Quantum Mechanics for Mathematicians

whereA,B,Caredbydreal matrices, withBandCsymmetric, i.e.,

B=BT, C=CT

Note that, replacing the block antisymmetric matrix by the unit matrix, in 16.10
one recovers the definition of an orthogonal matrix, in 16.11 the definition of
the Lie algebra of the orthogonal group.
The generalization of 16.7 is

Theorem 16.2.The Lie algebrasp(2d,R)is isomorphic to the Lie algebra of
order two homogeneous polynomials onM=R^2 dby the isomorphism (using a
vector notation for the coefficient functionsq 1 ,···,qd,p 1 ,···,pd)

L↔μL

where

μL=

1

2

(

q p

)

L

(

0 − 1

1 0

)(

q

p

)

=

1

2

(

q p

)

(

B −A

−AT −C

)(

q

p

)

=

1

2

(q·Bq− 2 q·Ap−p·Cp) (16.13)

We will postpone the proof of this theorem until section 16.2, since it is easier
to first study Poisson brackets between order two and order one polynomials,
then use this to prove the theorem about Poisson brackets between order two
polynomials. As ind= 1, the functionμLis the moment map function forL.
The Lie algebrasp(2d,R) has a subalgebragl(d,R) consisting of matrices
of the form

L=

(

A 0

0 −AT

)

or, in terms of quadratic functions, the functions

−q·Ap=−p·ATq (16.14)

whereAis any realdbydmatrix. This shows that one way to get symplectic
transformations is to take any linear transformation of the position coordinates,
together with the dual linear transformation (see definition 4.2) on momentum
coordinates. In this way, any linear group acting on position space gives a
subgroup of the symplectic transformations of phase space.
An example of this is the groupSO(d) of spatial rotations, with Lie algebra
so(d)⊂gl(d,R), the antisymmetricdbydmatrices, for which−AT=A. The
special cased= 3 was an example already worked out earlier, in section 15.4,
whereμLgives the standard expression for the angular momentum as a function
of theqj,pjcoordinates on phase space.