where 2 E(η, k) = |v ′ (η, k)| + ω 2 (η, k) |v(η, k)|2 , 2 F (η, k) = (v ′ (η, k)) + ω 2 (η, k)v 2 (η, k) , (11.212) and ω 2 (η, k) = k2 + m2eff (η). One could try to select the solutions v(η, k) so that F (η, k) ≡ 0. However, the resulting differential equation for v(η, k) is compatible with the basic equation (11.195) only if ω is η-independent. This simply reflects the fact that the (conformal)time-dependent Hamiltonian does not have stationary eigenstates. For η-independent ω the function F (η, k) vanishes identically for v(η, k) = p 1 2ω(k) eiω(k)η , (11.213) Q If the product k |α(k)|−1/2 is not convergent, the two sets of creation and annihilation operators simply act in two different Hilbert spaces providing nonequivalent representations of the same commutation relations. 44 361 and the operators χ(η, x) and Π(η, x) have the same form as (11.89) and (11.90), respectively in the ordinary Minkowski space-time. For ω 2(η, k) depending on η in a nontrivial way one possible choice of the functions v(η, k) is such that |0(v) i is the state in which the Hamiltonian taken at one particular instant η0 of the conformal time has the lowest expectation value. Minimizing h0(v) |H(η0 )|0(v) i with respect to possible choices of the functions v(η, k) one gets the conditions 1 , v ′ (η0 , k) = iω(η0 , k)v(η0 , k) , (11.214) v(η0 , k) = p 2ω(η0 , k) up to a phase factor common for v and v ′ . In this case H(η0 ) is diagonal, i.e. F (η0 , k) = 0. The state |0(v) i is therefore called the instantaneous vacuum. It should be clear that |0(v) i need not be neither the eigenstate nor the state giving the lowest energy expected value for the Hamiltonian H(η1 ) for η1 6= η0 . Moreover, it is important to remember, that the energy of the zero point fluctuations at η0 cannot be simply subtracted, because from the global point of view this contribution is eta-dependent and more complicated renormalization procedure is required to remove this kind of infinities in the curved space-time. Finally, in some cases it may happen that ω 2 (η, k) defined in (11.196) is negative for small |k| because of the last term in m2eff in (11.191). In such a case, the contribution to the vacuum expectation value h0(v) |H(η0 )|0(v) i of the modes corresponding to ω 2 (η, k) < 0 cannot be minimized and the instantaneous vacuum does not exist. In such cases one considers other choices of the functions v(η, k) defining other types of “vacuum” states (e.g. the Bunch-Davis vacuum, or the adiabatic vacuum) which we will not discuss here. A relatively simple situation occurs when the metric becomes Minkowskian in the far past (η → −∞) and in the far future (η → +∞), so that ω(η, k) → ωin (k) and ω(η, k) → ωout (k) in these limits. One can then define the vacua |0ini associated with the mode functions v(η, k) such that 1 lim v(η, k) = p eiωin (k)η , (11.215) η→−∞ 2ωin(k) and |0out i associated with the functions u(η, k) having the property 1 eiωout (k)η . (11.216) lim u(η, k) = p η→+∞ 2ωout (k) The states created by the operators a†v (k) related to the mode functions v(η, k) and by the operators a†u (k) related to the mode functions u(η, k) can be then interpreted as normal particles in the far past and in the far future, respectively. Using the Bogolyubov coefficients one can compute then probabilities of creating some number of particles by a variable gravitational field. 362 11.4 Lagrangian of the electromagnetic field Canonical quantization of fields transforming under changes of the reference frame as nontrivial representations of the Lorentz group requires special treatment because the assumption that generalized velocities (time derivatives of canonical variables) can be expressed as in (11.9) through the conjugated momenta is usually not fulfilled. This is, in particular, the case of the electromagnetic and Proca fields (describing after quantization massless and massive spin 1 particles, respectively). One method of dealing with this difficulty consists of eliminating some of the canonical variables (reducing the number of independent ones and therefore the number of conjugated momenta). This can be done either directly (as is possible for the Proca field - see Section 11.5) or through the use of the Dirac’s quantization formalism adapted to system with constraints, which will be presented in Section 11.6. We first discuss possible forms of the action I (11.1) governing classical dynamics of vector fields. We begin with the familiar case of the electromagnetic field. The Maxwell equations (in the Gauss’ system of units, in which the fields E and B have the same dimension - the relation of electromagnetic quantities in the Gauss system to their counterparts in the “official” SI of units is recalled in Appendix F) read 1 ∂B = 0, c ∂t 4π 1 ∂E = j, ∇×B − c ∂t c ∇×E + ∇·B = 0 , (11.217) ∇·E = 4πρ . (11.218) The first two are automatically satisfied if the scalar and vector potentials, forming together a four-vector Aµ = (ϕ, A) are introduced, in terms of which B = ∇×A , E = −∇ϕ − 1 ∂A . c ∂t (11.219) The two other equations, (11.218), follow from the Lagrangian density45 LEM = − 1 1 fµν f µν − eJµ Aµ . 16π c (11.220) The electromagnetic field strength tensor fµν is defined as fµν = ∂µ Aν − ∂ν Aµ . (11.221) ∂ 1 1 µν ∂LEM =− fλρ f λρ = − f . ∂(∂µ Aν ) 8π ∂(∂µ Aν ) 4π (11.222) As it is easy to find that 45 It is more natural to factorize e - the fundamental coupling constant - out of the four-current J ν . 363 The Euler-Lagrange equations (11.4) which here read ∂µ ∂LEM ∂LEM , = ∂(∂µ Aν ) ∂Aν (11.223) therefore give 4π ν eJ . (11.224) c Since the four-divergence of the left hand side of (11.224) vanishes by antisymmetry of the field strength tensor f µν (11.221), the current J ν to which the electromagnetic field couples must be conserved ∂µ J µ = 0 (otherwise the equations (11.224) are inconsistent). It is instructive to examine the content of (11.224). For ν = 0, since f 00 ≡ 0 by antisymmetry, on the left hand side we get46 1 ∂A i 0 0 i 0 i ∂i (∂ A − ∂ A ) = ∂i (−∂i A − ∂0 A ) = ∇· −∇ϕ − , (11.225) c ∂t ∂µ f µν = so that the second Maxwell equation (11.218) is recovered for e J 0 = cρ . (11.226) For ν = k from (11.224) we get ∂0 (∂0 Ak + ∂k A0 ) + ∂i f ik = 4π k eJ . c (11.227) Furthermore, it is easy to check, that f ij ≡ −∂i Aj + ∂j Ai = −ǫijk B k , so that (11.227) becomes 1∂ k 4π k 1∂ ∂k ϕ + A − ǫikl ∂i B l = eJ , c ∂t c ∂t c (11.228) which, if the second relation (11.219) is taken into account, is just the first equation (11.218), provided one identifies eJ with j. It is sometimes useful to have explicit forms of the tensors Fµν and F µν in terms of the Cartesian components of the ordinary three-vectors E and B: 0 −E x −E y −E z 0 Ex Ey Ez Ex −E x 0 −B z B y 0 −B z B y µν . , y f = fµν = z y z x E −E B 0 −B x B 0 −B E z −B y B x 0 −E z −B y B x 0 46 Since we want to keep contact with the ordinary three-dimensional notation, we convert all expressions to a form in which A0 and Ai have always upper indices whereas the derivatives ∂0 and ∂i have always lower indices, so that ∂i = ∂/∂xi is the ordinary gradient in the contravariant coordinates xi . 364 Having the explicit formulae it is easy to see that 1 1 E2 − B2 − eJ µ Aµ . (11.229) LEM = 8π c There exist also another term bilinear in the field Aµ , which, being a scalar with respect to proper ortochroneous Lorentz transformations,47 could be added to the Lagrangian density (11.220): ∆L ∝ f˜µν fµν , (11.230) where (recall we use ǫ0123 = +1) 1 f˜µν ≡ ǫµνλρ fλρ 2 0 B1 = B2 B3 −B 1 0 −E 3 E2 −B 2 E3 0 −E 1 −B 3 −E 2 . E1 0 (11.231) ∆L is, however, a total four-divergence: f˜µν fµν = ∂µ (2ǫµνλρ Aν ∂λ Aρ ) and, as such, it does not modify the Euler-Lagrange (i.e. Maxwell) equations. Being a total four-divergence, it has neither any impact on the dynamics of the quantized electromagnetic fields, because no topologically nontrivial configurations of the electromagnetic field exist classically (this statement cannot be given a justification here). Such topologically nontrivial configurations do exist, however, for nonabelian gauge fields (see Section 20) and the nonabelian analog of ∆L is not innocuous in the quantum theory of such fields. The Maxwell equations (11.217), (11.218) are written in the Gauss system of units, in which the Coulomb potential looks simple (it does not have the 1/4π factor nor the ǫ0 factor) and the fine structure constant (defined by the Thomson limit of the Compton scattering cross section) is αEM = e2 /~c = 1/137.035999679(94) ≈ 1/137. This system of units is not very convenient√for the quantum theory of the electromagnetic field because of the factor 4π which would appear in many places (e.g. in free field operators - see e.g. W.B.Bierestecki, J.M. Lifshitz, L.P.Pitaevski Quantum Electrodynamics). It is much more practical to go over to the Heaviside-Lorentz system of units in which electromagnetic quantities have the same physical dimensions as in √ the Gauss system but are rescaled by the appropriate factors of 4π: √ √ Aµ → 4πAµ , eJ µ → eJ µ / 4π , (11.232) √ (i.e. AGauss = 4πAHeaviside−Lorentz etc.), so that in the rescaled variables the µ µ Lagrangian density (11.220) takes the form LEM = − 1 1 fµν f µν − eJµ Aµ . 4 c 47 (11.233) As a result of the parity and time reversal transformations L′ changes sign; it is therefore neither P - nor T - (and, hence, neither CP -) invariant. 365 √ √ This amounts to the rescalings E → 4π E and B → 4π B. The corresponding changes of ρ and j in the Maxwell equations (11.217), (11.218), are, µ since the elementary charge e has been √ facorized out from the four-current √J , equivalent to the rescaling e → e/ 4π (i.e. eGauss = eHeaviside−Lorentz / 4π). As a result in the Heaviside-Lorentz units the fine structure constant αEM reads αEM = e2 . 4π~c (11.234) In the Lagrangian density (11.233) the current J µ couples to Aµ linearly. The task of quantizing the electromagnetic field coupled to a given external classical current seems therefore similar to the second example discussed in Section 11.3. If the electromagnetic field interacts with “matter” (particles or other fields), and both, the electromagnetic field and the matter, are treated as a single dynamical quantum system, the current j and the charge density ρ, that is J µ , are given in terms of the dynamical variables describing matter. In such a case the current J µ can, if the electromagnetic field is coupled to a system of scalar fields, even depend on the variable Aµ (but not on its derivatives); “matter” is then coupled nonlinearly to Aµ . In all these cases quantization of the system as a whole in the Schrödinger picture, as in (11.2), should proceed essentially as if the electromagnetic field was free - the interactions do not modify the canonical momenta. Such a canonical quantization of the electromagnetic field encounters, however, immediately an obstacle. If the four components Aµ are taken for independent dynamical variables, one finds that the momentum canonically conjugated to A0 vanishes identically: Πν ≡ ∂LEM 1 ∂LEM = − f0ν , = ∂(c ∂ 0 Aν ) c ∂ Ȧν (11.235) which implies that Π0 ∝ f00 ≡ 0. The electromagnetic field is therefore an example of a system subject to constraints. The peculiarity of the electromagnetic field is twofold: one is that Π0 = 0. The other one is its gauge invariance - the symmetry of the Lagrangian density (11.233) under (local) transformations Aµ (x) → A′µ (x) = Aµ (x) + 1 ∂µ θ(x) , e (11.236) in which θ is an arbitrary differentiable function of the space-time coordinates xµ . The invariance of the f µν fµν term of (11.233) is obvious as under the transformations (11.236) the tensor fµν does not change. Assuming that Jµ itself does not change, the eJµ Aµ term of (11.233) transforms into eJ µ Aµ + 366 R J µ ∂µ θ; the term J µ ∂µ θ does not contribute to the action I = d4 x LEM if the current is conserved, which is anyway the necessary condition for consistency of the field equations (11.224). Invariance of the current Jµ with respect to the transformations (11.236) and its conservation, ∂µ J µ = 0, are easy to check if it depends only, as in the case of nonrelativistic charged particles coupled to the electromagnetic field, discussed in Section 11.7, on dynamical variables describing “matter”. In more complicated cases, as for example in the case of scalar fields coupled to the electromagnetic field, the basic assuption is the gauge invariance of the complete action Z 1 d4 x fµν f µν + Irest [Aµ , ”matter”, ”fields”] , I[Aµ , ”matter”, ”fields”] ≡ − 4 by which one understands its invariance with respect to the transformation (11.236) of the electromagnetic field potentials supplemented with appropriate, dependent on θ(x), that is local, transformations of all other fields. For θ independent of x these transformations must form a U(1) group of global symmetries (which were discussed in Section 11.1) of the action Irest . For x-dependent θ this group becomes a local U(1) symmetry group of the complete action I[Aµ , ”matter”, ”fields”]. Gauge invariance of the complete action may be viewed as a fundamental principle, which must always be respected. This principle, generalized to nonabelian symmetry groups (see Section 20), underlies, in fact, the whole modern development of quantum field theory.48 In the case of the electromagnetic field coupled to matter or other fields, gauge invariance ensures that the generalized electromagnetic current J µ , which enters the right hand side of the Euler-Lagrange equation of the electromagnetic field ∂µ f µν = e ν J , c defined by49 −eJµ (x) ≡ δ δAµ (x) Irest [Aµ , ”matter”, ”fields”] , 48 (11.237) It will be seen, however, (Section 20.3) that to properly formulate quantum theories of gauge fields corresponding to nonabelian symmetry groups it is advantageous to take a more general approach and to start with the extended action I = Igauge inv + Inon−gauge inv which as a whole is not gauge invariant but posseses a continuous global symmetry of the supersymmetric type, called the BRST symmetry (Section 20.2). Quantization with the help of the Dirac method of Section 11.6 is then fairly easy and the BRST invariance of the quantum theory ensures that all the implications of gauge invariance are recovered in the properly defined physical subspace of the extended Hilbert space. 