Inmathematical physics, theBerezin integral, named afterFelix Berezin, (also known asGrassmann integral, afterHermann Grassmann), is a way to define integration for functions ofGrassmann variables (elements of theexterior algebra). It is not anintegral in theLebesgue sense; the word "integral" is used because the Berezin integral has properties analogous to the Lebesgue integral and because it extends thepath integral in physics, where it is used as a sum over histories forfermions.

Let${\displaystyle {\mathrm{\Lambda }}^n}$ 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.)

TheBerezin integral over the sole Grassmann variable${\displaystyle \theta ={\theta }_1}$ is defined to be a linear functional

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

where we define

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

so that :

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

These properties define the integral uniquely and imply

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

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.

TheBerezin integral on${\displaystyle {\mathrm{\Lambda }}^n}$ is defined to be the unique linear functional${\displaystyle {\int }_{{\mathrm{\Lambda }}^n}\cdot \text{d}\theta }$ with the following properties:

${\displaystyle {\int }_{{\mathrm{\Lambda }}^n}{\theta }_n\cdots {\theta }_1\mathrm{d}\theta =1,}$

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

for any${\displaystyle f\in {\mathrm{\Lambda }}^n,}$ where${\displaystyle \mathrm{\partial }/\mathrm{\partial }{\theta }_i}$ 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]

${\displaystyle {\int }_{{\mathrm{\Lambda }}^n}{\theta }_1\cdots {\theta }_n\mathrm{d}\theta :=1.}$

The formula

${\displaystyle {\int }_{{\mathrm{\Lambda }}^n}f(\theta )\mathrm{d}\theta ={\int }_{{\mathrm{\Lambda }}^1}\left(\cdots {\int }_{{\mathrm{\Lambda }}^1}\left({\int }_{{\mathrm{\Lambda }}^1}f(\theta )\mathrm{d}{\theta }_1\right)\mathrm{d}{\theta }_2\cdots \right)\mathrm{d}{\theta }_n}$

expresses the Fubini law. On the right-hand side, the interior integral of a monomial${\displaystyle f=g({\theta }^{\prime }){\theta }_1}$ is set to be${\displaystyle g({\theta }^{\prime }),}$ where${\displaystyle {\theta }^{\prime }=\left({\theta }_2,\dots ,{\theta }_n\right)}$; the integral of${\displaystyle f=g({\theta }^{\prime })}$ vanishes. The integral with respect to${\displaystyle {\theta }_2}$ is calculated in the similar way and so on.

Let${\displaystyle {\theta }_i={\theta }_i\left({\xi }_1,\dots ,{\xi }_n\right),i=1,\dots ,n,}$ be odd polynomials in some antisymmetric variables${\displaystyle {\xi }_1,\dots ,{\xi }_n}$. The Jacobian is the matrix

${\displaystyle D=\left\{\frac{\mathrm{\partial }{\theta }_i}{\mathrm{\partial }{\xi }_j},i,j=1,\dots ,n\right\},}$

where${\displaystyle \mathrm{\partial }/\mathrm{\partial }{\xi }_j}$ refers to theright derivative (${\displaystyle \mathrm{\partial }({\theta }_1{\theta }_2)/\mathrm{\partial }{\theta }_2={\theta }_1,\mathrm{\partial }({\theta }_1{\theta }_2)/\mathrm{\partial }{\theta }_1=-{\theta }_2}$). The formula for the coordinate change reads

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

Consider now the algebra${\displaystyle {\mathrm{\Lambda }}^{m\mid n}}$ of functions of real commuting variables${\displaystyle x=x_1,\dots ,x_m}$ and of anticommuting variables${\displaystyle {\theta }_1,\dots ,{\theta }_n}$ (which is called the free superalgebra of dimension${\displaystyle (m|n)}$). Intuitively, a function${\displaystyle f=f(x,\theta )\in {\mathrm{\Lambda }}^{m\mid n}}$ is a function of m even (bosonic, commuting) variables and of n odd (fermionic, anti-commuting) variables. More formally, an element${\displaystyle f=f(x,\theta )\in {\mathrm{\Lambda }}^{m\mid n}}$ 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${\displaystyle {\mathrm{\Lambda }}^n.}$ 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

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

Let a coordinate transformation be given by${\displaystyle x_i=x_i(y,\xi ),i=1,\dots ,m;{\theta }_j={\theta }_j(y,\xi ),j=1,\dots ,n,}$ 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:

where each even derivative${\displaystyle \mathrm{\partial }/\mathrm{\partial }{y}_j}$ commutes with all elements of the algebra${\displaystyle {\mathrm{\Lambda }}^{m\mid n}}$; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks${\displaystyle A=\mathrm{\partial }x/\mathrm{\partial }y}$ and${\displaystyle D=\mathrm{\partial }\theta /\mathrm{\partial }\xi }$ are even and the entries of the off-diagonal blocks${\displaystyle B=\mathrm{\partial }x/\mathrm{\partial }\xi ,C=\mathrm{\partial }\theta /\mathrm{\partial }y}$ are odd functions, where${\displaystyle \mathrm{\partial }/\mathrm{\partial }{\xi }_j}$ again meanright derivatives.

When the function${\displaystyle D}$ is invertible in${\displaystyle {\mathrm{\Lambda }}^{m\mid n},}$

So we have theBerezinian (orsuperdeterminant) of the matrix${\displaystyle \mathrm{J}}$, which is the even function

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

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

where${\displaystyle \epsilon =\mathrm{s}\mathrm{g}\mathrm{n}(det\mathrm{\partial }x(y,0)/\mathrm{\partial }y}$) 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${\displaystyle {\delta }_i,i=1,\dots ,m}$ are even nilpotent elements of${\displaystyle {\mathrm{\Lambda }}^{m\mid n}}$ and set

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

where theTaylor series is finite.

The following formulas for Gaussian integrals are used often in thepath integral formulation ofquantum field theory:

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

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

• 

with${\displaystyle M}$ being a complex skew-symmetric${\displaystyle n\times n}$ matrix, and${\displaystyle \mathrm{P}\mathrm{f}M}$ being thePfaffian of${\displaystyle M}$, which fulfills${\displaystyle (\mathrm{P}\mathrm{f}M{)}^2=detM}$.

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]) :

• ${\displaystyle \int \exp \left[{\theta }^{T}A\eta +{\theta }^{T}J+K^T\eta \right]d{\eta }_{1}d{\theta }_1\dots d{\eta }_{n}d{\theta }_n=detA\exp [-K^T{A}^{-1}J]}$

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

Berezin integral was probably first presented byDavid John Candlin in 1956.^[3] Later it was independently discovered byFelix Berezin in 1966.^[4]

Unfortunately Candlin's article failed to attract notice, and has been buried in oblivion. Berezin's work came to be widely known, and has almost been cited universally,^{[footnote 1]} becoming an indispensable tool to treat quantum field theory of fermions by functional integral.

Other authors contributed to these developments, including the physicists Khalatnikov^[9] (although his paper contains mistakes), Matthews and Salam,^[10] and Martin.^[11]

• Supermanifold
• Berezinian

• Theodore Voronov:Geometric integration theory on Supermanifolds, Harwood Academic Publisher,ISBN 3-7186-5199-8
• Berezin, Felix Alexandrovich:Introduction to Superanalysis, Springer Netherlands,ISBN 978-90-277-1668-2

• Multilinear algebra
• Differential forms
• Integral calculus
• Mathematical physics
• Quantum field theory
• Supersymmetry

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