indistinguishability of quanta and symmetry under interchange of quanta since
only the numbers of quanta appear in the description of the state. The separate
symmetry postulate needed in the conventional quantum mechanical description
of multiple identical particles by tensor products is no longer needed.
In chapter 43 we’ll see that in the case of relativistic scalar quantum field
theory the dual phase spaceMwill be the space of real solutions of an equation
called the Klein-Gordon equation. TheJneeded for Bargmann-Fock quantiza-
tion will be determined by the decomposition into positive and negative energy
solutions, andH 1 =M+J will be a space of states describing a single relativistic
particle.
In this chapter, we’ll consider a non-relativistic theory, with wave equation
the Schr ̈odinger equation. Here the dual phase spaceMof complex solutions
will already be a complex vector space, and we can use the version of Bargmann-
Fock quantization described in section 26.4, soH 1 =M. The “second quanti-
zation” terminology is appropriate, since we take as (dual) classical phase space
a quantum state space, and quantize that.
36.1.2 Fermions and the fermionic oscillator
For the case of fermionic particles, ifH 1 is the state space for a single particle,
an arbitrary number of particles will be described by the state space
Λ∗(H 1 ) =C⊕H 1 ⊕(H 1 ⊗H 1 )A⊕···where (unlike the bosonic case) this is a finite sum ifH 1 is finite dimensional.
One can proceed as in the bosonic case, using instead ofFdthe fermionic oscil-
lator state spaceHF=Fd+. This again has three isomorphic descriptions:
- Fd+has an orthonormal basis
|n 1 ,n 2 ,···,nd〉labeled by the eigenvalues of the number operatorsNjforj= 1,···,d
wherenj∈{ 0 , 1 }. This is called the “occupation number” basis ofFd+.- Fd+is the Grassmann algebraC[θ 1 ,θ 2 ,···,θd] (see section 30.1) of poly-
nomials indanticommuting complex variables, with orthonormal basis
elements corresponding to the occupation number basis the monomials
θn 11 θn 22 ···θndd (36.4)- Fd+is the algebra of antisymmetric multilinear forms
Λ∗(VJ+) =C⊕VJ+⊕(VJ+⊗VJ+)A⊕···discussed in section 9.6, with product the wedge-product (see equation
9.6). HereVJ+is the space of complex linear functions on a vector space
V=R^2 d(the pseudo-classical phase space), eigenvectors with eigenvalue
+ifor the complex structureJ.