49 In most cases this current coincides with the Noether current of the globa U (1) symmetry. 367 even if it depends on Aµ , is gauge invariant and conserved. To see this let us consider Irest of the general form Z Irest [Aµ , φi ] = d4 x Lrest (Aµ , φi , ∂φi ) , (11.238) depending on Aµ and a set of fields φi , which under the changes (11.236) of the gauge transform according to the rule φi (x) → φ′i (x) = e−iθ(x)T ij φj (x) ≈ φi (x) − iθ(x)Tij φj (x) , (11.239) in which T is a matrix representation of the U(1) group generator. From the assumed gauge invariance of Irest for an infinitesimal gauge transformation we get 0 = Irest [A′µ , φ′i ] − Irest [Aµ , φi ] Z 1 δIrest = d4 x ∂µ θ(x) e δAµ (x) Z ∂Lrest ∂Lrest 4 θ(x)Tij φj (x) + ∂µ [θ(x)Tij φj ] . −i d x ∂φi (x) ∂(∂µ φi (x)) The last line is zero, as can be seen by integrating by parts and using the Euler-Lagrange equations of motion50 (11.4) of the fields φi . In view of the arbitrarines of θ(x) one gets (integrating by parts also in the first term) therefore the condition of the conservation of the current (11.237). Gauge invariance of the current (11.237) is of course guaranteed by the gauge invariance of the action (11.238). But it is precisely the gauge invariance of the complete action I[Aµ , ”matter”, ”fields”] describing interactions of the electromagnetic field, which is responsible for additional difficulties which we will have to overcome quantizing the electromagnetic field in Section 11.7. 11.5 The Proca field and its quantization Before dealing with the difficulties of the electromagnetic field, it is instructive to quantize first a simpler system with constraints, the so-called Proca field V µ (a vector field with a nonzero mass) whose classical dynamics is governed by the Lagrangian density51 LProca = − 1 1 Vµν V µν + M 2 Vµ V µ − g Vµ J µ , 4 2 50 (11.240) The reasoning remains valid also if the action Irest depends on higher derivatives of the fields φi , provided one uses the appropriate generalization of the Euler-Lagrange equations of motion. 51 We denote this field Vµ , instead of Aµ , to distinguish it from the electromagnetic field. 368 in which Vµν ≡ ∂µ Vν − ∂ν Vµ and g is the coupling constant (we have set ~ = c = 1). In the following we assume that the current J µ is independent of Vµ . The transformation (11.236) is not a symmetry of the the Lagrangian (11.240) due to the mass term 21 M 2 Vµ V µ by which it differs from (11.233). Still (11.240) leads to the similar (at first sight) obstacle in the canonical quantization as (11.233): Π0 ≡ 0. Before we overcome this problem we comment on the form of the Lagrangian density (11.240). Since it is not invariant under the gauge transformations (11.236), there is a priori no reason for which its derivative part should take the restricted form as in (11.240). In principle, we could write down the most general Lagrangian density, at most quadratic in Vµ and with at most two derivatives in the form 1 1 1 LProca = − ∂µ Vν ∂ µ V ν − κ ∂µ Vν ∂ ν V µ + M 2 Vµ V µ − g Vµ J µ , (11.241) 2 2 2 with an arbitrary real κ (any other Lagrangian density satisfying our requirements differ from the above by a total derivative; an arbitrary real negative constant, which could multiply the first term, can always be made equal to −1/2 by an appropriate rescaling Vµ , κ, M 2 and g Jµ ). However, taking the four-divergence of both sides of the Euler-Lagrange equation ∂µ ∂ µ V ν + κ ∂ ν (∂µ V µ ) + M 2 V ν = g J ν , (11.242) resulting from (11.241), we discover that the four-divergence ∂ν V ν ≡ φ satisfies the independent equation (1 + κ)∂µ ∂ µ φ + M 2 φ = g J˜ , (11.243) in which J˜ = ∂µ J µ . Therefore, the Lagrangian density (11.241) describes among other things also the dynamics of a scalar field52 coupled to the scalar ˜ To remove this propagation of an independent scalar field (that current J. is, to remove the homogeneous part of the classical solution for φ, in the case of a conserved current J µ ) we set κ = −1; taking then the four-divergence of both sides of the Euler-Lagrange equation (11.242) we find M 2 ∂µ V µ = g ∂µ J µ , (11.244) which shows that φ can be expressed in terms of the external current (which depends on other fields). For κ = −1 the Lagrangian density (11.241) is just the Proca Lagrangian density (11.240). 52 That an unconstrained vector field Vµ describes in addition a massive spin zero particle was also revealed in Section 8.4 in the course of constructing the free-field operator transforming as a vector representation of the Lorentz group uder changes of the reference frame. 369 We now quantize the Proca field. From its Lagrangian density (11.240) the canonical momenta Πν conjugate to the variables V µ are obtained: Πi = ∂LProca = −V0i = V̇ i + ∂i V 0 , i ∂ V̇ Π0 = 0 . (11.245) As we have stressed, Π0 ≡ 0 is an obstacle in the canonical quantization. The problem lies in the fact that the naive Hamilton’s formalism constructed ignoring the difficulty does not lead to equations equivalent to the original Euler-Lagrange equations ∂µ V µν = −M 2 V ν + g J ν , (11.246) following from the Proca Lagrangian density (11.240), that is to the equations ∂i Vi0 = M 2 V 0 − g J 0 , (11.247) ∂0 (∂0 V k + ∂k V 0 ) + ∂i (−∂i V k + ∂k V i ) = −M 2 V k + g J k , (11.248) obtained by setting in (11.246) ν = 0 and ν = k, respectively. The Hamiltonian corresponding to the Proca Lagrangian should be constructed following the standard recipe: H = Π0 V̇ 0 + Πi V̇ i − LProca 1 1 = Π0 V̇ 0 + Πi V̇ i − (V̇ i + ∂i V 0 )(V̇ i + ∂i V 0 ) + ∂i V j (∂i V j − ∂j V i ) 2 2 1 2 i i 1 2 0 0 + M V V − M V V − g V i J i + g V 0 J 0 . (11.249) 2 2 Eliminating V̇ i by using (11.245), i.e. substituting V̇ i = Πi − ∂i V 0 , one gets 1 1 H = Πi Πi + Π0 V̇ 0 (V λ , Πλ ) − Πi ∂i V 0 + (∇×V)2 2 2 1 2 i i 1 2 0 0 + M V V − M V V − g V iJ i + g V 0J 0 . 2 2 (11.250) Of course, since Π0 ≡ 0, there is no way to express the generalized velocity V̇ 0 through the canonical variables V λ and Πλ . Let us therefore see, what one gets, setting naively Π0 = 0 in the Hamiltonian R 3 (11.250). The canonical equations following from the Hamiltonian H = d x H obtained in this way read ∂H , V̇ i = V i , H PB = ∂Πi ∂H V̇ 0 = V 0 , H PB = ≡ 0, ∂Π0 ∂H , (11.251) ∂V i ∂H = − 0 6= 0 , ∂V Π̇i = {Πi , H}PB = − Π̇0 = {Π0 , H}PB 370 and are evidently incompatible with the Euler-Lagrange equation (11.246): the latter imply firstly that Π0 = 0 for any time t, whereas here, even if we set Π0 = 0 for t = 0, a nonzero Π0 will be generated for t 6= 0 due to a nonzero derivative Π̇0 . Secondly, from (11.247) expressed in terms of Πi it follows that the time evolution of V 0 is fully determined in terms of J 0 and ∂i Πi : 1 (11.252) V 0 = 2 ∂i Πi + g J 0 . M However, the third canonical equation (11.251) implies that V 0 is constant. Thus, the dynamics of V µ following from the Hamiltonian (11.250) is not the same as the one generated by the Euler-Lagrange equations (11.247) and (11.248). The equivalence is restored if in the Hamiltonian (11.250), in addition to setting Π0 = 0, one substitutes for V 0 the expression (11.252), that is, if one “algebraically” eliminates one of the canonical variables (in other words, if one expresses its value at any t by a combination of other canonical variables taken at the same instant t). One then obtains (integrating by parts): 1 1 1 (∂i Πi )2 + (∇×V)2 H = Πi Πi + 2 2 2M 2 2 1 g g + M 2 V iV i − g V iJ i + J 0 J 0 + 2 J 0 ∂i Πi . 2 2 2M M The canonical equations now become (11.253) 1 g ∂H = Πi − 2 ∂i (∂j Πj ) − 2 ∂i J 0 , (11.254) ∂Πi M M ∂H (11.255) Π̇i = − i = −ǫijk ǫklm ∂j ∂l V m − M 2 V i + g J i . ∂V Their equivalence with the Euler-Lagrange (11.246) equations can be seen as follows: If we define an auxiliary variable V 0 as in (11.252); the equation (11.247) is then automatically satisfied. Next, the equation (11.254) can be written in the form Πi = V̇ i + ∂i V 0 . Differentiating this relation with respect to time and inserting Π̇i obtained in this way into (11.255) transforms the latter equation into (11.248). Thus, by eliminating “algebraically”one of the variables, we have constructed the Hamilton’s formalism which is equivalent to the field equations following from the original Lagrangian (11.240). V̇ i = Quantization is now simple: one promotes V i (x, t) and Πi (x, t) at t = 0 to operators in the Schrödinger picture and imposes the standard canonical commutation relations h i i V̂ (x), Π̂j (y) = i V i (x), Πj (y) PB = iδji δ(x − y) , h i h i V̂ i (x), V̂ j (y) = Π̂i (x), Π̂j (y) = 0 . (11.256) 371 on the operators representing the independent canonical variables. In this way quantized get only physical degrees of freedom of the system of fields. If the current J µ depends on field variables φa and Πa other than V i and Πi themselves, the states |Ψi of the system can be represented as wave functionals Ψ[V i (x), φa (x), t] = h [V i ], [φa ]|Ψ(t)i on which the operators V̂ i (x) and Π̂i (x) act as V̂ i (x) = V i (x) , Π̂i (x) = −i δ . δV i (x) (11.257) The next task is to find such a representation of the operators53 V i (x) and Πi (x) in terms of the creation and annihilation operators (having the standard commutation rules) that the free part H0 of the Hamiltonian H (that is, its part independent of the current J µ ) becomes diagonal. To simplify this task we can apply the method outlined in Section 11.3 allowing to easily quantize the free field (i.e. for J µ = 0) in the Heisenberg picture; the freefield Heisenberg picture operators VHi (t, x), ΠH i (t, x) taken at t = 0 will just provide the necessary representations of the Schrödinger picture operators of the interacting (i.e. for J µ 6= 0) Proca field. To this end first write down the the most general solution to the classical field equations (11.242) and (11.244) following from the Proca Lagrangian, which for J µ = 0 read (∂ν ∂ ν + M 2 )V µ (t, x) = 0 , ∂µ V µ (t, x) = 0 . (11.258) The most general classical solution of the first equation can be written in the form (11.116). In order to satisfy the second of these equations we make in (11.116) the replacement X a(k) → a(k, λ) ǫµ (k, λ) , (11.259) λ=0,±1 where ǫµ (k, λ) are three vectors satisfying the conditions kµ ǫµ (k, λ) = 0 . (11.260) (since k 2 = M 2 6= 0 there can be only three linearly independent such vectors). Two of them, corresponding to λ = ±1, can be chosen to have only space-like components (in the plane perpendicular to k) and will be normalized so that ǫµ (k, λ) ǫ∗µ (k, λ) = −1 . 53 We suppress hats on operators from now on. 372 (11.261) (i.e. ǫ(k, λ) · ǫ∗ (k, λ) = 1). For the third one, corresponding to λ = 0, we take the four-vector |k| k E(k) µ ǫ (k, λ = 0) = , (11.262) , M |k| M which also satisfies the conditions (11.260) and (11.261). The vectors ǫµ (k, λ) satisfy then the sum rule X λ=0,±1 ǫµ (k, λ) ǫ∗ν (k, λ) = −gµν + kµ kν . M2 (11.263) Thus, for J µ = 0 for the free Heisenberg picture field operators we take Z X µ VH (t, x) = dΓk a(k, λ) ǫµ (k, λ) e−iEt+ik·x λ=0,±1 +a† (k, λ) ǫµ∗ (k, λ) eiEt−ik·x , (11.264) √ where, as usually, dΓk = d3 k/(2π)3 2E and E = k2 + M 2 (of the four components of VHµ only VHi ’s are independent). We then recall that in the classical Proca theory Πi (t, x) = V̇ i (t, x) + ∂i V 0 (t, x). In agreement with this relation, for the free operator ΠH i (in the Heisenberg picture) we take the expression Z X 1 H a(k, λ) ǫ̃i (k, λ) e−iEt+ik·x dΓk E(k) Πi (t, x) = i λ=0,±1 −a† (k, λ) ǫ̃i∗ (k, λ) eiEt−ik·x , (11.265) where ǫ̃i (k, λ) ≡ ǫi (k, λ) − ki 0 ǫ (k, λ) . E (11.266) Obviously, ǫ̃i (k, λ) = ǫi (k, λ) for λ = ±1 and for λ = 0 one finds ǫ̃i (k, λ = 0) = (k i /|k|)(M/E). It is then easy to verify that VH0 (t, x) is correctly reproduced by (11.252) for J µ = 0. Furthermore, one can check that the canonical equal-time commutation rules (11.256) are satisfied if a(k, λ), a† (k′ , λ′ ) = (2π)3 2E(k)δλλ′ δ (3) (k − k′ ) , [a(k, λ), a(k′ , λ′ )] = a† (k, λ), a† (k′ , λ′) = 0 . (11.267) It follows that for the Schrödinger picture operators V i (x) and Πi (x), which at t = 0 are equal to the Heisenberg picture operators of the interacting Proca field (which for t 6= 0 do not have the forms (11.264) and (11.265)!) one can take VHi (t = 0, x) and ΠH (t = 0, x), respectively. In terms of the 373 creation and annihilation operators introduced in this way, the free part H0 of the Hamiltonian (11.253) takes the form Z X H0 = dΓk E(k) a† (k, λ) a(k, λ) , (11.268) λ=0,±1 plus an infinite constant which we discard. It is also easy to see that the operators (11.264) and (11.265), which in the theory of the interacting Proca field become the interaction picture (free) operators VIi (x) and ΠIi (x) (see Section 11.9), are obtained from the Schrödinger picture ones as VIi (t, x) = eiH0 t/~ V i (x) e−iH0 t/~ , ΠIi (t, x) = eiH0 t/~ Πi (x) e−iH0 t/~ , as they should. Finally, one can compute the commutators of V 0 (x) which for nonzero J 0 is given by (11.252) with J 0 depending on the canonical variables of systems coupled to the Proca field but not on the canonical variables V i or Πi of the vector field itself. We find 1 i V i (x), V 0 (y) = 2 V i (x), ∂jy Πj (y) = − 2 ∂ix δ (3) (x − y) , M M Πi (x), V 0 (y) = 0 . (11.269) Thus, the operator corresponding to the time component of V µ does not commute with the operators corresponding to the remaining components. 