wang
(Wang)
#1
of the strong interactions, though it is not well-understood why, as it would be natural to
violateCPalso’in the strong interactions. This is often referred to as thestrong CP problem,
and remains an active area of investigation in particle physics.
Parityis the operation which reverses an odd number of space-like direction. In 3 space
dimensions, it is convenient to take it to act byx→−x, a transformation which commutes
with rotations. On Hilbert space, the transformation acts by a linear operatorP, which is
self-adjoint and has unit squareP^2 =I. On states and operators,Pacts as follows,
PxP†=−x P|x〉=|−x〉
PpP†=−p P|p〉=|−p〉
PLP†= +L P|ℓ,m〉= (−)ℓ|ℓ,m〉 (9.82)
Because of the last equation, a state may be characterized by both its angular momentum
quantum numbersℓ,morj,mand its behavior under parity. It is important to note that
a state of any given integer spin may be even or odd under parity. For example, spin zero
states|α〉fall into two categories,
π|α+〉= +|α+〉 |α+〉= scalar
π|α−〉=−|α−〉 |α−〉= pseudo-scalar (9.83)
The need for both kinds of states may be seen direct from the existence of spin zero operators
which have, however, odd parity. For example,
[J,(x·S)] = 0 P(x·S)P†=−(x·S) (9.84)
Therefore, if|α+〉is a scalar state with even parity, and spin 0, then the state (x·S)|α+〉
must also be spin zero, but have odd parity, and is thus a pseudo-scalar. Similarly, spin 1
states that are odd under parity are referred to asvectorstates, while spin 1 states that
are even under parity are referred to as pseudo-vectors or more commonly asaxial vectors.
Parity provides with a single selection rule, given by
(1−πOπαπβ)〈β|O|α〉= 0 P|α〉=πα|α〉
P|β〉=πβ|β〉
POP†=πOO (9.85)
whereπα^2 =π^2 β=π^2 O= 1.
Charge conjugation,C, maps the electron into a state with the same mass and momen-
tum, but with opposite electric charge; this state is the positron. Similarly,C maps any
particle into its anti-particle. The existence of anti-particles is required by special relativity,
