XNOR gate

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.

Definition

The Berezin integral is defined to be a linear functional

$\int [af(\theta )+bg(\theta )]d\theta =a\int f(\theta )d\theta +b\int g(\theta )d\theta$

where we define

$\int \theta d\theta =1$

$\int d\theta =0$

so that :

$\int \frac{\partial }{\partial \theta }f(\theta )d\theta =0.$

These properties define the integral uniquely.

$\int (a\theta +b)d\theta =a.$

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:

$\int f_1({\theta }_1)\cdots f_n({\theta }_n)d{\theta }_1\cdots d{\theta }_n=\int f_1({\theta }_1)d{\theta }_1\cdots \int f_n({\theta }_n)d{\theta }_n.$

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

Suppose now we want to do a substitution:

${\theta }_i={\theta }_i({\xi }_j)$

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

$J_{ij}=\frac{\partial {\theta }_i}{\partial {\xi }_j}.$

the substitution formula now reads as

$\int f({\theta }_i)d\theta =\int f({\theta }_i({\xi }_j))\det (J_{ij}{)}^{-1}d\xi .$

Substitution formula

Consider now a mixture of even and odd variables, i.e. x_a and θ_i. Again we assume a coordinate transformation as $x_a=x_a(y_b,{\xi }_j),{\theta }_i={\theta }_i(y_b,{\xi }_j),$ 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

$J_{\alpha \beta }=\frac{\partial (x_a,{\theta }_i)}{\partial (y_b,{\xi }_j)}.$

This matrix consists of four blocks:

$J=\left[\begin{array}{cc}A& B\\ C& D\end{array}\right].$

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:

${\mathrm{Ber}}_{+-}{J}_{\alpha \beta }=\mathrm{sgn}\det A\mathrm{Ber}{J}_{\alpha \beta }.$

For further details see the article about the Berezinian.

The complete formula now reads as:

$\underset{U}{\int }f(x_a,{\theta }_i)d(x,\theta )=\underset{U^{\prime }}{\int }f(x_a,{\theta }_i){\mathrm{Ber}}_{+-}\frac{\partial (x_a,{\theta }_i)}{\partial (y_b,{\xi }_j)}d(y,\xi ).$

Gaussian integrals over Grassmann variables

The following formulas for Gaussian integrals are used often in the path integral formulation of quantum field theory:

$\int \exp \left[{\theta }^{T}A\eta \right]d\theta d\eta =\det A$

with $A$ being a $n\times n$ matrix.

$\int \exp [-{\theta }^{T}M\theta ]d\theta =\{\begin{array}{ll}{2}^{\frac{n}{2}}\sqrt{\det M},& n\text{ even}\\ 0,& n\text{ odd}\end{array}$

with $M$ being an $n\times n$ antisymmetric matrix. In the above formulas the notation $d\theta =d{\theta }_1\cdots d{\theta }_n$ is used.

From the above formulas, other useful formulas follow:

$\int \exp [{\theta }^{T}A\eta +{\theta }^{T}J+K^T\eta ]d\theta d\eta =\det A\exp [-K^T{A}^{-1}J]$

with $A$ being an invertible $n\times n$ matrix, $K$ and $J$ are Grassmann variables.

$\int \exp [-\frac{1}{2}{\theta }^{T}M\theta +{\eta }^T\theta ]d\theta =\{\begin{array}{ll}\sqrt{\det M}\exp [-\frac{1}{2}{\eta }^T{M}^{-1}\eta ],& n\text{ even}\\ 0,& n\text{ odd}\end{array}$

with $M$ being an invertible $n\times n$ antisymmetric matrix, and $\eta$ Grassmann variable.

See also

• Berezin integral

• Supermanifold

Literature

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