J 777711771 772 J dfxdf2 dridr2 1 771771 f2r2 jiiir2i2

= - J <¿771*71 J drji 771 j £¿772772 j <¿772772 = -1. (14.143)

Now, suppose we generalize the quadratic form tjt] to fjAr], which can be written t)AT) = f\iAijj]j . (14.144)

Then, the integral (14.143) generalizes to

J dfjidfj2dT]idri2 e"^ = \ J df\idfj2drndr}2 [fjxAl3 rj-j) [fjkAki ru) = \ J df}idfj2dr]idT]2 m tjj fjk r)t [Ai:j Ake]

= - det A , or, if an i is inserted in the exponent,

where n is the dimension of the matrix A. The result, which can be generalized to arbitrarily large matrices, will be very useful in the following sections. Note that this result for Grassmann integrals is very different from what would have been obtained from ordinary numbers:

Free Dirac Fields

Since we have successfully defined coherent states for Dirac annihilation operators and obtained a form for the resolution of unity which is identical to the one for scalar fields, the rest of the discussion for scalar fields may be carried over directly to Dirac fields. The generator for free Dirac fields is then

where 4>(x), tp(x), 77(1) and fj(x) are all four-component Dirac vectors, which are also infinite-dimensional Grassmann variables, with

We reduce Zo using the same steps carried out for the scalar field <f> in Sec. 14.4. In particular, transform the integrals to momentum space using


d4x (27T)2

Integrating by parts,

and substituting the Fourier transforms, we obtain

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