The Berezin integral on is defined to be the unique linear functional with the following properties:
for any where 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
expresses the Fubini law. On the right-hand side, the interior integral of a monomial is set to be where ; the integral of vanishes. The integral with respect to is calculated in the similar way and so on.
Consider now the algebra of functions of real commuting variables and of anticommuting variables (which is called the free superalgebra of dimension ). Intuitively, a function is a function of m even (bosonic, commuting) variables and of n odd (fermionic, anti-commuting) variables. More formally, an element is a function of the argument that varies in an open set with values in the algebra 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 where are even and are odd polynomials of depending on even variables The Jacobian matrix of this transformation has the block form:
where each even derivative commutes with all elements of the algebra ; the odd derivatives commute with even elements and anticommute with odd elements. The entries of the diagonal blocks and are even and the entries of the off-diagonal blocks are odd functions, where again mean right derivatives.
We now need the Berezinian (or superdeterminant) of the matrix , which is the even function
defined when the function is invertible in Suppose that the real functions define a smooth invertible map of open sets in and the linear part of the map is invertible for each The general transformation law for the Berezin integral reads
where ) is the sign of the orientation of the map The superposition is defined in the obvious way, if the functions do not depend on In the general case, we write where are even nilpotent elements of and set
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 in 1956. Other authors contributed to these developments, including the physicists Khalatnikov (although his paper contains mistakes), Matthews and Salam, and Martin.
^Matthews, P. T.; Salam, A. (1955). "Propagators of quantized field". Il Nuovo Cimento. Springer Science and Business Media LLC. 2 (1): 120–134. doi:10.1007/bf02856011. ISSN0029-6341.
^"The Feynman principle for a Fermi system". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 251 (1267): 543–549. 23 June 1959. doi:10.1098/rspa.1959.0127. ISSN2053-9169.