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QUANTUM FIELD THEORY
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ljI(x) -+ 1jI'(x) = S(A)IjI(A -lX) = IjI(X)
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(3-154)
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The operator r P= ia ap + i(xao P - xpo a) acts on the spinor indices as well as the configuration variables. Correspondingly,
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If bw ap is x independent we verify that the action is invariant. Thus when bwaP varies with x we find as a variation around the stationary point
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(jJ =
d4xil]l[{yJl, a;P} + i(xaa p - xpaa)y/l JIjI O/lb;a
) d4x (O/ljll a P b~ap = 0
A candidate for the generalized angular momentum tensor density is therefore
(3-155)
We can verify directly that O/lJ/l ap = O. This tensor has an orbital and a spin part. The above construction gives a similar role to IjI and 1]1. The space integrals of 0 /l and JO.ap will yield the quantum generators of the group when we find a commutation prescription for the fields. We can also add that the in variance under phase transformations IjI -+ eialjl, 1]1 -+ e-ial]l leads to a conserved current
(3-156)
which was used in the one-particle theory of Chap. 2 to normalize the states
QUANTIZATION-FREE FIELDS
when integrating JO. It will be necessary to give a more general interpretation in the quantized picture. To implement our program let us now expand the operators I/I(x) and l/I(x) in terms of the c-number plane wave solutions of the Dirac equation, with operator-valued amplitudes b, bt, d, and dt : I/I(x) = l/I(x) =
d \ m I [b,,(k)U<")(k) e- ik . x (2n) ko ,,= 1,2
+ dt(k)v(")(k) eik . X]
(3-157)
d \ ~ I (2n) ko ,,= 1,2
[b!(k)ii(")(k)e ik . x + dAk)v(")(k) e- ik . X]
The spinors u and v were given in the previous chapter, Eqs. (2-37) and (2-38). The operators band d must satisfy commutation rules such that I/I(x + a) = eiP'al/l(x) e- iP ' a or in differential form Ol'l/I(x) = i[P 1" l/I(x)] We express PI' using the decomposition (3-157) with the result that PI' (3-158)
d3 x
I' -
d k m '" t( t (2n)3 k O kl' ~ [ b" k)bik) - d,,(k)d,,(k) ]
(3-159)
from which we have to subtract the vacuum contribution. We read from (3-159) that if the vacuum is defined in such a way that b,,(k)/O) = d,,(k)/O) = 0, and if we were to quantize according to commutators, b particles and d particles would contribute with opposite signs to the energy. The theory would not admit a stable ground state. From (3-158) translational invariance is satisfied provided [PI" b,,(k)] = - kl'b,,(k) [PI" bJ(k)] = kl'b!(k) [PI',d,,(k)] = -kl'd,,(k) [PI" d!(k)] = kl'dJ(k) (3-160)
Replacing PI' by its expansion, this is equivalent to
[~[bl(q)bp(q) -
dp(q)dl(q)], b,,(k)] = -(2n)3 ': c5 3 (k - q)bik)
and three analogous relations. If we assume that [dl(q)dp(q), b,,(k)] = 0 this condition can be written
kO [bl(q){bp(q), bik)} - {bl(q), b,,(k)}bp(q)] = -(2n)3 - c5 3 (k - q)bik)
Therefore, we can insure the correct interpretation of energy and momentum using, as an alternative, anticommutators (the above curly brackets) instead of
QUANTUM FIELD THEORY
commutators between the fundamental creation and annihilation operators b t, dt, b, and d. This solves at once the stability question mentioned above since, after vacuum energy subtraction, we are left with only positive contributions to the energy. We set, therefore,
{b,,(k), b;(q)} = (21ll kO c5 3 (k - q)c5"p m
(3-161)
and alI other anticommutators vanish. As a consequence, {l/J ~(t, x), l/J~(t, y)} = c5 3 (x - y)c5~~
(3-162)
We may then easily verify that conditions (3-160) are fulfilled, together with the analogous ones stemming from the requirement of covariance under homogeneous transformations and involving the commutators of the generators j"P = d 3x JO,aP with the field. The interpretation of (3-160) is that bt(k) and dt(k) create [b(k) and d(k) destroy] a four-momentum k".
Wick products can be generalized to Fermi fields. When reordering the creation operators to the left of the annihilation ones, a sign corresponding to the parity ofthe permutation must be introduced. The correct definition of total energy momentum is therefore
f f(~:~3 ;
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