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Symmetry & Symmetry breaking © Maurits Cornelis Escher Paolo Finelli, Physics Department, University of Bologna, Nuclear Physics Course - 2012 Contents 1 Symmetry and Symmetry Breaking 3 1.1 Definition of Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . 5 1.2 Variational symmetry or symmetry as a unitary transformation? . . . . . . . . . 6 1.3 Coleman Theorem and Goldstone Theorem . . . . . . . . . . . . . . . . . . . . . 8 1.3.1 1.3.2 The invariance of the vacuum is the invariance of the world (Coleman Theorem [5]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Field theories with superconductor solutions (Goldstone Theorem [6, 7, 8]) 13 2 Examples of Spontaneous Symmetry Breaking 2.1 2.2 2.3 2.4 21 Complex scalar fields: U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 General background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Symmetry breaking potential . . . . . . . . . . . . . . . . . . . . . . . . . 24 Higgs mechanism [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.1 Local gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Real scalar fields: SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.1 Application of the Goldstone Theorem . . . . . . . . . . . . . . . . . . . . 36 Generalization to n-Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 A History of Spontaneous Symmetry Breaking 45 B Group Theory - a short introduction 48 B.1 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 B.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 B.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 C Wigner theorem 58 D Noether theorem 61 Chapter 1 Symmetry and Symmetry Breaking From Ref. [1], a general definition is The term symmetry derives from the Greek word συµµτ ρια (meaning with measure) and originally indicated a relation of commensurability (such is the meaning codified in Euclid’s Elements for example). It quickly acquired a further, more general, meaning: that of a proportion relation, grounded on (integer) numbers, and with the function of harmonizing the different elements into a unitary whole. Symmetry is closely related to harmony. In modern science (not only physics) The group-theoretic notion of symmetry is the one that has proven so successful in modern science. Symmetry remains linked to regularity and unity: by means of the symmetry transformations, distinct (but equal or, more generally, equivalent) elements are related to each other and to the whole, thus forming a regular unity. The definition of symmetry as invariance under a specified group of transformations allowed the concept to be applied much more widely, not only to spatial figures but also to abstract objects such as mathematical expressions in particular, expressions of physical relevance such as dynamical equations. In particular for quantum mechanics In general, if G is a symmetry group of a theory describing a physical system (that is, the fundamental equations of the theory are invariant under the transformations of G), this means that the states of the system transform into each other according 3 Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) Eigenvalue spectra of the invariants of the symmetry group Labels for classifying the irreducible representations Invariant properties of the physical system E SO(3) Casimir invariant L2=Lx2+Ly2+Lz2 l, angular momentum D (l=2) P (l=1) 2l+1 degeneracy S (l=0) Example to some representation1 of the group G. In other words, the group transformations are mathematically represented in the state space by operations relating the states to each other. In quantum mechanics, these operations are generally the operators acting on the state space that correspond to the physical observables, and any state of a physical system can be described as a superposition of states of elementary systems, that is, of states which transform according to the irreducible representations of the symmetry group. The observables representing the action of the symmetries of the theory in the state space, and therefore commuting with the Hamiltonian of the system, play the role of the conserved quantities. The eigenvalue spectra of the invariants of the symmetry group provide the labels for classifying the irreducible representations of the group: on this fact is grounded the possibility of associating the values of the invariant properties characterizing physical systems with the labels of the irreducible representations of symmetry groups, i.e. of classifying elementary physical systems by studying the irreducible representations of the symmetry groups. 1 Group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. They describe how the symmetry group of a physical system affects the solutions of equations describing that system. 4 [23/04/2012] Paolo Finelli 1.1 Corso di Teoria delle Forze Nucleari (2012) Definition of Spontaneous Symmetry Breaking Here we collect two definitions of Spontaneous Symmetry Breaking (SSB)2 : 1. Generally, the breaking of a certain symmetry does not imply that no symmetry is present, but rather that the situation where this symmetry is broken is characterized by a lower symmetry. In group theoretic terms, this means that the initial symmetry group is broken to one of its subgroups. It is therefore possible to describe symmetry breaking in terms of relations between transformation groups, in particular between a group (the unbroken symmetry group) and its subgroup(s) [1] 2. Spontaneous symmetry breaking (SSB) indicates a situation where, given a symmetry of the equations of motion, solutions exist which are not invariant under the action of this symmetry without the introduction of any term explicitly breaking the symmetry (whence the attribute spontaneous). When some parameter (order parameter) reaches a critical value, the lowest energy solution respecting the symmetry of the theory ceases to be stable under small perturbations and new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are asymmetric but are all related through the action of the symmetry transformations. In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory. SSB occurs both in classical and in quantum physics3 [1]. For history and relevant bibliography in Quantum Field Theory, please see Appendix A. 2 The adjective spontaneous differentiates symmetry breaking that arises due to the noninvariance of the vacuum state from that due to explicitly adding asymmetric terms to the Lagrangian. 3 A distinction has to be drawn is between finite and infinite physical systems. In the case of finite systems, SSB actually does not occur: tunnelling takes place between the various degenerate states, and the true lowest energy state or ground state turns out to be a unique linear superposition of the degenerate states. In fact, SSB is applicable only to infinite systems - many-body systems (such as ferromagnets, superfluids and superconductors) and fields - the alternative degenerate ground states being all orthogonal to each other in the infinite volume limit and therefore separated by a superselection rule. A superselection rule is a contraction of the Hilbert space. This means that the Hilbert space of the system is built up from one of the ground-states |Ωi, and other Hilbert spaces built on other ground-states become inaccessible because there are no local observables that can connect them [4]. 5 [23/04/2012] Paolo Finelli 1.2 Corso di Teoria delle Forze Nucleari (2012) Variational symmetry or symmetry as a unitary transformation? A typical definition of SSB is that the vacuum state of a broken symmetry theory is not invariant under all the symmetries of the underlying Lagrangian. Symmetry breaking results from a mismatch between a) variational symmetries of the Lagrangian and b) symmetries that can be defined as unitary transformations on the Hilbert space of states. The second sense of symmetry is familiar in quantum mechanics: a symmetry transformation preserves transition probabilities; that is, it is an (invertible) map f : |φi → |φ0 i defined on states |φi in a Hilbert space such that for all φ and ψ, |hφ|ψi| = |hφ0 |ψ 0 i|. Wigner proved that corresponding to any such mapping f there is a linear and unitary (or antilinear and antiunitary4 ) operator Û implementing the symmetry transformation. The mismatch between the two senses of symmetry, a) and b), occurs when there is no unitary operator corresponding to the Noether charge generating a variational symmetry. Noether’s first theorem establishes the existence of a conserved charge for every global variational symmetry of the Lagrangian5 . The theorem applies to the broad class of theories that R derive equations of motion via Hamilton’s principle from the action S = R d4 x L(φ, ∂µ φ, xµ ) where φ(x) are the dependent variables, xµ are the coordinates, and the Lagrangian density L is integrated over a compact space-time region R. A solution φ(xµ ) is a map from space-time to the space of field variables such that the equations of motion, the Euler-Lagrange equations for L, are satisfied. Suppose that there is an r−parameter Lie group G whose elements map (x, φ) → (x, φ0 ) such that S is invariant. Noether’s first theorem establishes that then there are r−conserved currents J µ (φ) such that ∂µ J µ (φ) = 0. The charge associated with the symmetry R is the integral of the time component of this conserved current, that is, Q(φ) = R d3 x J 0 ; it fol- lows from the vanishing divergence of the four vector that Q(φ) is constant and that dQ/dt = 0, if the current flux vanishes on the boundary of the region R. If the two senses of symmetry have to be matched, then in the quantized field theory based on this Lagrangian one would find a 4 Antiunitary operators correspond to symmetries that are not continuously connected to the identity, such as time reversal. Unitary and linear operators have to satisfy the following relations (U φ, U ψ) = (φ, ψ) (1.1) U (ξφ + ηψ) = ξU φ + ηU ψ , (1.2) whereas antilinear and antiunitary operators satisfy (U φ, U ψ) U (ξφ + ηψ) (φ, ψ)∗ = ∗ = (1.3) ∗ ξ Uφ + η Uψ , (1.4) where (·, ·) is the usual internal product. See Weinberg (1995, Sect 2.6, Vol. 1) for more details about Wigner’s theorem or Appendix C. 5 In the following we will consider only internal (no space-time) symmetries, see Appendix D for more details. 6 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) Symmetries as Unitary transformations (Wigner Theorem) Variational symmetries Noether Theorem � � �φ |ψ � = �φ|ψ� Conserved charges � Q = d3 J0 (x) U = eiχQ SSB but the set of degenerate vacua respects the symmetry of L � |Φ0 � |Φ0 � U = eiχQ Figure 1.1: Realization of a spontaneously broken symmetry. one-parameter family of unitary operators Û (ξ) = eiξQ̂ implementing the symmetry, where Q̂ is the operator corresponding to the Noether charge. It can be showed that if the vacuum state |0i is translationally invariant, then the vacuum is either invariant under the internal symmetry, Q̂|0i = 0, or there is no state corresponding to Q̂|0i in the Hilbert space. Fabri-Picasso theorem. There are only two possibilities: 1. Q̂|0i = 0 and |0i is an eigenstate of Q̂ with eigenvalue 0, so that |0i is invariant under Û (i.e. Û |0i = |0i). 2. @ Q̂|0i in the space (its norm is infinite). This statement is more accurate than more intuitive statements like Q̂|0i = 6 0 widely used in literature. By definition, an internal symmetry implemented by Q̂ commutes with the four-momentum operators P̂ µ , i.e. [Q̂, P̂ µ ] = 0, and, by translation invariance of the vacuum state, eiP̂ ·x |0i = |0i. 7 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) These two facts imply that h0|J0 (x)Q̂|0i = h0|eiP̂ ·x J0 (0)e−iP̂ ·x Q̂|0i = h0|J0 (0)Q̂|0i . (1.5) The norm of Q̂V can then be calculated by integrating the current h0|Q̂Q̂|0i = Z V 3 d x h0|J0 (x)Q̂|0i = Z V d3 x h0|J0 (0)Q̂|0i , (1.6) which diverges as V → ∞ unless Q̂|0i = 0 (Ref. [2] and Ref. [3] pgs. 197-198). The second case corresponds to SSB. The symmetry is hidden in that there is no unitary operator to map a physical state to its symmetric counterparts; instead, the symmetry is (roughly speaking) a map from one Hilbert space of states to an entirely distinct space. This is usually described as vacuum degeneracy although each distinct Hilbert space has a unique vacuum state. 