Editing Determinant (section)

== Berezin integral ==
The conventional definition of the determinant, as a sum over permutations over a product of matrix elements, can be written using the somewhat surprising notation of the [[Berezin integral]]. In this notation, the determinant can be written as
:<math>\int \exp\left[-\theta^TA\eta\right] \,d\theta\,d\eta = \det A </math>
This holds for any <math>n\times n</math>-dimensional matrix <math>A.</math> The symbols <math>\theta,\eta</math> are two <math>n</math>-dimensional vectors of anti-commuting [[Grassmann number]]s (aka "[[supernumber]]s"), taken from the [[Grassmann algebra]]. The <math>\exp</math> here is the [[exponential function]]. The integral sign is meant to be understood as the Berezin integral. Despite the use of the integral symbol, this expression is in fact an entirely finite sum.

This unusual-looking expression can be understood as a notational trick that rewrites the conventional expression for the determinant
:<math>\det A = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.</math>
by using some novel notation. The anti-commuting property of the Grassmann numbers captures the sign (signature) of the permutation, while the integral combined with the <math>\exp</math> ensures that all permutations are explored. That is, the [[Taylor's series]] for <math>\exp</math> terminates after exactly <math>n</math> terms, because the square of a Grassmann number is zero, and there are exactly <math>n</math> distinct Grassmann variables. Meanwhile, the integral is defined to vanish, if the corresponding Grassmann number does ''not'' appear in the integrand. Thus, the integral selects out only those terms in the <math>\exp</math> series that have exactly <math>n</math> distinct variables; all lower-order terms vanish. Thus, the somewhat magical combination of the integral sign, the use of anti-commuting variables, and the Taylor's series for <math>\exp</math> just encodes a finite sum, identical to the conventional summation.

This form is popular in physics, where it is often used as a stand-in for the Jacobian determinant. The appeal is that, notationally, the integral takes the form of a [[Functional integration|path integral]], such as in the [[path integral formulation]] for quantized [[Hamiltonian mechanics]]. An example can be found in the theory of [[Fadeev–Popov ghosts]]; although this theory may seem rather abstruse, it's best to keep in mind that the use of the ghost fields is little more than a notational trick to express a Jacobian determinant.

The [[Pfaffian]] <math>\mathrm{Pf}\,A</math> of a [[skew-symmetric matrix]] <math>A</math> is the square-root of the determinant: that is, <math>\left(\mathrm{Pf}\,A\right)^2=\det A.</math> The Berezin integral form for the Pfaffian is even more suggestive; it is
:<math>\int \exp\left[- \tfrac{1}{2} \theta^T A \theta\right] \,d\theta = \mathrm{Pf}\, A </math>
The integrand has exactly the same formal structure as a normal [[Gaussian distribution]], albeit with Grassman numbers, instead of real numbers. This formal resemblance accounts for the occasional appearance of supernumbers in the theory of [[stochastic dynamics]] and [[stochastic differential equation]]s.