Quantum Mechanics for Mathematicians

and a super-Jacobi identity

[X,[Y,Z]±]±= [[X,Y]±,Z]±+ (−1)|X||Y|[Y,[X,Z]±]±

where|X|takes value 0 for a usual generator, 1 for a fermionic generator.

Analogously to the bosonic case, on polynomials in generators with order of
the polynomial less than or equal to two, the fermionic Poisson bracket{·,·}+is
a Lie superbracket, giving a Lie superalgebra of dimension 1+n+^12 (n^2 −n) (since
there is one constant,nlinear termsξjand^12 (n^2 −n) quadratic termsξjξk).
On functions of order two this Lie superalgebra is a Lie algebra,so(n). We will
see in chapter 31 that the definition of a representation can be generalized to
Lie superalgebras, and quantization will give a distinguished representation of
this Lie superalgebra, in a manner quite parallel to that of the Schr ̈odinger or
Bargmann-Fock constructions of a representation in the bosonic case.
The relation between the quadratic and linear polynomials in the generators
is parallel to what happens in the bosonic case. Here we have the fermionic
analog of the bosonic theorem 16.2:

Theorem 30.1. The Lie algebraso(n,R)is isomorphic to the Lie algebra
Λ^2 (V∗)(with Lie bracket{·,·}+) of order two anticommuting polynomials on
V=Rn, by the isomorphism
L↔μL

whereL∈so(n,R)is an antisymmetricnbynreal matrix, and

μL=

1

2

ξ·Lξ=

1

2

∑

j,k

Ljkξjξk

Theso(n,R)action on anticommuting coordinate functions is

{μL,ξk}+=

∑

j

Ljkξj

or
{μL,ξ}+=LTξ

Proof.The theorem follows from equations 30.1 and 30.2, or one can proceed
by analogy with the proof of theorem 16.2 as follows. First prove the second
part of the theorem by computing







1

2

∑

j,k

ξjLjkξk,ξl







+

=

1

2

∑

j,k

Ljk(ξj{ξk,ξl}+−{ξj,ξl}+ξk)

=

1

2

(

∑

j

Ljlξj−

∑

k

Llkξk)

=

∑

j

Ljlξj (sinceL=−LT)