QUANTUM FIELD THEORY in VS .NET

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186 QUANTUM FIELD THEORY
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classical equivalent-this study should also give us some insight into the role of the Dirac equation for the quantized field l/J:
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[i - eJ(x) - m] l/J(x) = 0
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(4-77)
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Here AIl(X) is a given c number external field. This equation corresponds to the interaction lagrangian
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(4-78)
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All the steps leading to the interaction representation and to the S matrix in Sec. 4-1-4 may be repeated here. This results in the expression
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S = Texp [ - ie
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d4 x i/Jin(x)ylll/Jin(x)AIl(x) ]
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(4-79)
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But now, it is not trivial to put this expression into a normal form. This is because the interaction lagrangian is quadratic in the fields, rather than linear, and therefore the commutator of two currents i/Jylll/J is not a c number. Notice also that we do not know the general solution of the Dirac equation (4-77) in an arbitrary field A!l' In spite of these difficulties, we have all the machinery of the Wicks theorem at our disposal, and we can find a formal solution of the problem. Let us concentrate on the probability amplitude of emitting no pair:
So (A) =
<0 inl S 10 in)
00 (-iet n~o -----;;!
dXl'" dXn
<01 T[l/J(Xl)J(Xl)l/J(xd"
. l/J(Xn)J(xn)l/J(xn)]
0 1)
(4-80)
where "in" subscripts have been omitted. According to the Wick theorem, each term in (4-80) is a sum of products of contractions of the form
<01 T J(xd l/J(Xk) i/J(XI)
C(ab Xk; al. Xl)
For Xk and Xl given, let us introduce the 4 x 4 matrix
Ia <01 T[J"",.(Xk)l/Ja(Xk)i/Ja,(XI)] 10)
(4-81)
In terms of C, So(A) reads
So(A)
= ~ ---;;y- dXl'" dXn ~ 8p at. ~ ,an C(ab Xl; apI' Xp)'" C(am Xn; aPn' XP.)
(-1t
(4-82) It is convenient to consider the discrete indices ai and the continuous variables Xi on the same footing and to group them in a bracket notation lx, a). The space of these vectors lx, a) is nothing but the space of classical spinors. We
introduce the matrix r of generic element
<X, al r Iy, {3) =
C(x, a; y, {3)
(4-83)
INTERACTION WITH AN EXTERNAL FIELD
Using this matrix notation, we may write
So (A)
= Det (J - r) = exp [Tr In (I - r)]
(4-84)
These expressions are indeed true for any finite-dimensional matrix r: det (I - r)
exp [tr In (I - r)]
+ -- I
( _1)2
(raarbb - rabrba)
+ ...
(4-85)
permutations of(I, ... ,n)
as a result of the Cayley-Hamilton formula for a determinant. This expansion stops at the dth order for a finite d-dimensional matrix. The term of order d is, of course, (_I)d det r. This formula expresses the determinant as a sum of traces of antisymmetrized tensor products. There exists a similar formula for the inverse determinant
det- 1(I - r)
exp [ -tr In(I - r)]
1 + I raa
(raarbb
+ rabrba) + ...
(4-86)
permutations of(1, ... ,n)
that is, as a sum of traces of symmetrized tensor products. This latter expression would be relevant for the problem of quantized boson fields coupled to an external field.
These expressions extend to the case of infinite matrices in the framework of the Fredholm theory, under proper assumptions on the behavior of 1. We use capital letters for Det and Tr to recaII that these operations imply integrations over continuous variables. The use of normal products in (4-78) would amount to the assumption that laa = 0, and therefore to dropping the term L 1aa in (4-85). As in the last section of Chap. 2, it is convenient to introduce operators XJ1. and PJ1.' defined on the states lx, a): XJ1.IX, a) = XJ1.IX, a) <x, al pJ1.lo/) = i oxJ1. <x, alo/) They enjoy the canonical commutation relation [XJ1.,Pv]
= -i9J1.v
(4-87)
and eigenvectors of PJ1.' denoted Ip), are such that
e ip . x <pix) = <xlp)* = (2n)2
Using these matrix notations, together with the expression (3-174) of the Dirac
QUANTUM FIELD THEORY
propagator, we find
eJ(x) I'
1 . -m+/e
So(A) = Det [1 - eJ(x) I' -m+/e 1 .]
=Det{Cf-eJ(x)-m+ie]f 1 .} -m+le
(4-88)
exp { - Tr In
[(I' - m) I' _ eJ(x{ _ m + ie]}
The last expression exhibits the Feynman propagator in the external fi.eld AJl(x).
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