Berezin Integral - Integration On An Exterior Algebra

Integration On An Exterior Algebra

Let be the exterior algebra of polynomials in anticommuting elements over the field of complex numbers. (The ordering of the generators is fixed and defines the orientation of the exterior algebra.) The Berezin integral on is the linear functional with the following properties:

for any where 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 is set to be where ; the integral of vanishes. The integral with respect to is calculated in the similar way and so on.

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