11.6 The Dirac method In Section 11.5 the Proca vector field has been quantized by expressing “algebraically”one of its components (V 0 ) in terms of the remaining independent canonical variables (V i and Πi ). The form of the canonical commutation relations imposed on the independent canonical variables (that is the right quantization prescription) was then obvious and they determined also the commutation relations satisfied by the operator V 0 . Explicit solutions for dependent variables (like V 0 ) in terms of the other variables, chosen as independent, even if possible in principle, may not always be easy (in field theories solving “algebraically” may mean solving differential equations in the space variable x - the term “algebraic” refers only to the fact that no time derivatives are involved). Therefore, in this section we describe a systematic method proposed by Dirac, which allows to find the right commutation relations which must be imposed on the original set of canonical variables along with a set of identities which must be satisfied by operators representing these 374 variables, in order to properly quantize a system subject to constraints.54 In the case of systems subject to constraints (called second class) similar to the ones encountered in the Proca theory of the vector field the Dirac’s method allows to impose the right commutation relations directly on the original set of canonical variables without solving explicitly for dependent ones. The same method proves useful also for systems subject to another class of constraints (first class ones), which, as in the case of the electromagnetic field, reflect invariance of the (classical) physical state of the system with respect to changes of “gauge” - i.e. with respect to appropriate transformations of the canonical variables. Finally, the Dirac’s method allows for an easy canonical quantization of theories of non-Abelian Yang-Mills fields which in the BRST formulation become systems subject to second class constraints (see Section 20.3). Canonical quantization consists of the identification of the canonical variables q i , i = 1, . . . , n and the conjugated momenta pi and their subsequent promotion to the Schrödinger picture operators q̂ i and p̂i satisfying the commutation rule i q̂ , p̂j = i~ q i , pj PB = i~ δ i j . The method in this simple form is applicable if the canonical variables and their momenta are all independent. Classical systems may, however, be subject to constraints as a result of which not all their canonical variables are independent. The primary constraints ΦM (q, p) = 0, M = 1, . . . , n−r, follow from the structure of the Lagrangian and reflect the impossibility to solve for q̇ j , j = r + 1, . . . , n the equations pj = ∂ L(q, q̇) , ∂ q̇ j (11.270) for all q̇ j . In such a case the original Euler-Lagrange equations of motion are equivalent (see Appendix I) to the set of canonical equations which follow from the extended Hamiltonian HE of the form ! n n−r X X ˙ ∂L i HE (q, p) = q − L(q, q̇) + ΦM (q, p) uM i ∂ q̇ i i=1 M =1 q̇=q̇ (q,p,q̇) ≡ H(q, p) + n−r X ΦM (q, p) uM , (11.271) M =1 combined with the set of (primary) constraints ΦM = 0. The subscript q̇ = q̇ i (q, p, q̇) on the bracket in the first term of (11.271) means that r of the 54 The method itself, however, does not give any clues, how to satisfy these commutation relations and the identities by expanding field operators taken at t = 0 into creation and annihilation operators. 375 generalized velocities q̇ i (those for which the relations (11.270) can be solved for) are expressed in terms of the canonical variables and the remaining n − r generalized velocities.55 The sum in the second term runs over the set of n−r primary constraints ΦM (q, p) = 0. The factors uM , which can be viewed as Lagrange multipliers, are just those n − r generalized velocities q̇ i , for which the equations (11.270) cannot be solved. By construction (see Appendix I) primary constraints ΦM (q, p) = 0 have the structure pM − fM (q, pk ) = 0, M = r + 1, . . . , n, and k = 1, . . . , r, where pM are the canonical momenta corresponding to the velocities uM . The mentioned equivalence is to be understood in the sense that the constraints ΦM (q, p) = 0 differentiated with respect to time, combined with the evolution equations q̇ i = {q i , HE }PB and ṗi = {pi , HE }PB obtained from the extended Hamiltonian HE (11.271) yield second order differential equations for q i which are equivalent to the original Euler-Lagrange equations. The dynamics can be cast into a fully Hamiltonian form by giving the factors uM an appropriate dependence on pi and q i . This is done as follows. The constraints ΦM (q, p) = 0 should be preserved by the dynamics generated by the Hamiltonian (11.271) - in this case it will suffice to impose them on the initial data only. This means that Φ̇N (q, p), that is the Poisson brackets {ΦN , HE }PB , must be made vanishing. Investigating these Poisson brackets one can encounter different situations. Barring the case of obvious contradictions, like 1 = 0, the first possibility is that {ΦN , HE }PB vanishes if the already identified constraints are imposed after computing the Poisson bracket (the Poisson brackets are computed treating all variables q i and pi as independent). We denote this {ΦN , HE }PB ≃ 0 , where the symbol ≃ means “vanishes weakly” (Dirac’s terminology). The second possibility is that the investigated Poisson bracket of the constraint ΦN with the Hamiltonian (11.271) does not vanish, even weakly, and the result does not depend on the (yet) unknown coefficients uN . This means that the new constraint ΦK ≡ {ΦN , HE }PB = 0 has to be added to the list of constraints. Constraints identified in this way are called secondary. The obvious next step is to investigate the Poisson bracket of the newly identified constraint ΦK with the Hamiltonian (11.271) which can lead to a yet new constraint and so on. Proceeding in this way, one eventualy identifies all the constraints a given system is subject to. Finally, there are Poisson brackets of ΦN with the Hamiltonian HE which do not vanish weakly but depend on the coefficients uM . Requiring that all such Poisson brackets, obtained for 55 In Appendix I it is shown that the content of the bracket, that is H(q, p) in the second line of (11.271), is independent of the n − r velocities q̇ j for which the relations (11.270) cannot be solved. 376 primary and all secondary constraint (weakly) vanish one gets the system of linear equations {ΦN , H}PB + n−r X {ΦN , ΦM }PB uM ≃ 0 , (11.272) M =1 in which the sum runs over the n − r primary constraints. The solution of this system of linear inhomogeneous equations takes the general form u M M = c (q, p) + p X M v(a) (q, p) sa , (11.273) a=1 M in which v(a) (q, p), a = 1, . . . , p are all linearly independent solutions of the homogeneous system n−r X M {ΦN , ΦM }PB v(a) (q, p) ≃ 0 . (11.274) M =1 In this way the extended Hamiltonian (11.271) gets fixed and all the consistency conditions {ΦN , HE }PB ≃ 0 are satisfied. An important feature of the extended Hamiltonian (11.271) is its possible dependence on the coefficients sa , which may be arbitrary functions of time. They are present whenever there exist linear combinations Φa (q, p) ≡ M ΦM (q, p) v(a) (q, p) of the primary constraints which, as follows form (11.274), have (weakly) vanishing Poisson brackets with all other constraints. Linear combinations Φa (q, p) such that {ΦN , Φa }PB ≃ 0 for all N (11.275) (the Poisson brackets should be computed as if the variables were unconstrained; the constraints ΦM = 0 are imposed afterwards) are called first class constraints. The remaining ones, which have at least one nonzero Poisson bracket with the remaining constraints, are called second class. It should be noted at this point that the consistency conditions ensured by fixing the coefficient functions uM (q, p) as in (11.273) would not be spoiled if other second class constraints, formed as linear combinations which include also secondary constraints, are included into the Hamiltonian HE (11.271). Thus for the complete Hamiltonian HT of the system one may take the expression HT (q, p) = H(q, p) + 2k X ΦM (q, p) cM (q, p) + X Φa (q, p) sa (t) , (11.276) a M =1 in which H is defined by (11.271), the sum over M runs over all 2k ≤ n − r second class constraints (formed as linear combinations of the promary and 377 secondary constraints) and the sum over a runs over all primary first class constraints. Consistency of the constraints with the dynamics generated by HT is ensured by giving (solving the equations like (11.272)) the coefficients cM (q, p) the appropriate dependence on q and p (it is then sufficient to impose the constraints on the initial data only), while the factors sa (t) can be arbitrary functions of time without affecting this consistency. Due to the presence of the arbitrary functions sa (t), which is always the case if there are primary first class constraints, the time evolution of the canonical variables q i (t) and pi (t) generated by the Hamiltonian (11.276) is not unique - the initial data q i (t0 ) and pi (t0 ) set at some t = t0 do not specify uniquely the values of q i (t) and pi (t) at other times t. Another possible situation arises if there are secondary first class constraints (and no primary first class ones): the time evolution is then unique, once the initial data are given, but their choice is not. Thus, first class constraints are always associated with some kind of “gauge invariance” by which one means a situation in which the (classical) physical state of a system subject to such constraints does not change if the values of the variables used to describe it are changed appropriately. In other words, infinitely many different values of the same canonical variables correspond to the same (classical) physical state. It is easy to see, that the infinitesimal transformations of the canonical variables which do not change the physical state are generated by the primary (?) first class constraints Φa : δq i = θa {q i , Φa }PB , δpi = θa {pi , Φa }PB , with arbitrary time dependent parameters θa . There are two ways of quantizing systems subject to first class constraints. One consists of working in the extended Hilbert space of states |Ψi. The physical states |Ψphys i, forming a subspace of the full Hilbert space, are then selected by the conditions Φa (q̂, p̂)|Ψphys i = 0 , for all a. (11.277) This approach which in principle requires some care in ordering the operators q̂, p̂ so that [Φ̂a , Φ̂b ] = cd (q̂, p̂)Φ̂d (as is necessary for consistency of the conditions (11.277)), is used e.g. to quantize the electromagnetic field in the covariant Lorentz gauge ∂µ Aµ = 0 (this is very similar to the GuptaBleuler method of the electromagnetic field quantization - see Section 11.11) and in the quantization of the relativistic string (see the old review by C. Rebbi). The alternative way, which will be demonstrated on the example of the electromagnetic field in the next section, is to fix the gauge completely. 378 This is done by imposing additional constraints Φ(q, p) = 0 from outside,56 so that the entire system of constraints becomes second class and the time evolution of the variables becomes uniquely determined by the equations of motion and the initial data; the resulting classical systems subject to second class constraints can be quantized with the help of the Dirac prescription which we describe below and apply next to the Proca theory of the vector field. Finally, it may happen that the additional constraints do not fix the gauge completely and there remain some secondary first class constraints (but all primary constraints are already second class). In this case the classical time evolution is uniquely determined but there is a freedom in fixing the initial data. The quantum theory can be then constructed by treating the second class constraints á la Dirac (see below) and the remaining first class constraints should be imposed as subsidiary conditions selecting those vectors of the Hilbert space which represent quantum states of the physical system. To quantize a theory in which first class constraints are absent from the beginning or have been eliminated by imposing additional constraints from outside, one forms the matrix CN M : CN M ≡ {ΦN , ΦM }PB . (11.278) Since the matrix CN M is antisymmetric (the basic property of the Poisson brackets) and det(CN M ) 6= 0 (vanishing of this determinant would mean that one can form at least one more linear combination of the constraints which has zero Poisson brackets with all others and, hence, belongs to to the first class constraints which we have assumed to be already eliminated), it must be of even dimension. The Dirac quantization prescription for systems subject to second class constraints then reads [q̂, p̂] = i~ {q, p}D , ΦN (q̂, p̂) ≡ 0 , for all N , (11.279) that is, the commutators (or anticommutators in the case of fermionic fields - see section 11.8) are determined by the Dirac brackets (instead of Poisson brackets) and all the constraints ΦM = 0 are realized as strong operator identities. The Dirac bracket {A(p, q), B(p, q)}D of two functions of the canonical variables is defined as follows: X N M {A, B}D ≡ {A, B}PB − {A, ΦN }PB C −1 {ΦM , B}PB , (11.280) N,M 56 This is possible because, as explained above, the physical state of the system does not fix the values of the canonical variables uniquely; the added constraints can be then made consistent with the dynamics generated by HE (or HT ) from which arbitrary functions sa are now absent. 