Chapter 38

Symmetries and

Non-relativistic Quantum

Fields

In our study (chapters 25 and 26) of quantization using complex structures on
phase space we found that, using the Poisson bracket, quadratic polynomials of
the (complexified) phase space coordinates provided a symplectic Lie algebra
sp(2d,C), with a distinguishedgl(d,C) sub-Lie algebra determined by the com-
plex structure (see section 25.2). In section 25.3 we saw that these quadratic
polynomials could be quantized as quadratic combinations of the annihilation
and creation operators, giving a representation on the harmonic oscillator state
space, one that was unitary on the unitary sub-Lie algebrau(d)⊂gl(d,C).
The non-relativistic quantum field theory of chapters 36 and 37 is an infi-
nite dimensional version of this, with the dual phase space now the spaceH 1 of
solutions of the Schr ̈odinger equation. When a groupGacts onH 1 preserving
the Hermitian inner product (and thus the symplectic and complex structures),
generalizing the formulas of sections 25.2 and 25.3 should give a unitary repre-
sentation of such a groupGon the multi-particle state spaceS∗(H 1 ).
In sections 36.5 and 37.2 we saw how this works forG=R, the group of time
translations, which determines the dynamics of the theory. For the case of a
free particle, the field theory Hamiltonian is a quadratic polynomial of the fields,
providing a basic example of how such polynomials provide a unitary represen-
tation on the states of the quantum theory by use of a quadratic combination
of the quantum field operators. In this chapter we will see some other examples
of how group actions on the single-particle spaceH 1 lead to quadratic opera-
tors and unitary transformations on the full quantum field theory. The moment
map for these group actions gives the quadratic polynomials onH 1 , which after
quantization become the quadratic operators of the Lie algebra representation.