where it is convenient to introduce an exponential in the last step because of the property ex ev = ex+v. Note the form of (6|, which is dictated by the requirement mi = m.
The form of (14.133) suggests defining definite integrals of Grassmann numbers over the interval (-oo, oo) so that the resolution of unity assumes the familiar form
where, by convention, the factors of 27ri are omitted for fermions. First, consider the implications of this equation for the case of only one frequency. We have
1 = J db* dbi e-b'bi (l - 6<Bj) |0)(0| (1 - Bib*) . (14.135)
Expanding both sides of this equation gives
= Jdb'dbiil- 6*6,) ||0)(0|-6i|li>(0|-|0)(li|6*+ 64|li)<li|6;} = Jdb* d6j|(l - 6*6j) |0)(0| - 6i|lj)(0| — |0)(li|6J' + 6i|li)(li|6* J .
The equality of the first and last lines implies that
We leave it as an exercise to show that these conditions are also sufficient to prove (14.134).
Note that these conditions also preserve the translational invariance of Grassmann integrals, which is essential for the reduction of path integrals. Translational invariance implies that
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