Grassmann integral

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.^[1] Some earlier insights were made by the physicist David John Candlin^[2] in 1956.

Definition

The Berezin integral is defined to be a linear functional

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

where we define

${\displaystyle \int \theta \,d\theta =1}$

${\displaystyle \int \,d\theta =0}$

so that :

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

These properties define the integral uniquely.

${\displaystyle \int (a\theta +b)\,d\theta =a.}$

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:

${\displaystyle \int f_{1}(\theta _{1})\cdots f_{n}(\theta _{n})\,d\theta _{1}\cdots \,d\theta _{n}=\int f_{1}(\theta _{1})\,d\theta _{1}\cdots \int f_{n}(\theta _{n})\,d\theta _{n}.}$

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

Suppose now we want to do a substitution:

${\displaystyle \theta _{i}=\theta _{i}(\xi _{j})}$

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

${\displaystyle J_{ij}={\frac {\partial \theta _{i}}{\partial \xi _{j}}}.}$

the substitution formula now reads as

${\displaystyle \int f(\theta _{i})\,d\theta =\int f(\theta _{i}(\xi _{j}))\det(J_{ij})^{-1}\,d\xi .}$

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 ${\displaystyle x_{a}=x_{a}(y_{b},\xi _{j})\,,\;\theta _{i}=\theta _{i}(y_{b},\xi _{j})\;,}$ 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

${\displaystyle J_{\alpha \beta }={\frac {\partial (x_{a},\theta _{i})}{\partial (y_{b},\xi _{j})}}.}$

This matrix consists of four blocks:

${\displaystyle J={\begin{bmatrix}A&B\\C&D\end{bmatrix}}.}$

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:

${\displaystyle \operatorname {Ber} _{+-}J_{\alpha \beta }=\operatorname {sgn} \,\operatorname {det} A\,\operatorname {Ber} J_{\alpha \beta }.}$

For further details see the article about the Berezinian.

The complete formula now reads as:

${\displaystyle \int _{U}f(x_{a},\theta _{i})\,d(x,\theta )=\int _{U'}f(x_{a},\theta _{i})\operatorname {Ber} _{+-}\,{\frac {\partial (x_{a},\theta _{i})}{\partial (y_{b},\xi _{j})}}\,d(y,\xi ).}$

Gaussian integrals over Grassmann variables

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

$\int \exp\left[-\theta^TA\eta\right] \,d\theta\,d\eta = \det A$

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

${\displaystyle \int \exp \left[-\theta ^{T}M\theta \right]\,d\theta ={\begin{cases}2^{n \over 2}{\sqrt {\det M}},&n{\mbox{ even}}\\0,&n{\mbox{ odd}}\end{cases}}}$

with $M$ being an $n\times n$ antisymmetric matrix. In the above formulas the notation ${\displaystyle d\theta =d\theta _{1}\cdots \,d\theta _{n}}$ is used.

From the above formulas, other useful formulas follow:

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

with $A$ being an invertible

See also

• Berezin integral
• Supermanifold

Literature

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

References

1. A. Berezin, The Method of Second Quantization, Academic Press, (1966)
2. D.J. Candlin (1956). "On Sums over Trajectories for Systems With Fermi Statistics". Nuovo Cimento. 4: 231. doi:10.1007/BF02745446.