379 (again the Poisson brackets have to be computed as if there were no constraints). The Dirac bracket shares all the algebraic properties of the Poisson bracket: it is bilinear in canonical variables, antisymmetric and satisfies the Jacobi identity. It is also a matter of a straightforward algebra to check that the definition (11.280) is invariant with respect to linear changes N ΦM (q, p) = OM (q, p) Φ̃N (q, p) , N of the basis of constraints, provided the matrix OM (q, p) is nonsingular (invertible) on the surface of constraints. Finally, the Dirac bracket has the (easy to verify) property {A, ΦN }D = 0 , (11.281) which ensures compatibility of the commutation relation (11.279) with the constraints realized as strong operator identities. This also means that the Hamiltonian operator is obtained from the H part of the extended classical Hamiltonian HE (11.271): the sum of the primary constraints (which in the classical theory must be kept in order to ensure the compatibility of constraints with the dynamics) is in the quantum theory just a zero operator. The Dirac prescription can be justified as follows. First of all, one can notice that if the first class constraints are absent, that is the last sum is absent in the Hamiltonian HT (11.276), all the factors cM (q, p) are determined from the consistency conditions {ΦN , HT }PB ≃ 0 and are given by cM (q, p) = −(C −1 )M N {ΦN , HT }PB . The canonical equtions of motion q̇ i = {q i , HT }PB and ṗi = {pi , HT }PB take then just the form i i i q̇ = {q , HT }PB = {q , H}PB + 2k X {q i , ΦM }PB cM (q, p) ≡ {q i , H}D , M =1 and similarly for ṗi . Since in the quantum theory the equations of motion satisfied by the Heisenberg picture operators are given by their commutators with the Hamiltonian, the above form of the classical equations of motion strongly supports the Dirac quantization rule. More formally, it can be shown that for the even number of second class constraints ΦM , M = 1, 2, . . . , 2k there always exists a canonical transformation (qi , pi ) → (Qi (q, p), Pi (q, p)), i = 1, 2, . . . , n, such that in the new canonical variables the constraints ΦM take the simple form Qi = 0 , Pi = 0 for i = 1, . . . , k. (11.282) One can therefore quantize the system in the remaining new variables (Qi , Pi ) with i = k + 1, . . . , n forgetting about the first k + k = 2k variables which 380 are zero. The commutator of the quantum operators (q̂i , p̂j ) when expressed in terms of the commutators [Q̂i , P̂j ] = i~ δij for i, j > k are then given by the Dirac bracket (times i~). The advantage of the Dirac method for finding the right commutation relations which must be satisfied by the canonical variables is that one does not have to find the canonical transformation (qi , pi ) → (Qi (q, p), Pi (q, p)) explicitly nor does one have to solve the constraints for some variables in terms of others. In the case of the Proca field the primary constraint57 Φ1 ≡ Π0 (t, x) = 0 , (11.283) follow from the structure of the Lagrangian (11.240). The extended Hamiltonian takes therefore the form 1 1 1 HE = Πi Πi − Πi ∂i V 0 + (∇×V)2 + M 2 V i V i 2 2 2 1 2 0 0 − M V V + Π0 u1 − g V i J i + g V 0 J 0 , (11.284) 2 with an initially unknown function u1 . Since Φ̇1 = {Φ1 , HE }PB does not vanish (even weakly), the secondary constraint Φ2 ≡ M 2 V 0 (t, x) − ∂i Πi (t, x) − g J 0 (t, x) = 0 , (11.285) must be imposed as the consistency condition. There no more constraints vanishing of Φ̇2 = {Φ2 , HE }PB can be ensured by adjusting the function u1 . This completes the Hamiltonization of the classical dynamics of the Proca field. In the quantum theory Φ1 and Φ2 become operator identities; therefore, Π0 must be represented by the zero operator and through (11.285) the operator V 0 becomes completely determined by Πi and J 0 . The canonical commutation relations of the V i , Πi , V 0 and Π0 operators are given by the Dirac prescription (11.279) which ensures their compatibility with the operator relations Φ1 = 0 and Φ2 = 0. The matrix CN M (11.278) has in this case the form C1x,2y = {Φ1x , Φ2y }PB = {Φ1 (x), Φ2 (y)}PB = −M 2 δ (3) (x − y) , (11.286) and its inverse reads C −1 = − M12 0 (3) δ (x − y) 57 1 M2 δ (3) (x − y) 0 . (11.287) In field theory the index i of q i in the formulae (11.270)-(11.282) includes also the space variable x. Hence, each ΦM stands for an infinite set of constraints ΦMx ≡ ΦM (x). 381 The Dirac bracket of any two functions A(x) and B(y) of the canonical variables V µ and Πµ therefore reads {A(x), B(y)}D = {A(x), B(y)}PB Z 1 − 2 d3 z {A(x), Π0 (z)}PB M 2 V 0 (z) − ∂i Πi (z) − gJ 0 (z), B(y) PB M Z 1 + 2 d3 z A(x), M 2 V 0 (z) − ∂i Πi (z) − gJ 0 (z) PB {Π0 (z), B(y)}PB . M Computing using this formula the commutator i V (x), V 0 (y) = i~ V i (x), V 0 (y) D , (11.288) one finds the same result as was obtained in section 11.5 by treating the V 0 operator as constructed out of Πi and J 0 in agreement with the constraint Φ2 = 0. It is also easy to check that the Dirac prescription gives all the remaining commutators in their standard forms. The rest of the quantization procedure is then unmodified. 11.7 Quantization of the electromagnetic field We now deviate a little bit from the main logic of this Section (devoted mainly to quantization of systems of relativistic fields) and consider quantization of the system consisting of the classical electromagnetic field coupled to N classical nonrelativistic charged particles. The reason for this deviation is that in this way we will obtain a working theory, called nonrelativistic quantum electrodynamics (NRQED), which is applicable to a wide range of physical phenomena related to interaction of light with nonrelativistic matter, so long as spin effects can be neglected. It also constitutes a modern starting point for calculations of properties of bound states in fully relativistic quantum electrodynamics. The Lagrangian describing classical dynamics of such a system is given by the sum of three terms: L = LEM + Lmatt + Lint , where Z 1 d3 x fµν f µν , (11.289) LEM = − 4 N Lmatt N N X 1X 1X mi vi2 − V (ri ) − V (ri − rj ) , = 2 i=1 2 i=1 (11.290) i6=j Lint Z N h i X qi e = −e qi φ(ri ) − A(ri ) ·vi ≡ − d3 x Jµ (x)Aµ (x) , (11.291) c c i=1 382 where the current in the last formula has the form (qi are particle charges in units of e > 0) 0 J (t, x) = J(t, x) = N X i=1 N X i=1 qi c δ (3) (x − ri (t)) , qi vi δ (3) (x − ri (t)) . (11.292) For greater generality we have allowed in Lmatt also for 1-particle interactions V (ri ) with external potentials (which can be also of electromagnetic origin e.g. the electrostatic interactions binding electrons in atoms, which we may treat separately from the dynamical electromagnetic field) and two-particle interactions V (ri − rj ) of non-electromagnetic origin. To quantize this system we first try to take for its canonical variables and conjugated momenta ri (t), Pi (t), Aµ (t, x) and Πµ (t, x). In the matter part everything goes in the standard way: Pi (t) = qi ∂L = mi vi (t) + eA(t, ri (t)) . ∂vi (t) c (11.293) Expressing vi (t) in the matter part of the Hamiltonian through Pi (t) we get H = HE + N 2 X 1 qi Pi − eA(ri ) 2mi c i=1 +e N X qi φ(ri ) + i=1 N X V (ri ) + i=1 N 1X V (ri − rj ) , 2 i6=j (11.294) where HE is the current-independent (here “current-independent” before expressing φ in terms of J 0 !) part of the (extended) electromagnetic field Hamiltonian (11.319) which we derive below. Quantization of the matter part is standard: we promote ri (t) and Pi (t) taken at t = 0 to Schrödinger picture operators satisfying the commutation rules [r̂i , P̂j ] = i~ δij . We can then work either in the position representation in which P̂i = −i~ ∂/∂ri , or we can go over to the second quantization formalism (presented in Section 5), which is especially convenient if the matter particles are identical. The electromagnetic part of the system is more troublesome. Written in terms of the potentials the Lagrangian density L (11.233) of the electromagnetic field coupled to the current J µ reads (c is kept for decoration, φ ≡ A0 ): LEM 1 = 2 1∂ A + ∇φ c ∂t 2 − 1 1 1 (∇ × A)2 − eJ 0 φ + eJ·A . (11.295) 2 c c 383 R Of course, d3 x LEM = LEM + Lint in the notation of (11.289-11.291). As already checked in Subsection 11.4, the Euler-Lagrange equations following from this Lagrangian density are (ρ ≡ eJ 0 ) ∇×B − 1 1 ∂E = eJ , c ∂t c ∇·E = ρ . (11.296) As the canonical momenta Πi we get Πi = 1 ∂Ai 1 1 + ∂i φ = − E i , 2 c ∂t c c (11.297) and Π0 = 0, similarly as in the case of the Proca vector field. We have therefore the primary constraint Φ0 ≡ Π0 and we have to check whether it is compatible with the Hamilton’s canonical equations. To this end we form the Hamiltonian density HE ≡ HEM + Hint + Φ0 u0 according to the Dirac method expressing Ȧi through Πi by using the relation (11.297) and adding the primary constraint with an unknown coefficient u0 . This gives:58 HE = Πi Ȧi + u0 Φ0 − L 1 1 1 c2 = Πi Πi − c Πi ∂i φ + (∇×A)2 + Π0 u0 + eJ 0 φ − eJ·A 2 2 c c 1 1 2 2 0 E + B + E·∇φ + Π0 u + ρ φ − eJ·A , (11.298) = 2 c Using this Hamiltonian density we find59 Z ∂HE ∂u0 1 3 Φ̇0 ≡ Π0 , d x HE =− = −Π0 − c ∂i Πi − eJ 0 0 ∂φ ∂A c PB 1 (11.299) ≃ −c ∂i Πi − eJ 0 , c As this is not zero (even weakly), the secondary constraint Φ2 ≡ c2 ∂i Πi + eJ 0 has to be imposed. This is obviously the Gauss law, which is the second of the Euler-Lagrange equations (11.296), but is lost in the Hamilton’s formalism because of vanishing of Π0 . The constraint Φ2 is already compatible with the dynamics: {Φ2 , HE }PB = c ∂µ (eJ µ ) vanishes provided the current is conserved.60 58 Note, that J depends now on r̂i and P̂i . In computing Poisson brackets all variables and momenta have to be treated as independent; therefore, for any two functionals G and H we have Z δH δG δH δG µ µ 3 . − G[A , Πµ ], H[A , Πµ ] ≡ d x δAλ (x) δΠλ (x) δAλ (x) δΠλ (x) PB 59 60 To derive the relation {Φ2 , HE }PB = c ∂µ (eJ µ ) one must take into account the full Hamiltonian of the theory which in the case considered here includes, in addition to the terms displayed in (11.298), also the term Hmatt governing the dynamics of the nonrelativistic particles; one then finds that {J 0 , HE }PB = c ∂0 J 0 . 384 The system is therefore subject to two constraints, Φ0 and Φ2 , but they turn out to be the first class: {Φ0 , Φ2 }PB = 0 , and the Dirac prescription for quantization cannot be applied because the (classical) state of the electromagnetic field, fully characterized by E and B, does not determine uniquely the potentials Aµ . The mathematical reason for this is the gauge invariance of the Lagrangian (11.295). The Euler-Lagrange equations do not determine Aµ (t, x) uniquely for t 6= 0 from initial conditions Aµ (x) and Ȧµ (x) specified at t = 0: one can always imagine a function θ(t, x) such that ∂ µ θ(t, x) = 0 and ∂ µ θ̇(t, x) = 0 for t = 0, but nonvanishing for t 6= 0. If Aµ (t, x) solves the Euler-Lagrange equations with the given initial conditions, then so does Aµ (t, x) + ∂ µ θ(t, x) with the same initial conditions at t = 0. Since the constraints Φ0 = 0 and Φ2 = 0 essentially make the canonical Hamilton’s equations equivalent to the Euler-Lagrange equations, it is clear that they cannot help to determine Aµ (t, x) uniquely for all t as it was possible for the Proca vector field. Unique determination of the time evolution of Aµ (t, x) becomes possible if one chooses a gauge. In a fixed gauge the Euler-Lagrange equations can determine Aµ (t, x) for all t unambiguously and it should be possible to find an equivalent Hamilton’s canonical formulation too. One possibility is the Coulomb gauge ∇·A = 0 . (11.300) Another, convenient in some applications, is the temporal gauge A0 = 0. However, while the latter makes the time evolution unique, it does not fix uniquely the initial data: transformations with time independent function θ(x) are still possible. As a result, the constraint Φ2 = 0 (the Gauss law) remains in this approach first class and has to be imposed as a subsidiary condition selecting physical state-vectors of the Hilbert space. For this reason we here choose to work in the Coulomb gauge (11.300) which classically has a clear physical interpretation: it disentangles electrostatic effects from those due to radiation. We restart, therefore, the whole procedure with the same Lagrangian density (11.295) and one primary constraint Φ1 ≡ ∇·A ≡ ∂i Ai , (11.301) imposed from outside. As previously one finds Φ0 ≡ Π0 as the primary constraint and has to consider the extended Hamiltonian which now involves two initially unspecified functions u0 and u1 : HE = c2 1 1 1 Πi Πi − c Πi ∂i φ + (∇×A)2 + Φ0 u0 + Φ1 u1 + eJ 0 φ − eJ·A . 2 2 c c 385 The consistency condition {Φ0 , HE }PB ≃ 0, leads again to the secondary constraint (the Gauss law) Φ2 ≡ c2 ∂i Πi + eJ 0 . (11.302) In turn the requirement that {Φ1 , HE }PB ≃ 0 leads to yet another constraint Φ3 ≡ c2 ∂i Πi − c ∂i ∂i φ . (11.303) There are no more constraints: conservation of the current J µ implies vanishing of Φ̇2 , provided the function u1 is set to zero, while u0 can be adjusted to ensure Φ̇3 = 0. It is convenient to replace the constraint Φ3 (11.303) by the linear combination Φ3′ of (11.303) and (11.302): Φ3′ ≡ c ∂i ∂i φ + eJ 0 = 0 . (11.304) This shows that in the quantum theory the operator A0 (t, x) ≡ φ(t, x) is related to the operator J 0 by the identity:61 Z J 0 (t, y) e d3 y . (11.305) φ(t, x) = 4πc |x − y| Thus, the operator A0 (t, x) ≡ φ(t, x) becomes completely determined in terms of the matter canonical variables only (note, that J 0 given by (11.292) does not depend on vi and, hence, J 0 , unlike J, when expressed through r̂i and P̂i does not depend on Ai ). Since Π0 must be represented by the zero operator, the constraints Φ0 and Φ2 effectively eliminate these variables as independent operators from the quantum theory. The complete system of constraints Φ0 , Φ1 , Φ2 and Φ3′ is of second class. The corresponding 4×4 matrix CM N = −CN M is nonsingular - it has nonzero elements on the antidiagonal: C1x,2y = {Φ1 (x), Φ2 (y)}PB (x) (y) i (x) (x) = c2 ∂i ∂j A (x), Πj (y) PB = −c2 ∂i ∂i δ (3) (x − y) , (11.306) and C0x,3′ y = {Φ0 (x), Φ3′ (y)}PB = 1 (x) (x) C1x,2y = −c ∂k ∂k δ (3) (x − y) . (11.307) c The canonical commutation relation which must be imposed on the operators Ai , Πi , A0 and Π0 are determined by the Dirac prescription (11.279). 61 1 Recall that ∇2 |x| = −4π δ (3) (x). 386 The inverse (C −1 )N x,M y of the CM N matrix has also nonzero elements only on its antidiagonal. These are C −1 1x,2y =− 1 4πc2 |x − y| = 0x,3′ y 1 C −1 , c (11.308) Using this matrix one determines the basic commutator: 1 i A (x), Πj (y) = Ai (x), Πj (y) D ≡ Ai (x), Πj (y) PB (11.309) i~ Z Z 2z,1w − d3 z d3 w Ai (x), Φ2 (z) PB C −1 {Φ1 (w), Πj (y)}PB (terms with the remaining pairs of constraints give all zero). Using the Poisson brackets i (x) A (x), Φ2 (z) PB = −c2 ∂i δ (3) (x − z) , (w) {Φ1 (w), Πj (y)}PB = ∂j δ (3) (w − y) , it is easy to find that i (x) (x) A (x), Πj (y) D = δji δ (3) (x − y) + ∂i ∂j 1 . 