Quantum Mechanics for Mathematicians

Definition(Semi-direct product group).Given a groupK, a groupN, and an
actionΦofKonNby automorphisms

Φk:n∈N→Φk(n)∈N

the semi-direct productNoKis the set of pairs(n,k)∈N×Kwith group law

(n 1 ,k 1 )(n 2 ,k 2 ) = (n 1 Φk 1 (n 2 ),k 1 k 2 )

One can easily check that this satisfies the group axioms. The inverse is

(n,k)−^1 = (Φk− 1 (n−^1 ),k−^1 )

Checking associativity, one finds

((n 1 ,k 1 )(n 2 ,k 2 ))(n 3 ,k 3 ) =(n 1 Φk 1 (n 2 ),k 1 k 2 )(n 3 ,k 3 )

=(n 1 Φk 1 (n 2 )Φk 1 k 2 (n 3 ),k 1 k 2 k 3 )

=(n 1 Φk 1 (n 2 )Φk 1 (Φk 2 (n 3 )),k 1 k 2 k 3 )

=(n 1 Φk 1 (n 2 Φk 2 (n 3 )),k 1 k 2 k 3 )

=(n 1 ,k 1 )(n 2 Φk 2 (n 3 ),k 2 k 3 )

=(n 1 ,k 1 )((n 2 ,k 2 )(n 3 ,k 3 ))

The notationNoK for this construction has the weakness of not explicitly
indicating the automorphism Φ which it depends on. There may be multiple
possible choices for Φ, and these will always include the trivial choice Φk= 1
for allk∈K, which will give the standard product of groups.

Digression.For those familiar with the notion of a normal subgroup,Nis a
normal subgroup ofNoK. A standard notation for “Nis a normal subgroup
ofG” isN
G. The symbolNoKis supposed to be a mixture of the×and

symbols (note that some authors define it to point in the other direction).

The Euclidean groupE(d) is an example withN=Rd,K=SO(d). For
a∈Rd,R∈SO(d) one has
ΦR(a) =Ra

In chapter 42 we will see another important example, the Poincar ́e group which
generalizesE(3) to include a time dimension, treating space and time according
to the principles of special relativity.
The most important example for quantum theory is:

Definition(Jacobi group).The Jacobi group inddimensions is the semi-direct
product group
GJ(d) =H 2 d+1oSp(2d,R)

If we write elements of the group as
(((
cq
cp

)

,c

)

,k

)