QuantumPhysics.dvi

electro-magnetic interactions, which occurs fast, while theπ±cannot, resulting in typical

weak scale life-times. Other approximate selection rules resulting from the effects of the

weak interactions are as follows,

• individual quark number (up, down, charm, strange, top, bottom)

• individual lepton number (electron,μ,τ)

• parity

• CP (charge conjugation combined with parity)

In the remainder of this section, we shall explore the implications of the exact selection rule

of the conservation of angular momentum. The techniques developed here will be applicable,

however, also to selection rules that are only approximate.

9.8 Vector Observables

In practice, selection rules will often manifest their effect throughthe vanishing of certain

probability amplitudes, which may emerge as the matrix elements of certain observables.

Here, we shall be interested in selection rules associated with rotation invariance. The states

of the system may be organized as combinations of angular momentum eigenstates|j,m〉. It

readily follows from the action of the angular momentum generatorsJa,a= 1, 2 ,3 on these

states that

(1) 〈j′,m′|Ja|j,m〉= 0 if j′ 6 =j

(2) 〈j, m′|Ja|j,m〉= 0 if |m′−m|> 1 (9.57)

These results may be viewed as a simple form of angular momentum selection rules.

Analogous selection rules may be derived for operators other thanJa. The simplest

case is forvector observables. A vector observable is actually a tripletV= (V 1 ,V 2 ,V 3 ) of

observables which under rotations transform exactly as angular momentum itself does. Let

R= exp{−i~ω·J}be any finite rotation, then a vector observable is defined to obey the

transformation rule,

RVaR†=

∑^3

b=1

RabVb (9.58)

withRa real 3×3 orthogonal matrix, i.e. satisfyingRtR=I. It is more convenient to use

the infinitesimal version of this equation, given by

[Ja,Vb] =i ̄h

∑^3

c=1

εabcVc (9.59)