Quantum Mechanics for Mathematicians

spaceHto be a space of functions on a subspace of the classical phase space
which had the property that the basis coordinate functions Poisson-commuted.
Two examples of this are the position coordinatesqj, since{qj,qk}= 0, or the
momentum coordinatespj, since{pj,pk}= 0. Unfortunately, for symmetric
bilinear forms (·,·) of definite sign, such as the positive definite case Cliff(n,R),
the only subspace the bilinear form is zero on is the zero subspace.
To get an analog of the bosonic situation, one needs to take the case of
signature (d,d). The fermionic phase space will then be 2ddimensional, withd
dimensional subspaces on which (·,·) and thus the fermionic Poisson bracket is
zero. Quantization will give the Clifford algebra

Cliff(d,d,R) =M(2d,R)

which has just one irreducible representation,R^2

d

. This can be complexified to
get a complex state space
HF=C^2

d

This state space will come with a representation ofSpin(d,d,R) from expo-
nentiating quadratic combinations of the generators of Cliff(d,d,R). However,
this is a non-compact group, and one can show that on general grounds it can-
not have faithful unitary finite dimensional representations, so there must be a
problem with unitarity.
To see what happens explicitly, consider the simplest cased= 1 of one degree
of freedom. In the bosonic case the classical phase space isR^2 , and quantization
gives operatorsQ,Pwhich in the Schr ̈odinger representation act on functions
ofq, withQ=qandP =−i∂q∂. In the fermionic case with signature (1,1),
basis coordinate functions on phase space areξ 1 ,ξ 2 , with

{ξ 1 ,ξ 1 }+= 1, {ξ 2 ,ξ 2 }+=− 1 , {ξ 1 ,ξ 2 }+= 0

Defining

η=

1

√

2

(ξ 1 +ξ 2 ), π=

1

√

2

(ξ 1 −ξ 2 )

one gets objects with fermionic Poisson bracket analogous to those ofqandp

{η,η}+={π,π}+= 0, {η,π}+= 1

Quantizing, we get analogs of theQ,Poperators

η̂= Γ+(η) =

1

√

2

(Γ+(ξ 1 ) + Γ+(ξ 2 )), π̂= Γ+(π) =

1

√

2

(Γ+(ξ 1 )−Γ+(ξ 2 ))

which satisfy anticommutation relations

̂η^2 =̂π^2 = 0, ̂η̂π+̂πη̂= 1

and can be realized as operators on the space of functions of one fermionic
variableηas

̂η= multiplication byη, ̂π=

∂

∂η