Berezin integral

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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. It is named after the Russian mathematician Felix Berezin.

1 Definition
2 Multiple Variables
3 Substitution formula
4 Literature

[edit] Definition

The Berezin integral is defined to be a linear functional $\int af(\theta)+bg(\theta) d\theta = a\int f(\theta)d\theta +b\int g(\theta)d\theta$

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

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

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

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

[edit] Multiple Variables

Integration over multiple variables is defined by Fubini's theorem: $\int f_1(\theta_1)\ldots f_n(\theta_n) d\theta_1\ldots d\theta_n = \int f_1(\theta_1)d\theta_1\ldots \int f_n(\theta_n) d\theta_n \;.$

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

Suppose now we want to do a substitution: $θ i = θ i (ξ j)$ 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 $\int f(\theta_i)d\theta = \int f(\theta_i(\xi_j)) \operatorname{det}(J_{ij})^{-1} d\xi \;.$

[edit] 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 $x_a=x_a(y_b,\theta_j)\,,\; \xi_i=\xi_i(y_b,\theta_j)\;,$ 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 4 blocks $J=\begin{bmatrix}A&B\\ C&D \end{bmatrix}$ . 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: $Ber_{+-} J_{\alpha\beta} = sgn\, \operatorname{det} A\, \operatorname{Ber} J_{\alpha\beta}$

For further details see the article about the Berezinian.

The complete formula now reads as: $\int_U f(x_a,\theta_i) d(x,\theta) = \int_{U'} f(x_a,\theta_i) \operatorname{Ber}_{+-}\, \frac{\partial(x_a,\theta_i)}{\partial(y_b,\xi_j)} d(y,\xi)$