QuantumPhysics.dvi

where the composition is given by

G=−i[G 1 ,G 2 ] (9.25)

using the Jacobi identity. Thus the infinitesimal symmetry transformations naturally form

a Lie algebra.

9.5 Examples of quantum symmetries

Many of the most important examples have already been encountered in the special cases that

we have studied. Here, we shall limit the enumeration to continuous symmetries. Important

discrete symmetries will be studied later on.

• space translation invariance; charge = momentum;

• time translation invariance; charge = energy;

• space rotation symmetry; charge = angular momentum;

• boosts in (non) relativistic mechanics; charge = Lorentz generators;

• U(1) phase rotations associated with gauge invariance; charge = electric charge;

• SU(3) rotations of the strong interactions; charge = color;

• SU(3) approximate symmetry betweenu,d,squarks; charge = isospin and strangeness.

The existence of symmetries has important consequences;

* The states at a given energy level transform under a unitary representation of the sym-

metry group or of the symmetry Lie algebra;

* Selection rules imply relations between certain probability amplitudes and imply the van-

ishing of other probability amplitudes and expectation values.

In the subsequent subsections, we shall discuss examples of these phenomena.

9.6 Symmetries of the multi-dimensional harmonic oscillator

TheN-dimensional harmonic oscillator provides an excellent laboratory for learning about

symmetries and their realizations. The dynamical variables areqi(t) andpi(t), and canonical

commutation relations [qi,pj] =i ̄hδij, withi,j= 1,···,N. The Hamiltonian is

H=

∑N

i=1

( 1

2 m

p^2 i+

1

2

mω^2 q^2 i

)

(9.26)