Edward Green wrote:

Some characterisitic language used in the description of symmetries
existing or lacking in physical law seems to be rather misleading. For
example:

"The essence of Charge Parity (CP) is the concept of
symmetry. Both C and P are symmetries that are
conserved in most particle interactions."

"C represents swapping the electric charges of all the
sub-atomic particles in an interaction; in other words,
swapping particles and antiparticles. P is called parity
and it corresponds to looking in a mirror that reverses
all three spatial co-ordinates."

http://news.bbc.co.uk/1/hi/sci/tech/1330190.stm

The symmetries are said to be "conserved", but the conservation of a
physical quantity is not implied; there is no conserved physical
quantity called "parity". It so happens electric charge _is_ a
conserved quantity, but this merely adds to the confusion. What is
meant is that physical law is either invariant in form, or not, under
the reversal of charge or coordinates.

In QM a parity transformation is given by an operator P acting on the
Hilbert space H of the theory. The fact that P^2 = 1 implies that the
Hilbert space splits up as a direct sum of eigenspaces of P with
eigenvalues +-1. A state in one of these eigenspaces is said to have
parity charge +-1 depending on its eigenvalue.

A system has parity symmetry iff the Hamiltonian is invariant under
parity, i.e.:

PHP^(-1) = H

This is equivalent to [P,H] = 0. If this holds then the Heisenberg
equation for P is:

dP/dt = [P,H] = 0

i.e. P is a conserved quantity.

Or if you prefer the Schrodinger picture, note that [P,H] = 0 is
equivalent to the statement that e^(-iHt) P e^(iHt) = P and so given a
state |psi(t_0):

e^(-iHt) P e^(iHt) |psi(t_0) = P |psi(t_0)

i.e.

P |psi(t_0+t) = e^(iHt) P |psi(t_0)

In particular if |psi(t_0) has parity charge n = +-1 then:

P |psi(t_0+t) = e^(iHt) n |psi(t_0)
= n |psi(t_0+t)

so the final state |psi(t_0+t) also has charge n. Therefore if a given
reaction starts with say a state with parity charge 1 and ends with
parity charge -1 then you can rule out this reaction (as long as your
system is parity invariant).