20.3 Explicit calculations
As a balance to the abstract discussion so far in this chapter, in this section
we’ll work out explicitly what happens for some simple examples of subgroups
ofSp(2d,R) acting on phase space. They are chosen because of important later
applications, but also because the calculations are quite simple, while demon-
strating some of the phenomena that occur. The general story of how to explic-
itly construct the full metaplectic representation is quite a bit more complex.
These calculations will also make clear the conventions being chosen, and show
the basic structure of what the quadratic operators corresponding to actions of
subgroups of the symplectic group look like, a structure that will reappear in
the much more complicated infinite dimensional quantum field theory examples
we will come to later.
20.3.1 TheSO(2)action by rotations of the plane ford= 2
In the cased= 2 one can consider theSO(2) group which acts as the group
of rotations of the configuration spaceR^2 , with a simultaneous rotation of the
momentum space. This leaves invariant the Poisson bracket and so is a subgroup
ofSp(4,R) (this is just theSO(2) subgroup ofE(2) studied in section 19.1).
From the discussion in section 16.2, thisSO(2) acts by automorphisms on
the Heisenberg groupH 5 and Lie algebrah 5 , both of which can be identified
withM⊕R, by an action leaving invariant theRcomponent. The groupSO(2)
acts oncq 1 q 1 +cq 2 q 2 +cp 1 p 1 +cp 2 p 2 ∈Mby
cq 1
cq 2
cp 1
cp 2
→g
cq 1
cq 2
cp 1
cp 2
=
cosθ −sinθ 0 0
sinθ cosθ 0 0
0 0 cosθ −sinθ
0 0 sinθ cosθ
cq 1
cq 2
cp 1
cp 2
sog=eθLwhereL∈sp(4,R) is given by
L=
0 −1 0 0
1 0 0 0
0 0 0 − 1
0 0 1 0
Lacts on phase space coordinate functions by
q 1
q 2
p 1
p 2
→L
T
q 1
q 2
p 1
p 2
=
q 2
−q 1
p 2
−p 1
By equation 16.22, with