Chapter 26

Complex Structures and

Quantization

The Schr ̈odinger representation ΓSofH 2 d+1uses a specific choice of extra struc-
ture on classical phase space: a decomposition of its coordinates into positions
qjand momentapj. For the unitarily equivalent Bargmann-Fock representation
a different sort of extra structure is needed, a decomposition of coordinates on
phase space into complex coordinateszjand their complex conjugateszj. Such
a decomposition is called a “complex structure”J, and will correspond after
quantization to a choice that distinguishes annihilation and creation operators.
In previous chapters we used one particular standard choiceJ=J 0 , but in this
chapter will describe other possible choices. For each such choice we’ll get a
different version ΓJof the Bargmann-Fock construction of a Heisenberg group
representation. In later chapters on relativistic quantum field theory, we will
see that the phenomenon of antiparticles is best understood in terms of a new
possibility for the choice ofJthat appears in that case.

26.1 Complex structures and phase space

Quantization of phase spaceM =R^2 dusing the Schr ̈odinger representation
gives a unitary Lie algebra representation Γ′Sof the Heisenberg Lie algebrah 2 d+1
which takes theqjandpjcoordinate functions on phase space to operators−iQj
and−iPjonHS=L^2 (Rd). This involves a choice, that of taking states to be
functions of theqj, or (using the Fourier transform) of thepj. It turns out to be
a general phenomenon that quantization requires choosing some extra structure
on phase space, beyond the Poisson bracket.
For the case of the harmonic oscillator, we found in chapter 22 that quantiza-
tion was most conveniently performed using annihilation and creation operators,
which involve a different sort of choice of extra structure on phase space. There