1.3 Coleman Theorem and Goldstone Theorem Let brief review some basic results [4]. We start from a Lagrangian density (of real scalar fields for simplicity)6 L (φi (x), ∂µ φi (x)) , (1.7) which gives us by the principle of minimal action δ Z d4 x L = 0 (1.8) the Euler-Lagrange equations of motion ∂µ ∂L ∂(∂µ φi ) − ∂L =0. ∂φi (1.9) If the Lagrangian (1.7) is invariant under an n parametric transformation group φi (x) → Vij (ξ1 , ξ2 , . . . , ξn )φj (x) V = eiξk Ik , (1.10) (1.11) 6 From this point onwards we omit the hat (ˆ) over the operators whenever is possible for simplicity, to ease readability of the text. 8 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) where Ik are the infinitesimal generators, the equations of motion (1.9) will be automatically invariant respect to the same transformations (1.10). At the same time, Noether’s theorem gives us n conserved currents Jkµ = −i ∂L Ik φ ∂µ φ (1.12) (in matrix notation). Since the momentum canonically conjugate to φ is Π= ∂L , ∂(∂0 φ) (1.13) satisfying the well-known Poisson bracket relation7 [φ(x), Π(y)] = δ(x − y) (1.15) [φ(x), φ(y)] = 0 (1.16) [Π(x), Π(y)] = 0 , (1.17) we have, considering that Jk0 = −iΠIk φ, Jk0 (x), φ(y) = iIk φ(x)δ(x − y) 0 Jk (x), φi (y) = i (Ik )ij φj (x)δ(x − y) (1.18) (matrix notation) (1.19) or, introducing the conserved charges, Qk = Z d3 x Jk0 (x) [Qk , φi (y)] = i (Ik )ij φj (y) . we have (1.20) (1.21) Qk is therefore the generator of the infinitesimal canonical transformations corresponding to (1.10). It is natural to introduce an operator U (ξ) = eiξk Qk , (1.22) 7 In canonical coordinates (qi , pj ), on the phase space, given two functions f (pi , qi , t), and g(pi , qi , t), the Poisson bracket takes the form N X ∂f ∂g ∂f ∂g [f, g] = − . (1.14) ∂qi ∂pi ∂pi ∂qi i=1 9 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) in order to implement unitarily the symmetry8 U (ξ)φi (x)U −1 (ξ) = Vij φj (x) . (1.23) In fact Eq. (1.21) could also be obtained expanding (1.23) in the exponential representation. Let consider a unitary operator inducing the group tranformation U = eiξk Qk , where Qk are the conserved charges (or the group generators associated to the k Noether currents). If the vacuum9 is invariant under the group transformation (i.e. U |0i = |0i) then it is necessarily a singlet10 and it is annihilated by the symmetry generators, namely Qk |0i = 0 . (1.24) In fact if we consider the infinitesimal transformation U (ξk )|0i = (1 + iξk Qk )|0i = |0i, it imme- diately follows that the group generator Qk has the property to annihilate the vacuum (1.24). This is the so called Wigner-Weyl realization of the symmetry. The Hamiltonian H can be shown to remain invariant with respect to continuous transformations generated by the group G and the symmetry manifests itself directly in the spectrum of H as degenerate multiplets. The Wigner-Weyl realization is a sort of accounting symmetry since, for example, it allows to classify the particles according to the irreducible representations of the group G (like the isospin label for baryon multiplets in the hadron spectrum). It is easy to see that multiplet structures emerge naturally if the vacuum is left invariant under the symmetry transformation. To prove the last sentence, let consider two states |Ai and |Bi: |Ai = φ†A |0i , |Bi = φ†B |0i , (1.25) where φ†A and φ†B are supposed to relate to each other by a vector transformation [Q, φ†A ] = φ†B (1.26) 8 For every element g ∈ G, it is possible to, given any representation T in the space L, define a new represen0 tation TU (g) acting in the vector space L as follows TU (g) = U T (g)U −1 . T (g) and TU (g) are equivalent representations. We define, as usual, the group generators as Ik the derivatives of the operator T (ξk ) respect to the parameter ξk , taken at ξk = 0. 9 We define the vacuum |0i as the state of the system for which h0|H|0i = min. 10 A singlet is a one-dimensional representation. 10 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) for some generator Q of the symmetry group, such that [Q, H] = 0 . (1.27) U φ†A U −1 ' φ†A + iφ†B (1.28) Eq. (1.26) is equivalent to for an infinitesimal transformation U ' 1 + iQ. Thus φ†A is rotated into φ†B by U , and the operators will create states related by the symmetry transformation. Let’s assume that H|Ai = EA |Ai H|Bi = EB |Bi , (1.29) what assumption is necessary to prove that EA = EB is satisfied? We have EB |Bi = H|Bi = Hφ†B |0i = H(Qφ†A − φ†A Q)|0i . (1.30) Now if Q|0i = 0 we can rewrite the right-hand side of the previous equation as follows HQφ†A |0i = QHφ†A |0i = QH|Ai = EA Q|Ai = EA Qφ†A |0i = EA (φ†B + φ†A Q)|0i = EA |Bi . (1.31) If Q|0i = 0 then follows EA = EB , and multiplets appear naturally in the energy spectrum. 1.3.1 The invariance of the vacuum is the invariance of the world (Coleman Theorem [5]) If a generator Qa of a continuous symmetry group G is given as a space integral of some current density Jaµ (x, t), and if it has the property to annihilate the vacuum (so the vacuum is invariant under G), then the Hamiltonian remains invariant under transformations of the fields according to G and the current is conserved. Proof: If the vacuum is invariant under the group then the generator of the group must annihilate the vacuum. That is to say, Qa (t)|0i = Z d3 x Ja0 (x, t)|0i = 0 , 11 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) then hn| Z 3 d x Ja0 (x, t)|0i = Z d3 x hn|Ja0 (x, t)|0i = 0 , is certainly correct for any arbitrary state |ni, and, of course, for a state |ni with vanishing 3-momentum (p = 0) and non-zero energy11 (p0 6= 0). If the previous relation is valid then12 hn|Ja0 (x)|0i = 0 x = (x, t) is also valid (because of the way we have chosen the momentum), which is the same as hn|∂µ Jaµ (x)|0i = 0 . (1.32) Lorentz-invariance tells us that if Eq. (1.32) is true in one Lorentz frame, it is true in all Lorentz frames. Since any momentum eigenstate can be obtained by applying a Lorentz transformation to a state with zero 3-momentum, the latter equation is true for any momentum state on the left. This is to say that ∂µ J µ (x)|0i = 0 . In QFT there is a theorem (by Federbush and Johnson) which states that any local operator13 which annihilates the vacuum vanishes identically. Therefore ∂µ J µ = 0 , as it should be from Noether’s theorem. This implies that the generator Qa (t) is independent of time and commute with the Hamiltonian H, dQa = i[Qa , H] = 0. dt If the vacuum is not invariant under the symmetry operation associated with m ≤ n genera- tors Qa , then the corresponding symmetry operation applied to the vacuum leads to new states so that Qa |0i = 6 0 or, better ||Qa |0i|| = ∞ . (1.33) 11 In fact non zero energy modes are consequences of a spontaneously broken realization of the symmetry. It is easy to prove that the integral relation exists, at least with a dense set of states on the left. On the other hand if we work with momentum eigenstates this proof can not be considered rigorous, see [7]. 13 Charges are not local operator because Qa is defined as an integral over the space. 12 12 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) In the spectrum of the Hamiltonian there necessarily exists a branch of elementary gap-free excitations whose energies go to zero as the momentum goes to zero. This type of Hamiltonian invariance does not imply the existence of multiplet structure; we call it a Nambu-Goldstone realization of the symmetry. In cases in which the symmetry is spontaneously broken, the total charge of the conserved Noether current associated with the symmetry transformation is not identical to the generator of the corresponding unitary group. Such charges/generators are known as broken charges. In the following we will show that in this case the total charges indeed do not exist as Hermitian operators in a Hilbert space and that the states of the system do not transform according to a irreducible representation of the symmetry group. The phenomenon of vacuum non-invariance can be explained assuming the existence of some non-vanishing macroscopic averages of local operators in the ground-state (so called anomalous averages or order parameters). Each value of an anomalous average will then define a unique vacuum and a corresponding Hilbert space. 1.3.2 Field theories with superconductor solutions (Goldstone Theorem [6, 7, 8]) A spontaneously broken symmetry realization is identified by systems in which the ground state is not an eigenstate of some generators of the global symmetry of the Hamiltonian. Given the charge density J 0 (x), one introduces for an arbitrary finite space domain Ω the operator QΩ (t) = Z d3 x J 0 (x, t) (1.34) Ω The symmetry breaking condition can be restated as the existence of a (not necessarily local) operator Φ such that lim h0|[QΩ (t), Φ]|0i = 6 0 Ω→∞ (1.35) where |0i is a translationally invariant ground state. This expectation value is known as the order parameter. Clearly, this formal definition immediately implies the previous one: if the vacuum were an eigenstate of the charge operator, the expectation value of this commutator would have to be zero. It is customary to identify Q(t) = limΩ→∞ QΩ (t) formally with the integral charge operator. However, this operator strictly speaking does not exist because of the Fabri-Picasso theorem. The intuitive picture of spontaneous symmetry breaking, based on the observation that a symmetry transformation does not leave the ground state intact, suggests high degeneracy of equivalent ground states. Indeed, since the charge operator commutes with the Hamiltonian, so will a finite symmetry transformation generated by this operator. It will therefore transform the ground state into another state with the same energy. As long as the 13 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) symmetry group is continuous, we will find infinitely many degenerate ground states. On account of the fact that they are all connected by symmetry transformations, they must be physically equivalent and any one of them can serve as a starting point for the construction of the spectrum of excited states. The finite volume charge operator QΩ (t) induces a finite symmetry transformation, UΩ (θ, t) = exp[iθQΩ (t)], which in turn gives rise to a rotated ground state, |θ, tiΩ = UΩ† (θ, t)|0i. However, like the limit limΩ→∞ QΩ (t) does not exist, the operator exp[iθQ(t)] is not well defined either. In fact, it can be proved that lim h0|θ, tiΩ = lim h0| exp[−iθQΩ (t)]|0i = 0 . Ω→∞ (1.36) Ω→∞ It means that in the infinite volume (thermodynamic) limit, any two ground states, formally connected by a symmetry transformation, are actually orthogonal. The same conclusion holds for excited states constructed above these vacua. All these states therefore cannot be accommodated in a single separable Hilbert space, forming rather two distinct Hilbert spaces of their own. Any of these Hilbert spaces can, nevertheless, be taken as a basis for an equivalent description of the system, and the choice has no observable physical consequences. Unlike the transformations of physical states, finite symmetry transformations of observables can be consistently defined. Using the Baker–Campbell–Hausdorff formula one obtains for any operator A that 1 Aθ,t;Ω ≡ UΩ (θ, t)AUΩ† (θ, t) = A + iθ[QΩ (t), A] + (iθ)2 [QΩ (t), [QΩ (t), A]] + . . . 2 where [QΩ (t), A] = Z d3 x [J 0 (x, t), A] . (1.37) (1.38) Ω As long as the theory satisfies the microcausality condition14 , that is, the commutator of any two local operators separated by a spacelike interval vanishes, and as long as the operator A is localized in a finite domain of spacetime, there will be a region Ω0 such that the charge density 14 The requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≤ 10−14 cm and to times ≤ 10−24 sec. It is shown in the theory of relativity that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of light leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of light in the small. In quantum theory, where operators correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of space-time if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause). [Grigor0 ev] 14 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) outside this region does not contribute to the commutator, Z 3 0 d x [J (x, t), A] = 0 and R3 \Ω0 lim [QΩ (t), A] = Ω→∞ Z d3 x [J 0 (x, t), A] (1.39) Ω0 The transformation UΩ (θ, t)AUΩ† (θ, t) therefore has a well-defined limit as Ω → ∞. The expectation value of the rotated operator Aθ,t;Ω in the vacuum |0i can then be interpreted as the expectation value of A in the rotated vacuum |θ, tiΩ . Goldstone theorem can be proved under the following basic hypotheses: 1. The degenerate vacuum is invariant under a subgroup F of the symmetry group of the hamiltonian G. 2. Lorentz covariance of the theory (in non-relativistic approaches the theorem still applies but with non-trivial consequences about the counting of the Nambu-Goldstone modes). Under the previous hypotheses it follows the Goldstone theorem15 : For each broken symmetry (each broken charge) one massless mode (massless particle) appears in the energy spectrum. R Given QΩ (x0 ) = Ω d3 x J 0 (x) where Ω is the volume, if lim [H, QΩ ] = 0 Ω→∞ and lim ||QΩ |0i|| = ∞ Ω→∞ and if an operator A exists with lim h0|[QΩ (x0 ), A]|0i 6= 0 | {z } order parameter Ω→∞ then a massless excitation (particle) is present in the energy spectrum. We recall here a standard proof of the Goldstone theorem. We start showing that if a Lagrangian is invariant under a symmetry transformation then a current is conserved ∂µ J µ (x) = 0 , (1.40) 15 There are exceptions, the most important is the case of gauge theories. For any sponteously broken local symmetry, one Goldstone boson disappears from the physical spectrum of the states and the corresponding gauge bosons acquires a mass (due to the fact that is impossible to mantain at the same time manifest Lorentz-covariance and positivity of the Hilbert space), the so called Higgs mechanism (see Sect. 2.2). 