# Berezin integral

In mathematical physics, a Berezin integral, named after Felix Berezin, (or Grassmann integral, after Hermann Grassmann) is a way to define integration of elements of the exterior algebra (Hermann Grassmann 1844). It is called integral because it is used in physics as a sum over histories for fermions, an extension of the path integral.

## Integration on an exterior algebra

Let ${\displaystyle \Lambda ^{n}}$ be the exterior algebra of polynomials in anticommuting elements ${\displaystyle \theta _{1},\dots ,\theta _{n}}$ over the field of complex numbers. (The ordering of the generators ${\displaystyle \theta _{1},\dots ,\theta _{n}}$ is fixed and defines the orientation of the exterior algebra.) The Berezin integral on ${\displaystyle \Lambda ^{n}}$ is the linear functional ${\displaystyle \int _{\Lambda ^{n}}\cdot {\textrm {d}}\theta }$ with the following properties:

${\displaystyle \int _{\Lambda ^{n}}\theta _{n}\cdots \theta _{1}\,\mathrm {d} \theta =1,}$
${\displaystyle \int _{\Lambda ^{n}}{\frac {\partial f}{\partial \theta _{i}}}\,\mathrm {d} \theta =0,\ i=1,\dots ,n}$

for any ${\displaystyle f\in \Lambda ^{n},}$ where ${\displaystyle \partial /\partial \theta _{i}}$ means the left or the right partial derivative. These properties define the integral uniquely. The formula

${\displaystyle \int _{\Lambda ^{n}}f\left(\theta \right)\,\mathrm {d} \theta =\int _{\Lambda ^{1}}\left(\cdots \int _{\Lambda ^{1}}\left(\int _{\Lambda ^{1}}f\left(\theta \right)\,\mathrm {d} \theta _{1}\right)\,\mathrm {d} \theta _{2}\cdots \right)\mathrm {d} \theta _{n}}$

expresses the Fubini law. On the right-hand side, the interior integral of a monomial ${\displaystyle f=g\left(\theta ^{\prime }\right)\theta _{1}}$ is set to be ${\displaystyle g\left(\theta ^{\prime }\right),\ }$ where ${\displaystyle \theta ^{\prime }=\left(\theta _{2},...,\theta _{n}\right)}$; the integral of ${\displaystyle f=g\left(\theta ^{\prime }\right)}$ vanishes. The integral with respect to ${\displaystyle \theta _{2}}$ is calculated in the similar way and so on.

## Change of Grassmann variables

Let ${\displaystyle \theta _{i}=\theta _{i}\left(\xi _{1},...,\xi _{n}\right),\ i=1,...,n,}$ be odd polynomials in some antisymmetric variables ${\displaystyle \xi _{1},...,\xi _{n}}$. The Jacobian is the matrix

${\displaystyle D=\left\{{\frac {\partial \theta _{i}}{\partial \xi _{j}}},\ i,j=1,...,n\right\},}$

where the left and the right derivatives coincide and are even polynomials. The formula for the coordinate change reads

${\displaystyle \int f\left(\theta \right)\mathrm {d} \theta =\int f\left(\theta \left(\xi \right)\right)\left(\det D\right)^{-1}\mathrm {d} \xi .}$

## Berezin integral

Consider now the algebra ${\displaystyle \Lambda ^{m\mid n}}$ of functions of real commuting variables ${\displaystyle x=x_{1},...,x_{m}}$ and of anticommuting variables ${\displaystyle \theta _{1},...,\theta _{n}}$ (which is called the free superalgebra of dimension ${\displaystyle \left(m\mid n\right)}$). This means that an element ${\displaystyle f=f\left(x,\theta \right)\in \Lambda ^{m\mid n}}$ is a function of the argument ${\displaystyle x}$ that varies in an open set ${\displaystyle X\subset \mathbb {R} ^{m}}$ with values in the algebra ${\displaystyle \Lambda ^{n}.}$ Suppose that this function is continuous${\displaystyle \ }$and vanishes in the complement of a compact set ${\displaystyle K\subset \mathbb {R} ^{m}.}$ The Berezin integral is the number

${\displaystyle \int _{\Lambda ^{m\mid n}}f\left(x,\theta \right)\mathrm {d} \theta \mathrm {d} x=\int _{\mathbb {R} ^{m}}\mathrm {d} x\int _{\Lambda ^{n}}f\left(x,\theta \right)\mathrm {d} \theta .}$

## Change of even and odd variables

Let a coordinate transformation be given by ${\displaystyle x_{i}=x_{i}\left(y,\xi \right),\ i=1,...,m;\ \theta _{j}=\theta _{j}\left(y,\xi \right),j=1,...,n}$, where ${\displaystyle x_{i},y_{i}}$ are even and ${\displaystyle \theta _{j},\xi _{j}}$ are odd polynomials of ${\displaystyle \xi }$ depending on even variables ${\displaystyle y.}$ The Jacobian matrix of this transformation has the block form:

