Quantum Mechanics for Mathematicians

Poisson bracket, which on basis elements is

{ξj,ξk}+=±δjk, {ξj, 1 }+= 0,{ 1 , 1 }+= 0

Quantization is given by finding a representation of this Lie superalgebra. The
definition of a Lie algebra representation can be generalized to that of a Lie
superalgebra representation by:

Definition(Representation of a Lie superalgebra). A representation of a Lie
superalgebra is a homomorphismΦpreserving the superbracket

[Φ(X),Φ(Y)]±= Φ([X,Y]±)

This takes values in a Lie superalgebra of linear operators, with|Φ(X)|=|X|
and
[Φ(X),Φ(Y)]±= Φ(X)Φ(Y)−(−)|X||Y|Φ(Y)Φ(X)
A representation of the pseudo-classical Lie superalgebra (and thus a quan-
tization of the pseudo-classical system) will be given by finding a linear map Γ+
that takes basis elementsξjto operators Γ+(ξj) satisfying the relations

[Γ+(ξj),Γ+(ξk)]+=±δjkΓ+(1), [Γ+(ξj),Γ+(1)] = [Γ+(1),Γ+(1)] = 0

These relations can be satisfied by taking

Γ+(ξj) =

1

√

2

γj, Γ+(1) = 1

since then

[Γ+(ξj),Γ+(ξk)]+=

1

2

[γj,γk]+=±δjk

are exactly the Clifford algebra relations. This can be extended to a represen-
tation of the functions of theξjof order two or less by

Theorem.A representation of the Lie superalgebra of anticommuting functions
of coordinatesξjonRnof order two or less is given by

Γ+(1) = 1 , Γ+(ξj) =

1

√

2

γj, Γ+(ξjξk) =

1

2

γjγk

Proof.We have already seen that this is a representation for polynomials inξj
of degree zero and one. For simplicity just considering the cases= 0 (positive
definite inner product), in degree two the fermionic Poisson bracket relations
are given by equations 30.1 and 30.2. For 30.1, one can show that the products
of Clifford algebra generators

Γ+(ξjξk) =

1

2

γjγk

satisfy [
1
2

γjγk,γl

]

=δklγj−δjlγk