QuantumPhysics.dvi

where

T^0 =

∑^3

k=1

Tkk

Tk^1 =

∑^3

i,j=1

1

2

εkijTij

Tij^2 =

1

2

Tij+

1

2

Tji−

1

3

δijT^0 (9.79)

The componentT^0 is a scalar and transforms in thej = 0 representation of the rotation

algebra;Ti^1 is a vector observable and transforms in thej= 1 representation of the rotation

algebra. Finally, the symmetric traceless tensorTij^2 has 5 linearly independent components

T++^2 ,T+0^2 ,T+^2 −−T 002 ,T 02 −,T−−^2 , and form an irreducible representationj= 2.

The Wigner-Eckardt theorem may be generalized to tensor observables. Let us illustrate

this by discussion point (2) of this theorem. Since we have

〈j′,m′,α′|Uk|j′′,m′′,α′′〉= 0 |j′−j′′|≥ 2

〈j′′,m′′,α′′|Vℓ|j,m,α〉= 0 |j−j′′|≥ 2 (9.80)

it follows that

〈j′,m′,α′|Tkℓ|j,m,α〉= 0 |j′−j|≥ 3 (9.81)

This result follows by inserting the identity betweenUkandVℓand representing the identity

by the completeness relationI=

∑

j′′,m′′,α′′|j

′′,m′′,α′′〉〈j′′,m′′,α′′|.

9.11 P,C, andT

The operations of reversal of space,P, reversal of time,T, and reversal of charges,C, are

discrete transformations which are all symmetries of quantum electrodynamics (QED), but

not necessarily of the other interactions. It is a very general result that, in any quantum

theory invariant under special relativity (as QED and the StandardModel of Particle Physics

are), the combined transformationCPT is a symmetry. This result is often referred to as

theCPTtheorem, and goes back to Pauli, who gave a first proof of the theorem. Of course,

it is ultimately an experimental question as to whetherCPT is a symmetry of Nature, and

so far, no violations have been observed.

Parity is a symmetry of the strong interactions, but is violated (“maximally”) by the

weak interactions. The combined operationCP (equivalent to T in a theory withCPT

invariance) is also violated by the weak interactions. CP also appears to be a symmetry