38.4 Fermionic fields
It is an experimentally observed fact that elementary particles with spin^12
behave as fermions and are described by fermionic fields. In non-relativistic
quantum field theory, such spin^12 elementary particles could in principle be
bosons, described by bosonic fields as in section 38.3.3. There is however a
“spin-statistics theorem” in relativistic quantum field theory that says that spin
1
2 fields must be quantized with anticommutators. This provides an explanation
of the observed correlation of values of the spin and of the particle statistics,
due to the fact that the non-relativistic theories describing fundamental particles
should be low-energy limits of relativistic theories.
The discussion of the symplectic and unitary group actions onH 1 of section
38.1 has a straightforward analog in the case of a single-particle state space
H 1 with Hermitian inner product describing fermions, rather than bosons. The
analog of the infinite dimensional symplectic group action (preserving the imag-
inary part of the Hermitian inner product) of the bosonic case is an infinite
dimensional orthogonal group action (preserving the real part of the Hermitian
inner product) in the fermionic case. The multi-particle state space will be
an infinite dimensional version of the spinor representation for this orthogonal
group. As in the bosonic case, there will be an infinite dimensional unitary group
preserving the full Hermitian inner product, and the groups of symmetries we
will be interested in will be subgroups of this group.
In section 31.3 we saw in finite dimensions how unitary group actions on a
fermionic phase space gave a unitary representation on the fermionic oscillator
state space, by the same method of annihilation and creation operators as in
the bosonic case (changing commutators to anticommutators). Applying this to
the infinite dimensional case of the single-particle spaceH 1 of solutions to the
free particle Schr ̈odinger equation is done by taking
θj→Ψ(x) or A(p)θj→Ψ(x) or A(p)
{θj,θk}+=δjk→{Ψ(x),Ψ(x′)}+=δ(x−x′) or {A(p),A(p′)}+=δ(p−p′)Quantization generalizes the construction of the spinor representation from sec-
tions 31.3 and 31.4 to theH 1 case, taking
aFj→Ψ(̂x) or a(p)aF†j→Ψ̂†(x) or a†(p)[aFj,aF†k]+=δjk→[Ψ(̂x),Ψ̂†(x′)]+=δ(x−x′) or [a(p),a†(p′)]+=δ(p−p′)
Quadratic combinations of theθj,θjgive the Lie algebra of orthogonal trans-
formations of the phase spaceM. We will again be interested in the generaliza-
tion toM=H 1 , but for very specific quadratic combinations, corresponding to
certain finite dimensional Lie algebras of unitary transformations ofH 1. Quan-
tization will take these to quadratic combinations of fermionic field operators,