Berezin integral

In mathematical physics, a 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 for some analogue properties and since it is used in physics in a parallel manner to ordinary integration. The technique was invented by the physicist David John Candlin in 1956 [cite journal|journal= Nuovo Cimento | author= D.J. Candlin | volume =4 | title = On Sums over Trajactories for Systems With Fermi Statistics|year=1956|pages=224|doi= 10.1007/BF02745446] , but it is named after the Russian mathematician Felix Berezin, who included a treatment in his textbook [A. Berezin, "The Method of Second Quantization", Academic Press, (1966)] .

Definition

The "Berezin integral" is defined to be a linear functional $int\; af(\; heta)+bg(\; heta)\; d\; heta\; =\; aint\; f(\; heta)d\; heta\; +bint\; g(\; heta)d\; heta$

fulfilling the partial integration rule$int\; fracpartial\{partial\; heta\}f(\; heta)d\; heta\; =\; 0\; ;.$

These properties define the integral uniquely up to a multiplicative constant which we can set to 1 and translate into the rule

*$int\; (a\; heta+b)\; d\; heta\; =\; a\; ;.$

This is the most general function, because every homogeneous function of one Grassmann variable is either constant or linear.

Multiple Variables

Integration over multiple variables is defined by Fubini's theorem:$int\; f\_1(\; heta\_1)ldots\; f\_n(\; heta\_n)\; d\; heta\_1ldots\; d\; heta\_n\; =\; int\; f\_1(\; heta\_1)d\; heta\_1ldots\; int\; f\_n(\; heta\_n)\; d\; heta\_n\; ;.$

Note that the sign of the result depends on the order of integration.

Suppose now we want to do a substitution:$heta\_i=\; heta\_i(xi\_j)$where as usual (ξ_j) implies dependence on all ξ_j. Moreover the function θ_i has to be an odd function, i.e. contains an odd number of ξ_j in each summand. The Jacobian is the usual matrix$J\_\{ij\}=frac\{partial\; heta\_i\}\{partialxi\_j\}\; ;.$

the substitution formula now reads as$int\; f(\; heta\_i)d\; heta\; =\; int\; f(\; heta\_i(xi\_j))\; operatorname\{det\}(J\_\{ij\})^\{-1\}\; dxi\; ;.$

ubstitution formula

Consider now a mixture of even and odd variables, i.e. x_a and θ_i. Again we assume a coordinate transformation as $x\_a=x\_a(y\_b,\; heta\_j),,;\; xi\_i=xi\_i(y\_b,\; heta\_j);,$ where x_a are even functions and θ_i are odd functions. We assume the functions x_a and θ_i to be defined on an open set U in R^m. The functions x_a map onto the open set U' in R^m.

The change of the integral will depend on the Jacobian . This matrix consists of 4 blocks . A and D are even functions due to the derivation properties, B and C are odd functions. A matrix of this block structure is called even matrix.

The transformation factor itself depends on the oriented Berezinian of the Jacobian. This is defined as:

For further details see the article about the Berezinian.

The complete formula now reads as:$int\_U\; f(x\_a,\; heta\_i)\; d(x,\; heta)\; =int\_\{U\text{'}\}\; f(x\_a,\; heta\_i)\; operatorname\{Ber\}\_\{+-\},\; frac\{partial(x\_a,\; heta\_i)\}\{partial(y\_b,xi\_j)\}\; d(y,xi)$

Literature

* Theodore Voronov: "Geometric integration theory on Supermanifolds", Harwood Academic Publisher, ISBN 3-7186-5199-8

References