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Set r. = (r')- 1 and observe that yO(P)!yO = P in such a way that the forms IjIpl/t are hermitian. With these notations Tr
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r'rp =
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Any 4 x 4 matrix X has an expansion
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By identifying the coefficient of Xab we find the following identity where Latin indices run from 1 to 4:
(3-207)
Applying this to
r'rp we have
(3-208)
Writing r. = e.r where e. = 1, it is readily seen that P'Py = e.p py is invariant by cyclic permutation and that given the values of two of its indices it is non vanishing for only one value of the third. The only nonvanishing terms are
p'ps
p'sp
b' p
PV,A,P = -PA,V,P = PT"A.A p
-ig.,
P A.T"Ap
= i(g.qg,p - g.pg,q)
/.lVPU
(3-209)
PTp.yPTp<T = PPTp.vTp(T = PT"Tp.T"
= i[g.p(g"g,q - g"g,q)
+ g.,(g,qgp, Consider now the five Lorentz scalars
+ g.q(g"gp, - g"g,p) g,pgq,) + g.,(g,qg,p - gMg",)]
s = u(4)u(2)u(3)u(l)
v = u(4)y u(2)u(3)y.u(l)
t = iu(4)a 'u(2)u(3)a.,u(l)
a = u(4)y'y"u(2)u(3)y.y5 u(l)
(3-210)
P = u(4)y5 u(2)u(3)y5 u(l)
Here u(l) denotes U(Ph all, the spinor with momentum P! and polarization a!. We shall use the symbol s(4,2; 3, 1) to characterize the way in which the Dirac indices are contracted, and similarly for the other amplitudes. Anyone of these quantities may be written as
b(4, 2; 3, 1) = ua3(3)iia4(4)r~4a,r.,a3alua,(l)ua,(2)
How are these quantities related to the corresponding ones where 4 is contracted with 1 and 3 with 2 The theorem states that there exists a numerical 5 >< 5 matrix F relating the two sets of quantities. This matrix will then necessarily be equal to its inverse. Before proceeding, let us remark that we could ask a similar question in x space with operators of the type :1jI4(X)1/t2(X)1jI3(X)I/It(X):. The
QUANTUM FIELD THEORY
relevant matrix would then be - F due to Fermi field anticommutation. In any of the five quantities b(4, I) we can use Eq. to rewrite
2; 3,
(3-207)
ua,(2)ua,(I) = ,)a,u,,)a,,,,u,,,(2)u,,/I) = !rp.a2",r~,,,,u,,,(2)u,,,(I)
Thus, with the help of
(3-208),
b(4, 2; 3, I)
!ua4(4)ua3(3)r~4a2rp.a2",r . a3a,r~'''2U''2(2)ua,(I)
!p'pyp/ou(4)Pu(l)u(3)rou(2)
It remains to use
(3-209) to obtain the desired result:
( ~)(4'2;3'1)=![:
-: -: -2 : 2 0
-:J(~)(4'1;3'2)
pi-I
-lip
The reader may check that F2 = 1 and diagonalize the matrix. What is the behavior of the various amplitudes under the discrete symmetries
NOTES
The physical meaning of the quantum commutation relations in electrodynamics is analyzed in N. Bohr and L. Rosenfeld, Kg/. Danske Videnskab. Selsk. Mat.-Fys. Medd., vol. 12, p. 8, 1933, and Phys. Rev., vol. 78, p. 794, 1950. For limits on the photon mass see A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys., vol. 43, p. 277, 1971. Macroscopic effects of vacuum fluctuations were considered by H. B. G. Casimir, Proc. Kon. Ned. Akad. Wetenschap., ser. B, vol. 51, p. 793, 1948. See also M. Fierz, Helv. Phys. Acta, vol. 33, p. 855, 1960; T. M. Boyer, Annals of Physics (New York), vol. 56, p. 474, 1970; and R. Balian and B. Duplantier, Annals of Physics (New York), vol. 112, p. 165, 1978. Experimental evidence is discussed by M. J. Sparnaay, Physica, vol. 24, p. 751, 1958. A study on Van der Waals forces is found in the work of G. Feinberg and J. Sucher, Phys. Rev., ser. A, vol. 2, p. 2395, 1970. Quantum field theory at finite temperature is presented, for instance, in L. Dolan and R. Jackiw, Phys. Rev., ser. D, vol. 9, p. 3320, 1974. For a general axiomatic formulation offield theory and a derivation of some fundamental properties, see R. F. Streater and A. S. Wightman, "peT, Spin and Statistics, and All That," Benjamin, New York, 1964, and R. Jost, "The General Theory of Quantized Fields," AMS, Providence, R.I., 1965.
CHAPTER
FOUR
INTERACTION WITH AN EXTERNAL FIELD
The interaction with an external field yields a simple example of a dynamical system. We introduce the important concepts of interaction representation and Wick's identities. Applications include the radiation of a classical source and the infrared catastrophe, the physical counterpart of which in the fermionic case is the process of pair creation by a c-number electromagnetic field.
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