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".

Oh yes there is! Your ignorance does not influence reality's
content.

http://physics.nist.gov/GenInt/Parity/cover.html

http://pdg.lbl.gov/2005/tables/contents_tables.html
"TESTS OF CONSERVATION LAWS"

Metric theories of gravitation wth tensors are parity-symmetric.
[(+/-)x,(+/-)y,(+/-)z) and [(-/+)x,(-/+)y,(-/+)z] give identical
answers (also, Newtonian gravitation and Green's function).
Affine-teleparallel theories of gravitation with pseudotensors can be
parity-antisymmetric. [(+/-)x,(+/-)y,(+/-)z) and
[(-/+)x,(-/+)y,(-/+)z] can give different answers.

The disjoint overlap of metric and affine gravitation is the only
apparent way remaining to empirically falsify General Relativity. It
allows all prior physical observations to remain unmolested while
falsifying the mathematical structure of General Relativity and
validating the mathematical structure of affine-teleparallel
gravitation.

Edward Green wrote:

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

Oh yes there is! Your ignorance does not influence reality's
content.

http://physics.nist.gov/GenInt/Parity/cover.html

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.

Local symmetries create conservation laws through Noether's theorem. A
conserved quantity derives from each symmetry commuting with time, and
the reverse. A divergence-free current (conserved property) arises if
the Lagrangian or the action is invariant under continuous
transformation.

1) To each continuous symmetry of an action there corresponds a
conserved quantity because of the Euler-Lagrange equations of the
Lagrangian, and the reverse.
2) To each gauge symmetry of an action there corresponds an
identity among Euler-Lagrange equations of the Lagrangian, and the
reverse.

A physical system with a Lagrangian invariant with respect to the
symmetry transformations of a Lie group has, in the case of a group
with a finite (or countably infinite) number of independent
infinitesimal generators, a conservation law for each such generator,
and certain "dependencies" in the case of a larger infinite number of
generators (General Relativity and the Bianchi identities). The
reverse is true.

A symmetry can be broken explicitly - a term in the action or
equations of motion may not be invariant. A symmetry can be broken
anomalously - not all classical theory symmetries exist in the
corresponding quantum theory. Quantum field theory anomaly spoils
renormalizability. Anomaly absence in the Standard Model is crucial. A
symmetry can be broken spontaneously if it is an exact symmetry of the
equations of motion but not of a particular solution therein.
Noether's theorem holds if the symmetry is not broken explicitly.
Conservations can be relaxed in subsystems displaying reduced symmetry
(Born scattering approximation, Fermi's golden rule, Snell's law).

PARITY is unique for not being a Noetherian symmetry.

Inversion of
all coordinates is a discrete process that cannot be approximated by a
Taylor series. Parity the symmetry is linked to parity the property
by other strong correspondences. Parity is conserved by strong
interactions but commonly violated by weak interactions (including the
Weak Interaction). As gravitation is the weakest known force, one
might optimistically expect a metric (parity-conserving) vs. affine
(parity-violating) gravitation anomaly.

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).