Quantum Mechanics for Mathematicians

also begin usingxto denote a spatial variable instead of theqconventional
when this is the coordinate variable in a finite dimensional phase space. In
quantum field theory, position or momentum variables parametrize the fun-
damental degrees of freedom, the field variables, rather than providing such
degrees of freedom themselves. In this chapter, emphasis will be on the momen-
tum parametrization and the description of collections of free particles in terms
of quanta of degrees of freedom labeled by momenta.

36.1 Multi-particle quantum systems as quanta of a harmonic oscillator

It turns out that quantum systems of identical particles are best understood by
thinking of such particles as quanta of a harmonic oscillator system. We will
begin with the bosonic case, then later consider the fermionic case, which uses
the fermionic oscillator system.

36.1.1 Bosons and the quantum harmonic oscillator

A fundamental postulate of quantum mechanics (see chapter 9) is that given a
space of statesH 1 describing a bosonic single particle, a collection ofNidentical
such particles has state space

SN(H 1 ) = (H‘⊗···⊗H 1

︸ ︷︷ ︸

N−times

)S

where the superscriptSmeans we take elements of the tensor product invariant
under the action of the groupSNby permutation of theNfactors. To describe
states that include superpositions of an arbitrary number of identical particles,
one should take as state space the sum of these

S∗(H 1 ) =C⊕H 1 ⊕(H 1 ⊗H 1 )S⊕··· (36.1)
This same symmetric part of a tensor product occurs in the Bargmann-Fock
construction of the quantum state space for a phase spaceM=R^2 d, where the
Fock spaceFdcan be described in three different but isomorphic ways. Note that
we generally won’t take care to distinguish here betweenFdfin(superpositions
of states with finite number of quanta) and its completionFd(which includes
states with an infinite number of quanta). The three different descriptions of
Fdare:

• Fdhas an orthonormal basis
  |n 1 ,n 2 ,···,nd〉
  labeled by the eigenvaluesnjof the number operatorsNjforj= 1,···,d.
  Herenj∈{ 0 , 1 , 2 ,···}and

n=

∑d

j=1

nj