4π|x − y| (11.310) Proceeding in the similar way, one can check that all the remaining Dirac brackets among the system of variables Ai , Πj , A0 and Π0 vanish (vanishing of the Dirac brackets of Π0 ≡ Φ0 with all functions of the canonical variables of the entire theory is ensured by the property (11.281)). In particular, i A (x), Aj (y) D = {Πi (x), Πj (y)}D = 0 . (11.311) Therefore, the basic quantization conditions read 1 (x) (x) , Ai (x), Πj (y) = i~ δji δ (3) (x − y) + i~ ∂i ∂j 4π|x − y| i A (x), Aj (y) = [Πi (x), Πj (y)] = 0 . (11.312) The function on the right hand side of the first commutation rule (11.312) is called the transverse delta function. It has the following momentum space representation 1 4π|x − y| Z 3 d k ik·(x−y) ij k i k j ij δ − , e ≡ δtr (x − y) = (2π)3 |k|2 j i δ ij δ (3) (x − y) + ∂(x) ∂(x) 387 (11.313) which makes it explicit that (x) (x) x ∂i δji δ (3) (x − y) + ∂i ∂j 1 4π|x − y| = 0, (11.314) and, therefore, that the commutation rules (11.312) are compatible with the constraint (11.301). The Hamiltonian density operator c2 1 1 1 Πi Πi + (∇×A)2 − c Πi ∂i φ + eJ 0 φ − eJ·A , (11.315) 2 2 c c in which φ is now a shorthand for the operator (11.305) and Π0 has been set to zero, together with the quantization rules (11.312) specify in principle the dynamics of the quantized electromagnetic field. There is, however, one technical problem that the operator Πi (x) does not commute with matter sector canonical variables. This is because the Dirac brackets of Πi (x) with functions F depending on the matter sector canonical variables do not necessarily vanish.62 Indeed, according to the Dirac prescription such commutators are given by i~ times the Dirac brackets H= {F (mat), Πi (x)}D = {F (mat), Πi (x)}PB Z Z M z,N w 3 − d z d3 w {F (mat), ΦM (z)}PB C −1 {ΦN (w), Πi (x)}PB . The Poisson bracket in the first line is of course zero. To the second line contributes only the term with MN = 21. This line therefore gives Z Z 1 (w) 3 ∂i δ (3) (w − x) − d z d3 w {F (mat), Φ2 (z)}PB 2 4πc |z − w| Z 1 (x) = d3 z F (mat), eJ 0 (z) PB ∂i . 2 4πc |z − x| This is evidently nonzero, if F is a function of Pj because J 0 (z) given by (11.292) depends on ri , and {ri , Pj }PB = δij . Dealing with such an operator would be difficult. Therefore, one defines another operator 1 ∂i φ , (11.316) ΠT i = Πi − c which in fact is just (1/c2 )Ȧi (cf. (11.297)), i.e. the transverse (the divergenceless) part of the electric field, which has better properties. Firstly, 1 (x) − ∂i {F (mat), φ(x)}D c Z Z M z,N w 1 (x) 3 = ∂i d z d3 w {F (mat), ΦM (z)}PB C −1 {ΦN (w), φ(x)}PB c Z 1 1 (x) d3 z F (mat), eJ 0 (z) PB = − ∂i c 4πc|z − x| 62 It is easy to check that {F (mat), Ai (x)}D = 0. 388 ′ (only the element (C −1 )3 z,0w contributes to the sum over constraints; the Poisson bracket {F (mat), φ(x)}PB obviously vanishes), so this precisely cancels out the unwanted term in the Dirac bracket of Πi with F (mat). Hence, the operator ΠT i does commute with the matter variables. Secondly, since 1 (x) i A (x), φ(y) D = 0 − ∂i c (the Poisson bracket of Ai (x) with φ(y) vanishes, Ai has nonzero Poisson bracket only with Φ2 , while φ only with Φ0 and (C −1 )2z,0w = 0, so the second term in the Dirac bracket vanishes too), it follows that i (x (x) i (3) A (x), ΠT j (y) = i~ δj δ (x − y) + i~ ∂i )∂j 1 , 4π|x − y| (11.317) just as for Πi (y). Finally, one has to check the Dirac bracket T 1 (x) (y) Πi (x), ΠT j (y) D = {Πi (x), Πj (y)}D + 2 ∂i ∂j {φ(x), φ(y)}D c 1 (y) 1 (x) − ∂i {φ(x), Πj (y)}D − ∂j {Πi (x), φ(y)}D . c c As already has been established in (11.311), the first term on the right hand side is zero. To show that the remaining three terms also vanish, we note that {φ(x), φ(y)}D = {φ(x), φ(y)}PB Z Z M z,N w 3 {ΦN (w), φ(y)}PB = 0 : − d z d3 w {φ(x), ΦM (z)}PB C −1 the Poisson bracket of two φ’s vanishes and the second term vanishes because φ’s have nonzero Poisson bracket Π0 and (C −1 )0z,0w = 0. To com only with y plete the proof, we consider Πi (x), ∂j φ(y) D : the corresponding Poisson bracket is zero and Z Z M z,N w 3 ΦN (w), ∂jy φ(y) PB − d z d3 w {Πi (x), ΦM (z)}PB C −1 also vanishes because Πi has a nonzero Poisson bracket only with Φ1 , φ only with Φ0 and (C −1 )1z,0w = 0. We have thus shown, that the new operator ΠT i has all the properties required from the canonical momentum operator but in contrast to Πi , it commutes with operators representing the matter sector variables. In addition, ΠT i satisfies a simpler constraint than Φ2 = 0: Φ2′ ≡ ∂i ΠT i = 0. 389 (11.318) Expressing now the Hamiltonian density operator (11.315) through ΠT i we obtain H= 1 1 1 c2 T T 1 Πi Πi + (∇×A)2 − ∂i φ∂i φ + e J 0 φ − eJ·A 2 2 2 c c c2 T T 1 1 1 = Πi Πi + (∇×A)2 + eJ 0 φ − e J·A . 2 2 2c c (11.319) The noncovariantly looking term eJ 0 φ/2c (obtained after integrating by parts and using the constraint Φ3′ = 0 (11.304)) produces in the Hamiltonian the term representing energy of the ordinary electrostatic Coulomb interaction Z Z e2 J 0 (x)J 0 (y) 1 3 d x d3 y . (11.320) H⊃ 2 4πc2 |x − y| It remains to find the representation of the Schrödinger picture operators A and ΠT simi in terms of the creation and annihilation operators satisfying R ple commutation rules and diagonalizing the Hamiltonian H0EM = d3 x H0EM obtained from (11.319) for J µ = 0. In addition, these representations should automatically ensure that the constraints Φ1 = Φ2′ = 0 hold as operator identities. It is easy to guess that Z 3 s 2 Xh i ~c dk † i ik·x i∗ −ik·x , (11.321) ǫ (k, λ)a (k)e + ǫ (k, λ)a (k)e Ai (x) = λ λ (2π)3 2ωk λ=±1 i where ωk = c|k| and ǫi (k, λ) are two polarization vectors such that k·ǫ(k, λ) = 0 (11.322) (so that the constraint Φ1 ≡ ∇·A = 0 is satisfied) and ǫ(k, λ)·ǫ∗ (k, λ′ ) = δλλ′ . (11.323) Because together with k/|k| the two vectors ǫ(k, λ) form an orthonormal basis of the three-dimensional space, the following summation rule holds X λ=±1 ǫi (k, λ) ǫj∗ (k, λ) = δ ij − kikj . |k|2 For the operator ΠT i we then postulate the expansion r Z ~ωk X i 1 d3 k T ǫ (k, λ) aλ (k) eik·x Πi (x) = 3 2 i (2π) 2c λ=±1 (11.324) i − ǫi∗ (k, λ) a†λ (k) e−ik·x . (11.325) 390 It is then straightforward to check that with the commutation rules h i † ′ aλ (k), aλ′ (k ) = (2π)3 δλλ′ δ(k − k′ ) , i h (11.326) [aλ (k), aλ′ (k′ )] = a†λ (k), aλ′ (k′ ) = 0 , all the fundamental commutation relations between Ai (x) and ΠT i (x) are satisfied and the free part of the Hamiltonian (11.319) takes (after discarding an infinite constant) the form63 Z 3 X † dk ~ω aλ (k)aλ (k) . (11.327) HEM = k (2π)3 λ=±1 As the full Hamiltonian of the electromagnetic field coupled to a system of N charged particles we therefore get H= N 2 X X † 1 qi d3 k a (k)a (k) + P̂ − ~ω eA(r̂ ) (11.328) λ i k i λ (2π)3 2m c i i=1 λ=±1 Z Z N N X 1 X 1 e2 J 0 (x)J 0 (y) 3 + V (r̂i ) + V (r̂i − r̂j ) + d x d3 y . 2 i6=j=1 2 4πc2 |x − y| i=1 Z where A(r̂i ) is the operator (11.321) depending now on the charged particle position operators r̂i . This dependence is introduced into A by the integral in Lint (11.291) and the delta function in the current (11.292). For J 0 of the form (11.292) the last term in the Hamiltonian reduces to N 1 X e2 qi qj , 2 i,j=1 4π|r̂i − r̂j | which is just the Coulomb electrostatic energy of a system of charged pointlike particles (in the Heaviside system of units). Note however, that the √ By the rescaling 2~ωk aλ (k) → aλ (k) the first of the rules (11.326) can be brought into the standard relativistic form 63 [aλ (k), a†λ′ (k′ )] = δλλ′ δΓ (k − k′ ) . The expansions of the operators Ai and ΠT i then take the simple forms Z X Ai (x) = ~c dΓk ǫi (k, λ) aλ (k) eik·x + H.c. , λ=±1 i ΠT i (x) = − c Z dΓk ~ωk ǫi (k, λ) aλ (k) eik·x − H.c. , and d3 k/(2π)3 in (11.327) gets replaced by dΓk ≡ d3 k/(2π)3 2~ωk . 391 summation includes infinite terms arising for i = j which have to be subtracted by hand. There can be no question about the relativistic covariance of the theory just constructed, because matter particles are treated nonrelativistically. Therefore we cosider here covariance of the quantum theory of the free (not coupled to matter particles) electromagnetic field. In this case from the Lagrangian density (11.233) one obtains the canonical energy-momentum tensor (we set ~ = c = 1) µν Tcan = −f µλ ∂ νAλ + 1 µν λκ g f fλκ , 4 (11.329) which is not symmetric. The symmetric Belinfante energy-momentum tensor takes the form µν Tsymm = −f µλ f νλ + 1 µν λκ g f fλκ , 4 (11.330) and gives64 0 P = Z d 3 00 x Tsymm 1 = 2 Z i h d3 x Ȧi Ȧi + (∇×A)2 , which is just (11.319) for zero external current J µ and Z i P = d3 x Πk ∂ iAk . (11.331) (11.332) µνλ The tensor Mcan (11.61) of the electromagnetic field takes the form µνλ µλ µν νλ κ Mcan = xν Tcan − xλ Tcan + if µκ Jvec Aρ . (11.333) ρ The generators J ij and J 0i of the Lorentz transformations take then the form Z o n ij J = d3 x Ȧk (xi ∂ j − xj ∂ i )Ak − Ȧi Aj − Ȧj Ai , Z h i 2 0i i 3 i 1 k k J = tP − d x x Ȧ Ȧ + (∇×A) . (11.334) 2 The operators H ≡ P 0 , J i ≡ 12 ǫkl J kl , P i and K i ≡ J 0i can be shown to satisfy the Poincaré algebra commutation rules (6.21) by virtue of the canonical commutation relations (11.317). However, while µ µ eiaµ P Ak (x) e−iaµ P = Ak (x + a) , k i ij i ωij J ij k − 2i ωij J ij − 2 ωij Jvec 2 Al (Λ−1 (ω)·x) , A (x) e = e e l 64 i For the free electromagnetic field A0 = 0 and ΠT i = Πi = Ȧ . 392 as usually (here Λ is a pure rotation), performing the transformation of Ak (x) corresponding to a Lorentz boost one finds that k 0i 0i 0i eiω0i J Ak (x) e−iω0i J = e−iω0i Jvec Al (Λ−1 (ω)·x) + ∆Ak (x) ≡ A′k (x) , l where the extra term ∆Ak (x) ensures that the ∂k A′k (x) = 0 as it must be for consistency: the divergence of the left hand side of the above formula obviously vanishes. This means that a transformations of the photon field operator Ai (x) generated by K i consist of the corresponding Lorentz boost supplemented with a suitable gauge transformation. Thus, transformations generated by the conserved Noether charges (11.329) and (11.333) in the Coulomb gauge automatically preserve this gauge. Finally we consider the thermodynamical properties of the free electromagnetic field in equilibrium with the walls of a box of volume V = L3 . If the field is quantized in the box, the Hamiltonian takes the form XX HEM = ~ωk a†kλ akλ , (11.335) k λ=±1 where now the wave vectors k are discrete, k = (2π/L)n, ωk = c|k| and [ak′ λ′ , a†kλ ] = δk′ k δλ′ λ . (11.336) The Hilbert space is spanned by the HEM eigenstates |nk1 λ+ , nk1 λ− , nk2 λ+ . . .i, where each occupation number nki λi can run from 0 to infinity. This allows to easily compute the Gibbs Canonical Ensemble partition function ∞ ∞ Y X X −~ ωk /kB T nkλ+ −~ ωk /kB T nkλ− Zstat = e e nkλ+ =0 nkλ− =0 k 2 Y 1 . (11.337) = −~ ωk /kB T 1 − e k The Helmholtz free energy is then65 X 2 ln 1 − e−~ ωk /kB T , F (T, V ) = kB T k or, in the thermodynamical limit, Z 2V 3 −~ ωk /kB T d k ln 1 − e F (T, V ) = kB T (2π)3 Z ∞ V = kB T 2 3 dω ω 2 ln 1 − e−~ ω/kB T . π c 0 65 (11.338) Had we not subtracted the P (infinite) energy of the zero point oscillations, we would get in F (T, V ) an extra term k ~ωk . 393 All well known properties of the electromagnetic radiation in equilibrium, including the celebrated equation of state p = 31 u(T ), where u(T ) = U(T )/V is the radiation energy density, can be derived from the free energy F (T, V ). 11.8 Canonical construction of half-integer spin quantum fields As demonstrated in sections 11.2, 11.3, 11.5 and 11.7, the formal procedure of canonical quantization (supplemented in the case of systems subject to constraints by the Dirac method described in section 11.6) applied to classical (c-number) fields which under spatial rotations of the reference frame transform as integer spin representations of the rotation group allows to build theories of quantum fields whose elementary excitations (“quanta”) are bosons. In the interaction picture (see section 11.9), i.e. when the basic field operators of such quantum theories are represented in Fock spaces of eigenstates of the free Hamiltonians of such theories, one recovers the same Feynman rules for computing S matrix elemens, which in sections 8 and 9 were obtained in the approach based on quantum mechanics of relativistic particles, without any reference to classical fields. Apart from making (owing to the Noether theorem presented in section 11.1) extraction of consequences of various possible symmetries more straightforward, the most important virtue of the approach based on field quantization is that being essentially nonperturbative, it allows for a deeper insight into the structure of the quantum theory (see Section 13). We here want to lay similar foundations for relativistic interactions of halfinteger spin particles. Superficially, it could seem it should suffice to apply the same formalism as previously to classical c-number fields but transforming under changes of the Lorentz frame as half-integer spin representations of the Spin(1, 3) ≃ SL(2, C) group (the universal covering of the Lorentz SO(1, 3) group). This is indeed so for the best known kind of half-integer spin particles - massive charged fermions accompanied by their antifermions: as in many textbooks one can in this case start with a relativistically invariant Lagrangian density of a c-number complex Dirac field ψ transforming as the ( 12 , 21 ) representation of the SL(2, C) group arguing that to obtain a sensible quantum theory it is necessary to replace the Poisson brackets with the anticommutators instead of commutators (one usually passes freely over the presence of constraints in this case). Yet, not for all types of half-integer spin particles relativistic interactions can be obtained in this way. The simplest example is the massive neutral spin 1/2 particle (called Majorana fermion) whose field operator was constructed at the end of section 8.3. This is be394 cause already the corresponding free Lagrangian density cannot simply be written down: in the expression 1 L = λ̄iσ̄ µ ∂µ λ − m(λλ + λ̄λ̄) 2 (11.339) (see section 8.6 for the Lorentz transformation properties of λα and λ̄α̇ ) the term proportional to m is simply zero, if λα and λ̄α̇ are treated as c-number spinors. This shows that the consistent field-based approach to constructing quantum theories of fermions must start with Lagrangian densities which are (bosonic) functions of fields taking values in the Grassmann algebra generated by the infinite set of anticommuting generators. To get acquinted with these notions it is convenient to consider first a classical bosonic system having a finite number of degrees of freedom. Dynamics of such a system described by n generalized (real) variables q i can be formulated as a mapping from R (time) into the n-fold direct product F × . . . × F of an abstract algebra F of functions f (a1 , . . . , a2n ) of 2n variables a1 , . . . , a2n which are identified with initial values q01 , . . . , q0n , q̇01 , . . . , q̇0n of the variables q i and the corresponding generalized velocities q̇ i (t). Indeed, for a fixed instant t the values q i (t) can be treated as a set of n functions q i (t) = f i (t, a1 , . . . , a2n ) of the variables a1 , . . . , a2n . With an appropriate topology (to define convergence of sequences) such functions f i (t, a1 , . . . , a2n ) can be viewed as elements of an abstract commutative and associative algebra over R generated by the set of 2n commuting generators a1 , . . . , a2n (ak al = al ak ): f (a1 , . . . , a2n ) = f0 + X fk ak + k 1X fk k ak1 ak2 2 k ,k 1 2 1 2 1 X + fk1 k2 k3 ak1 ak2 ak3 + . . . 