Gauge transformation preserves what?

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In General Relativity, "gauge" transformations are basically coordinate transformations which preserve length.

In Electroweak and the gauge forces like EM.. what are being preserved? I forgot my lessons before and would like to refresh.
 
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  • #2
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In General Relativity, "gauge" transformations are basically coordinate transformations which preserve length.

In Electroweak and the gauge forces like EM.. what are being preserved? I forgot my lessons before and would like to refresh.
In EM, the basic preserved quantity is the EM field ##F_{\mu\nu}##, which implies that ##F^{\mu\nu}F_{\mu\nu}## is preserved too. In non-Abelian Yang-Mills theories this generalizes to the fact that ##{\rm tr}F^{\mu\nu}F_{\mu\nu}## is preserved.
 
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In EM, the basic preserved quantity is the EM field ##F_{\mu\nu}##, which implies that ##F^{\mu\nu}F_{\mu\nu}## is preserved too. In non-Abelian Yang-Mills theories this generalizes to the fact that ##{\rm tr}F^{\mu\nu}F_{\mu\nu}## is preserved.
Did you mean what is preserved is the gauge field (which came from the fact laws of physics be invariant under "local" gauge transformations)?

In General Relativity.. you can make the length not preserved.. I learnt :"The gauge transformation involved here is the Weyl transformation, which is a shape-preserving transformation (hence the name Shape Dynamics). Shape-preserving means that angles are unchanged, but overall scale can change.

Having a theory which is invariant under these transformations is attractive because it means that the theory is no longer sensitive to the choice of length or time coordinates.".

In the Standard Model.. how do you create other forms of gauge transformations akin to Weyl transformation in GR that can make the say EM field became no longer preserved?
 
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Did you mean what is preserved is the gauge field (which came from the fact laws of physics be invariant under "local" gauge transformations)?
For EM gauge field, yes. For non-Abelian Yang-Mills gauge field, no.

In the Standard Model.. how do you create other forms of gauge transformations akin to Weyl transformation in GR that can make the say EM field became no longer preserved?
It's easy to construct such transformations by hand, but I don't think that such transformations have any physical relevance in the Standard Model. One obvious and trivial example is a transformation ##A_{\mu}\rightarrow A_{\mu}+\xi_{\mu}##, where ##\xi_{\mu}\neq \partial_{\mu}\lambda##.
 
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For EM gauge field, yes. For non-Abelian Yang-Mills gauge field, no.


It's easy to construct such transformations by hand, but I don't think that such transformations have any physical relevance in the Standard Model. One obvious and trivial example is a transformation ##A_{\mu}\rightarrow A_{\mu}+\xi_{\mu}##, where ##\xi_{\mu}\neq \partial_{\mu}\lambda##.
If the EM field is no longer preserved.. can it become non-local? What other possible physical interpretations can it be if the em field is no longer preserved?
 
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If the EM field is no longer preserved.. can it become non-local? What other possible physical interpretations can it be if the em field is no longer preserved?
What do you mean by non-local?
 
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Em field as gauge field came about because spacetime is not non-local. I know a "local" gauge transformation can vary from point to point in spacetime.

So if you can produce other variables akin to Weyl transformation in GR.. can you recover the non-locality (influence that doesn't propagate point to point)? Note that in Weyl transformation in GR (specifically the concept of Shape Dynamics).. it's the angles that are preserved that can change the length.. so in EM field.. what is the that analogy to angles that you can change and why doesn't it have physical consequence?
 
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Note that in Weyl transformation in GR (specifically the concept of Shape Dynamics).. it's the angles that are preserved that can change the length.. so in EM field.. what is the that analogy to angles that you can change and why doesn't it have physical consequence?
In the geometrical interpretation of gauge theories the manifold is not the spacetime, but a fibre bundle that locally looks like a product of spacetime and another space. That other space does not have a simple physically intuitive interpretation, so I'm afraid that a simple intuitive answer to your question does not exist.

EDIT: Perhaps an intuitive interpretation is possible in some versions of string theory, where that other space can be related to additional dimensions of spacetime.
 
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In the geometrical interpretation of gauge theories the manifold is not the spacetime, but a fibre bundle that locally looks like a product of spacetime and another space. That other space does not have a simple physically intuitive interpretation, so I'm afraid that a simple intuitive answer to your question does not exist.

EDIT: Perhaps an intuitive interpretation is possible in some versions of string theory, where that other space can be related to additional dimensions of spacetime.
Are you familiar with Barbour Shape Dynamics? see: http://discovermagazine.com/2012/mar/09-is-einsteins-greatest-work-wrong-didnt-go-far

In GR. You have shape (angles), length and time.. you can make gauge transformations out of any of them.

I was just asking that in electromagnetism, what are the variables that can be changed too? I wasn't talking about the geometrical interpretation of gauge theories.. but the variables that can be changed. See or read the article first if you have time if you still don't know what I'm describing...
 
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In General Relativity, "gauge" transformations are basically coordinate transformations which preserve length.

In Electroweak and the gauge forces like EM.. what are being preserved? I forgot my lessons before and would like to refresh.
A gauge symmetry implies that there's some reduncy in the equations, i.e., a physical situation is equivalently described by different mathematical objects.

In the case of classical electromagnetism the physical system is completely described by the electromagnetic field components ##(\vec{E},\vec{B})## or equivalently with the manifestly covariant Faraday tensor components ##F_{\mu \nu}##.