15 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) and for a local observable A (no assumptions about it, except that it is localized in a finite region of space) lim Ω→∞ d QΩ (t), A = 0 dt (1.41) where QΩ (t) = Z d3 x J 0 (x, t) . (1.42) Ω In fact, we have (applying the Gauss law) Z d QΩ (t), A + dS · [J (x, t), A] . d x [∂µ J (x, t), A] = 0= dt S Ω Z 3 µ (1.43) For Ω → ∞ the surface integral vanishes because the fields are supposed to vanish at the boundaries (a common prescription in QFT). This is equivalent to say that lim [QΩ (t), A] = Φ Ω→∞ where Φ is time-indipendent dΦ =0 dt and Φ = Φ(r). The Nambu-Goldstone mechanism is realized if16 h0|Φ(r)|0i = 6 0, where |0i is a traslationally invariant vacuum state. This relation implies that |0i cannot be an eigenstate of Q, and it follows from exp(iξQ)|0i = 6 |0i that the corresponding operator U is not a unitary operator. Please note that in the space of the eigenvectors of the observable A the charge Qa is unobservable because it does not commute with A. Let consider now a set of local operators φi (x) not invariant under a continuous symmetry R generated by the charge QaΩ = Ω d3 x J0a , then, by definition, lim h0|[QaΩ (t), φi (x)]|0i = 6 0. Ω→∞ (1.44) 16 More precisely the anomalous average of Φ(r) corresponding to a given Hilbert space is defined as an average over the volume Ω in the following way Z Z 1 1 h0| lim d3 r Φ(r)|0i = lim d3 r h0|Φ(r)|0i 6= 0 . Ω→∞ Ω Ω Ω→∞ Ω Ω 16 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) Explicitly, we have h0|[QaΩ (t), φi (x)]|0i = h0| Z 3 d y Ω and inserting a density of states 1 = → XZ Ω n P [J0a (y), φi (x)]|0i = Z Ω d3 y h0|[J0a (y), φi (x)]|0i , n |nihn|: d3 y (h0|J0a (y)|nihn|φi (x)|0i − h0|φi (x)|nihn|J0a (y)|0i) . (1.45) Then we make use of translational invariance17 J0a (y) = eipy J0a (0)e−ipy to obtain from Eq. (1.45) → XZ Ω n d3 y h0|eipy J0a (0)e−ipy |nihn|φi (x)|0i − h0|φi (x)|nihn|eipy J0a (0)e−ipy |0i . Evaluation of the action of the momentum operator p over the states |ni (and including explicitly the limit Ω → ∞) gives lim Ω→∞ XZ n Ω h i d3 y h0|J0a (0)|nihn|φi (x)|0ie−ipn y − h0|φi (x)|nihn|J0a (0)|0ieipn y 6= 0 . Performing the spatial integration (we now take safely the limit Ω → ∞) Z 0 d3 y e−ipn y = (2π)3 δ 3 (pn )e−iEn y = (2π)3 δ 3 (pn )e−iEn t Ω we have X n h i (2π)3 δ 3 (pn ) h0|J0a (0)|nihn|φi (x)|0ie−iEn t − h0|φi (x)|nihn|J0a (0)|0ieiEn t = h0|Φ(r)|0i = v ∈ C . Now this equation is valid for all times t and, since we have shown that Φ(r) does not depend on t, it follows that the left-hand side of this equation must not depend on time. Clearly these conditions are consistent only if the left-hand side vanishes except for those states |ni where h0|φi (x)|nihn|J0a (0)|0i = 6 0 17 for En |pn →0 = 0 (1.46) Assuming that eipx |0i = |0i and keeping in mind that the operators are local. 17 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) It means that only those states |ni for which the energy vanishes as the 3-momentum goes to zero p→0 En = 0 m2n = p2 = 0 , contribute. In relativistic field theories this implies the existence of massless particles. The state |ni must have the same quantum number as φi (y) and J0a (x). (In particular this state must have the same Lorentz properties of the charge Qa ). The Goldstone particles have to be of zero energy since there is no accompanying change in the energy of the system in going from one vacuum to the other. One can also demonstrate the opposite of the Goldstone theorem: if H does not have a massless particle in its spectrum, the operator U (η) = lim exp iη Ω→∞ Z is unitary. Paolo PaoloFinelli Finelli Ω d x J (x) = lim UΩ (η, t) 3 0 Ω→∞ Corso Corso di di Teoria Teoria delle delle Forze Forze Nucleari Nucleari (2011) (2011) Wigner-Weyl realization Wigner-Weyl Wigner-Weyl realization realization Nambu-Goldstone realization Nambu-Goldstone Nambu-Goldstone realization realization Exact symmetry Spontaneous symmetry breaking Exact Exact symmetry symmetry Q|0� Q|0�==00 Degenerate multiplets multiplets EDegenerate Spontaneous Spontaneous symmetry symmetry breaking breaking ||Q|0�|| ||Q|0�||==∞ ∞ Massless Massless Goldstone Goldstone bosons bosons E Mass gap degenerate multiplets Massless mode as ItIt means means that that only only those those states states |n� |n� for for which which the the energy energy vanishes vanishes as the the 3-momentum 3-momentum goes goesto tozero zero 0 Some general remarks. 0 pp→ →00 EEnn ==00 m m2n2n ==pp22 ==00,, contribute. contribute. In In equations relativistic relativistic field fieldand theories theories this this implies implies the theofexistence existence of ofatmassless massless particles. 1) The of motion the currents involve products field operators the same particles. point and therefore are ill defined quantities whose properas meaning by particular The The state state |n� |n� must must have have the the same same quantum quantum number number as φφii(y) (y)should and and be JJ0a0aobtained (x). (x). (In (In particular limiting procedures starting from different space-time points. this this state state must must have have the the same same Lorentz Lorentz properties properties of of the the charge charge Q Qaa).). The The Goldstone Goldstone particles particles have have to to be be of of zero zero energy energy since since there there isis no no accompanying accompanying change change in in the the 18 [23/04/2012] energy energyofofthe thesystem system in in going going from from one one vacuum vacuum to to the the other. other. One One can can also also demonstrate demonstrate the the opposite opposite of of the the Goldstone Goldstone theorem: theorem: ifif H H does does not not have have aa Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) 2) The construction of the charge from R d3 x J 0 requires in the classical physics case the hypothesis that the fields should vanish at infinity to ensure convergence of the integral. This is physically reasonable. In the quantum case the existence of vacuum fluctuations occuring all over space (translation invariance) does not allow us to take the quantum analogue of the charge as a well defined operator even if a meaning has been given to the density. For this reason one must work with the vacuum expectation values. 3) Spontaneous breaking of a global continuous symmetry implies the existence of poles at p2 = 0 in certain Green’s functions (this poles are related to the presence of massless scalar particles in the physical spectrum, of course). Let us consider the following Green’s function Gaµ,k (x − y) = h0|T Jµa (x)φk (y)|0i , (1.47) where Jµa is the current corresponding to a generator Qa of the symmetry G and φk belongs to an irreducible multiplet of real scalar fields. This Green’s function satisfies a Ward identity that can be obtained by differentiating it (being careful with the derivative of the functions involved in the time ordering) µ a φj (x)δ(x − y)|0i ∂(x) Gaµ,k (x − y) = δ(x0 − y 0 )h0|[J0a (x), φk (y)]|0i = δ(x0 − y 0 )h0| − Tkj a a h0|φj (0)|0i h0|φj (y)|0i = −δ(x − y)Tkj = −δ(x − y)Tkj (1.48) We have used the transformation properties of the field as generated by the Noether current and translational invariance of the vacuum. The Fourier transform of the Ward identity reads a ipµ G̃aµ,k (p) = Tkj h0|φj (0)|0i , where Gaµ,k (x − y) = Z d4 p exp (−ip(x − y)) G̃aµ,k (p) . (2π)4 (1.49) (1.50) Plugging the most general form of the (Fourier-transformed) Green’s function as allowed by Lorentz invariance, G̃aµ,k (p) = pµ Fka (p2 ), in the Ward’s identity we get Fka (p2 ) = − i a T h0|φj (0)|0i , p2 kj (1.51) which implies that the Green’s function corresponding to a generator that does not annihilate the vacuum, a Tkj h0|φj (0)|0i = 6 0, 19 (1.52) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) has a pole at p2 = 0. Let us now consider the following matrix element h0|Jµa (x)|π k (p)i = ifka pµ e−ipx , (1.53) where |π k (p)i describes a particle of mass mk which is a quantum of the field φk . The reduction formula relates this matrix element to the Green’s function G̃sµ,k (p). Defining Gkk0 = = Z Z d4 q −δkk0 e−iq(x−y) 4 2 (2π) q − m2k + i d4 y G−1 (x − y)G(y − x) = δ(x − z) , (1.54) we get h0|Jµa (x)|π k (p)i = = Z d4 y d4 z e−ipz Gaµ,k0 (x − y)iG−1 kk0 (y − z) lim e−ipx G̃aµ,k0 (p)iG̃−1 kk0 (p) p2 →m2k = − lim e−ipx G̃aµ,k0 (p)i(p2 − m2k ) . p2 →m2k (1.55) Putting together Eq. (1.53) with this equation we get, lim G̃aµ,k0 (p)i(p2 − m2k ) = −fka pµ , p2 →m2k (1.56) which implies mk = 0, for those Green’s functions that have a massless pole (i.e. those corresponding to generators that don’t annihilate the vacuum), as we wanted to prove. This proof also gives us the value of fka , a fka = iTkj h0|φj (0)|0i . (1.57) a h0|φ (0)|0i, corresponding to Thus, there must be a massless boson, |Πa (p)i = i|π k (p)iTkj j each broken generator. For more details, see Ref. [4]. 20 [23/04/2012] Chapter 2 Examples of Spontaneous Symmetry Breaking 2.1 2.1.1 Complex scalar fields: U (1) General background Let us start with a complex scalar field Φ, described by a free Lagrangian 1 L = (∂µ Φ∗ )(∂ µ Φ) − M 2 Φ∗ Φ , 2 (2.1) that can be interpreted as composed by two real fields Φ1 and Φ2 1 Φ∗ = √ (Φ1 + iΦ2 ) 2 1 Φ = √ (Φ1 − iΦ2 ) , 2 (2.2) described by a free Lagrangian of two fields with the same mass M 1 1 1 1 L = (∂µ Φ1 )(∂ µ Φ1 ) − M 2 Φ21 + (∂µ Φ2 )(∂ µ Φ2 ) − M 2 Φ22 . 2 2 2 2 (2.3) Φ as a field operator (Φ̂) has the following expansion [13] Φ̂ = Z d3 k √ â(k)e−ik·x + b̂† (k)eik·x , (2π)3 2ω (2.4) where the creation/destruction operators are defined as â(k) = 1 √ (â1 − iâ2 ) 2 21 (2.5) Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) b̂† (k) = and ω = √ 1 √ (â†1 − iâ†2 ) , 2 (2.6) M 2 + k2 . The operators â, ↠, b̂, b̂† obey the commutation relations h i 0 0 â(k), ↠(k ) = (2π)3 δ 3 (k − k ) h i 0 0 b̂(k), b̂† (k ) = (2π)3 δ 3 (k − k ) , (2.7) (2.8) while all other commutators are vanishing. The Hamiltonian operator for the complex field Φ̂ is (dropping the zero point energy, i.e. normal ordering) Ĥ = while the Lagrangian is Z i d3 k h † † â (k)â(k) + b̂ (k) b̂(k) ω, (2π)3 1 L̂ = (∂µ Φ̂† )(∂ µ Φ̂) − M 2 Φ̂† Φ̂ . 