Download 7 Scattering theory and the S matrix

Scattering theory and the S matrix
7
In this section we introduce the S matrix which is the central object for calculating probabilities of various reactions. We will show how to compute it
iteratively, using either the old-fashioned perturbative calculus or the modern covariant expansion. We will also discuss in some details its general
properties such as unitarity, partial wave expansion and various symmetries.
The starting point will be a Hamiltonian of the form H = H0 + Vint , where
H0 is the Hamiltonian of a given set of free particles constructed as a sum
of Hamiltonians (6.113) and Vint is the interaction operator. The exposition
will be based on the scattering theory which in its general form applies to the
ordinary nonrelativistic as well as relativistic Quantum Mechanics. In fact,
despite some important differences between its simplest version - the theory
of scattering by an external potential based on nonrelativistic QM of a single
particle and the scattering theory based on Relativistic Quantum Mechanics of particles (that is Quantum Field Theory) developed here,1 keeping in
mind the former is helpful in understanding also the latter one. The S matrix describing interactions of relativistic particles has to transform in a well
defined way when the reference frame is changed. Whether it does so, that
is, whether it is Lorentz covariant, depends in turn (assuming that H0 is of
the appropriate form) on the structure of the interaction Vint . We will investigate this requirement in details and will formulate the sufficient conditions
under which the Hamiltonian H = H0 + Vint leads to a Lorentz covariant S
matrix. As will turn out, these conditions are not satisfied in all theories
of physical interest and the ultimate Poincaré covariance of their S matrices
must be ensured by additional special features of these theories; nevertheless,
the conditions formulated here constitute a useful reference point for further
constructions.
One-particle states described in Section 6 play an important role in the
S matrix description of elementary particle reactions. This is because we
expect that the theory we are developing is going to describe primarily (but
certainly not only!) collision-type processes in which long before and long
after the reaction particles look as (mutually) noninteracting. Therefore, it
should be possible to associate with a given scattering process the Hilbert
space state-vectors which, in a well defined way, correspond to multi-particle
states constructed as (symmetrized or antisymmetrized for identical parti1
There exist of course intermediate theories based on nonrelativistic Quantum Mechanics of many particles which, similarly to the relativistic one allow to consider multichannel
scattering process.
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cles) tensor products of one-particle states2
|p1 σ1 , p2 σ2 , . . . , pN σN i .
(7.1)
The states (7.1) are normalized so that
′
′
′ ′
hp′N σN
, p′N −1 σN
−1 , . . . , p1 σ1 |p1 σ1 , p2 σ2 , . . . , pM σM i
X
= δN M
(−1)P δΓ (p′1 − pP (1) ) . . . δΓ (p′N − pP (N ) ) . (7.2)
P
The sum in (7.2) is over permutations within groups of labels corresponding
to identical particles and (−1)P is the sign of the permutation of fermionic
labels in a given permutation P . The symbol δΓ (p′ − p) which here is assumed to include also the Kronecker delta of the spin variables σ, depends
on the normalization of the one-particle states; for the one usually adopted
in nonrelativistic applications δΓ (p′ − p) = (2π)3 δσ′ σ δ (3) (p′ − p); in relativistic theories more convenient is the normalization such that δΓ (p′ − p) =
(2π)3 2Ep δσ′ σ δ (3) (p′ − p), corresponding to 2Ep particles in the unit volume
(see Section 10.2). Because in most of considerations of this chapter we will
not be interested in the detailed composition of the multi-particle states, it
is convenient to introduce a compact notation, in which |αi stands for states
of the form (7.1) and the scalar product (7.2) is concisely written as
hβ|αi = δ(β − α) ≡ δβα .
The completeness relation
!
∞
X
XX
. . . δN,(N1 +N2 +...)
1=
N =0
N1
N2
N
X Z Y
σ1 ,...,σN
i=1
1
N1 !N2 ! . . .
(7.3)
(7.4)
dΓpi |p1 σ1 , . . . , pN σN ihpN σN , . . . , p1 σ1 | ,
in which the summation is over different numbers Ni of distinct types of
particles, will be then compactly written as
Z
Z
1 = dα |αihα| ,
i.e.
|Ψi = dα |αihα|Ψi .
(7.5)
where |Ψi is any state in the Hilbert space H.
2
If the state (7.1) represents Na particles of type a, Nb particles of type b, etc. (Na +
Nb + . . . = N ), different groups of labels, e.g. (pi σi , . . . , pi+Na σi+Na ) correspond to
different types of particles but we do not introduce any additional index to distinguish
which labels correspond to which type of particles. We assume the labels corresponding
to identical particles are symmetrized or antisymmetrized as described in section 5. Basis
N -particle states constructed as appropriate linear combinations of the states (7.1) are
also in use (see section 6.4 for examples of such alternative bases for N = 2).
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7.1
Time evolution and the S0 operator
Multi-particle states (7.1), which from now on will be denoted |α0 i, are nonnormalizable (if the theory is formulated in the continuum, that is, in the
space of infinite volume) eigenstates3 of a free Hamiltonian H0 constructed
as the sum of free Hamiltonians (6.113) for a finite number of distinct types
of particles. They are created and annihilated by the corresponding creation
and annihilation operators a†σ (p), aσ (p) from the H0 ground state |Ω0 i, which
(in the continuum) is the only normalizable eigenstate of H0 , and span the socalled Fock space. Energy of any of such states is simply the sum of energies of
the one-particle states from which it is constructed. The dynamics governed
by H0 is trivial. It should be kept in mind that even in the finite volume V ,
when allowed particle momenta form a discrete set (as a result of imposing
periodic boundary conditions) and all state vectors are normalizable, the
Hilbert space is not nuclear - the basis formed by the |α0 i vectors is not
countable.4
Interactions are introduced by adding to the free Hamiltonian H0 describing a fixed set of types of particles an operator Vint . We will make two
assumptions. Firstly, we will assume that the Hamiltonian H = H0 +Vint does
describe particles, i.e. that it possesses particle-like generalized eigenstates,
which in the sense specified below have properties similar to the free-particle
generalized eigenstates (7.1) of a sum H̃0 of free Hamiltonians of the type
(6.113), not necessarily the same as H0 we start with. Moreover we will
assume that, similarly to H0 , the Hamiltonian H = H0 + Vint has (in the
infinite space volume) only a single normalizable state, the vacuum, which
will be denoted |Ωi, and that the particle-like non-normalizable eigenstates
of H, which will be introduced in section 7.2, together with |Ωi form a complete set of states5 (a basis of the Hilbert space). Secondly, we will make a
“technical” assumption that Vint is “small” in the sense that the spectrum of
H = H0 + Vint eigenstates is the same as the spectrum of H0 , i.e. the full H
eigenstates have the properties of the eigenstates |α0 i of H0 . (In other words,
3
Non-normalizable state vectors are usually called generalized vectors; |α0 i are in this
respect similar to the plane waves ψp = eip·x of ordinary one-particle nonrelativistic
Quantum Mechanics which are generalized (non-normalizable) eigenvectors of the H0 =
P̂2 /2m and P̂ operators.
4
This follows from the mathematical facts that for integer M and N both limits
lim
lim M N
N →∞ M→∞
and
lim 2N ,
N →∞
(relevant for counting bosonic and fermionic basis states) are equal to the power of the
continuum.
5
This is not always true in the scattering theory based on (nonrelativistic) Quantum
Mechanics of a single particle in which the potential Vint added to H0 can lead to the
existence of normalizable eigenstates of H (i.e. bound states).
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we assume that H̃0 = H0 ). In the approach to quantum field theory based
on relativistic quantum mechanics of particles, which is developed in sections
7-9 these assumptions can be satisfied only for interactions Vint that are judiciously constructed. This will become clear in Section 9.7, where it will turn
out that observance of these assumptions (by appropriately adjusting Vint )
is crucial for avoiding some type of ill defined contributions to the transition
amplitudes (S matrix elements) that would otherwise occur in perturbative
calculations.
It is important to realize that these assumptions are neither a priori obvious, nor are they always fulfilled. It could happen that H = H0 + Vint
does not possess particle-like eigenstates at all (or not all of its eigenstates
are particle-like). This is indeed so in conformal field theory models or theories of “unparticles” discussed in the recent literature.6 Furthermore, even
if H = H0 + Vint does posses particle-like eigenstates, they can be similar
to the eigenstates of a free-particle Hamiltonian H̃0 which is very different
from H0 used to build H. The most prominent example is Quantum Chromodynamics (QCD) - the theory of strong interactions whose H0 describes
free spin 21 coloured (i.e. transforming nontrivially under the action of the
colour SU(3)c symmetry group) quarks, antiquarks and spin 1 coloured gluons, whereas the true H eigenstates represent colourless, i.e. SU(3)c singlets,
baryons, antibaryons and mesons. The assumptions formulated above could
of course be checked if the theory could be solved exactly. Unfortunately,
in most cases one has to rely on some sort of approximations which usually
hinge on (some of) the above assumptions. A method with the help of which
the true spectrum of H could, at least in principle, be investigated will be
outlined in Section 13.
In principle, the second assumption is not necessary to investigate some
particular aspects of H = H0 + Vint , such as its true spectrum or thermodynamic properties of the system described by it, provided one takes the
position that the theory if formulated in a finite space volume V and with a
(ultraviolet) cutoff Λ on the momenta of the generalized H0 eigenstates which
is removed only after adjustment, correlated with the cutoff, of the terms in
Vint (i.e. after the renormalization). The momentum cutoff restricting the
Hilbert space by making its basis countable is necessary to exclude nonequivalent representations of the (anti)commutation relations (of the algebra
of the annihilation and creation operators). To meaningfully distinguish scattering states transition to infinite volume is, however, mandatory: in finite
volume all H eigenstates are normalizable and scattering processes cannot
be sharply distinguished from the general time evolution of the system: all
reactions would occur multiply as time goes and it would not make sense
6
Free Hamiltonians H0 of such theories describes massless particles.
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to appeal to the infinite time limits in order to define measurable quantities
corresponding to what in the real world is observed as scattering processes.
Therefore, to meaningfully define particle reactions the theory must be formulated in the continuum.7 With the two assumptions spelled out above
it becomes possible to formulate the scattering theory based on relativistic Quantum Mechanics of particles (that is on Quantum Field Theory) in
the Fock space of H0 eigenstates. To the proper Hilbert space which we
will consider initially belong all possible normalizable states |Ψi that can be
constructed from the Fock space of generalized H0 eigenstates as
Z
Z
|Ψi = dα |α0ihα0 |Ψi ≡ dα |α0i ψ(α) ,
R
with integrable profiles ψ(α): dα |ψ(α)|2 = 1. One can then consider the
time evolution of such states defined at t = 0 generated either by H or
H0 . Guided by the physical intuition we expect that normalizable states |Ψi
representing particle reactions, whose time evolution is governed by H:
|Ψ(t)i = e−iHt |Ψi ≡ U(t)|Ψi ,
(7.6)
should converge as t → ∓∞ (in the sense of convergence in the Hilbert space
of sequences of vectors) to some states
in/out
in/out
in/out
|Ψas
(t)i = e−iH0 t |Ψas
i ≡ U0 (t)|Ψas
i,
(7.7)
because in experiments one prepares states representing particles which before the collision are well localized and separated in space and are therefore
almost non-interacting with each other; likewise, long after the collision particles are again well separated and again look as mutually non-interacting.
Thus one assumes that on the whole Hilbert space the family of operators
does have the limits8
Ω(t) ≡ eiHt e−iH0 t ,
(7.8)
lim Ω(t) = Ω(∓∞) ≡ Ω± .
(7.9)
t→∓∞
on any normalizable smooth superposition of the |α0 i states. Ω± are called
M6 oller operators. Since
d iHt −iH0 t
e e
= i U † (t)Vint U0 (t) ,
dt
7
A strong physical conviction is that physical processes can always be confined to a
finite volume and that their measurable characteristics (if properly defined) do not depend
on the size of the volume (if it is sufficiently large). It is therefore a matter of pure
calculational convenience that quantum field theory is formulated in the continuum.
8
It should be stressed that in the relativistic theory (QFT) this is true only for special
interactions Vint ; in particular for such interactions Ω(t) have also limit on |Ω0 i (which is
normalizable by itself) - Ω± acting on this state give the states |Ω± i, which can differ only
by a phase factor.
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and since Ω(0) = 1 the operator Ω(t) can alternatively be defined by the
integral relation
Z t
Ω(t) = 1 + i dt′ U † (t′ )Vint U0 (t′ ) .
(7.10)
0
Furthermore, as the operators Ω(t) are unitary for any fixed t, that is satisfy
Ω† (t) Ω(t) = 1, the M6 oller operators Ω± are at least isometric - are defined
on the whole Hilbert space H and preserve the norm: (Ω± Ψ|Ω± Ψ) = (Ψ|Ψ),
that is satisfy the relation
Ω†+ Ω+ = Ω†− Ω− = 1 .
They preserve, therefore, also the scalar product of normalizable states:
(Ω± Φ|Ω± Ψ) = (Φ|Ψ) .
(7.11)
In relativistic quantum mechanics of particles (i.e. in QFT) we assume that9
Ω± H = H, that is any H state can be represented as the image of the action
in/out
of Ω+ and Ω− on the appropriate states |Ψas i:
out
|Ψi = Ω+ |Ψin
as i = Ω− |Ψas i .
(7.12)
At this point one can already introduce the S0 operator. As usually in
quantum theory one is interested in scalar products SΦΨ ≡ hΦ|Ψi of normalized states. Expressing |Ψi as the Ω+ image of the appropriate |Ψin
as i and |Φi
i
one
gets
as the Ω− image of |Φout
as
in
SΦΨ ≡ hΦ|Ψi = hΦout
as |S0 |Ψas i ,
(7.13)
where the S0 operator is defined as the product
S0 ≡ Ω†− Ω+ .
