Chapter 16

Quadratic Polynomials and

the Symplectic Group

In chapters 14 and 15 we studied in detail the Heisenberg Lie algebra as the Lie
algebra of linear functions on phase space. After quantization, such functions
will give operatorsQjandPjon the state spaceH. In this chapter we’ll begin
to investigate what happens for quadratic functions with the symplectic Lie
algebra now the one of interest.
The existence of non-trivial Poisson brackets between homogeneous order
two and order one polynomials reflects the fact that the symplectic group acts by
automorphisms on the Heisenberg group. The significance of this phenomenon
will only become clear in later chapters, where examples will appear of inter-
esting observables coming from the symplectic Lie algebra that are quadratic in
theQjandPjand act not just on states, but non-trivially on theQjandPj
observables.
The identification of elementsLof the Lie algebrasp(2d,R) with order-
two polynomialsμLon phase spaceMis just the moment map for the ac-
tion of the symplectic groupSp(2d,R) onM. Quantization of these quadratic
functions will provide quantum observables corresponding to any Lie subgroup
G⊂Sp(2d,R) (any Lie groupGthat acts linearly onMpreserving the sym-
plectic form). Such quantum observables may or may not be “symmetries”,
with the term “symmetry” usually meaning that they arise by quantization of
aμLsuch that{μL,h}= 0 forhthe Hamiltonian function.
The reader should be warned that the discussion here is not at this stage
physically very well-motivated, with much of the motivation only appearing in
later chapters, especially in the case of the observables of quantum field theory,
which will be quadratic in the fields, and act by automorphisms on the fields
themselves.