local gauge symmetries and particle histories

[verdigris]

A local symmetry is one in which a transformation can be performed in
one region of space without affecting another region of space.I can
change the polarization of a photon in a certain local region of space
and not affect what happens in another local region.Sometimes photon
polarizations are coupled and I could change the polarization of a
photon in one region of space and instantaneously change another,
correlated, photon polarization, over a big distance.I would suspect
that a photon,with a specific polarization,interacts with vacuum
particles in a different way if its polarization in a particular
region of space is altered.How accurate a reflection of nature can a
local symmetry be? Clearly a local symmetry can be destroyed depending
on the history of the photons involved (coupled or uncoupled).Does
this mean, in general,that local symmetries and associated
conservation laws should take into account the history of the
particles under consideration,and not just introduce particles on an
ad hoc basis?

[verdigris]

On 5 Apr, 13:17, verdigris <verywells...@yahoo.co.uk> wrote:
Clearly a local symmetry can be destroyed depending
> on the history of the photons involved (coupled or uncoupled).Does
> this mean, in general,that local symmetries and associated
> conservation laws should take into account the history of the
> particles under consideration,and not just introduce particles on an
> ad hoc basis?

Verdigris:

I would think that to distinguish a coupled photon pair from an
uncoupled pair,in a symmetry theory, would require that individual
photons have some property when they are coupled that is different
from when they are uncoupled.This property would, according to Bell's
Theorem, be a "hidden variable" and its existence would mean that
instantaneous action at a distance (Einstein called it
spukhaftefernwirkung - ghostly action at a distance - to emphasise
that he thought it was not a real phenomenon) in quantum mechanics is
wrong.

[René Meyer]

I guess the point is that in our modern understanding of quantum field
theory all interactions are designed to be gauge invariant, and thus
also coupled photons are coupled in a gauge invariant way. This
building principle of QFTs worked very well for quantum
electrodynamics, so people sticked to it. Such gauge symmetries can,
however, be spontaneously broken by special interaction potentials, in
which case the action of the theory is gauge invariant, but the vacuum
is not.

On 5 Apr., 19:24, verdigris <verywells...@yahoo.co.uk> wrote:
> On 5 Apr, 13:17, verdigris <verywells...@yahoo.co.uk> wrote:
> Clearly a local symmetry can be destroyed depending
>
> > on the history of the photons involved (coupled or uncoupled).Does
> > this mean, in general,that local symmetries and associated
> > conservation laws should take into account the history of the
> > particles under consideration,and not just introduce particles on an
> > ad hoc basis?
>
> Verdigris:
>
> I would think that to distinguish a coupled photon pair from an
> uncoupled pair,in a symmetry theory, would require that individual
> photons have some property when they are coupled that is different
> from when they are uncoupled.This property would, according to Bell's
> Theorem, be a "hidden variable" and its existence would mean that
> instantaneous action at a distance (Einstein called it
> spukhaftefernwirkung - ghostly action at a distance - to emphasise
> that he thought it was not a real phenomenon) in quantum mechanics is
> wrong.

[Igor Khavkine]

On 2007-04-05, verdigris <verywellsaid@yahoo.co.uk> wrote:
> A local symmetry is one in which a transformation can be performed in
> one region of space without affecting another region of space.I can
> change the polarization of a photon in a certain local region of space
> and not affect what happens in another local region.

Locality and causality: local changes are confined to the emanating
light cone. Whether you have something more precise in mind is hard to
decipher.

> Sometimes photon
> polarizations are coupled and I could change the polarization of a
> photon in one region of space and instantaneously change another,
> correlated, photon polarization, over a big distance.

Nothing happens instantaneously. Correlation does not equal causation.
Hence, you should avoid making deductions about causation based merely
on correlation.

> I would suspect
> that a photon,with a specific polarization,interacts with vacuum
> particles in a different way if its polarization in a particular
> region of space is altered.

The vacuum is rotationally symmetric. It does not care about
polarizations. You never explicitly state this, but it seems that you
think of photons as extended objects with independent polarizations at
every point of space-time. It is your choice whether you call something
like that a "photon", but that is not what "photon" means to physicists
in general. Change your thinking from "photon" to "electromagnetic
field". This way, your ideas will be both more clear and more comparable
to those of others. It matters not whether you want to consider a
classical or a quantum situation, the electromagnetic field is still the
electromagnetic field.

> How accurate a reflection of nature can a
> local symmetry be?

Only as accurate as the predictions of a theory incoroporating such a
symmetry. The same is true of any theory.

> Clearly a local symmetry can be destroyed depending
> on the history of the photons involved (coupled or uncoupled).Does
> this mean, in general,that local symmetries and associated
> conservation laws should take into account the history of the
> particles under consideration,and not just introduce particles on an
> ad hoc basis?

First, it is not clear either from this post or your own follow up as to
what you mean by "coupled" vs "uncoupled". So I will ignore that part.
I can't answer your question as given, because I don't understand what
precisely you are asking. However, from it, I can glimpse a common
confusion that I can try to remedy. At this point, it is best to drop
into mathematical notation.