(7.14)
It, among other things, maps the asymptotic “incoming” states onto the corin
responding “outgoing”states: |Ψout
as i = S0 |Ψas i. The scalar products SΦΨ the matrix elements of S0 - have the natural interpretation of the probability
amplitude of finding the system in the state |Φi, which, if evolved in time,10
would become in the far future indistinguishable from an evolved state |Φout
as i
having direct interpretation in terms of noninteracting (and spatially separated in the far future) particles, if it is prepared as the state |Ψi which, if
9
This is not necessarily true in nonrelativistic Quantum Mechanics of a single particle.
See Appendix C.
10
Notice that the states are always identified at t = 0; that is we implicitly work in
the Heisenberg picture (see section 1.1) which in Quantum Field Theory allows to keep
relativistic invariance as manifest as possible.
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evolved in time, has in the far past a similar free-particle interpretation, being
indistinguishable from the evolved state |Ψin
as i. The scalar products SΦΨ thus
contain answers to a prevailing amount of experimentally accessible questions
which usually are formulated in the form “what is the probability that the
detectors will register a given free particle state produced as a result of an
interaction of particles which long before interaction were prepared (in the
accelerator) as another free particles state?”
It is also convenient to introduce an alternative notation (corresponding
to a slightly different labeling of states) and to call |Ψ+ i and |Ψ− i the two
different states which are images of the same state |Ψi under Ω+ and Ω− ,
respectively. Thus, in this notation
lim U(t)|Ψ± i = U0 (t)|Ψi .
t→∓∞
(7.15)
Matrix elements of the S0 operator in this notation are written as hΦ− |Ψ+ i
and since
|Ψ+ i = Ω+ Ω†− |Ψ− i ,
|Ψ− i = Ω− Ω†+ |Ψ+ i ,
(7.16)
they can be expressed as the matrix elements
SΦΨ ≡ hΦ− |Ψ+ i = hΦ− |S|Ψ− i = hΦ+ |S|Ψ+ i ,
(7.17)
of the S operator
S ≡ Ω+ Ω†− .
(7.18)
The S operator (which has been considered in section 1.2) which is different
from the S0 one11 will be of little use in what follows. However the notation
|Ψ± i will be useful.
The operators H = H0 + Vint and H0 satisfy the important intertwining
relation
H Ω± = Ω± H0 ,
(7.19)
which in particular imply12 that Ω†± HΩ± = H0 . Indeed,
eiHt Ω± = eiHt lim eiHτ e−iH0 τ = lim eiH(τ +t) e−iH0 (τ +t) eiH0 t = Ω± eiH0 t .
τ →∓∞
τ →∓∞
11
The difference between the S0 and S operators is particularly sharp in the nonrelativistic potential scattering theory, if H possesses bound states: while S0 acts nontrivially
on the whole Hilbert space H, S annihilates the whole subspace Hbound .
12
If Ω± are unitary, such relations imply that the spectra of H and H0 are identical
(which is one of our assumptions adopted here). This shows that in the case of ordinary
potential scattering Ω± cannot be unitary if H has bound states (normalizable eigenstates)
because H0 does not have such eigenstates. In such a case Ω± are only isometric operators.
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Differentiating this equality with respect to t at t = 0 yields the relation
(7.19). Exploiting this relation one can write13
S0 H0 = Ω†− Ω+ H0 = Ω†− H Ω+ = H0 Ω†− Ω+ = H0 S0 ,
(7.20)
[S0 , H0 ] = 0 .
(7.21)
that is,
The relation (7.19) means, in particular, that
in
out
out
hΨ|H|Ψi = hΨin
as |H0 |Ψas i = hΨas |H0 |Ψas i .
The energy is conserved in the scattering process.
Since any normalizable state |Ψi can be written as a superposition of the
non-normalizable generalized (i.e. not belonging to the proper Hilbert space)
H0 eigenstates |α0 i one can write
Z
out
in
hβ0 |Ψas i = hβ0 |S0 |Ψas i = dα hβ0|S0 |α0 ihα0 |Ψin
as i .
One is thus led to consider the matrix elements Sβα ≡ hβ0 |S0 |α0 i. From
(7.21) it follows that
0 = hβ0 |[H0 , S0 ]|α0 i = (Eβ − Eα )hβ0 |S0 |α0 i ,
which shows that hβ0 |S0 |α0 i ∝ δ(Eβ −Eα ) (because xδ(x) = 0). Furthermore,
because for Vint = 0 the S0 operator reduces to the unit operator, it is
convenient to write
S0 = 1 − iT0 ,
(7.22)
thereby introducing the operator T0 . Thus
Sβα ≡ hβ0 |S0 |α0 i = δαβ − 2πi δ(Eβ − Eα ) tβα (Eα ) ,
(7.23)
where 2πδ(Eβ − Eα ) tβα = hβ0 |T0 |α0 i. As will be shown in Section 10, it
is precisely the quantity tβα (Eα ) which is needed to compute the rate of
the process α → β. In the case of the nonrelativistic potential scattering
the quantity tβα ≡ t(p′ , p) is directly related to the standard scattering
amplitude f (θ) - see Appendix C. All measurable characteristics of scattering
processes can be obtained from the matrix elements Sβα of the S0 (or T0 )
operator for which a useful representation will be derived directly from the
13
See (C.2) for a more precise justification.
170
differential equation satisfied by the (interaction picture) evolution operator
introduced in section 1.1:
UI (t2 , t1 ) = eiH0 t2 e−iHt2 eiHt1 e−iH0 t1
= eiH0 t2 e−iH(t2 −t1 ) e−iH0 t1 = Ω†− (t2 )Ω+ (t1 ) ,
(7.24)
of which S0 is the double limit:
S0 = Ω†+ Ω− = lim
lim UI (t2 , t1 ) .
t2 →+∞ t1 →−∞
(7.25)
Notice also that Ω± = UI (0, ∓ ∞). Before deriving it one has, however, to
introduce the non-normalizable (generalized) H eigenstates and the resolvent
operators which allow to relate these to the H0 eigenstates |α0 i.
7.2
In and out states
A very important role in the formal scattering theory is played by the resolvent operators
G(z) ≡ (z − H)−1 ,
G0 (z) ≡ (z − H0 )−1 .
and
(7.26)
Their matrix elements between normalizable states are analytic functions of
the complex z plane except for isolated poles corresponding to normalizable
H (H0 ) eigenstates and branch cut along the continuous spectrum.
Substituting for A and B in the obvious operator identity
1
1
1
1
− = (B − A)
A B
B
A
the operators G(z) and G0 (z) (the operators G0 (z) and G(z)) one obtains
two relations
G(z) = G0 (z) + G0 (z) Vint G(z) ,
G(z) = G0 (z) + G(z) Vint G0 (z) ,
(7.27)
It is also easy to see that because H = H † (H0 = H0† ),
G(z ∗ ) = [G(z)]† ,
G0 (z ∗ ) = [G0 (z)]† .
(7.28)
Matrix elements of the resolvent operator G0 (z) between the non-normalizable
H0 eigenstates are explicitly given by
hβ0 |G0 (z)|α0 i = δβα
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1
.
z − Eα
(7.29)
Another very important operator is the T (z) operator defines as
T (z) ≡ Vint + Vint G(z) Vint .
(7.30)
It has the same analytic properties as G(z) and satisfies the following relations
G0 (z) T (z) = G(z) Vint ,
T (z) G0 (z) = Vint G(z) ,
(7.31)
which readily follow from the relations (7.27). They allow to express G(z)
through T (z): replacing in (7.27) Vint G(z) (or G(z) Vint ) using (7.31) one
gets
G(z) = G0 (z) + G0 (z) T (z) G0 (z) .
(7.32)
Using (7.31) in (7.30) leads instead to
T (z) = Vint + Vint G0 (z) T (z) ,
(7.33)
which is known as the Lippman-Schwinger equation for T (z). Iterating it
yields the series
T (z) = Vint + Vint G0 (z) Vint + Vint G0 (z) Vint G0 (z) Vint + . . .
(7.34)
Matrix elements of the S0 operator between generalized H0 eigenstates
can be expressed through the operator T (z). To this end, instead of representing S0 as in (7.25) as the double limit of the UI (t2 , t1 ) operator, it is
written as the single limit
S0 = Ω†− Ω+ = lim UI (τ, −τ ) = lim eiH0 τ e−2iHτ eiH0 τ .
τ →∞
τ →∞
Differentiating with respect to τ the operator UI (τ, −τ ) one obtains the differential equation satisfied by it, which, together with the obvious boundary
condition at τ = 0 allows to write for this operator an integral expression.
Using it, matrix element of the S0 operator between normalizable states can
be then written as
hΦ|S0 |Ψi = hΦ|Ψi
Z ∞
−i
dt e−εt hΦ|eiH0 t Vint e−2iHt eiH0 t + eiH0 t e−2iHt Vint eiH0 t |Ψi .
0
The factor e−εt is not necessary when the matrix element is taken between
two normalizable states, but when introduced,14 it allows to replace |Ψi and
14
It is usually (incorrectly) introduced from the beginning as a factor ensuring “adiabatic” switching on and off the interaction - something which certainly does not happen
in Nature! Notice also that if Vint was defined with such a factor, the evolution operator
U (t, t0 ) corresponding to the Hamiltonian H (explicitly time dependent then) would have
to have the form (1.9) instead of e−iH(t−t0 ) .
172
|Φi by the generalized H0 eigenstates |α0 i and |β0 i:
Z ∞
hβ0 |S0 |α0 i = δβα − i
dt hβ0 |Vint ei(Eβ +Eα−2H+i0)t + ei(Eβ +Eα −2H+i0)t Vint |α0 i
0
Eβ + Eα
Eβ + Eα
1
+ i0 + G
+ i0 Vint |α0 i .
= δβα + hβ0 |Vint G
2
2
2
Using the operator identities (7.31) one can replace here the operators G(z)
by G0 (z) ones which can act directly on the states |α0 i and hβ0 |. The second
term can be then cast in the form
1
Eβ + Eα
1
hβ0 |T
+
+ i0 |α0 i ,
Eβ − Eα + i0 Eα − Eβ + i0
2
which, upon using the Sochocki formula, leads to
hβ0 |S0 |α0 i = δβα − 2πi δ(Eβ − Eα ) hβ0 |T (Eα + i0)|α0 i .
(7.35)
This shows that the matrix element tβα of the T0 operator introduced in (7.23)
is given by the special limit of the general matrix element of the T (z) operator. Combining this with the truncated to its first term iterative solution
(7.34) of the Lippmann-Schwinger equation (7.33 for T (z), one immediately
obtains the formula known as the Born approximation
tβα (Eα ) ≈ hβ0 |Vint |α0 i .
(7.36)
If it is known how to compute the action of Vint on free particle states (which
is precisely the case, when Vint is expressed in terms of the creation and
annihilation operators of free particles) this formula provides the simplest
working approximation to amplitudes of particle reactions.15
We will now introduce nonnormalizable in and out eigenstates of H =
H0 + Vint and derive for them some useful representations. To this end we
consider the action of Ω± on a normalizable state |Ψi. One gets then the
scattering states |Ψ± i which, using the formula (7.10) can be written as
Z ∓∞
′
|Ψ± i = Ω± |Ψi = |Ψi + i
dt′ e−ε|t | U † (t′ )Vint U0 (t′ )|Ψi .
(7.37)
0
′
Again the factor e−ε|t | is not necessary for convergence when |Ψi is a normalizable state, but is necessary when |Ψi is decomposed into generalized
15
However, frequently in relativistic theories of interacting particles tβα (Eα ) = 0 in this
approximation. In Quantum Field Theory the name “Born approximation” is sometimes
also used to denote what otherwise is called the tree-level approximation (see section 9),
it coincides with (7.36) only for very special interactions Vint .
173
H0 eigenstates |α0 i:
Z
Z
∓∞
′
|Ψ± i = |Ψi + i dα
dt′ e−i(Eα −H±iε)t Vint |α0 ihα0 |Ψi
0
Z
= |Ψi + dα G(Eα ± i0) Vint |α0 ihα0 |Ψi .
(7.38)
One can now define the in and out generalized states |α± i by the formula
|α± i ≡ Ω± |α0 i ≡ lim eiHt e−iH0 t |α0 i .
t→∓∞
(7.39)
Owing to the intertwining relations (7.19), |α± i turn out to be just the
generalized (non-normalizable) eigenstates of the Hamiltonian H = H0 +Vint :
H|α± i = Eα |α± i ,
(7.40)
with the eigenvalue Eα equal to the energy (w.r.t. H0 ) of the corresponding
|α0 i states. With the assumption that the spectra of H and H0 are identical,
the vectors |α0 i, |α+ i and |α− i form three equivalent bases of the theory
Hilbert space H. From this point of view the S-matrix elements
Sβα = hβ− |α+ i = hβ0 |S0 |α0 i ,
form a collection of numbers, such that
Z
Z
|α+ i = dβ |β− i Sβα ,
hβ− | = dα hα+ | Sβα .
(7.41)
(7.42)
As a matrix connecting two complete sets of orthonormal states (it is just
the matrix of the change of bases) Sβα must be unitary:
Z
Z
∗
(7.43)
dβ Sβγ Sβα = dβ hγ+ |β− ihβ− |α+ i = hγ+ |α+ i = δγα .
This reflects also the unitarity of the S0 operator: S0−1 = S0† . The states |α+ i
and |α− i are, in turn, connected by the S operator defined in (7.18):
S|α− i = |α+ i ,
or
hβ+ |S = hβ− | ,
(7.44)
so that, in analogy to (7.17)
Sβα = hβ+ |S|α+ i = hβ− |S|α− i .
(7.45)
From the practical point of view (7.39) establishes the strict one-to-one
correspondence between the in and out eigenstates of H and the eigenstates
of H0 on which the formulation of the perturbative calculation of the S
174
matrix elements will be based. (This strict correspondence will be relaxed
only in Section 13 where a more flexible, nonperturbative in essence, way of
accessing S matrix elements will be formulated).