2 (2.9) (2.10) In the following we summarize some basic results (details can be easily found in any quantum field theory textbook). The classical real Lagrangian is symmetric respect to O(2) transformations1 , leading to a conserved (Noether) current N µ = Φ1 ∂ µ Φ2 − Φ2 ∂ µ Φ1 , (2.15) ∂µ N µ = 0 , (2.16) that satisfy and a charge (constant of motion) NΦ = 1 Z d3 x N 0 , (2.17) Rotation of coordinates about a predefined z-axis 0 φ1 0 φ2 = (cos α)φ1 − (sin α)φ2 (2.11) = (sin α)φ1 + (cos α)φ2 , (2.12) of an angle α, a real parameter. This is like a rotation of coordinates about the z- axis of ordinary space, but of course it mixes field degrees of freedom, not spatial coordinates. For operators we obtain 0 φ̂1 0 φ̂2 = (cos α)φ̂1 − (sin α)φ̂2 (2.13) = (sin α)φ̂1 + (cos α)φ̂2 . (2.14) 22 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) i.e. with the property that d NΦ = 0 , dt [NΦ , H] = 0 . (2.18) NΦ distinguishes particles from antiparticles. In fact, in terms of Φ, we have N µ = i(Φ∗ ∂ µ Φ − Φ∂ µ Φ∗ ) , (2.19) and the symmetry (operator) charge N̂Φ = Z 3 0 d x N̂ = Z i d3 k h † † â (k)â(k) − b̂ (k) b̂(k) . (2π)3 (2.20) In fact N̂Φ |0i = 0. From Eqs. (2.7, 2.8) some useful relations can be derived h h N̂Φ , Φ̂ N̂Φ , Φ̂† i i = −Φ̂ (2.21) = Φ̂† , (2.22) and, defining Û (α) = eiαN̂Φ and expanding the exponential, we obtain the transformation law 0 Û (α)Φ̂Û −1 (α) = e−iα Φ̂ = Φ̂ , (2.23) i.e., a U (1) rotation2 , a simple phase change. Consider now a state |NΦ i which is an eigenstate of N̂Φ with eigenvalue nφ . It is easy to show that N̂Φ Φ̂|NΦ i = (nΦ − 1)Φ̂|NΦ i , (2.25) 2 U (1) is the group of complex vectors of unit length. The elements of this group, g ∈ U (1), have the form g = eiα . They form a group in the sense that R 1. this set it is closed under complex multiplication i.e. g = eiα ∈ U (1) and g 0 = eiβ ∈ U (1) → g · g 0 = eiα+β ∈ U (1) (2.24) S1 2. there is an identity element, i. e. g = 1 3. for every element g = eiα there is an unique inverse element g −1 = e−iα . The elements of the group U (1) are in one-to-one correspondence with the points of the unit circle S1 . Consequently, the parameter α that labels the transformation (or element of this group) is defined modulo 2π, and it should be restricted to the interval (0, 2π]. However, transformations infinitesimally close to the identity 1 lie essentially on the straight line tangent to the circle at 1 and are isomorphic to the group of real numbers R. The group U (1), which is compact in the sense that the length of its natural parametrization is 2π, which is finite. In contrast the group R of real numbers is non-compact. For infinitesimal transformations the groups U (1) and R are essentially identical. 23 |z|=1 1 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) so the application of Φ̂ to a state lowers its nΦ eigenvalue by 1. This is consistent with the interpretation that the field Φ̂ destroys particles via the â piece or creates an antiparticle via the b̂† piece. In the same way, considering Φ̂† |NΦ i one easily verifies that Φ† increases nΦ by 1, creating a particle via ↠or destroying an antiparticle via b̂. The vacuum state (no particles or antiparticles) is defined by â(k)|0i = b̂(k)|0i = 0 (2.26) for all k. 2.1.2 Symmetry breaking potential If we consider a more general interaction term, the complex Lagrangian can be written as follows 1 L = (∂µ Φ∗ )(∂ µ Φ) − V (Φ) , 2 (2.27) 1 V ≡ λ(Φ∗ Φ)2 + µ2 (Φ∗ Φ) , 4 (2.28) where with µ2 , λ > 0 (λ must be positive to have a bounded energy spectrum). The Hamiltonian density is then H = (∂t Φ∗ )(∂t Φ) + ∇Φ∗ · ∇Φ + V (Φ) . (2.29) It is very easy to see that L is invariant under U (1) transformations. As usual, one first consider the classical case, where the absolute minimum can be reached for: i) Φ = constant and ii) Φ = Φ0 where Φ0 is the minimum of the classical potential. With the previous choice for µ2 and λ, the minimum is Φ = 0. In this case we have two degrees of freedom, both massive, and the vacuum expectation value of the corresponding operator is zero: h0|Φ̂|0i = 0 , (2.30) because â(k)|0i = b̂(k)|0i = 0. If we change the sign of µ2 , the potential will lead to spontaneous symmetry breaking (B stands for breaking) 1 V = VB ≡ λ(Φ∗ Φ)2 − µ2 (Φ∗ Φ) . 4 (2.31) In this case the point Φ1 = Φ2 = 0 is a stationary point, but unstable respect to small fluctuations. The minimum occurs when (Φ∗ Φ) = 24 2µ2 , λ (2.32) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) or, alternatively Φ21 + Φ22 = 4µ2 = v2 . λ (2.33) The symmetry breaking condition can be written as √ |Φ| = v/ 2 . (2.34) At this point, it is useful to introduce the polar variables ρ(x) and θ(x), in order to have ρ(x) Φ(x) = √ e−iθ(x)/v . 2 (2.35) The minimum condition is represented by the circle ρ = v: any point on this circle, at any value of θ, represents a possible classical ground-state (an infinitely degenerate set). If we consider the excitations about a point on the circle of minima, i.e. ρ = v and θ = 0, we obtain 1 Φ̂(x) = √ (v + ĥ(x))e(−iθ̂(x)/v) , 2 (2.36) for the field operator. The Lagrangian becomes3 1 1 µ4 ∂µ ĥ∂ µ ĥ − µ2 ĥ2 + ∂µ θ̂∂ µ θ̂ + + ... 2 2 λ 3 (2.43) We have for the kinetic terms 1 1 −i∂µ θ̂ −iθ̂/v e ∂µ Φ̂ = √ (∂µ ĥ)e−iθ̂/v + √ (v + ĥ) v 2 2 (2.37) and 1 i∂µ θ̂ iθ̂/v 1 ∂µ Φ̂† = √ (∂µ ĥ)eiθ̂/v + √ (v + ĥ) e , v 2 2 and so the terms which are quadratic in the fields are ∂µ Φ̂† ∂ µ Φ̂ = 1 1 ∂µ ĥ∂ µ ĥ + ∂µ θ̂∂ µ θ̂ . 2 2 (2.38) (2.39) The potential terms (up to quadratic powers in the field ĥ) are (no θ̂ degrees of freedom) = = µ2 1 1 − λ (v + ĥ)4 + (v + ĥ)2 4 4 2 λ µ2 2 µ2 2 − (v 4 + 4v 3 ĥ + 6v 2 ĥ2 + . . .) + v + µ2 v ĥ + ĥ 16 2 2 λv 4 λ 3 µ2 2 µ2 2 − − v 3 ĥ − λv 2 ĥ2 + v + µ2 v ĥ + ĥ + . . . 16 4 8 2 2 (2.40) (2.41) (2.42) If we substitute v = 2µ/λ1/2 the linear terms in ĥ cancel. 25 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) The new fields correspond to massive radial oscillations (ρ̂) and angle massless oscillations (θ̂). The particle spectrum in the spontaneously broken case is very different from that in the normal case: instead of two degrees of freedom with the same mass µ, one (θ) is massless and the other √ (h) has a finite mass: 2µ. The broken vacuum |0iB is, of course, annihilated by the operator âh and âθ . This implies B h0|Φ̂|0iB √ = v/ 2 . (2.44) This simple model contains the essence of spontaneous symmetry breaking: a non-zero vacuum value of a field which is not invariant under the symmetry group, zero mass bosons and massive excitations in a direction of the field space which is orthogonal to the degenerate ground states (see Fig. 2.1). This model has, of course, a phenomenological origin because the mechanism V(φ) Nambu-Goldstone massless boson Massive scalar boson § φ2 φ1 Figure 2.1: Massive and massless excitations in the U (1) model. µ2 → −µ2 has to be put in by hand. All the vacuum states are good starting point to build the excited states. If we choose another vacuum: θ = −α, then B h0, α|Φ̂|0, αiB v = e−iα √ = e−iα B h0|Φ̂|0iB . 2 (2.45) On the other hand 0 e−iα Φ̂ = Φ̂ = Ûα Φ̂Ûα−1 , 26 (2.46) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) where Ûα = eiαN̂φ . Eq. (2.45) becomes B h0, α|Φ̂|0, αiB =B h0|Ûα Φ̂Ûα−1 |0iB , (2.47) and we may interpret Ûα−1 |0iB as the alternative (rotated) vacuum |0, αiB (not in the infinite volume limit). For the symmetry current, in terms of ĥ and θ̂, N̂ µ = v∂ µ θ̂ + 2ĥ∂ µ θ̂ + ĥ2 ∂ µ θ̂/v . (2.48) The term involving just the single field θ̂ tells us that there is a non-zero matrix element of the form B h0|N̂ µ |θ, piB = −ipµ ve−ip·x , (2.49) where |θ, pi stands for the state with one θ−quantum state with momentum pµ . When the symmetry is spontaneously broken, the symmetry current connects the vacuum to a state with one Goldstone quantum, with an amplitude which is proportional to the symmetry breaking vacuum expectation value v. ∂µ N̂ µ = 0 only if p2 = 0, as it should be for a Goldstone boson. 27 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) 2.2 Higgs mechanism [3] 2.2.1 Local gauge symmetry Consider the local case: α = α(xµ )4 . If we consider infinitesimal transformations, it is easy to see that derivative terms change differently from fields: Φ → Φ + δΦ = Φ − iαΦ (2.50) Φ∗ → Φ∗ + δΦ∗ = Φ∗ + iαΦ∗ (2.51) ∂µ Φ → ∂µ Φ + δ(∂µ Φ) = ∂µ Φ − i(∂µ α)Φ − iα(∂µ Φ) (2.52) ∂µ Φ∗ → ∂µ Φ∗ + δ(∂µ Φ∗ ) = ∂µ Φ∗ + i(∂µ α)Φ∗ + iα(∂µ Φ∗ ) . (2.53) If we consider the variation of the free Lagrangian density L0 = (∂µ Φ∗ )(∂ µ Φ) , (2.54) δL0 = −i(Φ∗ ∂ µ Φ − Φ∂ µ Φ∗ )δ(∂µ α) + total divergence = j µ δ(∂µ α) . (2.55) we obtain The Lagrangian density is not invariant under local U (1) gauge transformation but its variation depends on the conserved current and the spatial derivatives of the gauge variable α. To mantain 4 te rac tio n In tr me ym al s Loc The gauge principle, which might also be described as a principle of local symmetry, is a statement about the invariance properties Gauge of physical laws. It requires that every continuous symmetry must field be a local symmetry. The key ideas leading up to the introduction of local gauge fields came from Noether, Weyl, and London. Noether was the first to understand the relation between symmetries and conservation laws. The first attempt to generalize continuous symmetry for local invariance instead is due to Weyl. The invariance that Weyl hoped to exploit was an invariance with respect to change of scale: the requirement that physical laws be the same if the scale of all length measurements is changed by Conserved Symmetry the same overall factor. Weyl wanted to require a local gauge inquantity Noether’s Theorem variance in which the scale changes are allowed to be different at different points in space and time, analogous to the curvilinear coordinate transformations of general relativity. In 1927, Fritz London pointed out that the symmetry associated with electric charge conservation was not a scale invariance, but a phase invariance, i.e. the invariance of quantum theory under an arbitrary change in the complex phase of the wavefunction. The invariance under a global phase change multiplication of the wavefunction by a constant phase factor eiθ was trivial in fact; the nontrivial fact was that the existence of the electromagnetic field allows a much broader kind of invariance, invariance under a local phase change, in which the phase factor varies arbitrarily from one point to another in space-time. That is, θ becomes an arbitrary function of x, y, z and t. The word ”gauge” historically refers to a choice of length scale, rather than to the assignment of complex phases. y 28 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) the gauge invariance we must introduce a xµ -dependent term L1 = −gj µ Aµ , (2.56) where g is a coupling constant and Aµ is the gauge field that transforms like 1 Aµ → Aµ + ∂ µ α . g (2.57) L2 = g 2 Aµ Aµ Φ∗ Φ , (2.58) Finally we have to add in order to have a locally gauge invariant Lagrange density δL = δL0 + δL1 + δL2 = 0 . (2.59) 1 L = (∂µ Φ + igAµ Φ) (∂ µ Φ∗ − igAµ Φ∗ ) − M 2 Φ∗ Φ − Fµν F µν , 4 (2.60) Collecting everything we have5 where the term involving Fµν = ∂µ Aν − ∂ν Aµ is clearly gauge invariant and looks like the Lagrangian density of the Maxwell field. Dµ Φ ≡ ∂µ Φ + igAµ Φ is called covariant derivative since it transforms under gauge transformations in the same way as the field Φ δΦ → −iαΦ , (2.61) δ(Dµ Φ) → δ(∂µ Φ) + ig(δAµ )Φ + igAµ (δΦ) 1 = −iα(∂µ Φ) − i(∂µ α)Φ + ig ∂µ α Φ + igAµ (−iαΦ) g = −iα(∂µ Φ) + igAµ (−iαΦ) = −iα(∂µ Φ + igAµ Φ) = −iα(Dµ Φ) . (2.62) From a geometric point of view we can picture the situation as follows. In order to define the phase of Φ(x) locally, we have to define a local frame or fiducial field with respect to which the phase of the field is measured. Local invariance is then the statement that the physical properties of the system must be independent of the particular choice of frame. In this model the field Φ can be associated with a particle of charge q = g = e and the conjugate field Φ∗ with a particle of charge q = −g = −e. The electromagnetic field can therefore be seen as a gauge field that 5 2 a non vanishing mass for the gauge field requires a term of the form MA Aµ Aµ , but such term is not gauge invariant, i.e. gauge fields must be massless. For massless fields there are only two degrees of freedom (transverse d.o.f.). 29 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) arises due to the local U (1) symmetry. The equation of motion for the gauge field is ∂ν F νµ = −ig(Φ∗ Dµ Φ − ΦDµ Φ∗ ) ≡ gJ µ , (2.63) and the source term of the electromagnetic field6 is now given by the covariant current J µ rather than the original current j µ , where ∂µ J µ = 0. We can always promote a global symmetry to a local (i.e. gauge) symmetry by replacing the derivative operator by the covariant derivative. Thus, we can make a system invariant under local gauge transformations at the expense of introducing a vector field Aµ (the gauge field) which plays the role of a connection. From a physical point of view, this result means that the impossibility of making a comparison at a distance of the phase of the field Φ(x) requires that a physical gauge field Aµ (x) must be present. This procedure, which relates the matter and gauge fields through the covariant derivative, is known as minimal coupling. Let add |Φ|4 self-interaction, in order to have 1 L = (∂µ Φ + igAµ Φ) (∂ µ Φ∗ − igAµ Φ∗ ) − µ2 Φ∗ Φ + λ|Φ∗ Φ|2 − Fµν F µν , 4 (2.64) where M 2 → µ2 is now a free parameter. As been done before a new (non trivial) vacuum is generated (if µ2 < 0) v h0|Φ|0i = √ 2 with v= r −µ2 . 