Quantum Mechanics for Mathematicians

quantization of the theory, while in the relativistic case it does not, and in that
case a different sort of complex structureJis needed, one not related to the
complex nature of the field values.
Instead of trying to complexifyM, we introduce a conjugate complex vector
spaceMand an antilinear conjugation operation interchangingMandM, with
square the identity. In the case ofMcomplex solutions to a field equation,M
will be solutions to the complex conjugate equation. Then, Bargmann-Fock
quantization proceeds with the decomposition

M⊕M

playing the role of the decomposition

M+J⊕M−J

in our previous discussion.
This determinesJ: it is the operator that is +ionM, and−ionM. Given a
Hermitian inner product〈·,·〉MonM, a symplectic structure Ω and indefinite
Hermitian product〈·,·〉onM⊕Mcan be determined as follows, using the
relation 26.7
〈u 1 ,u 2 〉=iΩ(u 1 ,u 2 )

Writing elementsu 1 ,u 2 ∈M⊕Mas

u 1 =u+ 1 +u− 1 , u 2 =u+ 2 +u− 2

whereu+ 1 ,u+ 2 ∈ Mandu− 1 ,u− 2 ∈M, Ω is defined to be the bilinear form such
that

• Ω(u+ 1 ,u+ 2 ) = Ω(u− 1 ,u− 2 ) = 0


• the Hermitian inner product is recovered onM

iΩ(u− 1 ,u+ 2 ) =〈u− 1 ,u+ 2 〉M

• Ω is antisymmetric, so

iΩ(u+ 1 ,u− 2 ) =−iΩ(u− 2 ,u+ 1 ) =−〈u− 2 ,u+ 1 〉M

Basis vectorszjofMorthonormal with respect to〈·,·〉Msatisfy

〈zj,zk〉=δjk

〈zj,zk〉=−δjk

〈zj,zk〉=〈zj,zk〉= 0

The symplectic form Ω satisfies the usual Poisson bracket relation

Ω(zj,zk) ={zj,zk}=iδjk