Quantum Mechanics for Mathematicians

Remarkably, an analog of calculus can be defined on such unconventional
spaces, introducing analogs of the derivative and integral for anticommuting
functions (i.e., elements of the Grassmann algebra). For the casen= 1, an
arbitrary function is
F(ξ) =c 0 +c 1 ξ

and one can take
∂
∂ξ

F=c 1

For larger values ofn, an arbitrary function can be written as

F(ξ 1 ,ξ 2 ,...,ξn) =FA+ξjFB

whereFA,FBare functions that do not depend on the chosenξj(one getsFB
by using the anticommutation relations to moveξjall the way to the left). Then
one can define
∂
∂ξj

F=FB

This derivative operator has many of the same properties as the conventional
derivative, although there are unconventional signs one must keep track of. An
unusual property of this derivative that is easy to see is that one has

∂

∂ξj

∂

∂ξj

= 0

Taking the derivative of a product one finds this version of the Leibniz rule
for monomialsFandG

∂

∂ξj

(FG) =

(

∂

∂ξj

F

)

G+ (−1)|F|F

(

∂

∂ξj

G

)

where|F|is the degree of the monomialF.
A notion of integration (often called the “Berezin integral”) with many of the
usual properties of an integral can also be defined. It has the peculiar feature
of being the same operation as differentiation, defined in then= 1 case by
∫
(c 0 +c 1 ξ)dξ=c 1

and for largernby
∫
F(ξ 1 ,ξ 2 ,···,ξn)dξ 1 dξ 2 ···dξn=

∂

∂ξn

∂

∂ξn− 1

···

∂

∂ξ 1

F=cn

wherecnis the coefficient of the basis elementξ 1 ξ 2 ···ξnin the expression ofF
in terms of basis elements.
This notion of integration is a linear operator on functions, and it satisfies
an analog of integration by parts, since if

F=

∂

∂ξj

G