Since any normalizable
R state |Ψi can be decomposed into the generalized
H0 eigenstates |Ψi = dα |α0 i ψ(α), from the relation |Ψ± i = Ω± |Ψi one
gets
Z
Z
|Ψ± i = Ω± dα |α0 i ψ(α) = dα |α± i ψ(α) ,
(7.46)
That is, normalizable state-vectors |Ψ± i decompose onto the generalized H
eigenstates |α± i with the same profile ψ(α) as do their Ω†± images onto
the generalized H0 eigenstates |α0 i. Moreover, from the fact that the Ω±
operators preserve the scalar product of normalizable states (cf. (7.11)) it
follows that
hβ± |α± i = hβ0 |α0 i = δβα .
(7.47)
Since the in and out states |α+ i and |α− i are in the one-to-one correspondence with the free particle states |α0 i, in addition to the operators a(k, σ),
a† (k, σ) (which build the states |α0 i) one can define also the in and out creation and annihilation operators ain (k, σ), a†in (k, σ) and aout (k, σ), a†out (k, σ)
which acting on the corresponding vacua16 |Ω± i = Ω± |Ω0 i build the in and
out states and satisfy the same commutation relations and transformation
properties (in the relativistic case with respect to the full Poincaré symmetry group generated by H = H0 + Vint , P, J and K = K0 + W - see section
7.3) as do the operators creating and annihilating the free particle states α0 i.
From (7.44) it then follows (cf. (1.28)) that
S † a†in (k, σ)S = a†out (k, σ) ,
S † ain (k, σ)S = aout (k, σ) .
(7.48)
Finally, it should be stressed that the states U(t)|α± i = e−iEα t |α± i do
not converge to U0 (t)|α0 i = e−iEα t |α0 i for t → ∓∞. The convergence holds
only for normalizable states built as smooth superpositions of such states.
The operator identities established above allow to derive useful representations for the in and out states |α± i either in terms of the resolvent G(z) or
in terms of the T (z) operator. To obtain these representations one rewrites
(7.38), decomposing |Ψi onto the |α0 i states, in the form
Z
|Ψ± i = dα |α0 i + G(Eα ± i0) Vint |α0 i ψ(α) .
16
Do not confuse the |Ω± i vacua with the M6 oller operators Ω∓ . For closed systems,
i.e. system whose Hamiltonians H do not depend on time, the vacua |Ω+ i and |Ω− i differ
only by a phase factor.
175
Comparing this with (7.46) one gets the representation
|α± i = |α0 i + G(Eα ± i0) Vint |α0 i .
(7.49)
Yet another representation can be obtained using the identity
T (Eα ± i0)|α0 i = Vint [1 + G(Eα ± i0)Vint ]|α0 i = Vint |α± i ,
(7.50)
which follows from the definition (7.30) of the T (z) operator and (7.49).
This relation combined with the result (7.35) immediately allows to write
the element tβα (Eα ) in the S0 matrix element (7.23) as17
tβα (Eα ) = hβ0 |T (Eα + i0)|α0 i = hβ0 |Vint |α± i .
(7.51)
The identity (7.50) applied to (7.49), after trading in this formula the product
G(Eα ± i0) Vint for G0 (Eα ± i0) T (Eα ± i0) in agreement with (7.31), leads
to the Lippmann-Schwinger equation for |α± i:
|α± i = |α0 i + G0 (Eα ± i0) Vint |α± i ≡ |α0 i +
or
|α± i = |α0 i +
Z
dβ |β0 i
1
Vint |α± i , (7.52)
Eα − H0 ± i0
tβα (Eα )
.
Eα − Eβ ± i0
(7.53)
Notice that the formula (7.52) agrees with the identification of the |α± i as
the eigenstates of H, if the relation (7.40) is rewritten in the form
(Eα − H0 )|α± i = Vint |α± i .
The ±i0 prescription specifies the way of inverting the operator (Eα − H0 )
which has |α0 i as its zero eigenvector.18 Iterating the Lippmann-Schwinger
equation (7.52) eg. for |α+ i gives the series
|α+ i = |α0 i + G0 (Eα + i0) Vint |α0 i
+ G0 (Eα + i0) Vint G0 (Eα + i0) Vint |α0 i + . . . ,
(7.54)
When closed from the left with hβ0 |Vint , it reproduces the Born series for
tβα (Eα ) = hβ0 |T (Eα + i0)|β0 i which can be obtained from (7.34); the latter
series, truncated to the first term, gives the Born approximation (7.36).
17
Similar representation of tβα (Eα ) in terms of the out state is obtained by taking the
Hermitian conjugation of T (Eα −i0)|β0 i = Vint |β− i and using the property T † (z) = T (z ∗ ).
This leads to tβα (Eα ) = hβ− |Vint |α0 i.
18
Weinberg in his book derives the formula (7.53) directly from this equality. His derivation (quicker than the one given here) suffers, however, from the incorrect application of
the residue method for computing integrals: the integral over Eβ is not over the whole
real axis as requires this method, but is restricted to Eβ > Mmin ≥ 0 (energy of the states
|α0 i representing particles is never negative).
176
π
π
K
strong int.
π
π
Figure 7.1: Strong interaction induced rescattering of pions produced in the
decay of Kaon.
Another useful approximation can be obtained if the interaction Vint consists of two parts: Vint = Vstrong +Vweak of which one is “strong” and the other
one “weak”. One is then interested in accounting for the strong interactions
exactly, while the effects of the weak ones can be treated in the simplest
approximation. To this end, in addition to the in and out eigenstates |α±i
of the full Hamiltonian H = H0 + Vstrong + Vweak one defines also the in and
out states with respect to the strong interaction
|β±strong i = |β0 i +
1
Vstrong |β±strong i ,
Eβ − H0 ± i0
(7.55)
1
.
Eβ − H0 ∓ i0
(7.56)
so that
hβ0 | = hβ±strong | − hβ±strong |Vstrong
The full matrix tβα (Eα ) (7.51) can be then written in the form
1
strong
strong
(Vstrong + Vweak )|α+ i
tβα = hβ− | − hβ− |Vstrong
Eβ − H0 + i0
= hβ−strong |Vweak |α+ i + hβ−strong |Vstrong |α0 i ,
(7.57)
where the formula (7.52) with Vint replaced by Vstrong +Vweak has been used (in
the denominator Eβ can be replaced by Eα because we need tβα for Eβ = Eα )
to replace the product [Eβ − H0 + i0]−1 (Vstrong + Vweak )|α+ i by |α+ i − |α0 i.
This (exact) formula is most useful if the strong interaction cannot induce the
α → β transition. The second term, which is just tβα in the absence of weak
interactions (just set Vweak to zero in the formula given in the footnote related
to the formula (7.51) to see it!) which describes all possible transitions α → β
induced by Vstrong alone, is then zero and, moreover, since the effects of Vweak
are small, one can approximate the full Hamiltonian in state |α+ i in the first
strong
strong
i is used
term by |α+
i. The resulting formula tβα ≈ hβ−strong |Vweak |α+
strong
e.g. in nuclear physics to compute rates of nuclear weak beta decays (|α+
i
strong
and |α−
i are then the initial and final nucleon states). Furthermore, using
then the property (7.42) of the S matrix, this formula can be rewritten as
Z
strong
strong
strong
tβα = dγ Sβγ
hγ+
|Vweak |α+
i.
(7.58)
177
In this form it is used to account for the strong interaction re-scattering
effects (shown graphically in figure 7.1) in hadronic weak decays; such effects
are crucial for CP violation effects in the Kaon system.
The Born formula (7.36) is the first term of the entire perturbative series
which is obtained either by sandwiching the series (7.34) between the states
hβ0 | and |α0 i and evaluating it for z = Eα + i0, or by using the LippmannSchwinger formula (7.53) for |α+ i in the exact expression (7.51) for tβα :
Z
Vβγ tγα (Eα )
,
(7.59)
tβα ≡ hβ0 |Vint |α+ i = Vβα + dγ
Eα − Eγ + i0
where Vβα ≡ hβ0 |Vint |α0 i. Iterating this equation yields the series:
Z
Vβγ Vγα
tβα = Vβα + dγ
E − Eγ + i0
Z
Z α
Vβγ Vγγ ′ Vγ ′ α
+ dγ dγ ′
+ ...,
(Eα − Eγ + i0)(Eα − Eγ ′ + i0)
(7.60)
This is the so-called “old-fashioned” perturbation calculus. While in some
situations it is convenient to investigate some specific issues, its main drawback is the lack of manifest Lorentz covariance.
An alternative, more satisfactory in this respect, approach to perturbative
calculation of the S-matrix elements is developed by starting directly from
the formulae (7.25) and (7.24). Differentiating the latter with respect to t2
we get as in section 1.1 the formula (1.22) and, hence,
Z +∞
I
S0 = UI (+∞, −∞) = T exp −i
dt Vint (t) ,
(7.61)
−∞
where the interaction operator in the Dirac picture reads
I
Vint
(t) ≡ eiH0 t Vint e−iH0 t .
(7.62)
The formula (7.61) is the basis of the commonly used time-dependent perturbation calculus.
The equivalence of the formula (7.61) and the formulae (7.23) and (7.60)
should be clear from the construction (at least at the formal level), but can
also be seen directly by making use of the identity
Z
1
1 +∞
=
dτ ei(Eα −Eβ +i0)τ ,
(7.63)
Eα − Eβ + i0
i 0
to represent the energy denominators in (7.60). For example, the first terms
in the expansion of (7.61) give
Z +∞
I
Sβα = hβ0 |S0 |α0 i = hβ0 |1 − i
dt Vint
(t) + . . . |α0 i
−∞
178
= δβα − i
Z
+∞
dt e−i(Eα −Eβ )t Vβα + . . .
−∞
= δβα − 2πi δ(Eα − Eβ ) Vβα + . . .
(7.64)
Z
Z
+∞
+∞
(−i)2
I
I
dt1
dt2 hβ0 |T Vint
(t1 )Vint
(t2 ) |α0 i + . . .
+
2! −∞
−∞
and so on. The advantage of the time-dependent perturbative expansion
based on the formula (7.61) lies in the fact that in relativistic theories it
allows to keep Lorentz invariance manifest at each stage of the calculations.
7.3
S matrix in relativistic quantum mechanics
We now ask the question, what are the conditions the interaction Vint should
satisfy in order to lead to a relativistically covariant S matrix. Obviously, the
first (but by no means sufficient) requirement is that the free particle states,
which describe particles long before and long after the reaction, transform
as described in Section 6 when the reference frame is changed. Thus, in the
Hilbert space there should act a representation of the Poincaré group (or
of its universal covering) by unitary operators U0 (Λ, a) whose action on the
multi-particle H0 generalized eigenstates constructed as tensor products of
the 1-particle states follows from the rules (6.111) with U0 (Λ, a) and reads
X
Λ
Λ
U0 (Λ, a)|(p1 σ1 , . . . , pN σN )0 i = e−ia·( p1 +...+ pN ) ×
(7.65)
σ̄1 ,...,σ̄N
Λ
|( p1 σ̄1 , . . . ,
Λ
(s )
(s )
pN σ̄N )0 i Dσ̄11σ1 (W (Λ, p1)) . . . Dσ̄NNσN (W (Λ, pN )) .
The subscript “0” denotes the free particle states |α0 i and Λ pi stand for the
spatial components of the four-vectors Λµν pνi . As explained at the end of
Section 6.5, the operators P0µ and J0µν , that is, H0 , P0 , J0 and K0 generating
U0 (Λ, a) satisfying the commutation rules (6.21) can be constructed given
the creation and annihilation operators of free particles (provided the energy
operator H0 has the appropriate relativistic form).
Furthermore, in a relativistic theory of interacting particles there should
also exist a set of unitary operators U(Λ, a), also furnishing a representation
of the Poincaré group, and acting on the in and out states, whose labels are
in one-to-one correspondence with the momentum and spin labels of the free
multi-particle states, according to the rule:
X
Λ
Λ
U(Λ, a)|(p1 σ1 , . . . , pN σN )± i = e−ia·( p1 +...+ pN ) ×
(7.66)
σ̄1 ,...,σ̄N
Λ
Λ
|( p1 σ̄1 , . . . , pN σ̄N )± i
(s )
(s )
Dσ̄11σ1 (W (Λ, p1 )) . . . Dσ̄NNσN (W (Λ, pN )) .
179
Because the in and out states are the two sets of eigenvectors of the same
Hamiltonian H, the operators U(Λ, a) must act exactly the same way on the
in and out states. This means that the in and out creation and annihilation
operators ain (p, σ), a†in (p, σ) and aout (p, σ), a†out (p, σ) satisfy the rules (6.111)
with U(Λ, a) (instead of U0 (Λ, a)). Of course, the Hamiltonian H itself is one
of the operators generating U(Λ, a).
The problem we want to investigate in this subsection can be reformulated
as follows: for what interactions Vint can one construct the generators P, J
and K which satisfy the commutation rules (6.21) and which act the same
way on the in and out states (thereby ensuring that also the operators U(Λ, a)
will have the required properties)? Since H = H0 + Vint 6= H0 , it follows from
the commutation rules
i
i
K , P j = iHδ ij ,
K , H = iP i ,
i
i
K0 , P0j = iH0 δ ij ,
K0 , H0 = iP0i ,
that either P 6= P0 or K 6= K0 (or both).
Assuming for the moment that such operators U(Λ, a) do exist, we can
write the condition for the Lorentz covariance of the S matrix. Let |α+ i =
′
)− i. Then
|(p1 σ1 , . . . , pN σN )+ i and |β− i = |(p′1 σ1′ , . . . , p′M σM
Sβα = hβ− |α+ i = hβ− |U † (Λ, a)U(Λ, a)|α+ i
X
Λ
Λ
Λ ′
Λ ′
= e−ia·( p1 +...+ pN − p1 +...+ pM ) ×
X
(s′ )∗
Dσ̄′1σ′ (W (Λ, p′1 )) . . .
1 1
′ σ̄1 ,...,σ̄
σ̄1′ ,...,σ̄M
N
(s′ )∗
(s )
(s )
Dσ̄′Mσ′ (W (Λ, p′M )) Dσ̄11σ1 (W (Λ, p1)) . . . Dσ̄NNσN (W (Λ, pN ))
M
M
′
h(Λ p′M σ̄M
, . . . ,Λ p′1 σ̄1′ )− |(Λ p1 σ̄1 , . . . ,Λ pN σ̄N )+ i .
