Berezin integral

In mathematical physics, a Berezin integral, named after Felix Berezin, (or Grassmann integral, after Hermann Grassmann) is a way to define integration of elements of the exterior algebra (Hermann Grassmann 1844). It is called integral because it is used in physics as a sum over histories for fermions, an extension of the path integral. In mathematical physics, a Berezin integral, named after Felix Berezin, (or Grassmann integral, after Hermann Grassmann) is a way to define integration of elements of the exterior algebra (Hermann Grassmann 1844). It is called integral because it is used in physics as a sum over histories for fermions, an extension of the path integral. Let Λ n {displaystyle Lambda ^{n}} be the exterior algebra of polynomials in anticommuting elements θ 1 , … , θ n {displaystyle heta _{1},dots , heta _{n}} over the field of complex numbers. (The ordering of the generators θ 1 , … , θ n {displaystyle heta _{1},dots , heta _{n}} is fixed and defines the orientation of the exterior algebra.) The Berezin integral on Λ n {displaystyle Lambda ^{n}} is the linear functional ∫ Λ n ⋅ d θ {displaystyle int _{Lambda ^{n}}cdot { extrm {d}} heta } with the following properties: for any f ∈ Λ n , {displaystyle fin Lambda ^{n},} where ∂ / ∂ θ i {displaystyle partial /partial heta _{i}} means the left or the right partial derivative. These properties define the integral uniquely. The formula expresses the Fubini law. On the right-hand side, the interior integral of a monomial f = g ( θ ′ ) θ 1 {displaystyle f=gleft( heta ^{prime } ight) heta _{1}} is set to be g ( θ ′ ) ,   {displaystyle gleft( heta ^{prime } ight), } where θ ′ = ( θ 2 , . . . , θ n ) {displaystyle heta ^{prime }=left( heta _{2},..., heta _{n} ight)} ; the integral of f = g ( θ ′ ) {displaystyle f=gleft( heta ^{prime } ight)} vanishes. The integral with respect to θ 2 {displaystyle heta _{2}} is calculated in the similar way and so on. Let θ i = θ i ( ξ 1 , . . . , ξ n ) ,   i = 1 , . . . , n , {displaystyle heta _{i}= heta _{i}left(xi _{1},...,xi _{n} ight), i=1,...,n,} be odd polynomials in some antisymmetric variables ξ 1 , . . . , ξ n {displaystyle xi _{1},...,xi _{n}} . The Jacobian is the matrix