In mathematical physics, the Berezin integral, named after Felix Berezin, (also known as Grassmann integral, after Hermann Grassmann), is a way to define integration for functions of Grassmann variables (elements of the exterior algebra). It is not an integral in the Lebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends the path integral in physics, where it is used as a sum over histories for fermions.

Definition

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.)

One variable

The Berezin integral over the sole Grassmann variable ${\displaystyle \theta =\theta _{1))$ is defined to be a linear functional

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

where we define

${\displaystyle \int \theta \,d\theta =1,\qquad \int \,d\theta =0}$

so that :

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

These properties define the integral uniquely and imply

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

Take note that ${\displaystyle f(\theta )=a\theta +b}$ is the most general function of ${\displaystyle \theta }$ because Grassmann variables square to zero, so ${\displaystyle f(\theta )}$ cannot have non-zero terms beyond linear order.

Multiple variables

The Berezin integral on ${\displaystyle \Lambda ^{n))$ is defined to be the unique 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.

Notice that different conventions exist in the literature: Some authors define instead^[1]

${\displaystyle \int _{\Lambda ^{n))\theta _{1}\cdots \theta _{n}\,\mathrm {d} \theta :=1.}$

The formula

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

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

Change of Grassmann variables

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

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

where ${\displaystyle \partial /\partial \xi _{j))$ refers to the right derivative (${\displaystyle \partial (\theta _{1}\theta _{2})/\partial \theta _{2}=\theta _{1},\;\partial (\theta _{1}\theta _{2})/\partial \theta _{1}=-\theta _{2))$). The formula for the coordinate change reads

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

Integrating even and odd variables

Definition

Consider now the algebra ${\displaystyle \Lambda ^{m\mid n))$ of functions of real commuting variables ${\displaystyle x=x_{1},\ldots ,x_{m))$ and of anticommuting variables ${\displaystyle \theta _{1},\ldots ,\theta _{n))$ (which is called the free superalgebra of dimension ${\displaystyle (m|n)}$). Intuitively, a function ${\displaystyle f=f(x,\theta )\in \Lambda ^{m\mid n))$ is a function of m even (bosonic, commuting) variables and of n odd (fermionic, anti-commuting) variables. More formally, an element ${\displaystyle f=f(x,\theta )\in \Lambda ^{m\mid n))$ 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 and vanishes in the complement of a compact set ${\displaystyle K\subset \mathbb {R} ^{m}.}$ The Berezin integral is the number

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

Change of even and odd variables

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

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

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 off-diagonal blocks ${\displaystyle B=\partial x/\partial \xi ,\ C=\partial \theta /\partial y}$ are odd functions, where ${\displaystyle \partial /\partial \xi _{j))$ again mean right derivatives.

We now need the Berezinian (or superdeterminant) of the matrix ${\displaystyle \mathrm {J} }$, which is the even function

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

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

${\displaystyle {\begin{aligned}&\int _{\Lambda ^{m\mid n))f(x,\theta )\,\mathrm {d} \theta \,\mathrm {d} x=\int _{\Lambda ^{m\mid n))f(x(y,\xi ),\theta (y,\xi ))\varepsilon \operatorname {Ber} \mathrm {J} \,\mathrm {d} \xi \,\mathrm {d} y\\[6pt]={}&\int _{\Lambda ^{m\mid n))f(x(y,\xi ),\theta (y,\xi ))\varepsilon {\frac {\det \left(A-BD^{-1}C\right)}{\det D))\,\mathrm {d} \xi \,\mathrm {d} y,\end{aligned))}$

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

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

where the Taylor series is finite.

Useful formulas

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

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

with ${\displaystyle A}$ being a complex ${\displaystyle n\times n}$ matrix.

• ${\displaystyle \int \exp \left[-{\tfrac {1}{2))\theta ^{T}M\theta \right]\,d\theta ={\begin{cases}\mathrm {Pf} \,M&n{\mbox{ even))\\0&n{\mbox{ odd))\end{cases))}$

with ${\displaystyle M}$ being a complex skew-symmetric ${\displaystyle n\times n}$ matrix, and ${\displaystyle \mathrm {Pf} \,M}$ being the Pfaffian of ${\displaystyle M}$, which fulfills ${\displaystyle (\mathrm {Pf} \,M)^{2}=\det M}$.

In the above formulas the notation ${\displaystyle d\theta =d\theta _{1}\cdots \,d\theta _{n))$ is used. From these formulas, other useful formulas follow (See Appendix A in^[2]) :

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

with ${\displaystyle A}$ being an invertible ${\displaystyle n\times n}$ matrix. Note that these integrals are all in the form of a partition function.

History

The mathematical theory of the integral with commuting and anticommuting variables was invented and developed by Felix Berezin.^[3] Some important earlier insights were made by David John Candlin^[4] in 1956. Other authors contributed to these developments, including the physicists Khalatnikov^[5] (although his paper contains mistakes), Matthews and Salam,^[6] and Martin.^[7]