2λ (2.65) Expading around the vacuum, we use polar variables as done before 1 Φ̂(x) = √ (v + ρ̂(x)) e−iθ̂(x)/v . 2 (2.66) With this parametrization, the covariant derivative reads 1 Dµ Φ̂ = √ e−iθ̂(x)/v ∂µ ρ̂ + ig(ρ̂ + v)B̂µ , 2 (2.67) where µ has been replaced by B̂µ = µ + g1 ∂µ α̂ because α̂ = −θ̂(x)/v . The Lagrangian can be written in the form 1 1 1 L = − F̂µν F̂ µν + MB2 B̂µ B̂ µ + (∂µ ρ̂)2 + µ2 ρ̂2 4 2 2 1 2 2 + g (ρ̂ + 2ρ̂v)B̂µ B̂ µ − λv ρ̂3 − λρ̂4 , 2 6 (2.68) (2.69) Nonetheless the charge associated to the conservation of electric charge is still given by Qch = pointed out by authors of Ref. [15]. 30 R d3 r j 0 as [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) where now Fµν is expressed in terms of B̂µ -fields. The massless vector field µ and the massless would-be Goldstone field α̂ have been replaced by a new massive vector field B̂µ (with MB = gv). √ The mass of the residual neutral Higgs particle, specified by the ρ̂ field, is 2|µ|. The reason that the would-be Goldstone boson is absorbed in the case of U (1) gauge symmetry is that the freedom which exists to choose the gauge field Aµ (up to a local phase change α(x)) is exploited choosing α(x) = −θ(x)/v to eliminate one of the two real scalar fields, α(x) (the other being ρ(x)). The Goldstone mode θ(x) has been converted into the longitudinal mode of the vector (gauge) field. One therefore end up with a massive vector boson Bµ (x) (3 degrees of freedom) plus a massive scalar (neutral) boson ρ(x) (1 degree of freedom): four degrees of freedom as before (two real scalar fields plus two polarizations for the massless gauge field). To show explicitly how a Goldstone boson disappears let’s study the equation of motion of the gauge field: ν Âν − ∂ ν (∂µ µ ) = Jˆem , (2.70) ν Jˆem = iq(Φ̂† ∂ ν Φ̂ − (∂ ν Φ̂† )Φ̂) − 2q 2 Âν Φ̂† Φ̂ . (2.71) where If we insert Eq. (2.66), we obtain ν Jˆem ∂ ν θ̂  − vq 2 2 ν =v q ! + higher order terms . (2.72) Retaining the linear terms the gauge field satisfies the following equation of motion ν ν µ 2 2  − ∂ (∂µ  ) = −v q ∂ ν θ̂  − vq ν ! , (2.73) where now a gauge transformation on Âν has the following form 1 0 µ (x) →  µ (x) = µ (x) + ∂µ α̂(x) q (2.74) for arbitray α̂. If we define 0  µ (x) = µ (x) − ∂ ν θ̂ , vq (2.75) we basically fix the gauge. The resulting equation for µ is 0 0 0  µ (x) − ∂ ν ∂µ  µ (x) = −v 2 q 2  ν (x) 31 (2.76) [23/04/2012] Paolo Finelli Symmetry Corso di Teoria delle Forze Nucleari (2012) Mechanism Original fields Physical fields Goldstone 1 complex scalar field (2 d.o.f) 1 massive real scalar (1 d.o.f.) 1 massless Goldstone mode (1 d.o.f.) Higgs 1 complex scalar field (2 d.o.f) 1 gauge field (2 d.o.f.) 1 massive real Higgs field (1 d.o.f.) 1 massive vector field (3 d.o.f.) U(1) that can be interpreted as an equation for a free vector massive field 0 0 ( + v 2 q 2 ) µ (x) − ∂ ν ∂µ  µ (x) = 0 , (2.77) with mass equal to vq. 32 [23/04/2012] Paolo Finelli 2.3 Corso di Teoria delle Forze Nucleari (2012) Real scalar fields: SO(3) We now extend the discussion to a system with a continuous, non-Abelian symmetry such as SO(3). To that end, we consider the Lagrangian ~ = L(Φ1 , Φ2 , Φ3 , ∂µ Φ1 , ∂µ Φ2 , ∂µ Φ3 ) ~ ∂µ Φ) L(Φ, 1 µ2 λ = ∂µ Φi ∂ µ Φi − Φi Φi − (Φi Φi )2 , 2 2 4 (2.78) where µ2 < 0, λ > 0, with Hermitian fields Φi . The Lagrangian of Eq. (2.78) is invariant under a global isospin rotation,7 g ∈ SO(3) : Φi → Φ0i = Dij (g)Φj = (e−iαk Tk )ij Φj . (2.79) For the Φ0i to also be Hermitian, the Hermitian Tk must be purely imaginary and thus antisymmetric. The iTk provide the basis of a representation of the so(3) Lie algebra and satisfy the commutation relations [Ti , Tj ] = iijk Tk . We will use the representation with the matrix elements given by tijk = −iijk . As already done we now look for a minimum of the potential which does not depend on x and find ~ min | = |Φ r −µ2 ≡ v, λ ~ = |Φ| q Φ21 + Φ22 + Φ23 . (2.80) ~ min can point in any direction in isospin space we now have a non-countably infinite Since Φ number of degenerate vacua. In analogy to the discussion of the last section, any infinitesimal external perturbation which is not invariant under SO(3) will select a particular direction which, by an appropriate orientation of the internal coordinate frame, we denote as the 3 direction, ~ min = vê3 . Φ (2.81) 7 Of course, the Lagrangian is invariant under the full group O(3) which can be decomposed into its two components: the proper rotations connected to the identity, SO(3), and the rotation-reflections. For our purposes it is sufficient to discuss SO(3). 33 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) ~ min of Eq. (2.81) is not invariant under the full group G = SO(3) since rotations about Clearly, Φ ~ min .8 To be specific, if the 1 and 2 axis change Φ 0 ~ min = v Φ 0 , 1 we obtain 0 0 0 ~ min = T1 Φ 0 0 −i 0 i 0 0 0 i ~ T2 Φmin = 0 0 0 ~ min T3 Φ 0 0 0 = v −i v 0 0 i 0 = v 0 −i 0 0 v 0 0 −i 0 0 = i 0 0 0 = 0 . 0 0 0 v ~ min invariant does not form a group, Note that the set of transformations which do not leave Φ ~ min is invariant under a subgroup because it does not contain the identity. On the other hand, Φ F of G, namely, the rotations about the 3 axis: h∈F : ~ 0 = D(h)Φ ~ = e−iα3 T3 Φ, ~ Φ ~ min = Φ ~ min . D(h)Φ (2.82) We expand Φ3 with respect to v, Φ3 = v + η, (2.83) where η(x) is a new field replacing Φ3 (x), and obtain the new expression for the potential Ṽ = 1 λ λ (−2µ2 )η 2 + λvη(Φ21 + Φ22 + η 2 ) + (Φ21 + Φ22 + η 2 )2 − v 4 . 2 4 4 (2.84) 8 i.e. T1 and T2 do not annihilate the ground state or, equivalently, finite group elements generated by T1 and T2 do not leave the ground state invariant. 34 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) (x2 + y 2 ) (x2 + y 2 )2 − V (x, y) = 2 4 Figure 2.2: Two-dimensional rotationally invariant potential Upon inspection of the terms quadratic in the fields, one finds after spontaneous symmetry breaking two massless Goldstone bosons and one massive boson: m2Φ1 = m2Φ2 = 0, m2η = −2µ2 . (2.85) The model-independent feature of the above example is given by the fact that for each of the two generators T1 and T2 which do not annihilate the ground state one obtains a massless Goldstone boson. By means of a two-dimensional simplification (see the “Mexican hat” potential shown in Fig. 2.2) the mechanism at hand can easily be visualized. Infinitesimal variations orthogonal to the circle of the minimum of the potential generate quadratic terms, i.e. restoring forces linear in the displacement, whereas tangential variations experience restoring forces only of higher orders. 35 [23/04/2012] Paolo Finelli 2.3.1 Corso di Teoria delle Forze Nucleari (2012) Application of the Goldstone Theorem ~ Given a Hamiltonian operator with a global symmetry group G = SO(3), let Φ(x) = (Φ1 (x), Φ2 (x), Φ3 (x)) denote a triplet of local Hermitian operators transforming as a vector under G 9 , g∈G: ~ ~ 0 (x) = ei Φ(x) 7→ Φ = e−i P3 k=1 P3 k=1 αk Q k ~ Φ(x)e−i P3 l=1 αl Q l αk Tk ~ ~ Φ(x) 6= Φ(x), (2.86) where the Qi are the generators of the SO(3) transformations on the Hilbert space satisfying [Qi , Qj ] = iijk Qk and the Ti = (tijk ) are the matrices of the three dimensional representation satisfying tijk = −iijk . We assume that one component of the multiplet acquires a non-vanishing vacuum expectation value: h0|Φ1 (x)|0i = h0|Φ2 (x)|0i = 0, h0|Φ3 (x)|0i = v 6= 0. (2.87) Then 1. the two generators Q1 and Q2 do not annihilate the ground state 2. to each such generator corresponds a massless Goldstone boson. In order to prove these two statements let us expand Eq. (2.86) to first order in the αk : ~0 = Φ ~ +i Φ 3 X k=1 ~ = (1 − i αk [Qk , Φ] 3 X ~ αk Tk )Φ k=1 Comparing the terms linear in the αk i[αk Qk , Φl ] = lkm αk Φm and noting that all three αk can be chosen independently, we obtain i[Qk , Φl ] = −klm Φm , which, of course, simply expresses the fact that the field operators Φi transform as a vector. Using klm kln = 2δmn , we find i − kln [Qk , Φl ] = δmn Φm = Φn . 2 9 The relation is equivalent to say [Qi , Φj (x)] = ijk Φk (x) , where ijk is a pure totally antisymmetric function of the three indices. 36 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) In particular, i Φ3 = − ([Q1 , Φ2 ] − [Q2 , Φ1 ]), 2 (2.88) with cyclic permutations for the other two cases. In order to prove that Q1 and Q2 do not annihilate the ground state, let us consider Eq. (2.86) for α ~ = (0, π/2, 0), π ~ = e−i 2 T2 Φ cos(π/2) 0 0 sin(π/2) Φ1 Φ3 Φ2 = Φ2 −Φ1 Φ3 cos(π/2) Φ1 Φ3 Φ2 = Φ2 1 0 − sin(π/2) 0 0 0 1 = 0 1 0 Φ3 −1 0 0 Φ1 −i π Q i π2 Q2 = e Φ2 e 2 2 . −Φ1 Φ3 From the first row we obtain π π Φ3 = ei 2 Q2 Φ1 e−i 2 Q2 . Taking the vacuum expectation value π π v = h0|ei 2 Q2 Φ1 e−i 2 Q2 |0i Q2 |0i = 6 0 is the only possible solution, since otherwise the exponential operator could be replaced by unity and the right-hand side would vanish. A similar argument shows Q1 |0i = 6 0. Let us now turn to the existence of Goldstone bosons, taking the vacuum expectation value of Eq. (2.88) i i 0 6= v = h0|Φ3 (0)|0i = − h0| ([Q1 , Φ2 (0)] − [Q2 , Φ1 (0)]) |0i ≡ − (A − B). 2 2 We will first show A = −B. To that end we perform a rotation of the fields as well as the generators by π/2 about the 3 axis [see Eq. (2.86) with α ~ = (0, 0, π/2)]: π ~ = e−i 2 T3 Φ −Φ2 Φ1 π π Φ1 = ei 2 Q3 Φ2 e−i 2 Q3 , Φ3 Φ3 37 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) and analogously for the charge operators −Q2 Q1 π π Q1 = ei 2 Q3 Q2 e−i 2 Q3 . Q3 Q3 We thus obtain π π π π 3 i 2 Q3 B = h0|[Q2 , Φ1 (0)]|0i = h0| ei 2 Q3 (−Q1 ) |e−i 2 Q{z e } Φ2 (0)e−i 2 Q3 1 π i π2 Q3 −i π2 Q3 i π2 Q3 −e Φ2 (0)e e (−Q1 )e−i 2 Q3 |0i = −h0|[Q1 , Φ2 (0)]|0i = −A, where we made use of Q3 |0i = 0, i.e., the vacuum is invariant under rotations about the 3 axis. In other words, the non-vanishing vacuum expectation value v can also be written as 0 6= v = h0|Φ3 (0)|0i = −ih0|[Q1 , Φ2 (0)]|0i Z = −i d3 x h0|[J01 (~x, t), Φ2 (0)]|0i. We insert a complete set of states 1 = v = −i Z Z X n R P n |nihn| (2.89) into the commutator10 d3 x h0|J01 (~x, t)|nihn|Φ2 (0)|0i − h0|Φ2 (0)|nihn|J01 (~x, t)|0i , and make use of translational invariance Z Z X = −i d3 x e−iPn x h0|J01 (0)|nihn|Φ2 (0)|0i − · · · = −i n Z X n (2π)3 δ 3 (Pn ) e−iEn t h0|J01 (0)|nihn|Φ2 (0)|0i −eiEn t h0|Φ2 (0)|nihn|J01 (0)|0i . Integration with respect to the momentum of the inserted intermediate states yields an expression of the form = −i(2π)3 X n0 e−iEn t · · · − eiEn t · · · , 10 R P The abbreviation n |nihn| includes an integral over the total momentum p as well as all other quantum numbers necessary to fully specify the states. 38 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) where the n0 indicates that only states with P = 0 need to be considered. Due to the Hermiticity of the symmetry current operators J µ,a as well as the Φl , we have cn := h0|J01 (0)|nihn|Φ2 (0)|0i = hn|J01 (0)|0i∗ h0|Φ2 (0)|ni∗ , such that v = −i(2π)3 X n0 cn e−iEn t − c∗n eiEn t . (2.90) From Eq. (2.90) we draw the following conclusions. 1. Due to our assumption of a non-vanishing vacuum expectation value v, there must exist 0 (0)|ni and hn|Φ states |ni for which both h0|J1(2) 1(2) (0)|0i do not vanish. The vacuum itself cannot contribute to Eq. (2.90) because h0|Φ1(2) (0)|0i = 0. 2. States with En > 0 contribute (ϕn is the phase of cn ) 1 1 cn e−iEn t − c∗n eiEn t = |cn | eiϕn e−iEn t − e−iϕn eiEn t i i = 2|cn | sin(ϕn − En t) to the sum. However, v is time-independent and therefore the sum over states with (En > 0, 0) must vanish. 3. The right-hand side of Eq. (2.90) must therefore contain the contribution from states with zero energy as well as zero momentum, i.e. zero mass. These zero-mass states are the Goldstone bosons. 