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The above proof ignores the difficulties associated with the construction of an interacting field theory. The latter has to give a precise meaning to otherwise undefined products of operators. Nevertheless, the algebraic properties implied in the peT theorem are preserved by this operation.
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As an application we discuss form factors, a relativistic generalization of charge distributions, Consider first the matrix elements of the current V"(x) of a free Dirac theory between two oneparticle states
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PI, V"(x) Ip,a) =
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The greek indices a,
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13, .. ,,15 denote polarizations, Ip, a) =
b!(p) 10), and
v"(0) = : IiJ(O)y"I/I(O): = ffdkl dk2
L iiY)(kdy"do)(k2)b;(kl)bo(k2) + ",
We have not explicitly given the terms with vanishing contribution to the matrix element. From
t ,3 <p,p b,(k l )b o(k 2) p, a) = (2n) 6 k kg 15 3(k l - p)i5 (k2 - p)i5 p,i5 .. -2
it follows that
This is in agreement with the definition of the charge
<p',
PI Q Ip, a) =
f d 3x <p', PI VO(O, x) Ip, a) f d 3x iiP)(p')yOda)(p)
ei(p'-p)'x
(2n)3i5 3(p - p')u(P)(p')yOda)(p) = <p',p p, a)
Recall that up(p')yOua(p) is normalized to i5 ap (pOjm), Assume now that j"(x) is an hermitian four-vector current density with the same transformation properties under discrete symmetries as V"(x), and let us write the general structure of its one-particle matrix element
<p', Plj"(x)
Ip, a) =
e -i(p- p'), x
iiP>(p')O"(p', p)u(a)(p)
In the sequel we always understand that the numerical matrix O"(p', p) is sandwiched between the two spinors u and u, From Lorentz covariance O"(p', p) must obey
S(A)O"(P', p)S-I(A)
(A -1)"vO'(Ap', Ap)
yOO"(p', p)!yO = O"(p', p)
from hermiticity, We can take into account the fact that the spinors satisfy the free Dirac equation by replacing any O"(p', p) by
O"(p', p) --> ~ O"(p', p)
p'+m
2;;-
In particular this entails the Gordon identity 1 y" --> 2m [(p'
+ p)" + i<1"V(p' -
p),]
f!60
QUANTUM FIELD THEORY
We also require the current j"(x) to be conserved
0= <p', PI o"j"(x) Ip,
a> =
i(p' - p)/p',Plj"(x) Ip,
Thus (p' - p)"O"(p', p) vanishes. Let us note q, the space-like difference p' - p; since p2 = p'2 = m2 the only scalar invariant is q2. Allowing for the most general decomposition of 0" on the basis of the 16 Dirac matrices, and using Lorentz invariance, conservation, and hermiticity, we find
(3-203)
with Fi(q2) being real form factors-functions of the square momentum transfer q2. Quantizing spin along a fixed axis we derive from Eqs. (3-178) and (3-199) that symmetry under parity implies
<p', PU"(x) Ip,
<p', PIMx) Ip,
Since u(a)(p) = yOu(a)(p) this means O"(p', p) = yOO"(fj', p)yO, the consequence of which is F 3(q2) = -F3(q2). Thus parity conservation alone yields F3 = O. However, it could happen that parity is violated by some interaction (in practice, weak interactions). It was long believed that parity times charge conjugation remains a symmetry of all interactions. From the PCT theorem this meant that T was also a symmetry. This imposes Ft(q2) = EiFi(q2) with E, = E2 = - E3 = 1. From hermiticity we know that the F functions are reaL Therefore again F3 = O. It is known, however, that T is not a symmetry of all interactions, since the discovery of its violation by Fitch and Cronin in the neutral K meson decays. As an example let us assume that j"(x) is the electromagnetic current of the interacting theory. By considering two normalized states close to rest, the reader will have no difficulty in showing that the following correspondence holds:
eFt{O) = charge Q [Fl(O) 2m
+ F2(0)]
magnetic moment J.I.
(nonrelativistic magnetic coupling - J.l.lT B)
- F3(0) = electric dipole moment d 2m
(nonrelativistic electric coupling - dlT E).
Extensive measurements of nucleon-form factors have been performed using electron scattering off hydrogen and deuterium targets. For this case Fl and F2 arc now known in a wide range of q>. However, up to now F 3 has not shown up. In particular, bounds for the static electric dipole moment of the neutron have been reported (Ramsey) showing d neutron/e ;s 10- 23 cm. The magnitude of this electric dipole moment is a sensitive test of the mechanism of CP violation. For a charged particle, the total gyromagnetic ratio is 2[Fl(0) + F 2(0)]/F 1(0). For the proton, for instance (with the convention e = lei), F , (O) = 1, F 2 (0) = 1.79, and the gyromagnetic ratio is 5.58. In the literature we often find the Sachs combinations GE = Fl + (q2/4m 2)F2 and GM = F, + F2 which are more convenient for displaying the experimental data on scattering. These form factors extend analytically to positive (time-like) values of q2 where they are related to the process photon --+ fermion + antifermion. In the case of scalar (or pseudoscalar) particles the corresponding matrix element of the current has the form
(3-204)
We close this section with an algebraic exercise. It deals with quartic products of Dirac fields organized in covariant scalar densities as they occur for instance in the effective Fermi lagrangian for low-energy weak interactions. Its purpose is to examine the effect of rearranging the order in which these fields are coupled together to obtain a Lorentz scalar, the basic blocks being the quadratic quantities occurring in table (3-199). This is the Fierz theorem. Denote collectively by ra the 16 Dirac matrices
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