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).
be the exterior algebra of polynomials in anticommuting elements
is fixed and defines the orientation of the exterior algebra.)
The Berezin integral over the sole Grassmann variable
cannot have non-zero terms beyond linear order.
is defined to be the unique linear functional
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] The formula expresses the Fubini law.
On the right-hand side, the interior integral of a monomial
be odd polynomials in some antisymmetric variables
The formula for the coordinate change reads Consider now the algebra
of functions of real commuting variables
(which is called the free superalgebra of dimension
is a function of m even (bosonic, commuting) variables and of n odd (fermionic, anti-commuting) variables.
Suppose that this function is continuous and vanishes in the complement of a compact set
The Berezin integral is the number Let a coordinate transformation be given by
The Jacobian matrix of this transformation has the block form: where each even derivative
; the odd derivatives commute with even elements and anticommute with odd elements.
are even and the entries of the off-diagonal blocks
define a smooth invertible map
The general transformation law for the Berezin integral reads where
and set where the Taylor series is finite.
The following formulas for Gaussian integrals are used often in the path integral formulation of quantum field theory: with
Note that these integrals are all in the form of a partition function.
Berezin integral was probably first presented by David John Candlin in 1956.
[3] Later it was independently discovered by Felix 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.