39 [23/04/2012] Paolo Finelli 2.4 Corso di Teoria delle Forze Nucleari (2012) Generalization to n-Lie group Let consider the model in the case of an arbitrary compact Lie group G 11 of order nG resulting in nG infinitesimal generators. Once again, we start from a Lagrangian ~ ∂µ Φ) ~ = 1 ∂µ Φ ~ · ∂µΦ ~ − V (Φ), ~ L(Φ, 2 (2.95) ~ is a multiplet of scalar (or pseudoscalar) Hermitian fields. The Lagrangian L and thus where Φ ~ are supposed to be globally invariant under G, where the infinitesimal transformations also V (Φ) of the fields are given by g∈G: Φi → Φi + δΦi , δΦi = −ia taij Φj . (2.96) The Hermitian representation matrices T a = (taij ) are again antisymmetric and purely imaginary. We now assume that, by choosing an appropriate form of V , the Lagrangian generates a spontaneous symmetry breaking resulting in a ground state with a vacuum expectation value ~ min = hΦi ~ which is invariant under a continuous subgroup F of G. We expand V (Φ) ~ with Φ 11 N −component real scalar field φa (x)(a = 1, . . . , N ). In this case the symmetry is the group of rotations in N-dimensional space φa (x) = Rab φb (x) (2.91) φa is said to transform like the N −dimensional (vector) representation of the Orthogonal group O(N ). The elements of the orthogonal group, R ∈ O(N ), satisfy R1 ∈ O(N ) and R2 ∈ O(N ) → R1 R2 ∈ O(N ) ∃I ∈ O(N ) such that ∀R ∈ O(N ) → RI = IR = R ∀R ∈ O(N ), ∃R−1 ∈ O(N ) such that R−1 = RT (2.92) where RT is the transpose of the matrix R. N −component complex scalar field φa (x)(a = 1, . . . , N ). If the N-component vector φa (x) is a complex field, it transforms under the group of (N × N ) Unitary transformations U φ0 a(x) = U ab φb (x) . (2.93) The complex N × N matrices U are elements of the Unitary group U (N ) and satisfy U1 ∈ U (N ) and U2 ∈ U (N ) → U1 U2 ∈ U (N ) ∃I ∈ U (N ) such that ∀U ∈ U (N ) → U I = IU = U ∀U ∈ U (N ), ∃U −1 ∈ U (N ) such that U −1 = U † (2.94) where U † = (U T )∗ . 40 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) ~ min , |Φ ~ min | = v, i.e., Φ ~ =Φ ~ min + ~η , respect to Φ 2 ~ ~ ~ = V (Φ ~ min ) + ∂V (Φmin ) ηi + 1 ∂ V (Φmin ) ηi ηj + · · · . V (Φ) ∂Φ 2 ∂Φi ∂Φj | {z i } | {z } 2 0 m (2.97) ij The matrix M 2 = (m2ij ) must be symmetric and, since one is expanding around a minimum, positive semidefinite, i.e., X i,j m2ij ηi ηj ≥ 0 ∀ ~η . (2.98) In this case, all eigenvalues of M 2 are non-negative. Making use of the invariance of V under the symmetry group G, ~ min ) V (Φ ~ min ) = V (Φ ~ min + δ Φ ~ min ) = V (D(g)Φ (2.97) ~ min ) + 1 m2ij δΦmin,i δΦmin,j + · · · , = V (Φ 2 (2.99) one obtains, by comparing coefficients, m2ij δΦmin,i δΦmin,j = 0. (2.100) Differentiating Eq. (2.100) with respect to δΦmin,k and using m2ij = m2ji results in the matrix equation ~ min = ~0. M 2δΦ (2.101) ~ min = −ia T a Φ ~ min , we conclude Inserting the variations of Eq. (2.96) for arbitrary a , δ Φ ~ min = ~0. M 2T aΦ (2.102) The solutions of Eq. (2.102) can be classified into two categories: 1. T a , a = 1, · · · , nF , is a representation of an element of the Lie algebra belonging to the subgroup F of G, leaving the selected ground state invariant. In that case one has ~ min = ~0, T aΦ a = 1, · · · , nF , such that Eq. (2.102) is automatically satisfied without any knowledge of M 2 . 2. T a , a = nF + 1, · · · , nG , is not a representation of an element of the Lie algebra belonging ~ min 6= ~0, and T a Φ ~ min is an eigenvector of M 2 to the subgroup F. In that case T a Φ 41 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) with eigenvalue 0. To each such eigenvector corresponds a massless Goldstone boson. ~ min 6= ~0 are linearly independent, resulting in nG − nH In particular, the different T a Φ independent Goldstone bosons12 . Let us check these results by reconsidering the example of Eq. (2.78). In that case nG = 3 and nF = 1, generating 2 Goldstone bosons [see Eq. (2.85)]. We conclude this section with a remark. The number of Goldstone bosons is determined by the structure of the symmetry groups. Let G denote the symmetry group of the Lagrangian, with nG generators and F the subgroup with nF generators which leaves the ground state after spontaneous symmetry breaking invariant. For each generator which does not annihilate the vacuum one obtains a massless Goldstone boson. The total number of Goldstone bosons equals nG − nH . 12 If they were not linearly independent, there would exist a nontrivial linear combination nG nG X X ~ min ) = ~ min , ~0 = ca (T a Φ ca T a Φ a=nF +1 a=nF +1 | {z := T } such that T is an element of the Lie algebra of H in contradiction to our assumption. 42 [23/04/2012] Bibliography [1] Symmetries in Physics, Philosophical Reflections, Edited by Katherine Brading and Elena Castellani, Cambridge Press (2003). [2] E. Fabri and L.E. Picasso, Quantum Field Theory and Approximate Symmetries, Phys. Rev. Lett. 16 (1966) 408. [3] I. J. R. Aitchinson and A. J. G. Hey, Gauge Theories in Particle Physics, Volume II, IOP (2004). [4] E. Witten, Lectures delivered at the Institute for Advanced studies (IAS), http://www.math.ias.edu/QFT/spring/index.html . J. A. Swieca, Goldstone’s theorem and related topics, Cargese Lectures in Physics 4 (1970) 215 (available on request from the lecturer). These lectures are extremely difficult. [5] S. Coleman, The Invariance of the Vacuum is the Invariance of the World, Jour. Math. Phys. 7 (1966) 787. [6] J. Goldstone, Field Theories with Superconductor Solutions, N. Cim. 19 (1961) 154. [7] J. Bernstein, Spontaneous Symmetry Breaking, Gauge Theories, the Higgs Mechanism and All That, Rev. Mod. Phys. 46 (1974) 7. [8] T. Brauner, Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum Many-Body Systems, Symmetry 2 (2010) 609. [9] Chuang Liu Classical Spontaneous Symmetry Breaking, Phil. of Sci. 70 (2003) 1219 and references therein. [10] S. Scherer and M. R. Schindler, A Chiral Perturbation Theory Primer, [arXiv:hepph/0505265]. 43 Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) [11] S. Gasiorowicz, Quantum Physics, Wiley Ed. [12] S. Weinberg, The Quantum Theory Of Fields. Vol. 2: Modern Applications (Cambridge University Press, Cambridge, 1996). [13] M. E. Peskin and D. V. Schroeder, An Introduction To Quantum Field Theory, (AddisonWesley, 1995). [14] David J. Gross, Symmetry in Physics: Wigner’s Legacy, Phys. Tod. 48 (1995) 46. [15] K. Brading, Which Symmetry? Noether, Weyl, and Conservation of Electric Charge, Studies in History and Philosophy of Science Part B 33 (2002) 3. More references [16] a very exhaustive review: G. S. Guralnik, C. R. Hagen, T. W. B. Kibble, Broken symmetries and the Goldstone theorem, Adv. in Part. Phys. 2 (1968) 567 (beyond the scope of this lecture). [17] a high-level book: F. Strocchi, Symmetry Breaking, Lect. Not. in Phys. 732 (2008) (beyond the scope of this lecture). [18] C. Quigg, Spontaneous symmetry breaking as a basis of particle mass, Rep. Prog. Phys. 70 (2007) 1019 (SSB in gauge theories). 44 [23/04/2012] Appendix A History of Spontaneous Symmetry Breaking From Ref. [1] and Wikipedia: Historically, the concept of SSB first emerged in condensed matter physics. The prototype case is the 1928 Heisenberg theory of the ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with spin-spin interactions between nearest neighbours such that neighbouring dipoles tend to align. Although the theory is rotationally invariant, below the critical Curie temperature Tc the actual ground state of the ferromagnet has the spin all aligned in some particular direction (i.e. a magnetization pointing in that direction), thus not respecting the rotational symmetry. What happens is that below Tc there exists an infinitely degenerate set of ground states, in each of which the spins are all aligned in a given direction. A complete set of quantum states can be built upon each ground state. We thus have many different possible worlds (sets of solutions to the same equations), each one built on one of the possible orthogonal (in the infinite volume limit) ground states. To use a famous image by S. Coleman, a little man living inside one of these possible asymmetric worlds would have a hard time detecting the rotational symmetry of the laws of nature (all his experiments being under the effect of the background magnetic field). The symmetry is still there the Hamiltonian being rotationally invariant but hidden to the little man. Besides, there would be no way for the little man to detect directly that the ground state of his world is part of an infinitely degenerate multiplet. To go from one ground state of the infinite ferromagnet to another would require changing the directions of an infinite number of dipoles, an impossible task for the finite little man. As said, in the infinite volume limit all ground states are separated by a superselection rule. The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state, and the role of the little man being played by ourselves. This means that there may exist symmetries of the laws of nature which are not manifest to us because the physical world in which we live is built on a vacuum state which is not invariant under them. In other words, the physical world of our experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the fundamental laws of nature. SSB offers a key for understanding (and utilizing) this physical possiblity. 45 Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) The concept of SSB was transferred from condensed matter physics to QFT in the early 1960s, thanks especially to works by Y. Nambu and G. Jona-Lasinio. Jona-Lasinio. The idea of SSB was introduced and formalized in particle physics on the grounds of an analogy with the breaking of (electromagnetic) gauge symmetry in the 1957 theory of superconductivity by J. Bardeen, L. N. Cooper and J. R. Schrieffer (the so-called BCS theory). The application of SSB to particle physics in the 1960s and successive years led to profound physical consequences and played a fundamental role in the edification of the current Standard Model of elementary particles. In particular, let us mention the following main results that obtain in the case of the spontaneous breaking of a continous internal symmetry in QFT. Goldstone theorem. In the case of a global continuous symmetry, massless bosons (known as Goldstone bosons) appear with the spontaneous breakdown of the symmetry according to a theorem first stated by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious problem since no particles of the sort had been observed in the context considered, was in fact the basis for the solution by means of the so-called Higgs mechanism (see the next point) of another similar problem, that is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields predicted unobservable massless particles, the gauge bosons. Higgs mechanism. In 1964 three teams proposed related but different approaches to explain how mass could arise in local gauge theories. These three, now famous, papers were written by Robert Brout and Franois Englert, Peter Higgs, and Gerald Guralnik, C. Richard Hagen, and Tom Kibble, and are credited with the prediction of the Higgs boson and Higgs mechanism (or Englert-Brout-Higgs-GuralnikHagen-Kibble mechanism) which provides the means by which gauge bosons can acquire non-zero masses in the process of spontaneous symmetry breaking. The mechanism is the key element of the electroweak theory that forms part of the Standard Model of particle physics, and of many models, such as the Grand Unified Theory, that go beyond it. Each of these papers is unique and demonstrates different approaches to showing how mass arise in gauge particles. Over the years, the differences between these papers are no longer widely understood, due to the passage of time and acceptance of end-results by the particle physics community. While first to publish by a couple months, Higgs, Brout and Englert solved half of the problem massifying the gauge particle. Guralnik, Hagen and Kibble, while published a couple months later, had a more complete solution. Additional bibliography: SSB1 L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev. 104 (1956) 1189. SSB2 J. Bardeen, L.N. Cooper and J. R. Schrieffer, Microscopic Theory of Superconductivity, Phys. Rev. 106 (1957) 162. SSB3 J. Bardeen, L. N. Cooper and J. R. Schrieffer, Theory of Superconductivity, Phys. Rev. 108 (1957) 1175. 46 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) SSB4 Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. I, Phys. Rev. 122 (1961) 345. SSB5 Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II, Phys. Rev. 124 (1961) 246. SSB6 J. Goldstone, A. Salam and S. Weinberg, Broken Symmetries, Phys. Rev. 127 (1962) 965. SSB7 P. W. Anderson, Plasmons, Gauge Invariance, and Mass, Phys. Rev. 130 (1963) 439. SSB8 A. Klein and B. W. Lee, Does Spontaneous Breakdown of Symmetry Imply Zero-Mass Particles?, Phys. Rev. Lett. 12 (1964) 266. SSB9 W. Gilbert, Broken Symmetries and Massless Particles, Phys. Rev. Lett. 12 (1964) 713. SSB10 F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett. 13 (1964) 321. SSB11 P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett. 13 (1964) 508. SSB12 G. S. Guralnik, C.R. Hagen and T.W.B. Kibble, Global Conservation Laws and Massless Particles, Phys. Rev. Lett. 13 (1964) 585. SSB13 P. W. Higgs, Broken Symmetries, Massless Particles and Gauge Fields, Phys. Lett. 12 (1964) 132. SSB14 P. W. Higgs, Spontaneous Symmetry Breakdown without Massless Bosons, Phys. Rev. 145 (1966) 1156. SSB15 S. Weinberg, Conceptual Foundations of the Unified Theory of Weak and Electromagnetic Interactions, Nobel Prize Lecture (1979). SSB16 P. W. Higgs, My Life as a Boson, www.kcl.ac.uk/nms/depts/physics/news/events/MyLifeasaBoson.pdf . SSB17 G. S. Guralnik, The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles, [arXiv:0907.3466]. As pedagogical introductions we suggest the following links 1. http://www.vega.org.uk/video/programme/76 2. http://www.hep.ucl.ac.uk/ djm/higgsa.html 47 [23/04/2012] Appendix B Group Theory - a short introduction A group G is a set of elements {g, h, k, . . .