(7.67)
(The last line in the above formula is just Sβ Λ αΛ ). Applying this formula for
Λ = 1, we infer that the total four-momentum has to be conserved because
otherwise the relation
′
′
Sβα = e−ia·(p1 +...+pN −p1 +...+pM ) Sβα ≡ e−ia·(Pα −Pβ ) Sβα ,
would imply Sβα = 0. It follows, that the formula (7.23) can be written in
the form
Sβα = δβα + (2π)4 δ (4) (Pβ − Pα ) (−iMβα ) ,
(7.68)
that is, the delta function expressing the overall three-momentum conservation can be factorized from the reaction matrix tβα introduced in the preceding subsection:
tβα = (2π)3 δ (3) (Pβ − Pα )Mβα .
180
(7.69)
The factor Mβα is frequently called invariant amplitude (despite being rather
covariant than invariant...)
The condition equivalent to (7.67) can also be derived for the S0 operator
defined by (7.14). Writing
Sβα = hβ0 |S0 |α0 i = hβ0 |U0† (Λ, a)U0 (Λ, a)S0 |α0 i ,
(7.70)
we see that with (7.65) the formula (7.67) will be recovered provided
[S0 , U0 (Λ, a)] = 0 ,
(7.71)
or, equivalently, provided the operator S0 given by (7.61) commutes with all
generators H0 , P0 , J0 and K0 . This can hold only for special forms of Vint
and we will see, that the existence of the generators P, J and K which satisfy
the commutation rules (6.21) and act the same way on in and out states and
the condition that S0 commutes with H0 , P0 , J0 and K0 are equivalent.
Let us now construct the generators P, J and K. Most of the interactions
Vint are such that they commute with P0 and J0 generators. In such cases
we can identify
P = P0 ,
and
J = J0 .
(7.72)
Indeed, since P0 and J0 all commute with H0 , their commutation with the
interaction operator Vint
[P0 , Vint ] = [J0 , Vint ] = 0 ,
(7.73)
implies that they commute also with H = H0 + Vint and, hence, also with
the M6 oller operator Ω(t) ≡ eiHt e−iH0 t (also for finite t). We have then
P|α± i = PΩ∓ |α0 i = P0 Ω∓ |α0 i = Ω∓ P0 |α0 i ,
(7.74)
(and the same for the operators J). This shows that the operators P and
J act the same way on the in and out states. It is also clear that for Vint
such that (7.73) holds, the operators P0 and J0 commute with the evolution
operator
[P0 , UI (τ2 , τ1 )] = [J0 , UI (τ2 , τ1 )] = 0 ,
(7.75)
for arbitrary finite τ1 and τ2 and, therefore, also with the operator S0 =
UI (−∞, +∞). Since the Poincaré group generator H acts the same way
on in and out states owing to the relations (7.19) and, as has been shown,
[H0 , S0 ] = 0, we conclude that those of the commutation rules (6.21) that
involve only H, P and J are easy to satisfy.
181
It remains to construct the operators K satisfying the commutation rules
(6.21) and to prove that [K0 , S0 ] = 0. This is the most tricky part of the
construction. Since P = P0 while H 6= H0 , it follows from the commutation
rule
[K, H] = iP = iP0 ,
(7.76)
that K 6= K0 . Therefore we write
K = K0 + W .
(7.77)
The operator W has to be such that
[K0 , Vint ] = − [W, H0 + Vint ] ≡ − [W, H] .
(7.78)
The operators K0 are known (we know their action on the free multi-particle
states |α0 i) and we have to find W satisfying the condition (7.78) and such,
that K act the same way on in and out states. One could try to define
(without imposing any conditions on the interaction Vint itself) W by simply
giving its matrix elements between a complete set of state vectors, say the
in (or out or free multi-particle) states, e.g. by
hβ+ |W|α+ i :=
hβ+ | [K0 , Vint ] |α+ i
,
Eβ − Eα
(7.79)
(so that it would automatically fulfill the condition (7.78)) but the matrix
elements defined in this way would not be smooth functions of energy and
K = K0 + W would not (as we will see) then act the same way on in and
out states.
We will now show that if there exist operators W satisfying the condition
(7.78) and whose matrix elements are smooth functions of energy, then K
act the same way on in and out states and, at the same time, the operators
K0 commute with the S0 operator. To this end we consider the commutator
[K0 , UI (τ2 , τ1 )] for finite τ1 and τ2 . From the commutation rule [K0 , H0 ] =
iP0 and the fact that P0 commute with H0 we get
K0 , eiH0 τ = −τ P0 eiH0 τ .
(7.80)
Similarly, from [K, H] = iP = iP0 and the fact that P0 commutes also with
Vint (which means that P commutes with H) we get
K, eiHτ = −τ P eiHτ = −τ P0 eiHτ .
(7.81)
Therefore,
[K0 , UI (τ2 , τ1 )] = K0 , eiH0 τ2 e−iH(τ2 −τ1 ) e−iH0 τ1
= −τ2 P0 UI (τ2 , τ1 ) + eiH0 τ2 K0 , e−iH(τ2 −τ1 ) e−iH0 τ1
+τ1 UI (τ2 , τ1 )P0 ,
182
where we have used (7.80). In the second term we put then K0 = K −
W which enables us to make use of (7.81), after which we find that the
term obtained from the commutator of K precisely cancels the two terms
containing P0 (recall that [P0 , UI (τ2 , τ1 )] = 0). We are therefore left with
[K0 , UI (τ2 , τ1 )] = −WI (τ2 )UI (τ2 , τ1 ) + UI (τ2 , τ1 )WI (τ1 ) ,
(7.82)
where we have defined the operator WI (τ ) in the Dirac picture
WI (τ ) = eiH0 τ W e−iH0 τ .
(7.83)
From the result (7.82) it is clear that K0 does not commute with the evolution
operator for finite times τ1 and τ2 . However, for the Lorentz-covariance of
the S matrix one only needs vanishing of the RHS of (7.82) for τ1 → −∞ and
τ2 → +∞. This is ensured if the matrix elements of W are smooth functions
of energy, for then, for any two smooth profiles ψ(α) and phi(β), we have
Z
Z
dα dβ φ∗ (β) ψ(α) hβ0|WI (τ )|α0 i
Z
Z
= dα dβ φ∗ (β) ψ(α) ei(Eβ −Eα )τ hβ0 |W|α0i . (7.84)
By the Riemann-Lebesgue theorem such a Fourier transform vanishes as
τ → ±∞ provided the integrand is smooth enough, in particular provided it
does not have poles such as the RHS of (7.79). Thus, if the matrix elements
of W between the complete set of free multi-particle states |α0 i are smooth
functions of energy, the matrix elements of WI (τ ) between smooth superpositions of such states, which form a dense set in the Hilbert space, all vanish
for τ → ±∞, that is, the operator WI (τ ) itself vanishes (in the weak sense)
in this limit. The right hand side of (7.82) then vanishes in the double limit
t2 → ∞, t1 → −∞ too, and K0 commutes with the S0 operator.
In this case also the operator K = K0 + W acts the same way on in and
out states. To see this, consider the commutator
[K0 , Ω∓ ] ≡ [K0 , UI (0, ∓∞)] = −WI (0)Ω∓ + Ω∓ WI (∓∞) ,
(7.85)
where we have use the result (7.82) for τ2 = 0 and τ1 = ∓∞. For matrix
elements of W which are smooth functions of energy the last term in (7.85)
vanishes and one obtains the intertwining relations
KΩ∓ = K0 Ω∓ + WΩ∓ = Ω∓ K0 − WΩ∓ + WΩ∓ = Ω∓ K0 ,
(7.86)
(where we have used the commutator (7.85) and the fact that W(0) = W)
analogous to (7.74) satisfied by the generators P and J and (7.19) relating
the Hamiltonians H and H0 . Note that if the matrix elements of W were not
183
smooth functions of energy, the last term in the commutator (7.85) would be
nonzero and different for τ → ∓∞; K would then act differently on in and
out states.
Thus, if the operators W with the required properties can be constructed,
all generators G0 of the Poincaré group acting on the states |α0 i commute
with the S0 operator and, simultaneously, hold the related intertwining relation:
i)
ii)
[G0 , S0 ] = 0
G Ω∓ = Ω∓ G0
→
→
[U0 (Λ, a), S0 ] = 0 ,
U(Λ, a) act the same way
on the in and out states.
(7.87)
Analogous intertwining relations will be also crucial for operator quantization
of theories of non-Abelian gauge fields (Section 20.3) based on the BRST
symmetry. The same scheme works also for parity and charge conjugation
operators P and C, as well as for generators of various possible internal
symmetries like isospin or the “eightfold way” SU(3) (see Section 12) etc.
Usually all generators Qa of internal symmetries are such that Qa = Qa0 (as
for P and J), because their free particle counterparts Qa0 commute with the
interaction operator Vint . Slightly more tricky is the the action of the time
reversal operator T because it is antiunitary and interchanges the in and out
states. We will explore consequences of these symmetries for the S matrix
in due course.
The question now is, for which interactions Vint can the operators W with
the required properties be constructed? It turns out that a rather broad class
of such interactions can easily be identified. Belong to it all interactions Vint
whose interaction picture counterparts
I
Vint
(t) = eiH0 t Vint e−iH0 t ,
(7.88)
appearing in (7.61), can be obtained as the space integral of a local interaction
Hamiltonian density Hint (t, x):
Z
I
(7.89)
Vint (t) = d3 x Hint (t, x) ,
which is such that (using now the four-dimensional notation xµ = (t, x))
U0 (Λ, a) Hint (x) U0−1 (Λ, a) = Hint (Λ·x − a) ,
(7.90)
(x − y)2 < 0 .
(7.91)
and
[Hint (x), Hint (y)] = 0
for
184
Note that for Λ = I and aµ = (t, 0) (7.90) is consistent with (7.88). It should
be noted also that with the condition (7.91) the formula (7.61), which for
Vint given by (7.89), reads
Z
Z
∞
X
(−i)N
4
S0 =
d x1 . . . d4 xN T {Hint (x1 ) . . . Hint (xN )} ,
N!
N =0
becomes fully covariant: the time ordering of two spacetime points x1 and x2
is Lorentz invariant only for (x1 − x2 )2 > 0; but the condition (7.91) makes
the time ordering irrelevant for (x1 − x2 )2 < 0.
To these requirements one must also add the one spelled out at the beginning of the subsection 7.2, namely that that structure of Hint (x) (i.e. Vint )
must be such that there is a one-to-one correspondence between the eigenstates of H0 and the in and out eigenstates of H. In section 9 we will construct
interactions Vint satisfying the conditions (7.89)-(7.91) using field operators
introduced in section 8). The additional condition will be investigated in
section 13.
The property (7.91) of Hint (x), called the local causality condition, has no
counterpart in nonrelativistic quantum mechanics. As it will turn out, it is
precisely this property of Hint (x) that is responsible for the fundamental spinstatistics connection (i.e. it enforces that the creation/annihilation operators
of integer and half-integer spin particles satisfy respectively the commutation
and anticommutation rules, thereby implying that these two kinds of particles
obey respectively the Bose-Einstein and the Fermi-Dirac statistics), as well
as for the non-conservation of the number of particles in relativistic theories
and, finally, for the existence of antiparticles.
To check that for Hint (x) satisfying (7.90) the operator W with the required properties can indeed be constructed, we specify (7.90) to the case
aµ = 0 and write
i
µν
U0 (Λ) = exp − ωµν J0
.
(7.92)
2
For infinitesimal ωµν from (7.90) we get
i
− ωµν [J0µν , Hint (x)] ≈ Hint (x + ω·x) − Hint (x)
2
∂Hint (x)
1
∂Hint (x) λ ν
ω ν x = ωµν g λµ xν − g λν xµ
, (7.93)
≈
λ
∂x
2
∂xλ
which, in view of the arbitrariness of ωµν , implies
−i [J0µν , Hint (x)] = g λµ xν − g λν xµ
185
∂Hint (x)
.
∂xλ
(7.94)
Take now J00i ≡ K0i (i.e. µ = 0 and ν = i in the above equality):
∂Hint (x)
.
(7.95)
−i K0i , Hint (x) = t ∇i Hint (x) + xi
∂t
For t = 0, integrating both sides over d3 x we find
Z
Z
i
∂Hint (x)
i
3
3
i
K0 , d x Hint (0, x) ≡ K0 , Vint = i d x x
∂t
t=0
Z
∂
3
i
iH0 t
−iH0 t = i d xx
(7.96)
e
Hint (0, x) e
∂t
t=0
Z
3
i
= H0 , − d x x Hint (0, x) .
This suggests that we can take
Z
W = − d3 x x Hint (0, x) .
Owing to the condition (7.91) we then have
Z
Z
3
[Vint , W] = − d y d3 x x [Hint (0, y), Hint (0, x)] = 0 .
(7.97)
(7.98)
The condition (7.78)
[K0 , Vint ] = − [W, H0 + Vint ] ,
(7.99)
is then satisfied. Thus, if the matrix elements of Hint (0, x) are smooth functions of energy, the operators K = K0 + W act the same way on in and out
states.
7.4
Unitarity of the S matrix
The S0 operator, being a limit of the unitary evolution operator UI (τ2 , τ1 )
(7.61), is itself unitary, i.e. satisfies S0† S0 = 1. This relation written in terms
of the S matrix elements Sβα and the amplitudes Mβα takes the form
Sβα = δβα + (2π)4 δ (4) (Pα − Pβ ) (−iMβα ) ,
one can therefore write
Z
∗
δβα =
dγ Sγβ
Sγα
Z
=
dγ δγβ + (2π)4 δ (4) (Pγ − Pβ ) +iM∗γβ
× δγα + (2π)4 δ (4) (Pγ − Pα ) (−iMγα )
= δβα + (2π)4 δ (4) (Pβ − Pα ) iM∗αβ − iMβα
Z
+
dγ (2π)8 δ (4) (Pγ − Pβ )δ (4) (Pγ − Pα )M∗γβ Mγα .
