It is a good exercise to check gauge invariance. Under a gauge transformation in Visual Studio .NET

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It is a good exercise to check gauge invariance. Under a gauge transformation
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A.(x) -. A.(x) + iJ.<I>(x)
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f The relations (4-87) imply that
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eJ(x) - m -.
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eJ(x) - e~<I>(x) - m
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e-ie<l>(x)[f - eJ(x) - m] eie<l>(x) =
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eJ(x) - e~4>(x) - m
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Therefore a gauge transformation amounts to a unitary transformation in the space of the classical spinors. It does not affect the determinants and leaves So (A) invariant.
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Let us examine the consequences of the unitarity of the S matrix. It is convenient to introduce a one-body scattering operator ff(A) defined by
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ff(A) = eJ(x)
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+ eJ(x) I' -m+/e ff(A) .
(4-89)
We assume the potential AJl(x) to be real; for every operator B on the space of vectors lx, a), we set jj == yO BtyO where the hermitian conjugation acts on the indices a and x. This operation changes the sign of ie in ff(A):
ff(A)
= eJ(x) + eJ(x) I' _ m _ ie ff(A) = eJ(x) + ff(J) I' _ m _ ie eJ(x)
(4-90)
Thus
eJ(x) = ff(A) - ff(A)
I' _ m _
ie eJ(x)
The insertion of this expression into (4-89) leads to
ff(A) = eJ(x) + [ ff(A) ff(A)
1 . eJ(x) ] -m-/e
1 . ff(A) -m+/e
INTERACTION WITH AN EXTERNAL FIELD 189
eJ(x)
+ ff(A) I'
1 . ff(A) - ff(A) -m+ze
-m-ze ff(A)
I' -m-ze [ff(A) .
eJ(x)]
= ff(A) I'
-[ 1 . - I' 1] .
-m+ze ff(A) - ff(A)
1 + eJ(x) + ff(A) ~-m-ze eJ(x) 11> /
ff(A{f_!+ ie
f_!_ie]ff(A)+ff(A)
Thus
ff(A)
1 . -m+ze 1 _ m + ie
1 )ff(A) f-m-ie
= ff(A) (
I' -
I' -
1 )ff(A) m - ie
m 2)] ff(A)
(4-91)
ff(A) [2n i
(I' + m)i5(p2 -
The operator 2n(f + m)i5(p2 - m 2) may be decomposed into a sum of projectors over positive and negative energy states
2n(f
+ m)i5(p2 -
m 2) =
p(+)
+ p(-)
+ m)e( pO)i5(p2 m 2)
(4-92)
p( )(P) == 2n(f
These operators which project on the mass shell will emerge naturally in the computation of physical quantities, such as ISo(AW. If we return to the original expression (4-82) of So(A), every term in the right-hand side may be decomposed into cycles:
C(lXb Xl; 1X2, X2)C(1X2, X2; 1X3, X3) C(lXb
IXb Xl)
Had we used the retarded propagator instead of the Feynman one, all these cyclic terms would vanish, since the retarded Green function has its support inside the future cone and thus a cyclic term has an empty support. As noted in Chap. 2, the retarded propagator has the form
f+m (P
+ ie)2 -
I' -
1 m + i
where ell is an infinitesimal positive time-like four-vector. We conclude that
(4-93)
QUANTUM FIELD THEORY
where PP means principal part. Therefore,
f-m+ie
1 = Det
- -1 - = -ip(-) -
m + i
Det Det Det
= Det
[I [I {I {[I [I =
eJ(x) eJ(x)
f _ ~ + i ] f f
-m+ Ie
1 1 . - ieJ(X)p(-)]
eJ(x)
-m+le
. - i
eJ(x)
f _ ~ + ie
-m+w
[I ][1 -
eJ(x)
-m+w
. ]Y(A)P(-)}
iY(A)P(-)]}
eJ(x)
. ]Det [I - iY(A)p(-)]
(4-94)
From these expressions and from (4-91), it follows that
[So(A)r 1
Det [I - iY(A)p(-)]
ISo(A)I- Z = Det {[I
iY(A)p(-)] [I
+ i (A)p(-)]}
+ Y(A)p(-) (A)p(-)}
(4-95)
Det {I
+ i[ (A) -
Y(A)]p(-)
= Det [I - Y(A)p(+) (A)p(-)]
exp {Tr In [I - Y(A)p(+) (A)p(-)]}
We write (4-96) with
W(x) = tr
<xl In [I -
Y(A)p(+) (A)p(-)] Ix>
(4-97)
(where the trace now refers only to the Dirac indices), and we interpret w(x) as a probability density for pair creation. If we divide the four-dimensional space into small cells of volume LlXi about the points Xi, we may approximate
ISo(AW ~ 11 [1 i
LlXiW(Xi)]
which confirms the previous interpretation. Each cell gives an independent contribution W(Xi) to the emission probability. This may also be seen by an explicit
INTERACTION WITH AN EXTERNAL FIELD
computation of the probability of emission of one, two, etc., pairs. Because of the presence of the projectors p( ), the probability w(x) d4 x is expressed in terms of on-shell matrix elements of :!7 between -a particle and a hole state. This is made clearer by the introduction of a reduced operator t in the following way. Using the normalized positive and negative energy solutions da)(P), v(a)(p), introduced in Chap. 2, we define the matrix t on the mass shell by
<pa 1 1 t p'b
>= 2n L wp w p' ii~a)(p) <p, rL 1 p', 13> v}f')( 1/': l/Z 3"1
a,{J
(4-98)
with wp == (pZ
+ mZ)l/Z. From the expressions.of the projectors
a= 1,Z
u(a)(p)ii(a)(p) =
P+ m
for pO> 0 for po < 0
a=l,Z
p+m v(a)( - p)v(a)( - p) = - - 2m
we then find
Wtot =
d4 x w(x) =
d 3 p tr In (1
+ ttt)
(4-99)
This exhibits clearly the positivity of Wtot We shall now compute explicitly this probability in two special instances.
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