} for which a multiplication is defined which assigns to every two elements g, h ∈ G an element g · h which is again an element of the group. In addition the following properties should hold: 1. The multiplication is associative, which means that we have (g · h) · k = g · (h · k) for all g, h, k ∈ G. In the special case that the group multiplication is commutative, g · h = h · g for all g, h ∈ G, the group is called abelian. 2. The set of elements of G contains the identity I, for which we have I · g = g · I for all g ∈ G, as well as the inverse elements g −1 for every g ∈ G, i.e. g −1 · g = g · g −1 = I. A subset H of elements contained in G is called a subgroup of G if H itself is also a group according to the definition given above. Symmetry transformations always form a group. One can make an obvious distinction between discrete and continuous transformations. Discrete symmetries usually constitute a finite group, i.e. a group consisting of a finite number of elements. Continuous symmetries depend on one or more parameters in a continuous fashion. Clearly a group of such transformations contains an infinite number of elements. The dimension of a continuous group is defined as the number of independent parameters on which the group elements depend. If the dependence on these parameters is analytic then we are dealing with a so called Lie group. If there is a mapping from a group G to a set of matrices D(G) which preserves the group multiplication then D(G) is called a representation of the group G. In that case, to any element g ∈ G there belongs a matrix D(g) ∈ D(G) such that to the product g · h of two elements g and h of G there belongs a matrix D(g · h) such that D(g · h) = (D(g) · D(h)) ∈ D(G). 48 Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) The mapping between G and D(G) is called a homomorphism. If the mapping is one-to-one then it is called an isomorphism and D(G) is a faithful representation. The number of such matrices is equivalent to the rank of the group and it is independent of the dimension N of the representation that will be chosen. In case the mapping is not into a set of matrices but into some other algebraic structure, it is called a realization. In general a group can have many different representations. Example. As an example we recall the representations of the rotation group, which are wellknown from quantum mechanics. These representations are characterized by an integer l(l = 0, 1, 2, . . .) and consist of (2l+1)×(2l+1) matrices acting on states with total angular momentum L2 = ~2 l(l + 1); the latter are labelled by their value of angular momentum projected along a certain axis (e.g. Lz = −~l, −~(l − 1), . . . , ~l). For each rotation g (which is a 3 × 3 orthogonal matrix) there is a (2l + 1) × (2l + 1) matrix D(g), which specifies how the 2l + 1 states transform among themselves as a result of the rotation. The quantity 2l + 1 is called the dimension of the representation. It is rather obvious that combining two representations of dimension 2l1 + 1 and 2l2 + 1 leads to another representation of dimension 2(l1 + l2 + 1). The latter representations are called reducible as they can be reduced to smaller representations. Evidently not much new is to be learnt from studying reducible representations, so that one usually restricts oneself to irreducible representations. It can be shown that finite groups must have a finite number of irreducible representations. Continuous groups have infinitely many representations. B.1 Lie Groups Consider a one-parameter Lie group with elements g(ξ). Because of the analyticity of g(ξ) it is always possible to choose a so-called canonical parametrization, which satisfies g(ξ 1 )g(ξ 2 ) = g(ξ 1 + ξ 2 ) . (B.1) Consequently g(0) = I , g −1 (ξ) = g(−ξ) . (B.2) Using this parametrization and the fact that g(ξ) is analytic we can write an element in a neighbourhood of the identity element as g(ξ) = I + ξt + O(ξ 2 ) , (B.3) where t is an operator which generates the infinitesimal group transformation. Using (B.1) we can formally construct finite elements g(ξ) by making an infinite series of infinitesimally small 49 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) steps away from the identity element: n g(ξ) = {g(ξ/n)} = lim n→∞ n ξ I + t... = exp(ξt) , n (B.4) where the exponentiation is defined by its series expansion. The result can directly be extended to an n-parameter Lie group: g(ξ 1 , ξ 2 , . . . ξ n ) = exp(ξ a ta ) , (B.5) where we have adopted the summation convention The quantities ta , which characterize the infinitesimal transformations that are linearly independent, are called the generators of the Lie group. Example. As a second example consider all two-dimensional rotations, which obviously form a Lie group with the angle of rotation ξ as a natural canonical parameter. Using polar coordinates with x = r cos θ , y = r sin θ a (clockwise) rotation g(ξ) changes the value into θ − ξ. Infinitesimally one has x y ! → x y ! +ξ y −x ! + O(ξ 2 ) . (B.6) According to the previous relation the generator t can be written as a 2 × 2 matrix 0 1 −1 0 ! . (B.7) Using t2 = −I a finite rotation g(ξ) can be written as ∞ X 1 n n ξ t g(ξ) = exp(ξt) = n! n=0 ∞ X 1 1 2 n n 2n+1 = (−ξ ) I + (−1) ξ t (2n)! (2n + 1)! n=0 ! cos ξ sin ξ = cos ξI + sin ξt = , − sin ξ cos ξ (B.8) which indeed constitutes a general two-dimensional rotation. The above group is called SO(2). It is the group of all orthogonal 2 × 2 matrices with unit determinant. This is a special case of the group O(N ) which consists of all orthogonal N × N matrices, and the group SO(N ) for the subgroup of elements of O(N ) with unit determinant. Similarly, U (N ) is the group of unitary N ×N matrices, and SU (N ) is the group of elements of U (N ) with unit determinant. Obviously, 50 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) O(N ) and SO(N ) are subgroups of U (N ) and SU (N ), respectively. B.2 Lie Algebra Consider an n-parameter Lie group G with elements g(ξ 1 , . . . ξ n ) and generators ta (a = 1, . . . , n). According to (B.5) we may write g(ξ 1 , . . . ξ n ) = exp(ξ a ta ) . (B.9) A product of two such elements can be expressed by means of the Baker-Campbell-Hausdorff formula g(ξ 1 , . . . ξ n ) · g(χ1 , . . . χn ) = exp(ξ a ta ) · exp(χb tb ) 1 1 a b c = exp{ξ a ta + χa ta + ξ a χb [ta , tb ] + ξ ξ χ + χa χb ξ c [ta , [tb , tc ]] 2 12 + higher order commutators of t} . (B.10) Because G is a group, the product (B.10) must again be an exponential form of the generators, so there must be coefficients η 1 , . . . , η n such that g(ξ 1 , . . . , ξ n ) · g(χ1 , . . . χn ) = exp(η a ta ) . (B.11) This is possible if and only if any commutator of generators can again be written as a linear combination of generators. In other words, the generators must close under commutation: c [ta , tb ] = fab tc , (B.12) c are constants, antisymmetric in their lower indices (since we assume real parameters where fab ξ a , these constants are real). With this property the generators ta form the basis of the so-called Lie algebra g associated with the Lie group G. Therefore they are called the structure constants of the group (abelian groups have zero structure constants). Example. As an example consider the group SU (2), defined as the set of all unitary 2 × 2 matrices with unit determinant. Elements of this group can be written as (for SU (2) we prefer to include a factor i in order to have hemitean instead of antihermitean generators) ~ = exp(iξa ta ) . gSU (2) (ξ) (B.13) ~ is a unitary matrix with unit determinant leads to the following The requirement that gSU (2) (ξ) 51 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) conditions for the generators: 1 ta = τa . 2 (B.14) τi τj = δij 1 + iijk τk (B.15) These τ matrices obey and are hermitean (τi = τi† ) and traceless (Trτi = 0). Since [τi , τj ] = i2ijk τk is easy to prove that [ta , tb ] = iabc tc . (B.16) For infinitesimal transformations ~ ' 1 + iB + O(B 2 ) g(ξ) B = ξi τi 2 (B.17) where B † = B and TrB = 0. The finite transformation is found by exponentiation of Eq. (B.17): n τ ξ τ i i i ~ = lim 1 + i g(ξ) . = exp iξi n→∞ n 2 2 The matrices 12 τi are the generators of the rotations for the l = ~ = g(ξ) ∞ X n=0 1 (2n)! iξj τj 2 2n + ∞ X n=0 1 (2n + 1)! 1 2 (B.18) representation. By definition iξj τj 2 2n+1 . (B.19) With the help of (iξj τj )2 = ξj ξk τj τk = −ξ 2 1 , (B.20) one can show that (iξj τj )2n = (−)n ξ 2n 1 and (iξj τj )2n+1 = (−)n ξ 2n (iξj τj ) . (B.21) With the previous relations we can define the SU (2) transformation element as ~ = cos ξ 1 + i sin ξ ξj τj , g(ξ) 2 2 ξ where ξ = p (B.22) ξ12 + ξ22 + ξ32 . Every 2 × 2 matrix can be decomposed in th unit matrix 1 and τi : g = c0 1 + ici τi , 52 (B.23) [23/04/2012] Paolo Finelli where Corso di Teoria delle Forze Nucleari (2012) 1 c0 = Tr(g) 2 In our case the coefficients are c0 = cos 1 and ci = Tr(gτi ) . 2 (B.24) ξ 2 (B.25) ci = ξi ξ sin ξ 2 with the properties c20 + c2i = 1. Eq. (B.23) can also be written in terms of two complex parameters a and b g= a b −b∗ a∗ ! , (B.26) with |a|2 + |b|2 = 1. Such matrices of this form form elements of the group SU (2), the group of unitary 2 × 2 matrices with determinant 1 because they obey g † = g −1 detg = 1 . (B.27) The parameter space parametrized by ξ can be restricted to the inside of a 2-dimensional sphere of radius 2π, i.e. (ξ 1 )2 + (ξ 2 )2 + (ξ 3 )2 ≤ (2π)2 . (B.28) The parameter space of SU(2) can be divided into two parts: an inside region where ξ ≤ π and an outer shell with π < ξ ≤ 2π. To each point ξ in the first region one can assign a point ξ 0 in the second region, such that both are located on a straight line passing through the origin and separated by a distance 2π (so that they are in opposite directions); explicitly 2π − ξ ~ ξ 0 ≤ ξ ≤ π , π ≤ ξ 0 ≤ 2π . ξ~0 = − ξ (B.29) The SU (2) elements corresponding to ξ~ and ξ~0 are then related by ~ . gSU (2) (ξ~0 ) = −gSU (2) (ξ) B.3 (B.30) Representations Suppose that one can find n matrices Ya with the same commutation relations as the elements of some Lie algebra g: c [Ya , Yb ] = fab Yc 53 (B.31) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) ξ2 ξ3 =0 ■ ● ▲ g=1 ▲ ■ ● π 2π ξ1 g=-1 Figure B.1: Parameter space with ξ3 = 0. The origin corresponds to the identity 1. The boundary is a circle with radius 2π which corresponds to the SU (2) element g = −1. Each element inside the inner circle, which has radius π has a corresponding element in the outer region which differs by an overall minus sign. The curve connecting the two filled circles corresponds to a continuous set of SU (2) transformations differing by an overall sign. The parameter space of SO(3) can be imbedded in the same plot and covers only the inside of the circle with radius π. Two opposite points on the inner circle corresponds to the identical element of SO(3). The SO(3) elements corresponding to the solid curve describe a closed continuous set of SO(3) transformations. 