Quantum Mechanics for Mathematicians

Proof.An outline of the construction goes as follows:

1. Define a positive inner product onMby (u,v)J = Ω(u,Jv) onM. By
   Gram-Schmidt orthonormalization there is a basis of span{qj}⊂Mcon-
   sisting ofdvectorsqJj satisfying

(qJj,qJk)J=δjk

2. The vectorsJqJj will also be orthonormal since

(JqJj,JqkJ)J= Ω(JqJj,J^2 qJj) = Ω(qjJ,JqjJ) = (qJj,qJk)J

They will be orthogonal to theqjJsince

(JqJj,qkJ)J= Ω(J^2 qJj,qkJ) =−Ω(qJj,qJk)

and Ω(qJj,qJk) = 0 since any Poisson brackets of linear combinations of the

qjvanish.

3. Define
   zJj=

1

√

2

(qJj−iJqjJ)

ThezJjgive a complex basis ofM+J, their complex conjugateszJja complex

basis ofM−J.

4. The operators

Γ′J(zJj,0) =−ia†j=−izj, Γ′J(zJj,0) =−iaj=−i

∂

∂zj

satisfy the desired commutation relations and give a unitary representation

on linear combinations of the (zjJ,0) and (zJj,0) in the real subalgebra

h 2 d+1.

26.4 Complex vector spaces with Hermitian in-

ner product as phase spaces

In many cases of physical interest, the dual phase spaceMwill be a complex vec-
tor space with a Hermitian inner product. This will occur for instance whenM
is a space of complex solutions to a field equation, with examples non-relativistic
quantum field theory (see chapter 37) and the theory of a relativistic complex
scalar field (see chapter 44.1.2). In such cases, the Bargmann-Fock quantization
can be confusing, since it involves complexifyingM, which is already a complex
vector space. One way to treat this situation is as follows, takingM+J =M. In
the non-relativistic quantum field theory this gives a consistent Bargmann-Fock