Quantum Mechanics for Mathematicians

just a configuration space, and one does not need to introduce new momentum
variables. One could try and quantize this system by path integral methods, for
instance computing the propagator by doing the integral
∫
Dψ(x,t)e
~iS[ψ]

over paths inH 1 parametrized byt, taking values fromt= 0tot=T. This
is a highly infinite dimensional integral, over paths in an infinite dimensional
space. In addition, recall the warnings given in chapter 35 about the problematic
nature of path integrals over paths in a phase space, which is the case here.

37.5 Fermion fields

Most everything discussed in this chapter and in chapter 36 applies with little
change to the case of fermionic quantum fields using fermionic instead of bosonic
oscillators, and changing commutators to anticommutators for the annihilation
and creation operator. This gives fermionic fields that satisfy anticommutation
relations
[Ψ(̂x),Ψ̂†(x′)]+=δ(x−x′)

and states that in the occupation number representation havenp= 0,1, while
also having a description in terms of antisymmetric tensor products, or polyno-
mials in anticommuting coordinates. Field operators will in this case generate
an infinite dimensional Clifford algebra. Elements of this Clifford algebra act on
states by an infinite dimensional version of the construction of spinors in terms
of fermionic oscillators described in chapter 31.
For applications to physical systems in three dimensional space, it is often
the fermionic version that is relevant, with the systems of interest for instance
describing arbitrary numbers of electrons, which are fermionic particles so need
to be described by anticommuting fields. The quantum field theory of non-
relativistic free electrons is the quantum theory one gets by taking as single-
particle phase spaceH 1 the space of solutions of the two-component Pauli-
Schr ̈odinger equation 34.3 described in section 34.2 and then quantizing using
the fermionic version of Bargmann-Fock quantization. The fermionic Poisson
bracket is determined by the inner product on thisH 1 discussed in section 34.3.
More explicitly, this is a theory of two quantum fieldsΨ̂ 1 (x),Ψ̂ 2 (x) satisfying
the anticommutation relations

[Ψ̂j(x),Ψ̂†k(x′)]+=δjkδ^3 (x−x′)

These are related by Fourier transform

Ψ̂j(x) =^1

(2π)

(^32)

∫

R^3

eip·xaj(p)d^3 p

Ψ̂†j(x) =^1

(2π)

(^32)

∫

R^3

e−ip·xa†j(p)d^3 p