${\displaystyle \mathrm {J} ={\frac {\partial \left(x,\theta \right)}{\partial \left(y,\xi \right)}}=\left({\begin{array}{cc}A&B\\C&D\end{array}}\right),}$

where each even derivative ${\displaystyle \partial /\partial y_{j}}$ commutes with all elements of the algebra ${\displaystyle \Lambda ^{m\mid n}}$; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks ${\displaystyle A=\partial x/\partial y}$ and ${\displaystyle D=\partial \theta /\partial \xi }$ are even and the entries of the offdiagonal blocks ${\displaystyle B=\partial x/\partial \xi ,\ C=\partial \theta /\partial y}$ are odd functions, where ${\displaystyle \partial /\partial \xi _{j}}$ mean right derivatives. The Berezinian (or the superdeterminant) of the matrix ${\displaystyle \mathrm {J} }$ is the even function

${\displaystyle \mathrm {Ber~J} =\det \left(A-BD^{-1}C\right)\det D^{-1}}$

defined when the function ${\displaystyle \det D}$ is invertible in ${\displaystyle \Lambda ^{m\mid n}.}$ Suppose that the real functions ${\displaystyle x_{i}=x_{i}\left(y,0\right)}$ define a smooth invertible map ${\displaystyle F:Y\rightarrow X}$ of open sets ${\displaystyle X,\ Y}$ in ${\displaystyle \mathbb {R} ^{m}}$ and the linear part of the map ${\displaystyle \xi \mapsto \theta =\theta \left(y,\xi \right)}$ is invertible for each ${\displaystyle y\in Y.}$ The general transformation law for the Berezin integral reads

${\displaystyle \int _{\Lambda ^{m\mid n}}f\left(x,\theta \right)\mathrm {d} \theta \mathrm {d} x=\int _{\Lambda ^{m\mid n}}f\left(x\left(y,\xi \right),\theta \left(y,\xi \right)\right)\varepsilon \mathrm {Ber~J~d} \xi \mathrm {d} y}$
${\displaystyle =\int _{\Lambda ^{m\mid n}}f\left(x\left(y,\xi \right),\theta \left(y,\xi \right)\right)\varepsilon {\frac {\det \left(A-BD^{-1}C\right)}{\det D}}\mathrm {d} \xi \mathrm {d} y,}$

where ${\displaystyle \varepsilon =\mathrm {sgn~\det } \partial x\left(y,0\right)/\partial y}$ is the sign of the orientation of the map ${\displaystyle F.}$ The superposition ${\displaystyle f\left(x\left(y,\xi \right),\theta \left(y,\xi \right)\right)}$ is defined in the obvious way, if the functions ${\displaystyle x_{i}\left(y,\xi \right)}$ do not depend on ${\displaystyle \xi .}$ In the general case, we write ${\displaystyle x_{i}\left(y,\xi \right)=x_{i}\left(y,0\right)+\delta _{i},}$ where ${\displaystyle \delta _{i},\ i=1,...,m}$ are even nilpotent elements of ${\displaystyle \Lambda ^{m\mid n}}$ and set

${\displaystyle f\left(x\left(y,\xi \right),\theta \right)=f\left(x\left(y,0\right),\theta \right)+\sum _{i}{\frac {\partial f}{\partial x_{i}}}\left(x\left(y,0\right),\theta \right)\delta _{i}+{\frac {1}{2}}\sum _{i,j}{\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}\left(x\left(y,0\right),\theta \right)\delta _{i}\delta _{j}+...,}$

where the Taylor series is finite.

## History

The mathematical theory of the integral with commuting and anticommuting variables was invented and developed by Felix Berezin. Some important earlier insights were made by David John Candlin. Other authors contributed to these developments, including the physicists Khalatnikov [3] (although his paper contains mistakes), Matthews and Salam [4], and Martin [6].

## References

[1] F.A. Berezin, The Method of Second Quantization, Academic Press, (1966)

[2] F.A. Berezin, Introduction to superanalysis. D. Reidel Publishing Co., Dordrecht, 1987. xii+424 pp. ISBN 90-277-1668-4.

[3] I.M. Khalatnikov (1954), "Predstavlenie funkzij Grina v kvantovoj elektrodinamike v forme kontinualjnyh integralov" (Russian). JETP, 28, 635.

[4] P.T. Matthews, A. Salam (1955), "Propagators of quantized field". Nuovo Cimento 2, 120.

[5] D.J. Candlin (1956)."On Sums over Trajectories for Systems With Fermi Statistics". Nuovo Cimento 4:231. doi:10.1007/BF02745446.

[6] J.L. Martin (1959), "The Feynman principle for a Fermi System". Proc. Roy. Soc. A 251, 543.