3! k ,k .k 1 2 (11.340) 3 Differentiation can in the algebra F be defined as a linear algebraic operation by the basic rules ∂ i a = δji , j ∂a and ∂ 1 ∂ k 1 k l a . . . a . . . a = a a . . . al = a1 . . . al . k k ∂a ∂a and their extension to all elements of F using linearity. Thus each q i (t) is for fixed t an element of F ; a classical trajectory (q 1 (t), . . . , q n (t)) can be, therefore, viewed as a mapping t → F × . . . × F . A Lagrangian which determines the dynamics of such a system is in this picture a mapping of the 2n-fold direct product of the algebra F into F itself: F × . . . × F → L(f 1 , . . . , f 2n ) ∈ F and the action functional I[q(t)] is given 395 by Z t2 dt L(a1 , . . . , a2n )|a1 =q1 (t),...,a2n =q̇2n (t) t Z t2 Z 1t2 1 2n dt L(q 1 (t), . . . , q̇ 2n (t)) . (11.341) dt L(a , . . . , a ) ≡ ≡ I[q(t)] = t1 t1 Its variation with respect to the trajectory is to be understood as Z t2 X n ∂L d i ∂L i δq (t) + n+i δq (t) , (11.342) δI[q(t)] = dt i ∂a ∂a dt t1 i=1 where each δq i (t) is again a mapping from R into F reducing to the zero element of F for t = t1 and t = t2 . The fermionic counterpart of the algebra F of functions described above is the Grassmann algebra G over C generated by 2m elements (generators) ξ 1 , . . . , ξ 2m which anticommute: ξ α ξ β = −ξ β ξ α . Each element of G is of the form X X gα1 α2 ξ α1 ξ α2 + . . . g = g0 + gα ξ α + α + (11.343) α1 <α2 X gα1 ...α2m ξ α1 . . . ξ α2m . (11.344) α1 <...<α2m In contrast to (11.340), the number of terms in the above sum is finite owing to the property (11.343) which implies that ξ αξ α = 0 (the zero element of the algebra G). In the natural way the Grassmann algebra splits into the direct sum of Geven and Godd (the only common element of these two subspaces being the zero element of G) whose elements have the general forms X X gα1 ...α2m ξ α1 . . . ξ α2m , gα1 α2 ξ α1 ξ α2 + . . . + geven = g0 + α1 <...<α2m α1 <α2 godd = X α α gα ξ + . . . + X gα1 ...α2m−1 ξ α1 . . . ξ α2m−1 , α1 <...<α2m respectively. Elements of Geven and Godd can be distinguished by assigning them the G-algebra parities Pg which assume values 0 and 1 respectively (a general element of G has no definite parity). In the algebra G one defines two linear operations called left- and rightderivatives with respect to the generator ξ α by the rules ∂ ∂ β ξ = ξ β = δαβ , α α ∂L ξ ∂R ξ 396 ∂ ∂ ξ γ . . . ξ α . . . ξ σ = (−1)L ξ αξ γ . . . ξ σ = (−1)L ξ γ . . . ξ σ , α ∂L ξ ∂L ξ α ∂ ∂ ξ γ . . . ξ α . . . ξ σ = (−1)R ξ γ . . . ξ σ ξ α = (−1)R ξ γ . . . ξ σ , α α ∂R ξ ∂R ξ where L and R are the numbers of interchanges of the generators needed to place ξ α to the left or right of the string of the generators. Analogously to the bosonic case, “classical” dynamics of a fermionic system described by m variables ψ α is a mapping from R into an m-fold direct product Godd × . . . × Godd with the generators ξ 1 , . . . , ξ m and ξ m+1 , . . . , ξ 2m having the interpretation of the initial “values” of ψ α (t) and ψ̇ α (t). The Lagrangian is in this case a mapping of the 2m-fold product Godd × . . . × Godd into Geven : L(ψ 1 (t), . . . , ψ̇ 1 (t), . . .) ≡ L(ξ 1 , . . . , ξ 2m )|ξ1=ψ1 (t),..., ξm+1 =ψ̇1 (t),... . (11.345) Dynamics (the equations of motion) follows from the condition δI[ψ(t)] = 0 where the variation is written in terms of the right derivatives66 Z t2 X m ∂L α ∂L d α dt δI[ψ(t)] = δψ (t) + δψ (t) , (11.346) α m+α dt ∂ ξ ∂ ξ R R t1 α=1 where each δψ α (t) is an arbitrary mapping from R into Godd reducing to the zero element of Godd for t = t1 and t = t2 . The resulting “classical” dynamics is then rather abstract and, as has been discussed in the introduction to this chapter, no picture of fluctuating fields can be associated with the corresponding quantum theory. Finally, the mathematical structure allowing for a uniform treatment of mixed bosonic and fermionic systems is the Bieriezin algebra B obtained from the Grassmann algebra by treating the coefficient functions gα1 ... in (11.344) as elements of the algebra F of functions.67 Thus, the Bieriezin algebra is generated by 2n + 2m elements (generators) z a , z i = ai , i = 1, . . . , 2n and z 2n+α = ξ α, α = 1, . . . , 2m, which have the property z a z b = (−1)Pa Pb z b z a , (11.347) with the natural assignments of the B-algebra parities Pa (which as in the Grassmann algebra can assume values 0 and 1). Obviously, the parity of a product b1 b2 of two elements of B having well-defined parities Pb1 and Pb2 66 Alternatively it can be written in terms of the left derivatives with the variations placed to the left - see the firmula (11.349). 67 Of course real and imaginary parts of each gα1 ... are treated as two independent elements of F . 397 is Pb1 + Pb2 (mod 2) and b1 b2 = (−1)Pb1 Pb2 b2 b1 . Similarly as the Grassmann algebra, the Bieriezin algebra naturally splits into the direct sum of the even and odd subspaces Beven and Bodd whose elements have well defined parities (0 and 1, respectively). Left- and right derivatives with respect to the generators z a are linear operations defined by the rules similar to the ones in the Galgebra k X ∂ b1 bk z . . . z = (−1)Li δabi z b1 . . . (no z bi ) . . . z bk , ∂L z a i=1 k X ∂ b1 bk z . . . z = (−1)Ri δabi z b1 . . . (no z bi ) . . . z bk , ∂R z a i=1 (11.348) where Li (Ri ) are the numbers of odd generator interchanges needed to place z bi on the extreme left (right) of the string of the generators. Left- and rightderivatives with respect to bosonic generators ai are, of course, identical. A useful property of derivatives with respect to fermionic generators ξ α are ∂ ∂ (bodd ) = (bodd ) , α ∂L ξ ∂R ξ α ∂ ∂ (beven ) = − (beven ) . α ∂L ξ ∂R ξ α (11.349) Dynamics of a mixed bosonic-fermionic system is determined by the action principle Z t2 ∂L ∂L d a a dt δI[q, q̇] = δq + δq = 0 , ∂R q a ∂R q̇ a dt t1 in which L(z 1 , . . . , z 2n+2m ) is a function taking values in Beven and its derivatives with espect to q a and q̇ a have to be understood as in (11.342) and (11.346). The equations of motion take the standard form ∂L d ∂L = a dt ∂R q̇ ∂R q a a = 1, . . . , n + m . (11.350) Hamiltonization of the dynamics of a bosonic-fermionic system is analogous to the one of the ordinary bosonic systems except that one has to fix a convention (which is arbitrary) for the definition of the canonical momenta. We chose to define the momenta pa (t) as right derivatives pa (t) = ∂L . ∂R q̇ a (t) 398 (11.351) Consequently, the Hamiltonian must take the form X H= pa q̇ a − L , (11.352) a in order for its variation to be independent of the variations δ q̇ a of the velocities. Indeed, with this definition, using ∂L ∂ ∂L ∂ a δq + δ q̇ a, δpb = ∂R q a ∂R q̇ b ∂R q̇ a ∂R q̇ b one can write the variation of H ∂H a Pa Pb ∂ δH = δq + (−1) ∂R q a ∂R q̇ a ∂L ∂R q̇ b δ q̇ a q̇ b, in the form ∂H ∂ δH = − a ∂R q ∂R q a ∂L ∂R q̇ b q b δq a + q̇ b δpb . The Poisson bracket of the two functions F and G of definite B-algebra parities must be defined in agreement with the adopted convention. The right definition is X ∂F ∂G PF PG ∂F ∂G − (−1) , (11.353) {F, G}PB = a∂ p a∂ p ∂ q ∂ q R L a R L a a with the left derivatives with respect to the momenta. It has the easy to verify properties {F, G}PB = −(−1)PF PG {G, F }PB , {F, GH}PB = {F, G}PB H + (−1)PF PG G {F, H}PB . With the definition (11.353) of the Poisson bracket in the B-algebra the canonical equations of motion q̇ a = {q a , H}PB , ṗa = {pa , H}PB , are, for systems not subject to constraints, equavalent to the Euler-lagrange equations derived from the action principle δI[q(t)] = 0. Canonical quantization of such systems consists in promoting their canonical variables q i , pi and ψ α , πα to Schrödinger picture operators q̂ i , p̂i , ψ̂ α and π̂α satisfying the (anti)commutation relations: [q̂ i , p̂j ] = i~{q i , pj }PB , {ψ̂ α , π̂β }+ = i~{ψ α , πβ }PB , (11.354) in agreements with the symmetry/antisymmetry properties of the corresponding Poisson brackets. The resulting equations satisfied by the Heisenberg picture operators q̂ i (t) and ψ̂ α (t) are then formally identical with the classical Euler-Lagrange equations (11.350). 399 The Dirac procedure allowing to handle systems subject to (second class) constraints extends to the case of mixed bosonic-fermionic systems essentially without modifications. The only difference is that the C matrix is in this case a supermatrix and has more complicated symmetry/antisymmetry properties (the Poisson bracket in the B-algebra is not simply antisymmetric). Transition from classical mechanics to classical field theory is achieved, as usually, by passing with the number of variables q a (and therefore also with the number of generators of the Bieriezin algebra) to continuum, that is by assscibing a certain number of independent variables to each space point x. As an example of the application of this formalism we consider here quantization of the relativistic Grassmann field ψ transforming as the spinor representation of the Lorentz group, or more precisely, as the representation of the group Spin(1, 3) - the universal covering of the Lorentz SO(1, 3) group. The extension to Spin(1, 3) is possible because (what is clearly reflected in the formalism of the Grassmann and Bieriezin algebras) such fields are not classically measurable; all observables must be bosonic (i.e. belong to Geven or Beven ) that is bilinear or quadrilinear etc. in the fields ψ and transform as representations (scalar, vector, tensors) of the true Lorentz group.68 We assume therefore that under the change of the reference frame x → x′ = Λ·x the Grassmann algebra valued field ψ transforms as µν i ψ ′ (x′ ) = e− 2 ωµν Jspin ψ(x) , (11.355) µν where the matrices Jspin of the spinor representation satisfy the commutation rule h i µν λρ µρ νλ µλ νρ νρ µλ νλ µρ Jspin, Jspin = i g Jspin − g Jspin − g Jspin + g Jspin . (11.356) µν As discussed in Section 8.3 the matrices Jspin can be constructed either by means of the Clifford algebras or by exploiting the isomorphism of Spin(1, 3) and SL(2, C). Here, as in (8.53) we take µν Jspin = 1 µν i µ ν , [γ , γ ] ≡ σ4×4 4 (11.357) with the 4 × 4 matrices γ µ satisfying the basic Clifford algebra relation (8.52) in the representations (8.64) or (8.66). 68 At the “classical” level this is the only argument why to consider fields transforming under a larger symmetry group than SO(1, 3) despite the fact that only the Lorentz group of symmetries can be inferred from physical experience. In the quantum case, as was discussed in Section 4, going over to the universal covering of the SO(1, 3) group is a natural consequence of the possible occurrence of the projective representations of SO(1, 3). 400 The most frequently encountered Lagrangian density (leading to the theory of noninteracting fermions) has form69 L = iψ † γ 0 γ µ ∂µ ψ − m ψ † γ 0 ψ ≡ ψ̄(iγ µ ∂µ − m)ψ (11.358) (Lorentz invariance of this Lagrangian density can be shown using the formulae given in Section 8). The Grassmann algebra-valued fields ψα and ψα† are four-component spinors. They are all treated as an independent Grassmann variables. The Euler-Lagrange equations derived from (11.358) γ 0 (iγ µ ∂µ − m) ψ = 0 , ← ψ̄ −iγ µ ∂ µ − m = 0 , (11.359) are just the Dirac equations. To set up the canonical formalism for this system we first find the canonical momenta (recall that the derivatives are the right ones) Πψ = ∂L = iψ † , ∂R ψ̇ Πψ† = ∂L = 0. ∂R ψ̇ † (11.360) It is clear that it is impossible to express ψ̇ and ψ̇ † through Πψ and Πψ† . The system is therefore subject to constraints (we suppress spinor indices) Φ1 (x) ≡ Πψ (x) − iψ † (x) = 0 , Φ2 (x) ≡ Πψ† (x) = 0 , (11.361) which are of second class. Although in this case the Dirac procedure described in section 11.6 is not really indispensable, because the canonical equations when restricted to ψ and Πψ are fully equivalent to the original Euler-Lagrange equation (11.359), it is instructive to go through this procedure to see how it leads to the well established results. Constructing the extended Hamiltonian density according to the rules of section 11.6 one gets HE = Πψ ψ̇ + Πψ† ψ̇ † − ψ † iψ̇ − ψ̄iγ i ∂i ψ + m ψ̄ψ = ψ † γ 0 −iγ i ∂i + m ψ + Φ1 u1 + Φ2 u2 . (11.362) The functions u1 and u2 are just the velocities ψ̇ and ψ̇ † . Equating to zero the Poisson brackets ← {Φ1 , HE }PB = −ψ̄(iγ k ∂ k + m) − iu2 , {Φ2 , HE }PB = γ 0 (−iγ k ∂k + m)ψ − iu1 , 69 Another, less frequently encountered, is the symmetric form: i † 0 µ i ψ γ γ ∂µ ψ − ∂µ ψ † γ 0 γ µ ψ − m ψ † γ 0 ψ . 