186
Hence the matrices Mβα satisfy the following important unitarity condition
Z
∗
−i Mαβ − Mβα = dγ (2π)4 δ (4) (Pγ − Pα )M∗γβ Mγα ,
(7.100)
in which both sides are to be taken for pβ = pα . Recall that the integral
over dγ involves also summation over different numbers of particles in the
state |γ0 i and includes appropriate factors 1/ni ! for each set of ni identical
particles of type i in this state.
In the perturbative expansion of the S matrix, when the amplitudes Mαβ
are computed as power series in some (small) coupling constant(s), the importance of the condition (7.100) stems from the fact that it relates contributions
to Mαβ which are of different orders in the couplings. One is therefore able to
say something about higher order contributions to Mαβ knowing it in lower
orders. This will be exploited in Section ??. Here we discuss some consequences of the condition (7.100) which do not rely on perturbative expansion
and have therefore a general character.
The first useful relation is obtained by setting in (7.100) β = α. One then
gets
Z
−2 ImMαα = dγ (2π)4 δ (4) (Pγ − Pα ) |Mγα |2 .
(7.101)
This can for example be used to prove that in the framework of relativistic
quantum field theory the total decay widths of a particle and of its antiparticle
are equal. To this end we write the condition (7.101) for the CPT transformed
state |αi of a single unstable particle at rest19
Z
2
−2 ImM(CPT α)(CPT α) = dγ (2π)4 δ (4) (Pγ − Pα ) Mγ(CPT α) , (7.102)
µ
µ
(PCPT
α = Pα because if the CPT operator commutes with the Hamiltonian,
the states |αi and CPT |αi have the same energy and the action of the CPT
operator does not reverse the three-momenta), and use the fact that because
the CPT operation is a valid symmetry of any relativistic, unitary quantum
field theory, one always has
S(CPT α)(CPT β) = Sβα ,
(7.103)
which in turn implies that
M(CPT α)(CPT β) = Mβα .
19
(7.104)
Since the S matrix elements can, strictly speaking, be defined only for absolutely
stable particles, the “proof” we present here cannot be considered truly rigorous.
187
Thus, the left hand sides of (7.101) and (7.102) are equal (for α = β all phase
factors resulting from the CPT action mutually cancel out) and so are the
RHSs. Up to a multiplicative constant factor (see Section 10.2) the expressions on the right hand sides of (7.101) and (7.102) represent the total decay
widths of a particle and its antiparticle with reversed spin, respectively. However, by rotational invariance the full decay rate (integrated over all possible
directions of the final state particles and summed over possible projections
of their spins or over their helicities) cannot depend on the spin projection
of the decaying particle represented by |αi. This proves the proposition.
Another general consequence of the unitarity of the S matrix can be
obtained by taking for |α0 i in (7.101) some particular two-particle state.
Dividing both sides by the initial state flux factor F (to be defined in section
10.2) we obtain the optical theorem in the form
σtot (α → anything) = −
2
Im Mαα .
F
(7.105)
Further consequences of (7.101) can be explored by going over to the
helicity basis for interacting particles introduced in section 6.4. Consider
a collision process of two particles which are represented by the state |α0 i
and a two-particle final state represented by |β0 i in the center of mass (CM)
frame Inserting in the formula
hβ0 |T0 |α0 i = (2π)4 δ (4) (Pα − Pβ )Mβα .
(7.106)
in which T0 = i(S0 −1) (cf. (7.22)), the states |α0 i and hβ0 | decomposed as in
(6.110) into the states with definite total angular momentum, and denoting
the labels of the state hβ0 | with primes, we get
′
′
hP , p
×
, λ′1 , λ′2 |T0 |0, p, λ1 , λ2 i
p
=
∞ X
∞ XX
X
j′
m′j
j
(j ′ )∗
(j)
Dm′ λ′ −λ′ (Ωp′ ) Dmj λ1 −λ2 (Ωp )
j 1
2
mj
√
(2j ′ + 1)(2j + 1) ′ √ ′ ′ ′ ′ ′
hP , s , λ1 , λ2 , j , mj | T0 |0, s, λ1 , λ2 , j, mj i . (7.107)
4π
Since the total angular momentum as well as its z-axis projection are conserved by the interaction, we must have
√
√
hP′, s′ , λ′1 , λ′2 , j ′ , m′j | T0 |0, s, λ1 , λ2 , j, mj i
√
(j)
= (2π)4 δ (3) (P′ )δ(P 0′ − s) 64π 2 Tλ′ ,λ′ ;λ1 ,λ2 (s) δj ′ j δm′j mj , (7.108)
1
2
(j)
with the factor 64π 2 introduced for further convenience. Tλ′ ,λ′ ,λ1 ,λ2 (s), de1 2
fined by (7.108), are called the partial wave amplitudes. We recall (see the
188
√
formula (6.107)) that for identical particles the states |P, s, λ1 , λ2 , j, mj i
vanish for j odd if λ1 = λ2 and so must do the corresponding partial wave
amplitudes (for j odd and λ1 = λ2 or λ′1 = λ′2 ). Comparing with (7.106) we
get
Mβα = 16π
∞ X
X
mj
j
(j)
(j)
(j)∗
(2j + 1) Tλ′ ,λ′ ;λ1 ,λ2 (s) Dmj λ′ −λ′ (Ωp′ ) Dmj λ1 −λ2 (Ωp ) (. 7.109)
1
2
1
2
The choice of the angular momentum quantization axis in the direction of
(j)
the momentum p reduces Dmj λ1 −λ2 (Ωp ) to δmj λ1 −λ2 , so that:
Mβα = 16π
∞
X
j
(j)
(j)∗
(2j + 1) Tλ′ ,λ′ ;λ1 ,λ2 (s) Dλ1 −λ2 ,λ′ −λ′ (Ωp′ ) .
1
2
1
2
(7.110)
In the following we will take the state |β0 i to have the same two-particle con(j)
tent as |α0 i; the factors Tλ′ ,λ′ ;λ1 ,λ2 (s) will therefore be the elastic scattering
1 2
partial wave amplitudes. We will also need the formula
Mγβ = 16π
∞ X
X
j
mj
(j)∗
(j)
(j)
(2j + 1) T̃λa ,λb ;λ′ ,λ′ (s)Dmj λa −λb (Ωp̃ )Dmj λ′ −λ′ (Ωp′ ) ,
1
2
1
2
with the state |γ0 i = |0, p̃, λa , λb i ≡ |P = 0, p̃, λ1 λ2 i of two particles a
and b (not necessarily the same as in |α0 iand |β0 i) characterized by the
helicities λa , λb and the momentum p̃ (in their CM frame); we denote the
(j)
corresponding partial wave amplitudes by T̃λa ,λb ;λ′ ,λ′ (s). For |γ0 i = |α0 i, that
1 2
is, for λa = λ1 , λb = λ2 and p̃ = p, we get from this formula
M∗αβ
= 16π
∞
X
j
(j)∗
(j)∗
(2j + 1) Tλ1 ,λ2 ;λ′ λ′ (s)Dλ1 −λ2 ,λ′ −λ′ (Ωp′ ) ,
1 2
1
2
(7.111)
(j)
with the same partial wave amplitudes Tλ1 ,λ2 ;λ′ ,λ′ (s) as in (7.110) but with
1 2
the helicity labels interchanged.
These formulae allow to single out the contribution of two-particle states
|γ0 i = |p̃a , λa , p̃b , λb i to the unitarity condition (7.100) specified to the elastic
scattering amplitude,
i.e. for |α0 i and |β0 i with the same particle content.
R
The integral dγ in (7.100) involves the following contribution of a a twoparticle state |γ0 i:
Z
Z
X
√
Nλa ,λb dΓp̃a dΓp̃b (2π)4 δ(Ẽa + Ẽb − s) δ (3) (p̃a + p̃b )M∗γβ Mγα
λa ,λb
Z
X
1
∗
1/2
2
2
N
dΩ
M
M
λ
(s,
m
,
m
)
=
λ
λ
p̃
γα
a b
γβ
a
b
on
32π 2 s
λ ,λ
a
b
189
shell
, (7.112)
where ma and mb are the masses of the particles a and b in the state |γ√
0 i,
the subscript “on shell” means p̃a = −p̃b ≡ p̃ with |p̃| determined by s
(and ma and mb ), and the function λ(s, m2a , m2b ) (do not confuse it with the
helicity labels!) reads
λ(x, y, z) = x2 + y 2 + z 2 − 2xy − 2xz − 2yz .
(7.113)
The factor Nλa ,λb = 1/2 for a = b (i.e. for identical particles) with λa = λb ,
and Nλa ,λb = 1 otherwise, follows from the explicit form of the completeness
relation (7.5). Using now the property (6.109) of the D-functions we arrive
at
h
i
X
(j)∗
(j)
(j)∗
−i
(2j + 1) Dλ1 −λ2 ,λ′ −λ′ (Ωp′ ) Tλ1 λ2 ;λ′ λ′ (s) − Tλ′ λ′ ;λ1 λ2 (s)
1
1 2
2
1 2
j
=
X X
(j)∗
(2j + 1) Dλ1 −λ2 ,λ′ −λ′ (Ωp′ )
1
2
j
(ab)
×
X 2
(j)∗
(j)
λ1/2 (s, m2a , m2b ) Nλa λb T̃λa λb ;λ′ λ′ (s) T̃λa λb ;λ1 λ2 (s)
1 2
s
λa ,λb
1
dγ (2π)4 δ (4) (Pγ − Pα ) M∗γβ Mγα ,
(7.114)
16π
where the sum in the second line is over all kinematically allowed final states
with two20 particles (ab) and the integral in the last line includes all kinematically allowed three- and more particle final states.
+
Z
If the three- and more-particle channels are kinematically inaccessible (or
forbidden by some conservation laws) the last line is absent and integrating
(j)
both sides of (7.114) over dΩp′ with Dλ1 −λ2 ,λ′ −λ′ (Ωp′ ) we obtain the unitarity
1
2
condition in the form
h
i
(j)
(j)∗
−i Tλ1 λ2 ;λ′ λ′ (s) − Tλ′ λ′ ;λ1 λ2 (s)
1 2
1 2
X
2 1/2
(j)
(j)∗
(7.115)
Nλ̃1 λ̃2 Tλ̃ λ̃ ;λ′ λ′ (s) Tλ̃ λ̃ ;λ λ (s)
= λ (s, m21 , m22 )
1 2 1 2
1
2
1
2
s
λ̃1 ,λ̃2
+
X
(ab)6=(12)
X
2 1/2
(j)∗
(j)
Nλa λb T˜λa λb ;λ′ λ′ (s) T̃λa λb ;λ1 λ2 (s) ,
λ (s, m2a , m2b )
1 2
s
λ ,λ
a
b
where in the second line we have explicitly singled out the contribution of the
elastic channel. For scattering with no change of helicities, i.e. for λ′1 = λ1 ,
λ′2 = λ2 , this can be rewritten in the form
2
h
i2 sλ−1/2
s2 λ−1
(j)
(j)
Re Tλ1 λ2 ;λ1 λ2 (s) + Im Tλ1 λ2 ;λ1 λ2 (s) +
=
− Rj2 (s) ,(7.116)
2Nλ1 λ2
4Nλ21 λ2
20
Pairs (ab) and (ba) must be treated here as one and the same state i.e. only one of them
should be included in the sum. Alternatively, one can sum over a and b independently,
including in the sum both states, (ab) and (ba), and setting Nλa λb = 21 .
190
in which
Rj2 (s) =
2
X X Nλ′ λ′ (j)
1 2
Tλ′1 λ′2 ;λ1 λ2 (s)
N
λ1 λ2
′
′
λ1 6=λ1 λ2 6=λ2
+
X
(ab)6=(12)
2
X Nλ λ λ̃1/2 (j)
a b
T̃λa λb ;λ1 λ2 (s) ,
1/2
Nλ1 λ2 λ
λ ,λ
a
(7.117)
b
and we have used the notation λ ≡ λ(s, m21 , m22 ) and λ̃ ≡ λ(s, m2a , m2b ).
It is easy to see that inelastic processes leading to multi-particle final
states can also be included in Rj2 (s). To show this, using the formula (6.110)
we write the amplitudes of transitions from the |α0 i and |β0 i two-particle
states with the same particle content into any three- or more-particle state
|γ0 i as
∞ Xr
X
2j ′ + 1 (j ′ )
(j ′ )
Tγ; λ′ ,λ′ ,m′ (γ; s′ ) Dm′ λ′ −λ′ (Ωp′ ) ,
Mγβ =
1
2
2
j
j 1
4π
j ′ m′j
∞ r
X
2j ′′ + 1 (j ′′ )
Mγα =
Tγ; λ1 ,λ2 ,λ1 −λ2 (γ; s) ,
(7.118)
4π
′′
j
(j)
λ1 ,λ2 ,mj (γ; s)
where the amplitudes Tγ;
(2π)4δ (3) (Pγ )δ(Pγ0 −
√
are defined by the equality
(j)
λ1 ,λ2 ,mj (γ; s)
s) Tγ;
≡ hγ0 |T0 |0,
√
s, λ1 , λ2 , j, mj i .
(7.119)
The symbol γ used as the argument of T (j) is used to remind that this
amplitude depends, apart from s, also on the variables (relative momenta and
helicities) needed to specify the multiparticle state |γ0i. In Mγα , similarly as
(j )
in (7.110), the equalityDmαj λ1 −λ2 (Ωp ) = δmj λ1 −λ2 has been used. With these
formulae, and setting λ′1 = λ1 , λ′2 = λ2 , the last line of (7.114) takes the form
p
∞ ∞
1 X X X (2j ′′ + 1)(2j ′ + 1) (j ′ )∗
Dm′ λ1 −λ2 (Ωp′ )
j
16π ′′ ′
4π
′
j
j
mj
Z
(j ′ )∗
(j ′′ )
× dγ (2π)4 δ (4) (Pγ − Pα ) Tγ; λ1 ,λ2 ,m′ (γ; s) Tγ; λ1 ,λ2 ,λ1 −λ2 (γ; s) .
j
(j)
Integrating now as previously both sides of (7.114) with Dλ1 −λ2 ,λ′ −λ′ (Ωp′ )
1
2
over dΩp′ we get in (7.115) an extra term
s
Z
∞
1 X 2j ′′ + 1
dγ (2π)4 δ (4) (Pγ − Pα )
+
16π ′′
2j + 1
j
(j)∗
(j ′′ )
λ1 ,λ2 ,λ1 −λ2 (γ; s) Tγ; λ1 ,λ2 ,λ1 −λ2 (γ; s) .
