Berezin integration
In mathematical physics, a Grassmann integral, or, more correctly, Berezin integral, is a way to define integration for functions of Grassmann variables. It is not an integral in the Lebesgue sense; it is called integration because it has analogous properties and since it is used in physics as a sum over histories for fermions, an extension of the path integral. The technique was invented by the Russian mathematician Felix Berezin and developed in his textbook. Some earlier insights were made by the physicist David John Candlin in 1956.
The Berezin integral is defined to be a linear functional
where we define
so that :
These properties define the integral uniquely.
This is the most general function, because every homogeneous function of one Grassmann variable is either constant or linear.
Integration over multiple variables is defined by Fubini's theorem:
Note that the sign of the result depends on the order of integration.
Suppose now we want to do a substitution:
where as usual (ξj) implies dependence on all ξj. Moreover, the function θi has to be an odd function, i.e. contains an odd number of ξj in each summand. The Jacobian is the usual matrix
the substitution formula now reads as
Consider now a mixture of even and odd variables, i.e. xa and θi. Again we assume a coordinate transformation as where xa are even functions and θi are odd functions. We assume the functions xa and θi to be defined on an open set U in Rm. The functions xa map onto the open set U' in Rm.
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