Quantum Mechanics for Mathematicians

is finite. This is called the “occupation number” basis ofFd. In this basis,

the annihilation and creation operators are

aj|n 1 ,n 2 ,···,nd〉=

√

nj|n 1 ,n 2 ,···,nj− 1 ,···,nd〉

a†j|n 1 ,n 2 ,···,nd〉=

√

nj+ 1|n 1 ,n 2 ,···,nj+ 1,···,nd〉

• Fdis the space of polynomialsC[z 1 ,z 2 ,···,zd] indcomplex variables
  zj, with inner product theddimensional version of equation 22.4 and
  orthonormal basis elements corresponding to the occupation number basis
  the monomials
  1
  √
  n 1 !···nd!

zn 11 zn 22 ···zdnd (36.2)

Here the annihilation and creation operators are

aj=

∂

∂zj

, a†j=zj

• Fdis the algebra

S∗(M+J) =C⊕M+J⊕(M+J⊗M+J)S⊕··· (36.3)

with product the symmetrized tensor product given by equation 9.3. Here

M+J is the space of complex linear functions onMthat are eigenvectors

with eigenvalue +ifor the complex structureJ. Using the isomorphism

between monomials and symmetric tensor products (given on monomials

in one variable by equation 9.4), the monomials 36.2 provide an orthonor-

mal basis of this space. Expressions for the annihilation and creation

operators acting onS∗(M+J) can be found using this isomorphism, using

their action on monomials as derivative and multiplication operators.

For each of these descriptions ofFd, the choice of orthonormal basis elements
given above provides an inner product, with the annihilation and creation oper-
atorsaj,a†jeach other’s adjoints, satisfying the canonical commutation relations

[aj,a†k] =δjk

We will describe the basis state|n 1 ,n 2 ,···,nd〉as one containingn 1 quanta of
type 1,n 2 quanta of type 2, etc., and a total number of quantan.
Comparing 36.1 and 36.3, we see that these are the same state spaces if
H 1 =M+J. This construction of multi-particle states by taking as dual clas-
sical phase spaceMa space of solutions to a wave equation, then quantizing
by the Bargmann-Fock method, withH 1 =M+J the quantum state space for
a single particle, is sometimes known as “second quantization”. Choosing a
(complex) basis ofH 1 , for each basis element one gets an independent quan-
tum harmonic oscillator, with corresponding occupation number the number of
“quanta” labeled by that basis element. This formalism automatically implies