×Tγ;
191
Im T
Im T
Re T
Re T
Figure 7.2: Argand circles: if inelastic channels are closed, i.e. if Rj2 (s) = 0,
(left) the radius is sλ−1/2 /2Nλ1 λ2 ; if inelastic channels are open (right) the
radius is smaller. Partial amplitudes of the elastic scattering must lie on
the Argand circle. Short-dashed lines show possible partial elastic scattering
amplitudes in a weakly coupled theory (small corrections in the perturbative
expansion) whereas the long-dashed ones illustrate and elastic scattering amplitudes typical for a strongly coupled (nonperturbative) theory.
However, angular momentum conservation implies that only j ′′ = j can contribute to the sum and therefore the last term in (7.114) adds to Rj2 given
by (7.117) a strictly nonnegative contribution proportional to
Z
2
1
(j)
4 (4)
(7.120)
dγ (2π) δ (Pγ − Pα ) Tγ; λ1 ,λ2 ,λ1 −λ2 (γ; s) .
16π
(j)
The relation (7.116) demonstrates that the amplitude Tλ1 λ2 ;λ1 λ2 (s) of the
elastic scattering with no change of helicities must lie on a circle, called the
Argand circle, whose radius in not grater than sλ−1/2 /2Nλ1 λ2 and the center is
at the point (0, −sλ−1/2 /2Nλ1 λ2 ) in the complex plane, as shown graphically
in Figure 7.2. This shows, that the elastic scattering amplitude must have
a nonzero imaginary part
√ which grows as more and more inelastic channels
open up with increasing s (at high energies elastic scattering amplitudes are
therefore predominantly imaginary). From (7.116) it also follows, that the
fully elastic (with no change of helicities) scattering partial wave amplitude
(j)
Tλ1 λ2 ;λ1 λ2 (s) can be represented in terms of the (complex) phase shift δj (s) +
iβj (s)
(j)
Tλ1 λ2 ;λ1 λ2 (s) = i
where 0 < δj (s) < π and
sλ−1/2 2iδj (s)−2βj (s)
e
−1 ,
2Nλ1 λ2
4Nλ21 λ2 2
1
βj (s) = − ln 1 − 2 −1 Rj .
4
sλ
192
(7.121)
(7.122)
Of course, if only the elastic channel with no helicity change is open βj (s) = 0
and the phase shifts are real numbers.
These results, combined with the formula (6.108), allow to write down
the S matrix element corresponding to elastic (with no change of helicities)
scattering in the basis of states with definite angular momentum in the form
√
√
(7.123)
hP′ , s′ , λ1 , λ2 , j ′ , m′j |1 − iT0 |P, s, λ1 , λ2 , j, mj i
= (2π)4 δ (4) (P ′ − P ) 64π 2 δj ′ j δm′j mj
sλ−1/2 2iδj (s)−2βj (s)
e
,
2Nλ1 λ2
where for identical particles in the initial state the factor Nλ−1
should be
1 λ2
j
written as 1 +√(−1) δλ1 λ2 to account for the fact that for identical particles
the states |P, s, λ1 , λ2 , j, mj i vanish for j odd and λ1 = λ2 . We have also
used the fact that |p| (the length of the momentum of the first particle in
the CM frame) in the formula (6.108) is given by
1
|p| = √ λ1/2 (s, m21 , m22 ) .
2 s
(7.124)
The restriction to the elastic scattering amplitude with no helicity flip i.e.
to λ′1 = λ1 , λ′2 = λ2 can be removed by diagonalizing the elastic scattering
amplitude in the spin space
(j)
δkl T (j,k) (s) = (U † )k,(λ′1 λ′2 ) Tλ′ ,λ′ ;λ1 ,λ2 (s) U(λ1 λ2 )l ,
1
2
(7.125)
with the help of a (2s1 + 1) × (2s2 + 1) unitary (s dependent) matrix U
(j)
and writing the unitarity condition (7.100) in the basis in which Tλ′ ,λ′ ;λ1 ,λ2 is
1 2
diagonal (it suffices to sandwich the relation (7.115) between (U † )k,(λ′1 λ′2 ) and
Ul(λ1 λ2 ) ). One then gets the representations (7.116), and consequently, also
(j)
(j)
(7.121) with appropriate phase shifts δk (s) + iβk (s), for each of the elastic
2
scattering amplitudes T (j,k) (s) with Rjk
(s) which now does not include the
contribution from the elastic channel.
In principle, by appropriately choosing the basis of states, i.e. the unitary
transformations U, the whole scattering matrix could be diagonalized leading
to only purely elastic diagonal amplitudes T (jk) with purely real phase shifts
δjk (s). In such a basis,21 the S matrix can, extending the formula (7.123),
succinctly be written as
k
S(j ′ k′ )(jk) = δj ′ j δk′ k e2iδj (s) .
21
(7.126)
The basis in which the S matrix is diagonal may not, for general values of the Mandelstamm variable s, consist of experimentally realizable states: for example in the case of
two particle processes they are usually linear combinations of states representing different
pairs of two particles.
193
This makes clear that the phase shift factors are just the S matrix eigenvalues
on such states (since S matrix is unitary its eigenvalues must be numbers of
unit modulus).
Since not all elements of the scattering matrix are known, symmetries of
the interactions can be in practice be exploited
to diagonalize the scattering
√
amplitudes at least for low energies (small s), for which only a limited number of channels can be reached from a given initial state due to kinematical
restrictions. A canonical example is provided by low energy strong interactions of pions (the lightest strongly interacting spinless particles), if the
electromagnetic and weak interactions are neglected: due to the isospin invariance of the strong interactions (see Section 12) their S matrix is diagonal
in the isospin basis
hI ′ , I3′ , p, −p|S0 |I, I3, k, −ki = δI ′ I δI3′ I3 S (I) (s) ,
(7.127)
so that in the isospin basis |I, I3 , j, mi the formula (7.126) takes the form
I
S(j ′ I ′ I3′ )(jII3 ) = δj ′ j δI ′ I δI3′ I3 e2iδj (s) .
(7.128)
where the (purely real) isospin-angular momentum phase shifts δjI (s) parametrize the corresponding isospin partial wave amplitudes T (j)I . The formulae (7.128) and (7.58) are the basis for accounting for the final state rescattering effects in decay processes induced by the weak interactions. They
are particularly important in the analysis of the CP violation in the kaon
system (see section 12.4).
From the relation (7.116) or from Figure 7.2 we get two unitarity bounds
pertaining to elastic scattering (with no change of helicities) partial wave
amplitudes:
s
(j)
Nλ1 λ2 Tλ1 λ2 ;λ1 λ2 (s) ≤ 1/2
,
λ (s, m21 , m22 )
s
(j)
Nλ1 λ2 Re Tλ1 λ2 ;λ1 λ2 (s) ≤
.
(7.129)
1/2
2λ (s, m21 , m22 )
Moreover, since Rj2 cannot exceed s2 λ−1 /4Nλ21 λ2 (the right hand side of
(7.116) must be positive) we get also the bounds22 on partial wave amplitudes
of any two body (not necessarily elastic) scattering:
p
s
(j)
. (7.130)
Nλa λb Nλ1 λ2 T̃λa λb ;λ1 λ2 (s) ≤
2 1/4
1/4
2
2λ (s, m̃a , m̃b )λ (s, m21 , m22 )
22
It is clear that the quantities (7.120) related to the contribution of multi-particle
production to the total cross section are also bounded by this requirement.
194
Notice, that at the reaction threshold, where λ1/2 (s, m2a , m2b ) = 0, the bounds
(7.130) and (7.129) disappear. For s larger than any of the masses involved,
the unitarity bounds become23
(j)
Nλ1 λ2 Tλ1 λ2 ;λ1 λ2 (s) ≤ 1 ,
1
(j)
Nλ1 λ2 Re Tλ1 λ2 ;λ1 λ2 (s) ≤ ,
(7.131)
2
p
1
(j)
Nλa λb Nλ1 λ2 T˜λa λb ;λ1 λ2 (s) ≤ .
2
The bounds (7.129) and (7.130) have been derived assuming only that
the evolution of the quantum system is unitary.24 In particular they do not
rely on any perturbative expansion. Scattering amplitudes derived from local quantum field theory models which (are believed to) give rise to unitary
S matrices should in principle, respect these bounds. Since elastic scattering partial wave amplitudes computed in the lowest order of perturbation
expansion in quantum field theories are (usually) real (i.e. lie on the horizontal axis in Figure 7.2), they cannot satisfy the unitarity relation (7.116).
Higher order contributions must therefore bring elastic amplitudes back on
the Argand circle. Two distinct situations can be then encountered. If the
(absolute value of the) real part of the lowest order amplitude is bounded
by 1/2Nλ1 λ2 , higher order contributions required to restore unitarity can be
relatively small (short dashed lines in Figure 7.2) and the perturbative expansion is likely to be reliable. In contrast, if the real part of the lowest order
amplitude greatly exceeds 1/2Nλ1 λ2 , the necessary higher order contributions
must be comparable or even larger than the lowest order term and the perturbation expansion evidently fails. In specific quantum field theory models
the
√ magnitude of the lowest order amplitudes depends usually on the energy
s. In renormalizable
theories (see Section 14) the lowest order amplitudes
√
are bounded for s → ∞ by some constants and reliability of the perturbation expansion depends on their magnitude (whether such a constant is
smaller or bigger than 1/2Nλ1 λ2 ). In√nonrenormalizable theories the lowest
order amplitudes usually grow with s and above some critical energy the
perturbation expansion unavoidably breaks down.
p
(j)
In the literature it is customary to include the factor Nλa λb Nλ1 λ2 in Tλa λb ;λ1 λ2 (s). In
this way the Nλ1 λ2 factors disappear altogether from the formulae (7.114)-(7.121) and the
unitarity bounds (7.131) for distinct and identical particles look the same. We preferred
not
p to do so, in order to keep control over such factors and, moreover, because the factors
Nλ1 λ2 should not be included (for the cross section calculation) in the amplitudes of
scatterings of identical particles in the initial state.
24
Another assumption is that the partial wave expansion of amplitudes makes sense.
This may not be true in the presence of massless particles which produce long range
interactions.
23
195
Example of the latter situation is provided by the phenomenological Fermi
theory of weak interactions (introduced in Section 12). Amplitudes computed
in this model in the√
lowest order grow linearly with s and violate the unitarity
> 600 GeV indicating that for such energies either the
bounds (7.131) for s ∼
theory of weak interaction becomes strongly coupled (and the perturbation
expansion cannot be applied to it) or that the Fermi theory is only an effective
model which should be replaced by a more fundamental theory in which
exchanges of new particles restore unitarity of amplitudes computed in the
lowest order. It is the second option that is realized in the Nature - the Fermi
theory turned out to provide only an effective, low energy approximation to
the results obtained in the Standard Theory of electroweak interactions.
Unitarity bound derived in this section were also important in discussing
(before 2012) possible versions of the extension (ultraviolet completion in
the modern parlance) of the Fermi theory. Scattering amplitudes of longitudinally polarized massive spin 1 (vector) bosons - particles which are
experimentally now well known to mediate weak interactions - computed in
the lowest order grow linearly with s and violate these bounds if the contributions of the sector of the theory responsible for electroweak symmetry
breaking is not taken into account. This sector was, before 2012, experimentally unexplored (and remains largely such even now). The Standard Model the concrete realization of such an extension, in which which the electroweak
gauge symmetry is broken by a single doublet of scalar fields - predicted the
existence, without fixing its mass, of a single neutral spinless particle, h0 ,
whose contribution to the discussed scattering amplitudes cuts their rising
with s - they reach a constant value proportional to the mass squared of h0 .
As a result, the bounds would be violated (in a milder way, but still) if the
mass of h0 was greater than ∼ 1 TeV. The discovery of h0 with mass equal
125 GeV, giving strong support for the mechanism of electrowek symmetry
breaking realized in the Standard Model, cuted (forever?) the speculations
concerning more exotic possibilities of saving unitarity in the scattering of
longitudinally polarized electroweak vector bosons.
7.5
Other symmetries of the S matrix
Discrete symmetries: parity, time reversal and charge conjugation may or
may not be exact symmetries of a given model of relativistic quantum mechanics (a quantum field theory model). Even if they are not exact symmetries of the real world (we know they are not), it is still interesting to
consider them because in physics, in contrast to pure mathematics in which
the statement that some operation is not a symmetry closes the issue, we are
interested not only in whether they are symmetries but also, how they are
196
violated, i.e. by which type of interactions and in which processes.
The action of the parity operator on one-particle states has been discussed
in Section 6.3. Multi-particle states transform of course as tensor products:
P0 |(p1 σ1 , . . . , pN σN )0 i = η1 . . . ηN |(−p1 σ1 , . . . , −pN σN )0 i
(7.132)
(we have assumed that all particles are massive; modifications for massless
particles are obvious). The same formula, with P instead of P0 is also true
for the in and out states.
If P0 commutes with the interaction operator Vint we can set P = P0 .
The S matrix satisfies then the following identity
Sp′1 σ1′ ,p′2 σ2′ ...;p1 σ1 p2 σ2 ... = η1∗′ η2∗′ . . . η1 η2 . . . S−p′1 σ1′ ,−p′2 σ2′ ...;−p1 σ1 −p2 σ2 ... . (7.133)
This shows that if parity is conserved in elementary processes, then for
η1∗′ η2∗′ . . . η1 η2 . . . = +1 the S matrix is an even function of the particle momenta, whereas for η1∗′ η2∗′ . . . η1 η2 . . . = −1 it must be an odd function.