2 2 Using the Dirac formalism it is easy to check that it leads to the same quantum theory as (11.358). L= 401 of the constraints with the Hamiltonian determines these functions. No new (secondary) constraints are needed. Combining u1 and u2 obtained in this way with the canonical equations ψ̇ = {ψ, HE }PB = u1 , ψ̇ † = {ψ † , HE }PB = u2 one obtains equations which are fully equivalent to the Euler-Lagrange ones (11.359). In order to quantize the theory one computes the Dirac brackets. The nonzero elements of the C matrix are C1x,2y = C2x,1y ≡ {Φ1 (x), Φ2 (y)}PB = −iδ (3) (x − y) (it is in this case symmetric). Its inverse reads (C −1 )1x,2y = (C −1 )2x,1y = iδ (3) (x − y) . (11.363) The Dirac brackets computed as in (11.280) read {ψ(x), Πψ (y)}D = δ (3) (x − y) , {ψ(x), ψ † (y)}D = −iδ (3) (x − y) , {ψ(x), Πψ† (y)}D = 0 , {ψ † (x), Πψ (y)}D = 0 , {Πψ† (x), Πψ (y)}D = 0 , {ψ † (x), Πψ† (y)}D = 0 . The canonical quantization rule is now {A, B}+ = i~ {A, B}D which leads to the canonical anticommutator {ψα (x), ψβ† (y)}+ = ~ δαβ δ (3) (x − y) , (11.364) {ψα (x), ψβ (y)}+ = {ψα† (x), ψβ† (y)}+ = 0 . Since in the quantum theory the constraints are realized as operator identities, Πψ† becomes the zero operator and Πψ can everywhere be eliminated in favour of iψ † . The Hamiltonian density operator reads H = ψ † γ 0 (−iγ k ∂k + m)ψ . (11.365) The next step is to expand the basic operators ψα (x) and ψα† (x) into creation and annihilation operators, satisfyng simple anticommutation rules. Since ψα has four independent components, there must be four annihilation and four creation operators. As the Grassmann variables ψα and ψα† are not real c-numbers, there corresponding operators ψα and ψα† need not be Hermitian. We write therefore their expansions in the forms Z X ψα (x) = dΓp eip·x [uα (p, σ) au (p, σ) + vα (−p, σ) av (−p, σ)] , (11.366) σ ψα† (x) = Z X dΓp e−ip·x u∗α (p, σ) a†u (p, σ) + vα∗ (−p, σ) a†v (−p, σ) , σ 402 using the functions uα (p, σ) p and vα (p, σ) constructed in section 8.3 and with dΓp = (2π)3 2Ep , Ep = p2 + m2 . Computing the anticommutators (11.364) and anticipating that {au , av }+ = {a†u , a†v }+ = 0 we get Z Z XX ′ † {ψα (x), ψβ (y)}+ = dΓp dΓp′ ei(p·x−p ·y) σ σ′ ∗ ′ ′ uα (p, σ)uβ (p , σ ) {au (p, σ), a†u (p′ , σ ′ )}+ +vα (−p, σ)vβ∗ (−p′ , σ ′ ) {av (−p, σ), a†v (−p′ , σ ′ )}+ . Recalling the summation rules (8.105): X X uα (p, σ)u∗β (p, σ) = uα (p, σ)ūβ (p, σ)γ 0 = (Ep γ 0 − p · γ + m)γ 0 αβ , s σ X σ vα (−p, σ)vβ∗ (−p, σ) = (Ep γ 0 + p · γ − m)γ 0 αβ , it is easy to check that the commutation rules (11.364) are satisfied if {au (p, σ), a†u (p′ , σ ′ )}+ = δΓ (p − p′ ) δσσ′ , {av (p, σ), a†v (p′ , σ ′ )}+ = δΓ (p − p′ ) δσσ′ , (11.367) and the other anticommutators are zero as anticipated. To express the Hamiltonian through the operators au etc. we use the equality70 γ 0 (−iγ i ∂i + m)ψ(x) Z X = dΓp Ep eip·x [au (p, σ) u(p, σ) − av (−p, σ) v(−p, σ)] , σ and the normalization conditions u† (p, σ) · u(p, σ ′ ) = 2Ep δσσ′ , v † (p, σ) · v(p, σ ′ ) = 2Ep δσσ′ , u† (p, σ) · v(−p, σ ′ ) = v † (−p, σ) · u(p, σ ′) = 0 , 70 (11.368) It follows from the equalities γ 0 (−iγ k ∂k + m) u(p, σ) eip·x = γ 0 (γ ·p + m) u(p, σ) eip·x = Ep u(p, σ) eip·x γ 0 (−iγ k ∂k + m) v(−p, σ) eip·x = γ 0 (γ ·p + m) v(−p, σ) eip·x = −Ep v(−p, σ) eip·x , which in turm follow from the formulae (8.101) written in the form (γ 0 Ep −γ·p−m)u(p) = 0 and (γ 0 Ep + γ ·p + m)v(−p) = 0. 403 which can be derived from the rules of constructing the functions u(p, σ) and v(p, σ) given in section 8.3. In this way one gets Z H = d3 x ψ † (x)γ 0 (−iγ i ∂i + m)ψ(x) Z X = dΓp Ep a†u (p, σ) au (p, σ) − a†v (p, σ) av (p, σ) . (11.369) σ If the basic rules (11.364) and consequently (11.367) involved commutators (instead of anticommutators) this form of the Hamiltonian would be a disaster: from the commutation rules it would unambiguously follow (at least if the system is quantized in a finite box) that there is a state |Ω0 i annihilated by all au and av and that all other states are created by acting on |Ω0 i with a†u and a†v . But the states created by a†v would then have negative energy! In the Fock space there would exist states with arbitrarily large negative energy and, therefore, the spectrum of the Hamiltonian would not be bounded from below. Fortunately, with the anticommutators such a conclusion does not follow: since the anticommutation relations are symmetric with respect to interchanges a ↔ a† we can call av the creation operator. More formally, we can substitute au (p, σ) = b(p, σ) , av (p, σ) = d† (p, σ) , a†u (p, σ) = b† (p, σ) , a†v (p, σ) = d(p, σ) , (11.370) which allows to rewrite the Hamiltonian (discarding the infinite constant) as Z X H = dΓp Ep b† (p, σ) b(p, σ) + d† (p, σ) d(p, σ) . (11.371) σ The Hamiltonian (11.371) is positive semidefinite and the operators b† (p, σ), b(p, σ) and d† (p, σ), d(p, σ) satisfying the rules b(p, σ), b† (p′ , σ ′ ) + = δΓ (p − p′ ) δσσ′ , d(p, σ), d† (p′ , σ ′ ) + = δΓ (p − p′ ) δσσ′ , (11.372) (with all other anticommutators zero) are the creation and annihilation operators of particles and antiparticles, respectively. This interpretation follows from the fact that states created by b† (p, σ) and d† (p, σ) have the same energies (and, hence same masses). Moreover, the Lagrangian (11.358) is invariant under the global transformations ψ → ψ ′ = e−iQθ ψ ψ † → ψ †′ = eiQθ ψ † , (11.373) forming a U(1) symmetry group with Q being the charge and θ the transformation parameter. The corresponding conserved Noether current is j µ = ψ†γ 0 γ µψ , 404 (11.374) and the states created by b† and d† are eigenstates of Z Z X 3 0 Q̂ = d x j (x) = Q dΓp b† (p, σ) b(p, σ) − d† (p, σ) d(p, σ) , (11.375) σ with opposite eigenvalues Q̂ b† (p, σ)|Ω0 i = Q b† (p, σ)|Ω0 i , Q̂ d† (p, σ)|Ω0 i = −Q d† (p, σ)|Ω0 i . (11.376) Quantization of the theory defined by the Lagrangian (11.339) can be performed along the same lines. Quantization of theories of fermionic and mixed fermionic-bosonic systems of fields whose complete Lagrangians consist of (11.358) (or (11.339)) plus terms involving higher powers of fields than the second (examples of realistic such theories will be discussed in section 11.12), also proceed along the same lines. As in the case of the scalar field ϕ discussed in section 11.2, one can now introduce time dependent field operators ψ(t, x) ≡ ψ(x), ψ † (t, x) ≡ ψ † ((x) by the formulae ψ(t, x) = eiHt ψ(x) e−iHt , ψ † (t, x) = eiHt ψ † (x) e−iHt , (11.377) which satisfy the canonical equations of motion d ψ(t, x) = i[H, ψ(t, x)] , dt d † ψ (t, x) = i[H, ψ † (t, x)] , dt (11.378) which take the form formally identical with the classical equations of motion derived from the underlying Lagrangian (that is with the equations (11.359) in the case of the Lagrangian (11.358)). The time-dependent operators ψ(t, x) and ψ † (t, x), which in the theory of free fields are the Heisenberg operators and in the theory of interacting fields in which the Hamiltonian (11.365) or (11.371) play the role of the free part H0 of the full Hamiltonian, are the interaction picture operators, take the forms Z X ψα (x) = dΓp e−ipx uα (p, σ) b(p, σ) + eipx vα (p, σ) d† (p, σ) , σ ψα† (x) = Z dΓp X σ e−ipx vα∗ (p, σ) d(p, σ) + eipx u∗α (p, σ) b† (p, σ) . One can also find the canonical energy-momentum tensor µν Tcan = ∂L ∂ ν ψ − g µν L . ∂R (∂µ ψ) 405 As long as the Lagrangian is linear in ψ and ψ † (as in (11.358)), the time dependent (Heisenberg picture) operators satisfy the equations of motion µν which make L vanishing (as the operator) and Tcan simplifies to the fierst term only. Thus, in the theory (11.358) µν Tcan = iψ † γ 0 γ µ ∂ ν ψ . (11.379) R 00 When expressed through the field operators d3 xTcan is just the Hamiltonian (11.371) and Z Z X i 3 0i P = d x Tcan = dΓp pi b† (p, σ)b(p, σ) + d† (p, σ)d(p, σ) , (11.380) σ is the momentum operator commuting with H. Finally, the tensor µκ µν M µνκ = xν Tcan − xκ Tcan + ∂L νκ −iJspin ψ, ∂R (∂µ ψ) gives the operators generating rotations and boosts Z i νκ ν κ κ ν νκ 3 † J = i d x ψ x ∂ − x ∂ − σ4×4 ψ . 2 (11.381) (11.382) To see that the particles which are “quanta” of the field ψ have spin s = 1/2 it is enough to act with J 12 = J z on the one particle states representing particles at rest 1 J z b† (0, ± 12 )|Ω0 i = ± b† (0, ± 21 )|Ω0 i , 2 1 J z d† (0, ± 12 )|Ω0 i = ± d† (0, ± 21 )|Ω0 i . 2 (11.383) More generally, one can check the commutation rules of the operators H, P, J and K and verify that the one particle states b† (p, ± 12 )|Ω0 i and d† (p, ± 21 )|Ω0 i transform under the action of U(Λ) ≈ 1 − 2i ωµν J µν + . . . operators in the way appropriate for spin 1/2 particles. The statistical properties of the quantized fermionic field can also be readily obtained. If the it is quantized in the box of volume V = L3 the Hamiltonian (before subtracting the infinite constant) is XX H= Ep b†pσ bpσ + d†pσ dpσ − 1 , (11.384) p σ and the Hilbert space is spanned by the states |nbpσ , . . . , ndpσ , . . .i with nbpσ , ndpσ = 0 or 1. Since there is a conserved charge Q̂, one computes the Grand Statistical Ensemble sum Ξ(T, V, µ) = Tr e−β(H−µQ̂) 406 = Y = Y p p 1 X nbp =0 e2βEp 1 b 2 e 2 βEp e−β(Ep −µ)np × Y p 1 X ndp =0 1 d 2 e 2 βEp e−β(Ep +µ)np 2 2 1 + e−β(Ep −µ) 1 + e−β(Ep +µ) , (11.385) (the squares come from the two spin states). Thus, for Ω(T, V, µ) = −kB T ln Ξ, after going over to the continuous normalization, one gets Z d3 p −β(Ep −µ) Ω(T, V, µ) = −2V E + k T ln 1 + e p B (2π~)3 +kB T ln 1 + e−β(Ep +µ) , (11.386) p where Ep = c2 p2 + m2 c4 and the factor of 2 accounts for two spin states per each p. 11.9 Transition to the interaction picture As in the approach to quantum field theory based on quantum mechanics of relativistic interacting particles, also in the approach relying on quantizing fields one can assume that the full Hamiltonians of interacting fields possesses in and out particle-like eigenstates |α± i. One is then primarily interested in obtaining the S matrix elements Sβα = outhβ|αiin , allowing to compute various transition rates using the recipies formulated in section 10. We have therefore to discuss how to recover in the approach based on quantization of interacting classical fields the perturbative expansion and Feynman rules which in section 9 was formulated within the approach based on constructing interactions of relativistic particles. As we will see, the procedure by which this is achieved, called transition to the interaction picture, when applied to classical relativistic fields automatically produces all the additional interactions which in the previous approach had to be included by hands in order to obtain a Lorentz covariant S matrix. We begin with the simplest example of the theory described by the Lagrangian density 1 1 L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) . 2 2 (11.387) where Hint (ϕ) is some interaction density. After quantization described in section 11.2 the Hamiltonian of the system, expressed through the Schrödinger 407 picture (time independent) operators ϕ(x), Π(x) satisfying the rules (11.77) is the sum of the free Hamiltonian Z 1 d3 x Π2 (x) + (∇ϕ(x))2 + M 2 ϕ2 (x) , (11.388) H0 = 2 which is quadratic in field operators and the interaction71 Z Vint = d3 x Hint (ϕ(x), Π(x)) . The formula for S-matrix elements Z Sβα = hβ0 |T exp −i +∞ −∞ I dt Vint (t) |α0 i , (11.389) (11.390) established in section 7.2 using rather general setting of quantum mechanics should apply also to the field theory case. In (11.390), as in (7.41), Z I iH0 t d3 x Hint (ϕ(x), Π(x)) e−iH0 t Vint (t) = e Z Z 3 I = d x Hint (ϕI (t, x), ΠI (t, x)) ≡ d3 x Hint . (11.391) and for the states |α0 i one should take the eigenstates of H0 (11.388) constructed in section 11.2. The time-dependent operators ϕI (x) ≡ ϕI (t, x) and ΠI (x) ≡ ΠI (t, x) given by ϕI (t, x) = eiH0 t ϕ(x) e−iH0 t , ΠI (t, x) = eiH0 t Π(x) e−iH0 t , (11.392) are called in this context the interaction picture operators and have the structure of free-field operators (discussed in Section 8). In the theory defined by the lagrangian (11.387) they take the form Z ϕI (x) = dΓk a(k) e−ikx + a† (k) eikx , Z 1 dΓk E(k) a(k) e−ikx − a† (k) eikx . (11.393) ΠI (x) = i with kx ≡ Ek t − k·x. They satisfy (by construction) the equal-time commutation rules [ϕI (t, x), ΠI (t, y)] = iδ (3) (x − y) , 71 (11.394) To be more general we write the formulae as if Hint depended also on Π(x). 408 etc. and the equation (∂µ ∂ µ + M 2 )ϕI (t, x) = 0 . (11.395) It should also be clear that the free Hamiltonian H0 (11.388) can be also written in terms of the time dependent interaction picture operators Z 1 iH0 t −iH0 t d3 x Π2I (t, x) + (∇ϕI (t, x))2 + M 2 ϕ2I (t, x) . H0 = e H0 e = 2 In the case of the theory (11.387) it is obvious that ϕ̇I (t, x) = ΠI (t, x). In cases like the one of electromagnetic field interacting with other fields (to be discussed below) it is useful to remember that ϕI (t, x) and ΠI (t, y) satisfy the equations ϕ̇I (t, x) = i[H0 , ϕI (t, x)] , Π̇I (t, x) = i[H0 , ΠI (t, x)] , (11.396) of which the first one unambigously fixes the relation of ϕ̇I (t, x) with ΠI (t, x). In this way the perturbation expansion for the S matrix (which leads to the Feynman rules discussed in sections 9.3-9.6) derived from principles of quantum mechanics in the framework of Sections 8 and 9 is recovered in the approach based on quantization of relativistic fields. It should be obvious, that this perturbative expansion is based on the same assumptions as those formulated in section 7.1, namely that the particle-like in and out states of the full Hamiltonian are in the strict one-to-one correspondence (7.10) with the eigenstates |α0 i of the free Hamiltonian (11.388). Transition to the interacting picture is more subtle in the case of vector (massive and massless) fields. Let us first consider the Lagrangian density 1 1 L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − Hint (ϕ) − J µ ∂µ ϕ + . . . , (11.397) 2 2 in which J µ is some four-vector constructed out of fields other than ϕ itself (the ellipses stands for terms depending on these other fields). In the Hamilton’s formalism ∂L Π(t, x) = = ∂0 ϕ − J0 , (11.398) ∂ ϕ̇ and Z Z d x Π(x)ϕ̇(x) − d3 x L(ϕ(x), ϕ̇(x)) Z 1 1 3 = d x Π (Π + J0 ) − (Π + J0 )2 + (∇ϕ)2 2 2 1 2 0 + M ϕ(t, x) + Hint + J·∇ϕ + J (Π + J0 ) . 