Uncle Al wrote:

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

"Between Christmas of 1956 and New Year's Day, the first exciting
results emerged from a difficult but fundamental scientific experiment
at the National Bureau of Standards (NBS) in Washington, DC [currently
the National Institute of Standards and Technology (NIST)]. The
experiment showed, strikingly and convincingly, that in at least one
fundamental physical process, our world is distinguishable from its
mirror image."

Thus far I have no quarrel.

"Physicists had long assumed the opposite. They constructed their
theories so as to ensure that the corresponding mathematical property,
called parity, remains unaltered - is conserved - in all subatomic
processes. Thus this experiment brought about the fall of parity from
its exalted position alongside such well conserved physical quantities
as energy, momentum, and electric charge."

Well then, Al, either this is simply another example of poor language,
or I am indeed mistaken. I have no doubt many professionals _say_
"parity is conserved" or "parity is not conserved", but I think they
speak this way out of habit rather than reflection.

...

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.

N.B. _continuous_ transformation

Charge reversal, time reversal and parity reversal are not continuous
transformations, and as far as I know no conserved quantities attach to
discrete symmetries through Noether's theorem.

I know just enough to be dangerous. ;-)

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.

You should have read your own crib more carefully. :-)

And no, I do not think it is "unique". Charge reversal, time reversal
.... and for that matter any of the finite groups are not "Noetherian"
symmetries.

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.

As usual, you have loaded your canon with every bit of scrap you found
lying on the field, and some of it I do not recognize nor can I answer.
It is an old technique. Nonetheless, I have the conviction of a
simple conceptual clarity: A physical law is invariant with respect to
a certain symmetry if the corresponding change of variables leaves the
form of the law unchanged. Now, it might be that every physicist in
the world speaks of a physical law or interaction having this property
with respect to coordinate reflection as "conserving parity", but I
would respectfully point out, sir or ma'am, that I honor the weight of
culture, but your habitual language is inexact.

By the way, the Lorentz symmetry group is a continuous symmetry group,
so we might expect that it _is_ a "Noether symmetry". So what is the
corresponding conserved quanity?