What are the internal parities η of various known elementary particles?
For a small set particles η can be fixed by convention. This is because in the
real world the parity operator P = P0 can be redefined
P ′ = Pe−ic1 Q̂−ic2 B̂−ic3 L̂−... ,
(7.134)
where Q̂, B̂ and L̂ are the operators of electric charge, baryon and lepton
numbers and other quantum numbers conserved in parity-symmetric interactions.25 If P commutes with the Hamiltonian so does P ′ . This freedom in
the definition of the parity operator allows to assign (by convention) η = +1
to p, n and e− (these particles have all different combinations of B̂, L̂ and Q̂
and one can choose c1 , c2 and c3 so that the action of P ′ is consistent with
this assignment). In the same way using the factor with the strangeness or
charm operators S, C one can by convention assign η = 1 to one strange
baryon and one charmed baryon.
Internal parities of the remaining particles should be assigned in such
a way that parity is conserved in as broad class of processes as possible.
Consider for example the process of radiative capturing of π − from the “π
meson deuteron” i.e. in the dπ − bound state: d+π − → n+n. The capturing
occurs from the orbital l = 0 ground state of such an atom. The initial state
has j = 1 (deuteron is predominantly an l = 0 proton-neutron bound state
25
Even if the lepton and baryon numbers are not strictly conserved - at least for the lepton number there are strong indications that it is indeed violated by neutrino interactions
- this does not hurt, because they are conserved in the same interactions which preserve
parity.
197
with the total spin s = 1) and, therefore, the final state should also have
j = 1. Since the final state, being the state of the two identical fermions,
must be antisymmetric in their labels, it must have s = 1, l = 1 (other
possibilities allowing for j = 1: s = 0, l = 1 or s = 1, l = 0 or s = 1, l = 2
all lead to symmetric final states). Thus, if parity is conserved, we must have
ηd ηπ− = −ηn2 .
(7.135)
Since ηd = +1 (ηp = ηn = +1 by convention and l = 0), it follows that
ηπ− = −1 .
(7.136)
From the isospin symmetry (see section 12) it then follows that also
ηπ+ = ηπ0 = −1 .
(7.137)
Negative parity of the π mesons has important consequences. One is that
if parity is conserved, a particle decaying into three pions must necessarily
have negative parity. Indeed, in the rest frame of the decaying particle the
S matrix element describing its decay into three pions can depend only on
p1 · p2 , p1 · p3 or p2 · p3 (as pions are spinless, no other vectors are available; moreover, since in this frame p1 + p2 + p3 = 0 the triple product
(p1 × p2 ) · p3 = 0). Hence, parity of the final state is negative and so must
be parity of the initial state. Similarly, a particle decaying into two pions
must necessarily have positive parity. Non-conservation of parity in weak
interactions indicated first by the the observation of K + decays into two
and three pions became evident around 1950 (see Section 12). It is by now
firmly established and is one of the cornerstones of the theory of electroweak
interactions (Section ??).
The action of the time reversal operator on one-particle states has been
given in Section 6.3. Its action on free multi-particle states therefore reads
T0 |(p1 σ1 , . . . , pN σN )0 i
= ζ1 (−1)s1 −σ1 . . . ζN (−1)sN −σN |(−p1 −σ1 , . . . , −pN −σN )0 i . (7.138)
(as for parity we have assumed that all particles are massive; modifications for
massless particles are obvious). If T0 commutes with the interaction operator
Vint , we can set T = T0 . It is also easy to see that the action of T changes
the in states into the out states and vice-versa:
T |(p1 σ1 , . . . , pN σN )± i
= ζ1 (−1)s1 −σ1 . . . ζN (−1)sN −σN |(−p1 −σ1 , . . . , −pN −σN )∓ i . (7.139)
198
This can be seen from the formal expression (??):
T |α± i = lim T0 eiHτ e−iH0 τ |α0 i = lim e−iHτ eiH0 τ T0 |α0 i
τ →∓∞
τ →∓∞
iHτ −iH0 τ
= lim e
τ →±∞
e
|(T α)0 i ≡ |(T α)∓ i ,
(7.140)
(recall that T is antilinear!), where we have introduced the short-hand notation |(T α)i for the state (including its phase factors) on the RHS of (7.139):
e.g. for |α0 i = |(p, σi)0 we denote |(T α)0 i ≡ ζ(−1)s−σ |(−p, −σ)0 i. The
same conclusion follows also from the expression (7.52).
If the time reversal is a symmetry operation, from the properties of antiunitary operators (see Section 4) it follows that26
Sβα = (β− |α+ ) = (β− |T −1 T α+ ) = (β− |T † T α+ )
= (T β− |T α+ )∗ = (T α+ |T β− ) = ((T α)− |(T β)+ ) ≡ S(T α)(T β) , (7.141)
where for Sβα ≡ Sp′1 σ1′ ,p′2 σ2′ ...;p1 σ1 p2 σ2 ... the symbol on the RHS should be understood as
S(T α)(T β) ≡ ζ1∗′ (−1)s1′ −σ1′ ζ2∗′ (−1)s2′ −σ2′ . . . ζ1 (−1)s1 −σ1 ζ2 (−1)s2 −σ2 . . .
× S−p1 −σ1 −p2 −σ2 ...;−p′1 −σ1′ ,−p′2 −σ2′ ...
(7.142)
Notice, that in general the time reversal does not imply that the rate of the
reaction α → β is the same as of the reaction T α → T β. There are however
some special situations in which it does imply this. This is when the S matrix
strong
can be split into two parts (Sβα
can be viewed as the zeroth order term in
the expansion of Sβα in some small parameter)
strong
weak
Sβα = Sβα
+ Sβα
,
(7.143)
strong
weak
where |Sβα
| ≪ |Sβα
| because S strong is due to the strong interactions,
weak
whereas S weak is due to the weak ones. In the first order in Sβα
the unitarity
condition for S then reads
1 = S † S = S strong† S strong + S strong† S weak + S weak† S strong + . . .
(7.144)
So, approximately,
S weak ≈ −S strong S weak† S strong ,
because S strong† S strong = 1. More concretely,
Z
Z
weak† strong
weak
S
S strong
Sβα ≈ − dγ dγ ′ Sβγ
γγ ′ γ ′ α
Z
Z
weak ∗ strong
strong
Sγ ′ α
Sγ ′ γ
= − dγ dγ ′ Sβγ
Z
Z
∗
strong weak
.
S(T γ)(T γ ′ ) Sγstrong
= − dγ dγ ′ Sβγ
′α
26
We have to abandon for a while the Dirac bra-ket notation.
199
(7.145)
(7.146)
This relation is particularly useful if, as in the case of the nuclear β destrong
cays, Sβα
∝ δβα for the relevant states α and β (the process cannot occur
through the strong interactions). In the basis of states with fixed total angustrong
lar momentum using the formula (7.126) Sβα
= δβα exp(2iδα ) we get from
(7.146)
weak
∗
weak
Sβα
≈ −e2i(δβ +δα ) S(T
.
(7.147)
β)(T α)
which does imply that the rate of the process α → β is approximately27 the
same as of the reaction (T α) → (T β).
Action of the charge conjugation operator C0 on free one-particle states
was defined in Section 6.3. If the interaction operator Vint commutes with
C0 , one can take C = C0 and the charge conjugation symmetry implies that
Sp′1 σ1′ ,p′2 σ2′ ,...;p1 σ1 p2 σ2 ,... = ξ1∗′ ξ2∗′ . . . ξ1 ξ2 . . . Sp̄′1 σ̄1′ ,p̄′2 σ̄2′ ,...;p̄1 σ̄1 ,p̄2 σ̄2 ,... , (7.148)
where we have denoted antiparticle momentum and spin labels by bars.
Charge conjugation parities of different particles are assigned in a similar
way as the intrinsic parities. First of all, C can always be redefined
C ′ = C e−ia1 Q̂−ia2 B̂−ia3 L̂ ,
(7.149)
so that for three particles that have different Q, B and L charge conjugation
parities C can be fixed by a convention. Furthermore, charge conjugation
parities of neutral particles, which like photon or neutral pion do not carry
any conserved quantum numbers, are uniquely determined. As will become
evident in Section 8, a state of a (massive) fermion-antifermion pair has negative charge conjugation parity (more precisely, the electromagnetic current
µ
µ
µ
operator JEM
is such that CJEM
C −1 = −JEM
). Therefore, the photon which
couples to such pairs must also have negative charge conjugation parity (i.e.
the photon field operator must have the property CAµ C −1 = −Aµ ). Since π 0
decays into two photons, ξπ0 = +1 and, by isospin symmetry, the same must
be also true for π ± . It then follows that the process π 0 → 3γ is forbidden
(experimentally Br(π 0 → 3γ) < 3.1 × 10−8 ).
7.6
The cluster decomposition principle
In this subsection we shall briefly describe, without entering into details, the
so-called cluster decomposition principle, which any physically sensible S
matrix should satisfy in order the theory predictions for measurements made
27
Since the weak interaction is much much weaker than the strong one, this is in fact
an almost perfect approximation.
200
in spatially remote laboratories be uncorrelated. The cluster decomposition
principle imposes a simple but nontrivial constraint on the general structure
of quantum field theory Hamiltonians expressed in terms of the creation and
annihilation operators of free particles in the momentum representation.
We begin with the almost obvious statement that any operator O acting in the multi-particle Hilbert space spanned by the states (7.1) can be
represented in the form28
O=
∞ X
∞ Z
X
N =0 M =0
dΓp1 . . .
Z
dΓpM
Z
dΓq1 . . .
Z
dΓqN
(7.150)
CM N (p1 , . . . , pM ; q1 , . . . , qN )a† (p1 ) . . . a† (pM )a(q1 ) . . . a(qN ) ,
where for simplicity of the notation we consider only one type of spinless
particles. This means that given an operator O, or equivalently, given all
possible matrix elements of O between multi-particle states, we can always
choose the functions CM N (p1 , . . . , pM ; q1 , . . . , qN ) in (7.150) to reproduce
the matrix elements defining O. The proof is inductive: one adjusts first the
value of C00 so as to reproduce hΩ|O|Ωi, then one considers matrix elements
of O between the vacuum and the one-particle states to fix C01 and C10 , then
one adjusts the functions C11 to reproduce matrix elements of O between two
one-particle states and so on.
Consider now the S matrix in the position representation
Z
Z
Z
Z
Sy1 ,...,yM ;x1 ,...,xN = dΓp1 . . . dΓpM dΓq1 . . . dΓqN
(7.151)
e−ip1 y1 . . . e−ipM yM eiq1 x1 . . . eiqN xN Sp1 ,...,pM ;q1 ,...,qN ,
and imagine a process α → β (where α and β stand for collections of
positions and spins of the initial and final state particles, respectively),
in which particles in the initial and final states exhibit some clustering:
α = (α1 )(α2 )(α3 ) . . . (αn ), β = (β1 )(β2 )(β3 ) . . . (βn ). By clustering we mean
that the subprocesses α1 → β1 , α2 → β2 , . . .,αn → βn are measured separately in remote laboratories (like e.g. CERN, SLAC and FNAL). The S
matrix satisfies the cluster decomposition principle if it factorizes in such a
case, that is, if
Sβα = Sβ1 α1 Sβ2 α2 Sβ3 α3 . . . Sβn αn ,
(7.152)
when the distances |xi − xj | and |yi − yj | are large for xi , yi and xj , yj
28
The multi-particle states considered here can be the free multi-particle states but can
also be the in or the out states, for which the corresponding creation and annihilation
operators a†in (p), ain (p) and a†out (p), aout (p) can also be introduced (see section 8.7).
201
belonging to different clusters29 (large compared to the typical distances in
the same cluster). This in turn ensures independence of the theory predictions for experiments performed in different laboratories: if we are interested
in the probability of a concrete final state |β1 i in an experiment performed
e.g. at CERN, we sum over all possible final states which can be found in
far-away laboratories:
X
∗
Sβα Sβα
(7.153)
P (α1 → β1 ) =
β2 ,β3 ,...
= Sβ1 α1 Sβ∗1 α1
X
Sβ2 α2 Sβ∗2 α2
X
Sβ3 α3 Sβ∗3 α3 . . . = Sβ1 α1 Sβ∗1 α1 .
β3
β2
where in the last step unitarity of the S matrix has been used.
To implement the factorization of the S matrix let us first define its
C
connected part Sβα
by using a combinatoric trick30 . We write
X
(±)SβC1 α1 SβC2 α2 . . . ,
(7.154)
Sβα =
partitions
where the sum is over all possible divisions of the individual particle labels in
α and β into clusters (α1 )(α2 )(α3 ) . . . (αn ), (β1 )(β2 )(β3 ) . . . (βn ) (not treating
as different those partitions which differ only by a permutation of labels
within the same group, or differ by a permutation of the clusters as wholes).
The sign ± depends on whether the number of interchanges of fermionic
labels is even or odd. The definition (7.154) is recursive:
X
C
Sβα = Sβα
+
(±)SβC1 α1 SβC2 α2 . . . ,
(7.155)
partitions′
where now the sum goes over the partitions in which all clusters (αi ), (βi )
contain less particles than α and β. (We assume here that no one of the
C
clusters (αi ) and/or (βi ) is empty; this requires that in (7.154) S00
is set to
zero). It is easier to understand this on the example. We have by definition31
Z
Z
(3)
C
Sy,x ≡ Sy,x ∝ dΓp dΓq e−ipy eiqx δΓ (p − q) ≡ δy,x .
(7.156)
29
The factorization property of the S matrix is not in conflict with the phenomenon of
entanglement and the well known Einstein Podolski Rosen correlations: entangled (correlated) particles must have interacted (e.g. they originate from a decay of another particle)
before the measurements of their properties are made and, hence, must belong to the same
cluster βi .
