Appendix A

Conventions

I’ve attempted to stay close to the conventions used in the physics literature,
leading to the choices listed here. Most of the time, units are chosen so that
~=c= 1.

A.1 Bilinear forms

Parentheses
(·,·)

will be used for a non-degenerate symmetric bilinear form (inner product) on a
vector spaceRn, with the same symbol also used for the complex linear exten-
sion to a bilinear form onCn. The group of linear transformations preserving
the inner product will beO(n) orO(n,C) respectively.
Angle brackets
〈·,·〉

will be used for non-degenerate symmetric sesquilinear forms (Hermitian inner
products) on vector spacesCn. These are antilinear in the first entry, linear
in the second. When the inner product is positive, it will be preserved by
the groupU(n), but the indefinite case withn= 2dand groupU(d,d) will
also occur. The quantum mechanical state space comes with such a positive
Hermitian inner product, but may be infinite dimensional and a Hilbert space.
Non-degenerate antisymmetric forms (symplectic forms) will be denoted by

ω(·,·) or Ω(·,·)

with the first used for the symplectic form on a real phase spaceMof dimension
2 d, and the second for the corresponding form on the dual phase spaceM. The
group preserving these bilinear forms isSp(2d,R). The same symbol will also be
used for the complex linear extension of these forms toC^2 d, where the bilinear
form is preserved by the groupSp(2d,C).