Quantum Mechanics for Mathematicians

So
A∈gl(d,C)→UA′

is a Lie algebra representation ofgl(d,C)onHF
One also has (for column vectorsaFwith componentsaF 1 ,...,aFd)

[UA′,a†F] =ATa†F, [UA′,aF] =−AaF (27.1)

Proof.The proof is similar to that of 25.1, except besides the relation

[AB,C] =A[B,C] + [A,B]C

we also use the relation

[AB,C] =A[B,C]+−[A,B]+C

For example

[UA′,aF†l] =

∑

j,k

[aF†jAjkaFk,aF†l]

=

∑

j,k

aF†jAjk[aFk,aF†l]+

=

∑

j

aF†jAjl

The Hamiltonian is

H=

∑

j

(NFj−

1

2

1 )

which (up to the constant^12 that doesn’t contribute to commutation relations) is
justUB′ for the caseB= 1. Since this commutes with all otherdbydmatrices,
we have
[H,UA′] = 0

for allA∈gl(d,C), so these are symmetries and we have a representation of
the Lie algebragl(d,C) on each energy eigenspace. Only forA∈u(d) (Aa
skew-adjoint matrix) will the representation turn out to be unitary.

27.3 For further reading

Most quantum field theory books and a few quantum mechanics books contain
some sort of discussion of the fermionic oscillator, see for example chapter 21.3
of [81] or chapter 5 of [16]. The standard discussion often starts with consid-
ering a form of classical analog using anticommuting “fermionic” variables and
then quantizing to get the fermionic oscillator. Here we are doing things in
the opposite order, starting in this chapter with the quantized oscillator, then
considering the classical analog in chapter 30.