30
The term “connected” derives from the form of the Feynman diagrams (to be introduced in section 9) which contribute to this part of the S matrix.
(3)
31
In the presence of interactions Sp,q differs from δΓ (p − q) by a phase factor which is
not relevant for what follows.
202
Then, for the 2 → 2 element of the S matrix we have
Sy1 y2 ;x1 x2 = SyC1 y2 ;x1 x2 + δy1 ,x1 δy2 ,x2 ± δy1 ,x2 δy2 ,x1 .
Similarly,
Sy1 y2 y3 ;x1 x2 = SyC1 y2 y3 ;x1 x2 + SyC2 y3 ;x2 δy1 ,x1 ± SyC1 y3 ;x2 δy2 ,x1 + SyC1 y2 ;x2 δy3 ,x1
±SyC2 y3 ;x1 δy1 ,x2 + SyC1 y3 ;x1 δy2 ,x2 ± SyC1 y2 ;x1 δy3 ,x2 ,
for the 2 → 3 transitions,
Sy1 y2 y3 ;x1 x2 x3 = SyC1 y2 y3 ;x1 x2 x3
+ δy1 ,x1 SyC2 y3 ;x2 x3 ± permutations
+ δy1 ,x1 δy2 ,x2 δy3 ,x3 ± permutations ,
for the 3 → 3 transitions, and
Sy1 y2 y3 y4 ;x1 x2 x3 x4 = SyC1 y2 y3 y4 ;x1 x2 x3 x4
+ SyC1 y2 ;x1 x2 SyC3 y4 ;x3 x4 ± permutations
+ δy1 ,x1 SyC2 y3 y4 ;x2 x3 x4 ± permutations
(7.157)
δy1 ,x1 δy2 ,x2 SyC3 y4 ;x3 x4
+
± permutations
+ δy1 ,x1 δy2 ,x2 δy3 ,x3 δy4 ,x4 ± permutations .
for the 4 → 4 transitions etc.
The main point is that Sβα (7.155) satisfies the cluster decomposition
principle (7.152) if the connected matrices SβCi αi vanish when at least one
of the particles in clusters (αi ), (βi ) is spatially separated from the other
particles in the same cluster. To see it on an example, consider the process
4 → 4 and the corresponding S matrix element (7.157). Let us assume that
the initial and final particles 1 and 2 are far away from the initial and final
particles 3 and 4. Then we get from (7.157) throwing out all vanishing terms
Sy1 y2 y3 y4 ;x1 x2 x3 x4 = SyC1 y2 ;x1 x2 SyC3 y4 ;x3 x4
+ (δy1 ,x1 δy2 ,x2 ± δy1 ,x2 δy1 ,x1 )SyC3 y4 ;x3 x4
+ (δy3 ,x3 δy4 ,x4 ± δy3 ,x4 δy4 ,x3 )SyC1 y2 ;x1 x2
+ (δy1 ,x1 δy2 ,x2 ± δy1 ,x2 δy1 ,x1 )(δy3 ,x3 δy4 ,x4 ± δy3 ,x4 δy4 ,x3 ) ,
which is precisely Sy1 y2 ;x1 x2 Sy3 y4 ;x3 x4 .
To see what form of SpC1 p2 ...;q1 q2 ... should take in order to have this property, let us note that for
Z
Z
Z
Z
C
Sy1 y2 ...;x1 x2 ... = dΓp1 dΓp2 . . . dΓq1 dΓq2 . . .
e−ip1 y1 e−ip2 y2 . . . eiq1 x1 eiq2 x2 . . . SpC1 p2 ...;q1 q2 ... , (7.158)
203
p
the Riemann-Lebesgue theorem says that if |SpC1 p2 ...;q1 q2 ... |/ Ep1 . . . Eq1 . . .
is a Lebesgue integrable function then SyC1 y2 ...;x1 x2 ... vanishes if one of the
|xi | and/or |yi | is large. This is, however, too strong a condition because
SyC1 y2 ...;x1 x2 ... should not vanish if all xi and yi are simultaneously shifted by
the same vector a (no matter how large |a| is). By translational invariance
SyC1 y2 ...;x1 x2 ... should depend only on differences of the positions of initial and
final state particles. This means that SpC1 p2 ...;q1 q2 ... has to be proportional to
the single delta function (therefore it cannot be Lebesgue integrable) expressing the conservation of the total 3-momentum (and, by Lorentz covariance,
also to the delta function expressing the conservation of the total energy):
SpC1 p2 ...;q1 q2 ... = δ (3) (p1 + p2 + . . . − q1 − q2 − . . .)
×δ(Ep1 + Ep2 + . . . − Eq1 − Eq2 −
(7.159)
. . .)S̃pC1 p2 ...;q1 q2 ... .
The function S̃pC1 p2 ...;q1 q2 ... cannot then contain any additional delta functions.
If S̃pC1 p2 ...;q1 q2 ... was an analytic function of the momenta, SyC1 y2 ...;x1 x2 ... would
vanish exponentially fast with growing differences of the particle positions.
We can however, allow also for poles (they appear in theories with massless
particles) in S̃pC1 p2 ...;q1 q2 ... which leads to power-like fall-off of SyC1 y2 ...,x1 x2 ... with
growing differences of particle positions (such a fall-off signals the presence
of long-range forces).
The final question is what interactions Vint lead to S matrices satisfying
the cluster decomposition principle. The answer is simple if Vint is built
out of the creation and annihilation operators of the eigenstates |α0 i of a
free-particle Hamiltonian H0 . The cluster decomposition principle is then
satisfied if the coefficient functions hM N (p1 , . . . , pM ; q1 , . . . , qN ) of various
terms of the interaction
Z
Z
Z
∞ X
∞ Z
X
dΓp1 . . . dΓpM dΓq1 . . . dΓqN
Vint =
(7.160)
M =0 N =0
hM N (p1 , p2 , . . . ; q1 , q2 , . . .)a† (p1 ) . . . a† (pM )a(q1 ) . . . a(qN ) ,
contain only a single overall delta function
hM N (p1 , . . . , pM ; q1 , . . . , qN ) = δ (3) (p1 + . . . + pM − q1 − . . . − qN )
×h̃M N (p1 , . . . , pM ; q1 , . . . , qN ) .
(7.161)
This is automatically ensured when the quantum field theory Hamiltonians
are built from field operators (to be introduced in the next Section).
204
Appendix C
Potential scattering
In this Appendix we briefly recall standard scattering theory based on nonrelativistic Quantum Mechanics of a single (for simplicity spinless) particle
moving in an external spherically symmetric potential V (r) (playing the role
of the interaction operator Vint ) on which some restrictions are usually imposed.
The Hilbert space H is in this case L2 (R3 ) - the space of all square
Lebesque square-integrable functions. H0 = P2 /2m has in H no normalizable eigenvectors. In contrast, H = H0 + V (r) may have discrete normalizable eigenstates |φn i. Thus the spectrum of H typically consists of isolated
discrete values En < 0 (we assume V (∞) = 0) and a continuous part starting
from E = 0.
In this case the convergence of the operators Ω(t) = eiHt e−iH0 t can be
proved rigorously (for a class of potentials V (r)). This reduces to showing
that the integral term in the formula (7.10) has a finite limit for t → ±∞
on any normalizable vector |ψi. This is so, if the sequence of Hilbert space
vectors
Z t
|ψt i =
dτ U † (τ )Vint U0 (τ )|ψi ,
0
is a Cauchy sequence for t → ±∞ which in turns is ensured if1
Z ±∞
||
dτ U † (τ )Vint U0 (τ )|ψi || < 0 .
0
†
Since U (t) is unitary this requres that
Z ±∞
dτ ||Vint U0 (τ )|ψi|| < 0 .
0
Since the time evolution governed by H0 ultimately drives any localized (normalizable) wave packet (the vector |ψi) outside the domain in which V (r)
acts, the condition is (for appropriate potentials) satisfied.
A priori vectors of the form Ω± |ψi span two subspaces H± of the Hilbert
space H. One can easily show that normalizable eigenstates of H are orthogonal to all vectors of this form. Indeed, let |φn i be such a vector corresponding
to En and |ψ± i = Ω± |ψi. Then
hφn |ψ± i = hφn |U † (t)U(t)|ψ± i = eiEn t hφn |U(t)|ψ± i .
1
This is because the usual rule “tails contribute nothing to convergent integrals whose
domains extend to ±∞” applied to this integral turns out to be just the condition for the
sequence of vectors being a Cauchy sequence.
870
As this holds for any time, t can be pushed to ±∞ in which limit U(t)|ψ± i can
be replaced by U0 (t)|ψi. This vector represents a wave packet moving freely,
which ultimately, for t → ±∞ leaves the region in which |φn i is localized
and the scalar product must be zero. Furthermore, using a reasoning similar
to the one applied above, one shows that the operator Ω† (t) = eiH0 t e−iHt
does not have the t → ±∞ limits when acting on normalizable vectors |φn i.
Thus, while the operator limits limt→∓∞ Ω(t) do exist, if H has normalizable
eigenstates (bound states) |φn i the limits limt→∓∞ Ω† (t) do not. Notice that
this is not in conflict with the fact that Ω† (t) satisfy Ω(t)Ω† (t) = 1 for any
finite t, because the existence of the operator limits of Ω(t) does not imply
the same for Ω† (t); the operators Ω†± (defined below) are then not limits of
Ω† (t).
An important assumption (whose validity can be rigorously established
for some classes of potentials V (r)) is the one about the asymptotic completeness. It states that H+ = H− = Hscatt and that H = Hscatt ⊕ Hbound .
Defining the action of Ω†± on vectors φn ∈ Hbound one must apply the rule
(section 4.1)
(ψ|Ω†± φn ) = (Ω± ψ|φn ) = (ψ± |φn ) = 0 .
Since ψ is an arbitrary vector of H, the operators Ω†± necessarily annihilate
the whole subspace Hbound (on Hscatt they act as operators inverse to Ω± :
Ω†± ψ± = ψ). Thus while Ω†± Ω± = 1,
Ω± Ω†± = 1 − Πbound ,
(C.1)
where Πbound is the projector onto Hbound . The proof (7.20) that S0 H0 =
H0 S0 goes however unmodified:
Ω†− H Ω+ = Ω†− H (Ω− Ω†− + Πbound ) Ω+ = Ω†− H Ω− Ω†− Ω+ ,
(C.2)
because Πbound Ω+ = 0.
The states |α0 i are in the case considered here simply the states |pi, that
is the plane waves ψp (x) = eip·x = hx|pi, and the formula (7.23) takes the
form
Sp′ ,p ≡ hp′ |S0 |pi = (2π)3 δ (3) (p′ − p) − 2πi δ(Ep′ − Ep ) t(p′ , p) .
(C.3)
The usual scattering amplitude f (p′ , p) ≡ f (θ) (θ is the angle between p′
and p) is related to t(p′ , p) by
f (p′ , p) = −
m
t(p′ , p) .
2π
871
(C.4)
This can be justified by analysing scattering of localized wave packets peaked
around a well defined momentum p - one finds that the differential elastic
scattering cross section is just given by σ(θ) = |f (θ)|2 .
The formula (7.36) with the definition (C.4) gives the standard Born
approximation
Z
m ′
m
′
f (p , p) = − hp |Vint |pi = −
d3 x eiq·x V (|x|) ,
(C.5)
2π
2π
with q ≡ p′ − p.
The in and out states |α± i, denoted |p± i in the case considered here,
satisfy the equation (7.49) which, when written in the standard position
representation, reads
Z
ip·x
hx|p± i ≡ ψp± (x) = e
+ d3 y hx|G0 (Ep + i0)|yi V (|y|) hy|p±i . (C.6)
The matrix element hx|G0 (Ep + i0)|yi can be given explicitly:
Z
Z 3
im ∞
p eip|x−y|
d p eip·(x−y)
=
dp
.
hx|G0 (z)|yi =
(2π)3 z − Ep
2π −∞ p2 − 2mz
The remaining integral
√ can be computed2 by the residue method. It has two
simple poles at p = 2m w± , where w± = z and Imw+ > 0, Imw− < 0.
According to the Jordan lemma, the integration contour must be closed with
a large semicircle in the upper half plane. This gives
√
m exp(i 2m w+ |x − y|)
hx|G0 (z)|yi = −
.
2π
|x − y|
To find the asymptotic form of ψp± (x) for |x| → ∞, when V (|y|) vanishes
sufficiently fast for |y| → ∞, it is sufficient to approximate
|x − y| = r (1 − n·y) + . . . ,
√
where r ≡ |x| and n ≡ x/|x| = x/r. For z = Ep ± i0, so that 2m w+ =
±|p| + i0 The formula (C.6) takes then the form
Z
e±i|p|r
m
ip·x
3
∓in|p|·y
ψp± (x) ≈ e
+
−
d ye
V (|y|) ψp± (y) .
r
2π
The factor in the bracket is just −m/2π times the matrix element
h±p′ |Vint |p± i ≡ t(±p′ , p) ,
where p′ ≡ n|p| which, upon using the definition (C.4) allows to identify
ψp± (x) (playing here the role of the the |α± i states) as the (generalized)
872
eigenfunctions of H commonly used in the ordinary stationary scattering
theory
f (p′ , p) +i|p|r
e
,
r
f (−p′ , p) −i|p|r
e
,
ψp− (x) ≈ eip·x +
r
ψp+ (x) ≈ eip·x +
and representing asymptotically ψp+ (ψp− ) the incoming plane wave and the
outgoing (incoming) spherical wave.The convergence (7.15) of wave packets
(well localized in space) built on the in or out states to the wave packets
built with the same profile g(k) on the plane waves takes here the form
Z 3
Z 3
dp
dp
−iHt
−iH0 t
e
g(p)
ψ
(x)
→
e
g(p) ψp (x) .
p
±
(2π)3
(2π)3
Heuristically, it can be justified by appealing to the fact that for t → ∓∞
the localized wave packets formed from ψp± (x) are driven far from x = 0,
into the region in which ψp± (x) effectively do not differ much from the plane
waves ψp (x) ≡ eip·x .
873