Suppose we have an expression, say F(x). The notation is highly
schematic, x could be a number, a vector, or even a function, while F
could be a function, a linear operator, or even a differential/integral
operator. Now, introduce a transformation T, acting on x. Again, I'm
being intentionally vague about what kind of transformation T could be;
it does help to assume that it's invertible. Let x = y be a solution
of the equation F(x) = 0, it need not be unique. Let z = Ty (T applied
to y). We say that T is a symmetry of the equation F(x) = 0, if x = z is
necessarily a solution of the same equation (no matter which solution
x = y represents).

Here's the punchline. Just because T is a symmetry of F(x) = 0 and x = y
is a solution of this equation, it does not mean that Ty = y. There
could exist such a y, but that need not even be the case. Just because T
acting on y does not leave y fixed, it does not mean that T ceases to be
a symmetry of nor that x = y ceases to be a solution of F(x) = 0.

A simple example is the Laplace equation and its solutions. F would be
the laplacian operator, while x would be a scalar function. It is well
known that rigid transformations of Euclidean space leave the Laplacian
invariant, i.e. are its symmetries. So, let T be a rotation. There do
exist solutions of the Laplace equation that are rotationally invariant
(invariant under T). But there also exist solutions which are not. The
latter simply means that such a function rotated by T is also a solution
to the same equation. Although the point as expressed here may seem
trivial, in other situations it may be much more subtle.

If you are curious about conserved quantities and their relation to
symmetries in physics, you should read up on Noether's theorem and its
applications. The terms "symmetry" and "conserved quantity" are well
defined mathematically. And so is their relation, through Noether's
theorem. If you are for some reason unsatisfied with this relation, it
is unfortunately very hard to argue with a theorem. You would have to
come up with a specific physical scenario where either the theorem's
hypothesis is satisfied while its conclusion is undesired, or, on the
contrary, the theorem's hypothesis is not satisfied while its conclusion
is desired. Your comment about "particle histories", beside being vague,
does not fall into either category of observations.

Hope this helps.

Igor

[Tom Roberts]

verdigris wrote:
> A local symmetry is one in which a transformation can be performed in
> one region of space without affecting another region of space.I can
> change the polarization of a photon in a certain local region of space
> and not affect what happens in another local region.Sometimes photon
> polarizations are coupled and I could change the polarization of a
> photon in one region of space and instantaneously change another,
> correlated, photon polarization, over a big distance.

No. You cannot "change" the spin of a photon at all. What you can do:
for a coupled pair, by _measuring_ the spin of one photon you can
_discover_ what the other's must be, independent of their spatial
separation. That is _quite_different_ from "changing" either spin.

Those photons' spins are _correlated_, but correlation is not causation,
and your description in terms of "change" (causation) is invalid.


> How accurate a reflection of nature can a
> local symmetry be?

Experiments show it is not a good model for all phenomena. But for most
activities it is an exceedingly good approximation.


> Clearly a local symmetry can be destroyed depending
> on the history of the photons involved (coupled or uncoupled).

You assume that two coupled photons still obey a local symmetry. They don't.


> Does
> this mean, in general,that local symmetries and associated
> conservation laws should take into account the history of the
> particles under consideration,and not just introduce particles on an
> ad hoc basis?

No. It means that naive notions of locality are not valid in the
universe we inhabit.


> I would think that to distinguish a coupled photon pair from an
> uncoupled pair,in a symmetry theory, would require that individual
> photons have some property when they are coupled that is different
> from when they are uncoupled.This property would, according to Bell's
> Theorem, be a "hidden variable" and its existence would mean that
> instantaneous action at a distance (Einstein called it
> spukhaftefernwirkung - ghostly action at a distance - to emphasise
> that he thought it was not a real phenomenon) in quantum mechanics is
> wrong.

Quantum mechanics accurately describes the experiments, with no such
hidden variables, and no "spooky action at a distance". The situation is
far more subtle than you describe.


Tom Roberts

[Aaron Denney]

On 2007-04-07, Tom Roberts <tjroberts137@sbcglobal.net> wrote:
> verdigris wrote:
>> A local symmetry is one in which a transformation can be performed in
>> one region of space without affecting another region of space.I can
>> change the polarization of a photon in a certain local region of space
>> and not affect what happens in another local region.Sometimes photon
>> polarizations are coupled and I could change the polarization of a
>> photon in one region of space and instantaneously change another,
>> correlated, photon polarization, over a big distance.
>
> No. You cannot "change" the spin of a photon at all. What you can do:
> for a coupled pair, by _measuring_ the spin of one photon you can
> _discover_ what the other's must be, independent of their spatial
> separation. That is _quite_different_ from "changing" either spin.

Except, of course, that the Bell inequalities seem to mean that there is
no underlying property that is "merely" being discovered.

> Those photons' spins are _correlated_, but correlation is not causation,
> and your description in terms of "change" (causation) is invalid.

Quite right.

--
Aaron Denney
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