Quantum Mechanics for Mathematicians

Theorem 16.3.Thesp(2d,R)action onh 2 d+1=M⊕Rby derivations is

L·(cq·q+cp·p+c) ={μL,cq·q+cp·p+c}=c′q·q+c′p·p (16.22)

where (
c′q
c′p

)

=L

(

cq

cp

)

or, equivalently (see section 4.1), on basis vectors ofMone has

{

μL,

(

q

p

)}

=LT

(

q

p

)

Proof.One can first prove 16.22 for the cases when only one ofA,B,Cis non-
zero, then the general case follows by linearity. For instance, taking the special
case

L=

(

0 B

0 0

)

, μL=

1

2

q·Bq

the action on coordinate functions (the basis vectors ofM) is

{

1

2

q·Bq,

(

q

p

)

}=LT

(

q

p

)

=

(

0

Bq

)

since

{

1

2

∑

j,k

qjBjkqk,pl}=

1

2

∑

j,k

(qj{Bjkqk,pl}+{qjBjk,pl}qk)

=

1

2

(

∑

j

qjBjl+

∑

k

Blkqk)

=

∑

j

Bljqj (sinceB=BT)

Repeating forAandCgives in general
{
μL,

(

q

p

)}

=LT

(

q

p

)

We can now prove theorem 16.2 as follows:

Proof.
L→μL

is clearly a vector space isomorphism of matrices and of quadratic polynomials.
To show that it is a Lie algebra isomorphism, the Jacobi identity for the Poisson
bracket can be used to show

{μL,{μL′,cq·q+cp·p}}−{μL′,{μL,cq·q+cp·p}}={{μL,μL′},cq·q+cp·p}