54 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) then by definition, these matrices form the basis of a representation of this Lie algebra. Exponentiation of linear combinations of Ya leads to a representation of the corresponding Lie group G: g(ξ 1 , . . . , ξ n ) → exp(ξ a Ya ) , (B.32) c completely determine the Lie group. Thus, each representation of the Lie because the fab algebra induces a representation of the corresponding group. For each Lie group there is a special representation called the adjoint representation, which has the same dimension as the group itself. This follows from the Jacobi identity which holds for any three matrices A, B and C: [[A, B], C] + [[B, C], A] + [[C, A], B] = 0 . (B.33) Choosing A = ta , B = tb and C = tc , one obtains the Jacobi identity for the structure constants e d e d e d fea + fca feb = 0 . fec + fbc fab (B.34) If now regard the structure constants as elements of n × n matrices fa according to c (fa )cb ≡ fab , (B.35) we can rewrite Eq. (B.34) as a matrix identity −(fc )de (fa )eb + (fa )de (fc )eb − (fa )ec (fe )db = 0 (B.36) or, after relabeling of indices, c ([fa , fb ])de = fab (fc )de . (B.37) Consequently the matrices fa generate a representation of the Lie algebra and therefore of the Lie group; clearly, the adjoint representation has dimension n, like the Lie group itself. Obviously an abelian group has vanishing structure constants, so that its adjoint representation is trivial, i.e. it consists of the identity element. As an example consider again SU (2). As shown above this c equal to i group has three generators ta and structure constants fab abc . Therefore the adjoint representation is 3-dimensional with generators Sa , given by (Sa )cb = iabc 55 (B.38) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) or, explicitly 0 0 0 S1 = 0 0 −i , 0 i 0 0 0 i 0 0 , −i 0 0 S2 = 0 0 −i 0 S3 = i 0 0 0 0 . 0 (B.39) which are just the generators of SO(3). Hence SU (2) and SO(3) have the same structure constants and are therefore locally equivalent. This fact is of physical importance because rotations of spatial coordinates are governed by SO(3), while spin rotations are described by SU (2); hence spatial and spin rotations form different representations of the same group. It is interesting to compare the parameter space of SO(3) to that of SU (2) The parameter space of SU(2) covers the group SO(3) twice, because two points ξ and ξ 0 satisfying Eq. B.29 correspond to the same element of SO(3). For this reason the parameter space of SO(3) can be restricted to the region with ξ ≤ π. Opposite points on the 2-dimensional sphere with radius π that forms the boundary of this region correspond to the same SO(3) element. The corresponding SU (2) elements differ by a sign (B.30). Finite transformations in the adjoint representation can be obtained by exponentiation. Hence one has n × n transformation matrices defined by gadj (ξ) = exp(ξ a fa ) . (B.40) Quantities transforming in this representation are n−dimensional vectors φa . A convenient way of dealing with such vectors is based on a matrix notation Φ = φa ta , (B.41) where ta are the group generators in some arbitrary representation. The transformation Φ → Φ0 = gΦg −1 , (B.42) with g in the same representation as ta (so that g = exp(ξ a ta )), now induces the same transformation on the φa , i.e. g(φa ta )g −1 = φ0a ta , (B.43) φ0a = (gadj φ)a . (B.44) with 56 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) Spin 0 (scalar) SO(3) Symmetry group: rotations SU(2) Covering group Spin 1/2 (spinor) Spin 1 (vector) ... ... Irreducible representations Types of particles Figure B.2: Because quantum mechanical states are rays in a Hilbert space, Wigner was led from the group of rotations SO(3) to its covering group SU (2). The irreducible representations of SU (2) then correspond to types of particles. Casimir operator The Casimir operator is a quantity which is invariant in any representation, i.e. gCg −1 = C , (B.45) where g denotes any group element in the representation corresponding to ta . For the rotation group SO(3) the Casimir operator is just the total angular momentum operator (modulo factor 2~2 ), and one has 1 C = l(l + 1)I 2 (B.46) where l labels the representations. Bibliography. 1. H. Georgi, Lie Algebras in Particle Physics, (Benjamin/Cummings, Reading, MA, 1982). 2. G. ‘t Hooft, Lie Groups in Physics, http://www.phys.uu.nl/~thooft/lectures/lieg07.pdf 57 [23/04/2012] Appendix C Wigner theorem From Symmetry in Physics: Wigner’s Legacy by D.J. Gross [14]. Wigner started from the description of quantum mechanical states as unit rays1 in a Hilbert space: [eiθ ]ψ, where |ψ|2 = 1 and θ ranges from 0 to 2π. Noting that physical predictions are given by transition probabilities, |hψ, φi|, he defined a symmetry transformation as a map preserving |hψ, φi|, i.e. T : ψ → ψ0 = T ψ , (C.1) |hψ|φi|2 |hψ 0 |φ0 i|2 = hψ 0 |ψ 0 ihφ0 |φ0 i hψ|ψihφ|φi (C.2) This does not completely define the operator T since we can always rescale T by a complex number Zφ and still obtain the same physical state, i. e. T |φi and Zφ T |φi represent the same physical state. What Wigner showed was that if U is an operator satisfying |hT ψ|T φi|2 |hψ|φi|2 = , hT ψ|T ψihT φ|T φi hψ|ψihφ|φi (C.3) then one can always adjust the phases so that T is either a unitary operator or an anti-unitary operator. A unitary operator T is such that hT ψ|T φi = hψ|T † T |φi = hψ|φi 1 (C.4) Consider a quantum theory formulated on a Hilbert space H. A physical state corresponds to a ray R in the Hilbert space, where a ray is defined as a set of normalized vectors (hψ|ψi = 1), where |ψi and |ψ 0 i belong to the same ray if they are equal up to a phase (i.e., if |ψ 0 i = eiθ |ψi for some real θ). The notation |ψi ∈ R indicates that |ψi belongs to the ray R and we define R(ψ) to denote the ray that contains the vector |ψi. We will consider a transformation T defined on physical states, so T maps one ray onto another. The abbreviation T (ψ) denotes T (R(ψ)), the image under T of the ray that contains the vector |ψi. 58 Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) for all states |φi, |ψi, so that T †T = T T † = 1 (C.5) T (α|φi + β|ψi) = αT |φi + βT |ψi (C.6) and is linear so that where α, β are arbitrary complex numbers. In that case, |hψ|T † T φi|2 |hT ψ|T φi|2 |hψ|φi|2 = = hT ψ|T ψihT φ|T φi hψ|ψihφ|φi hψ|T † T ψihφ|T † T φi (C.7) as required. An anti-unitary operator is such that hT ψ|T φi = hψ|φi∗ = hφ|ψi (C.8) T (α|φi + β|ψi) = α∗ T |φi + β ∗ T |ψi (C.9) and is anti-linear so that for arbitrary complex numbers α, β. In that case, |hφ|ψi|2 |hψ|φi|2 |hT ψ|T φi|2 = = hT ψ|T ψihT φ|T φi hψ|ψihφ|φi hψ|ψihφ|φi (C.10) as required. The second possibility is required for the description of time-reversal invariance, in which the symmetry transformation interchanges initial and final states. In classical mechanics, symmetries of the equations of motion can be used to derive new solutions. Thus if the laws of motion are invariant under spatial rotations and x(t) is a solution of the equations of motion, say an orbit of the Earth around the Sun, then Rx(t), the spatially rotated x(t), is also a solution. Wigner understood that in quantum theory, invariance principles permit even further reaching conclusions than in classical mechanics. In quantum mechanics there is a new and powerful feature due to the linearity of the symmetry transformation and the superposition principle. Thus if |Ψi is an allowed state then so is R|Ψi, where R is the operator in the Hilbert space corresponding to the symmetry transformation R. So far this is similar to classical mechanics. However, we can now superpose these states, that is, construct a new allowed state: |Ψi + R|Ψi. There is no classical analog for such a superposition of, say, two orbits of the earth. As Wigner pointed out, the superposition principle means that we can construct linear combinations of states that transform in a simple way under the symmetry transformations. Thus superimposing P all states that are related by rotations, we obtain a state |Φi = R R|Ψi that is rotationally invariant: R|Φi = X R0 RR0 |Ψi = 59 X R00 R00 |Ψi = |Φi . (C.11) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) The state |Φi forms a singlet representation of the rotation group. Other superpositions of rotated states will yield other irreducible representations of the symmetry group. lrreducible representations are special: they cannot be further subdivided, any subset of states gets mixed by the symmetry group with all the other states of the representation. Furthermore any state can be written as a sum of states transforming according to irreducible representations of the symmetry group. Wigner realized that these special states can be used to classify all the states of a system possessing symmetries, and play a fundamental role in the analysis of such systems. 60 [23/04/2012] Appendix D Noether theorem The existence of continuous symmetries implies the existence of conservation laws. One important consequence is the existence of locally conserved currents. Noether’s theorem: For every continuous global symmetry there exists a global conservation law. Before showing the proof of Noether theorem, let’s discuss the connection that exists between locally conserved currents and constants of motion. Let J µ (x) be some locally conserved current, i.e. ∂µ J µ (x) = 0 . (D.1) Let Ω be a bounded 4−volume of space-time with boundary ∂Ω. The Gauss Theorem tells us that 0= Z 4 µ d x ∂µ J (x) = Ω I dSµ J µ (x) , (D.2) ∂Ω where the right hand side is a surface integral on the oriented closed surface ∂Ω (a 3−volume). Let Ω be a 4−volume which extends all the way to infinity in space but which has a finite extent in time ∆T . If there are no currents in the large volume limit (lim|x|→∞ J µ (x, x0 ) = 0), then only the top and the bottom of the boundary ∂Ω contribute to the surface integral. Hence 0= Z V (T +∆T ) 3 0 d x J (x, T + ∆T ) − Z d3 x J 0 (x, T ) , (D.3) V (T ) where we assumed dS0 ≡ d3 x. Thus we see the quantity Q(T ) Q(T ) = Z d3 x J 0 (x, T ) ; V (T ) is a constant of motion, i.e. Q(T + ∆T ) = Q(T ) for every ∆T . 61 (D.4) Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) V(T+ΔT) ∂Ω Time T+ΔT (surface) Ω T (volume) V(T) Space (R3) The existence of a locally conserved current ∂µ J µ (x) = 0 (D.5) implies the existence of a conserved charge Q Q= Z d3 x J 0 (x, T ) , (D.6) which is a constant of motion. The proof of Noether’s theorem reduces to the proof of the existence of a locally conserved current. We proceed with a simple proof using complex scalar field φ(x) 6= φ∗ (x) and considering internal symmetry. The system has the continuous global symmetry φ(x) → φ0 (x) = eiα φ(x) ∗ 0∗ −iα ∗ φ (x) → φ (x) = e φ (x) , (D.7) (D.8) where α is a real number. The system is invariant if L satisfies L(φ0 , ∂µ φ0 ) ≡ L(φ, ∂µ φ) . (D.9) In particular for infinitesimal transformations we have φ0 (x) = φ(x) + δφ = φ(x) + iαφ(x) 62 (D.10) [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) φ0∗ (x) = φ∗ (x) + δφ∗ = φ(x) − iαφ(x) . (D.11) If L is invariant, then variation δL δL = δL δL δL δL + + ∗+ δφ δ∂µ φ δφ δ∂µ φ∗ (D.12) must be zero. Using the equations of motion δL − ∂µ δφ δL δ∂µ φ =0 (D.13) and its complex conjugate, the variation of L can be written in the form of a total divergence δL δL ∗ δL = ∂µ i φ− φ α . δ∂µ φ δ∂µ φ∗ (D.14) Since α is arbitrary, δL = 0 if and only if the current µ J =i δL δL ∗ φ− φ δ∂µ φ δ∂µ φ∗ (D.15) is conserved, i.e. ∂µ J µ (x) = 0. Let’s discuss some consequences of the Noether’s theorem in the quantum context. We consider linear Hermitian operators in quantum mechanics. Each operators could play a double role: on one hand they can represent dynamical variables of the theory1 hψ|Â|ψi = hAi (D.16) and on other hand they can serve as generators of a class of transformations |ψi → |ψ 0 i = e−iλ |ψi , (D.17) where λ is a real parameter. Hermitian linear operators generally do not commute with each other and whether or not two operators commute has deep physical significance. If  and B̂ commute then 1. The transformations generated by  and B̂ commute with each other (i.e. translations commute, rotations do not). 1  is the operator, A is the dynamical variable and |ψi is vector state. 63 [23/04/2012] Paolo Finelli Corso di Teoria delle Forze Nucleari (2012) 2. The dynamical variable A is invariant under the transformations generated by B̂: hψ 0 |Â|ψ 0 i = hψ|Â|ψi = hAi |ψ 0 i = e−iλB̂ |ψi . (D.18) The same is true if we exchange A and B. 3. The dynamical variables A and B are simultaneously measurable with arbitrary precision, i.e. there is no uncertainty principle relating them (they have a complete set of simultaneous eigentates). Let’s suppose that B̂ = Ĥ, the Hamiltonian operator. The transformations generated by Ĥ are time displacements i~ d |ψi = Ĥ|ψi dt |ψ(t)i = e−iĤt |ψ(0)i . → (D.19) In this case we can make the following statements: 1. If the dynamical variable A is invariant under the transformations generated by Ĥ is equivalent to say that A is a constant of motion, i.e. its expectation value in any state will be invariant under any time displacement. 2. If H is invariant under transformations generated by  is equivalent to say that A represents a symmetry of dynamical laws. 3. The dynamical variable A is conserved if and only if the dynamical laws (generated by Ĥ) are invariant under the transformations generated by Â. 64 [23/04/2012]