2 H= 3 409 (11.399) Upon canonical quantization ϕ(x) and Π(x) become operators satisfying the standard canonical commutation rules. Splitting the full Hamiltonian into the free part H0 and the interaction Vint we get H0 (Π(x), ϕ(x)) as in (11.388) and Z Vint = d3 x [Π(x)J 0 (x) + J(x) ·∇ϕ(x) + 1 0 2 J (x) + Hint (ϕ(x))]. 2 (11.400) As in the preceding case H0 gets diagonalized upon introducing the creation and annihilation operators and the Fock basis of H0 eigenstates is constructed to play the role of the |α0 i states in the formula (11.390). The interation picture operators ϕI (t, x) and ΠI (t, x) are obtained as in (11.392). The important point is that the first of the equations (11.396) tells that ΠI (t, x) = ∂0 ϕI (t, x) , (11.401) and not ∂0 ϕI (t, x) − JI0 (t, x) (as for the corresponding Heisenberg picture operator). For this reason the interaction Vint (t) used in the perturbation expansion for the S matrix takes the form Z 2 1 0 µ I 3 Vint (t) = d x JI (t, x) ∂µ ϕI (t, x) + J (t, x) + Hint (ϕI (t, x)) , (11.402) 2 I i.e. it acquires a noncovariant term (JI0 )2 , so that the expression under the integral over d3 x in (11.402) is not a Lorentz invariant interaction density (as was assumed in (7.70)). However, as we have seen in Section 9.5, precisely such a noncovariant term had to be added (by hands in the approaches of Sections 7, 8) to compensate for the noncovariant term in the propagator of the vector field ∂µ ϕI (x) (arising due to the singular nature of products of field operators taken at the same space-time point) and to restore the Poincaré covariance of the S matrix. Similarly, in the case of the Proca theory the same procedure gives V̇Ii (t, x) = ΠIi (t, x) − 1 ∂i ∂j ΠIj (t, x) , M2 as in the free theory, which again leads to the presence of the noncovariant term (9.67) in Hint (VIi (t, x), ΠIi (t, x)). In electrodynamics in which the electromagnetic field couples to the conserved current J µ one considers the free field part Z n o 2 1 d3 x ΠT (x) + [∇×A(x)]2 + H0rest = H0EM + H0mat , (11.403) H0 = 2 410 of the full Hamiltonian consisting of (11.319) plus H mat which is built out ou the operators forming the current J µ , and expands the operators A(x), ΠT (x) into the creation and annihilation operators a† (k, λ), a(k, λ) associated with eigenstates of H0EM : Z i Xh † ik·x ∗ −ik·x A(x) = dΓk aλ (k) ǫ(k, λ) e + aλ (k) ǫ (k, λ) e , (11.404) λ=±1 1 Π (x) = i T Z i Xh † ik·x ∗ −ik·x aλ (k) ǫ(k, λ) e − aλ (k) ǫ (k, λ) e , dΓk |k| λ=±1 so that (up to a c-number constant) Z X EM H0 = dΓk |k| a† (k, λ) a(k, λ) . (11.405) λ=±1 The interaction picture operators AI (t, x) = eiH0 t A(x) e−iH0 t , ΠI (t, x) = eiH0 t ΠT (x) e−iH0 t , (11.406) which enter the formula (11.391), are related by ΠI (t, x) = ȦI (t, x). This again follows from computing the commutator Z ij i i iH0 t d3 y δtr (y − x) ΠTj (y) e−iH0 t = ΠIi (t, x) , ȦI (t, x) = i[H0 , A (x)] = e because ∂j ΠTj = 0. Notice that because A0 (x) is given by (11.305), the interaction picture operator A0I (t, x) is identically equal to Z e JI0 (t, y) 0 AI (t, x) = d3 y , (11.407) 4π|x − y| and can everywhere be replaced by this combination. This allows to introduce a dummy operator A0I (t, x) = 0, with the help of which the time dependent interaction (7.41) Z Z Z 0 0 e2 3 3 J (x)J (y) 3 I iH0 t dx dy e−iH0 t −e d x J(x)·A(x) + Vint (t) = e 2 4π|x − y| Z Z 2Z 0 e J (t, x)JI0 (t, y) = −e d3 x JI (t, x)·AI (t, x) + d3 x d3 y I , (11.408) 2 4π|x − y| 411 can be written in a formally covariant form72 Z Z Z J 0 (t, x)JI0 (t, y) e2 3 µ I 3 I d x d3 y I , (11.409) Vint (t) = e d x Jµ (t, x)·AI (t, x) + 2 4π|x − y| (H0 = H0EM + H0mat in this formula is the free part of the Hamiltonian of the entire system of fields, including the fields forming the currents JI0 ). As explained in (9.6), the Coulomb part of the interaction (11.409) can be dropped provided one drops at the same time noncovariant pieces in the photon propagator. 11.10 Corrections to external lines In the preceding section we have shown that quantizing classical fields one arrives essentially at the same scheme for calculating S matrix elements (assuming that particle-like in and out eigenstates of the full Hamiltonian H exist) as in the approach of sections 7 and 9 based on constructing interactions Vint of relativistic particles. Therefore, also in the approach discussed here one has to solve the problem what to do with the corrections to external lines like the ones discussed in subsection 9.7. There the problem was solved using the freedom in constructing interaction terms. Here it seems superficially we are not given such a freedom: quantized is a well defined classical theory with a fixed Lagrangian density and no new terms should be added to it. However, in quantizing fields there is another freedom which allows to find the solution of the problem73 (from the practical point of view this solution is equivalent to the one given in subsection 9.7). As usually, we will illustrate this on the example of the theory of a real scalar field defined by 72 Of course, if one is interested in computing amplitudes with insertions of the A0H (x) operator (see section 13) R 0 0 −i dt Vint (t) |α0 i outhβ|T (AH (x) . . .)|αiin = hβ0 |T AI (x) . . . e one has to use as A0I (x) in the pre-exponential factor the combination (11.407), and not the dummy operator A0I (x) = 0. 73 Recall, that the problem of external lines, we are now concerned with, has nothing to do with the question of finitness of integrals corresponding to loops in Feynman diagrams. One can therefore think that an ultraviolet cutoff is imposed on all loop integrals. Nevertheless, as it will become clear in section 14, only a special class of classical theories gives rise to quantum theories (called renormalizable) in which avoiding ill defined expressions of the sort discussed here and all other divergences (arising from integrations over loop four-momenta) indeed does not require adding new terms to the original Lagrangian. 412 the Lagrangian density 1 λ 1 L = ∂µ ϕ∂ µ ϕ − M 2 ϕ2 − ϕ4 . 2 2 4! (11.410) To quantize this theory we have first to choose the canonical field variable. Instead of ϕ (to which corresponds the canonical momentum Π equal, by classical Hamilton’s equations of motion, ∂0 ϕ), we can chose another variable ϕ̃ related to ϕ by ϕ = Z 1/2 ϕ̃. We have then the canonical momentum Π̃, by classical equations of motion equal Z∂0 ϕ̃ = Z 1/2 Π, which implies that the canonical equal time commutation rules satisfied by the corresponding operators ϕ̃H and Π̃H are the same as the ones satisfied by ϕH and ΠH : h i ϕ̃H (t, x), Π̃H (t, y) = iδ (3) (x − y) . (11.411) We will call the operator ϕH (singled out by the canonical form of the Lagrangian kinetic term in (11.410)) the canonical or, in the context of renormalization, the bare field operator and ϕ̃H = Z −1/2 ϕH with arbitrary Z 6= 1 the renormalized (although from the point of view of the Lagrangian dynamics ϕ̃ is also a “canonical” field variable; in this context renormalized simply means rescaled) field operator74 and the factor Z will be called the field renormalization constant (the name “wave function renormalization constant” is also in use). Working with ϕ̃H , Π̃H and performing the standard steps we arrive at the Hamiltonian density 1 1 1 λ H = Z −1 Π̃2H + Z (∇ϕ̃H )2 + ZM 2 ϕ̃2H − Z 2 ϕ̃4H , 2 2 2 4! which we rewrite in the form of H0 Z h i 1 2 H0 = d3 x Π̃2H (0, x) + (∇ϕ̃H (0, x))2 + Mph ϕ̃2H (0, x) , 2 (11.412) (11.413) and the interaction Hamiltonian density Hint = 1 1 −1 Z − 1 Π̃2H + (Z − 1) (∇ϕ̃H )2 2 2 1 λ 2 + ZM 2 − Mph ϕ̃2H + Z 2 ϕ̃4H . 2 4! (11.414) In this approach we have to admit that due to the self-interaction the physical mass Mph of the particles described by H is not equal to the bare mass parameter M in the Lagrangian (11.410). 74 For this reason in section 14 the operators like ϕ̃H will be denoted ϕR . 413 Performing now the transition to the interaction picture as in section 11.9, i.e. solving the commutation relation (11.411) at t = 0 with the help of the standard creation and annihilation operators diagonalizing H0 (as in subsection 11.2), we introduce the interaction picture field operators75 Z iH0 t −iH0 t ϕI (t, x) ≡ e ϕ̃H (0, x) e = dΓk a(k) e−ik·x + a† (k) eik·x , ΠI (t, x) ≡ eiH0 t Π̃H (0, x) e−iH0 t = ∂0 ϕI (t, x) . (11.415) Expressing the interaction Hamiltonian density similarly as in (11.391) in terms of ϕI (x) we get 2 1 1 I 2 Hint = − (Z − 1)∂µ ϕI ∂ µ ϕI + ZM 2 − Mph ϕI 2 2 1 −1 λ Z + Z − 2 ∂0 ϕI ∂0 ϕI . + Z 2 ϕ4I + 4! 2 (11.416) Thus, we get the additional interaction terms of the same structure76 as in (9.100). They allow, by properly adjusting, order by order in the perturbative 2 expansion, the factors δZ ≡ Z − 1 and δ 2 ≡ M 2 − Mphys , to cancel the dangerous corrections to external lines in the way explained in section 9.7. Notice, that quantizing the theory defined by the Lorentz scalar Lagrangian density (11.410) using the variable ϕ̃ we automatically obtain in the resulting interaction Hamiltonian density the noncovariant term ∝ ∂0 ϕI ∂0 ϕI whose necessity in the approach of subsection 9.7 could only be guessed at.77 As will become clear in section 13, the choice of δZ and of δM 2 such that the corrections to the external lines are absent altogether is necessary to satisfy the conditions underlying the scheme, developed in section 7.2 for direct calculation of S matrix elements.78 The choice of δM 2 is obvious: the eigenstates of H0 must correspond to particles whose mass is the same as the mass of the 1-particle eigenstates of the full H. The choice of Z is more subtle. In principle, any choice of the canonical field variable is good for quantization. However, the postulated correspondence (7.10) between the H0 eigenstates and the in and out eigenstates of H singles out, through the 75 Notice that ΠI = ∂0 ϕI does not conflict with the fact that by the operator equations of motion Π̃H = Z∂0 ϕ̃H : the limit limt→0 (∂0 ϕ̃H (t, x)) may differ from limt→0 (∂0 ϕI (t, x)). 76 We will make the notation for coefficients of these terms more systematic in sections 13 and 14. 77 Its coefficient is also automatically fixed in terms of the coefficient of the first, covariant term. Moreover, if Z = 1+O(λ), i.e. deviates from 1 starting with one-loop approximation only (in fact, in the considered example it starts only with two loops), the noncovariant term whose coefficient is Z −1 (Z − 1)2 contributes starting from yet higher order. 78 Strictly speaking Hint should be also supplemented with a constant term allowing to renormalize to zero energy of the vacuum state. This is equivalent to the usual practice of discarding all vacuum graphs. 414 argument presented in (9.99), one particular variable ϕ̃ which as the quantum operator ϕ̃H creates, for t → ∓∞, from the physical vacuum |Ω± i (the ground state of H) the physical one particle states |(p)± i with the amplitude equal to h(p)0 |ϕI (t, x)|Ω0 i. As will be shown in section 13, this is equivalent to working with the particular renormalized operator ϕ̃H , which can be called physical (or renormalized in the physical way) ϕph , for which the splitting of 2 H into H0 and Vint is such that the residue of the simple pole at p2 = Mph of the (connected) Green’s function G̃(2) (p2 ) defined by the formula Z Z ′ 4 (4) ′ (2) 2 4 (2π) δ (p + p )G̃ (p ) ≡ d x d4 y e−ip·x e−ip ·y hΩ− |T [ϕ̃H (x)ϕ̃H (y)]|Ω+ i , is exactly equal to i.79 In the perturbative expansion this is ensured by adjusting Z and δM 2 , order by order, in such a way that they cancel the two first 2 terms in the Taylor expansion of the self-energy Σ(p) around p2 = Mph . From the perspective of the general renormalization procedure (to be discussed in detail in section 14) the direct method of calculating S matrix elements is intimately related to the so-called On-Shell renormalization scheme. In section 13 it will be shown that in the approach to quantum field theory based on Green’s functions, which is more general, it is possible to work directly with the canonical (bare) field operator ϕH (or any other renormalized operator ϕ̃H that differs from the physical one). This approach does not rely on the assumptions of section 7.1; S matrix elements are then not computed directly, but must be obtained with the help of the Lehmann-SymanzikZimmermann (LSZ) prescription (see section 13.4) applied to appropriate Green’s functions of the theory. However, within the perturbative expansion the LSZ prescription reduces to a small modification of the rules for external lines of Feynman diagrams. 11.11 Indefinite metric quantization of the electromagnetic field Quantization of the electromagnetic field in the Coulomb gauge described in section 11.7 may appear inconvenient, as it is not manifestly covariant. It is also technically rather complicated. It is possible to quantize the electromagnetic field in a fully covariant way but this requires working in the 79 Superficially it may seem that the free particle Hilbert spaces spanned by the action of the creation and annihilation operators contained in ϕI (x) obtained from ϕH (0, x) and from ϕ̃H (0, x) are “identical”, so there should be no difference in the choice of the canonical variable. The mathematical subtlety here is that for systems with infinite number of degrees of freedom the canonical commutation relations can have unitarily inequivalent representations. Thus, the free particle Hilbert spaces constructed from ϕH (0, x) and ϕ̃H (0, x) are in fact “different”. 415