Often, it's more convenient to use the electromagnetic potentials, or in the covariant formalism the four-vector potential ##A^{\mu}##. Now, since
$$F_{\mu \nu}=\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu},$$
two vector potentials just differing in a four-gradient of a scalar field are physically indistinguishable and must be identified as one and the same field, i.e., ##A_{\mu}## and any "gauge transformed field"
$$A_{\mu}'=A_{\mu}+\partial_{\mu} \chi$$
for any scalar field ##\chi## is identical with ##A_{\mu}##.
 
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In General Relativity, "gauge" transformations are basically coordinate transformations which preserve length.

In Electroweak and the gauge forces like EM.. what are being preserved? I forgot my lessons before and would like to refresh.
Phase invariance. Take the Dirac lagrangian and change psi by a phase: psi -> psi exp(ia(x)) where a(x) is some arbitrary function of the coordinates. When you take the derivative, you do not get the original Dirac lagrangian back. You have an additional term. That term destroys the phase invariance of the theory. However, by repeating the procedure starting with this new form of the dirac lagrangian containing this extra term, you can force the phase to be invariant at the cost of this additional term, which turns out to be the electromagnetic field. The form you end up with is phase invariant and is the qed lagrangian. It works the same way for the weak interaction and strong interaction except that the phases are not simple scalars.

Phases are not observables in quantum mechanics so the equations must be invariant under a change of phase.
 
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  • #12
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Phase invariance. Take the Dirac lagrangian and change psi by a phase: psi -> psi exp(ia(x)) where a(x) is some arbitrary function of the coordinates. When you take the derivative, you do not get the original Dirac lagrangian back. You have an additional term. That term destroys the phase invariance of the theory. However, by repeating the procedure starting with this new form of the dirac lagrangian containing this extra term, you can force the phase to be invariant at the cost of this additional term, which turns out to be the electromagnetic field. The form you end up with is phase invariant and is the qed lagrangian. It works the same way for the weak interaction and strong interaction except that the phases are not simple scalars.

Phases are not observables in quantum mechanics so the equations must be invariant under a change of phase.
I understood this additional term being electromagnetic field thing as explained in great detail in the book "Deep Down Things". But it didnt talk abt General Relativity. So i guess in GR there is none of this "additional term" thing being the coordinate or length? So the Standard Model gauge uses the true gauge while in General Relativity.. its like a wanna be gauge or fake one? how do you describe the.difference between Standard model.gauge vs GR gauge in professional terms?
 
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  • #13
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I understood this additional term being electromagnetic field thing as explained in great detail in the book "Deep Down Things". But it didnt talk abt General Relativity. So i guess in GR there is none of this "additional term" thing being the coordinate or length? So the Standard Model gauge uses the true gauge while in General Relativity.. its like a wanna be gauge or fake one? how do you describe the.difference between Standard model.gauge vs GR gauge in professional terms?
Diffeomorphisms are not genuine gauge tranformations, but I don't think the difference matters to your question. In the standard model, the field strength tensor comes from the commutator of the gauge covariant derivatives, D_u = d_u -ieA_u, which gives you the field strength tensor F_uv, or in geometric language, the curvature. In general relativity, you have something similar: D_uV^a = d_u V^a + C^a_uv V^v. You get the riemann curvature tensor from the commutator. In gauge theory, the field A_u is the vector potential In general relativity, that role is played by the connection coefficients. The difference of course is that the curvature in the electromagnetic case is not a spacetime curvature. If that isn't the answer to your question, I am unsure of what you are asking.
 
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I understood this additional term being electromagnetic field thing as explained in great detail in the book "Deep Down Things". But it didnt talk abt General Relativity. So i guess in GR there is none of this "additional term" thing being the coordinate or length? So the Standard Model gauge uses the true gauge while in General Relativity.. its like a wanna be gauge or fake one? how do you describe the.difference between Standard model.gauge vs GR gauge in professional terms?
It is possible to produce a theory of gravity (Teleparallel gravity) by 'gauging' the generators of the translation group. See for instance the attached.
 

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  • #15
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Diffeomorphisms are not genuine gauge tranformations, but I don't think the difference matters to your question. In the standard model, the field strength tensor comes from the commutator of the gauge covariant derivatives, D_u = d_u -ieA_u, which gives you the field strength tensor F_uv, or in geometric language, the curvature. In general relativity, you have something similar: D_uV^a = d_u V^a + C^a_uv V^v. You get the riemann curvature tensor from the commutator. In gauge theory, the field A_u is the vector potential In general relativity, that role is played by the connection coefficients. The difference of course is that the curvature in the electromagnetic case is not a spacetime curvature. If that isn't the answer to your question, I am unsure of what you are asking.
I got it thanks. If u have time amidst the hurricane nearby.. pls read this... http://discovermagazine.com/2012/mar/09-is-einsteins-greatest-work-wrong-didnt-go-far so you understand the context of my question. You see... in General Relativity, you can make new gauge transformation which is called the Weyl transformation, which is a shape-preserving transformation (hence the name Shape Dynamics). Shape-preserving means that angles are unchanged, but overall scale can change. And the theory is no longer sensitive to the choice of length or time coordinates as Kim put it. The "gauge" transformations in General Relativity are basically coordinate transformations which preserve length. Weyl transformations are length-changing transformations. Now in basic electrodynamic or electroweak theory.. there is no other choices where you can create another gauge akin to using another gauge like Weyl transformation in GR?
 
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It is possible to produce a theory of gravity (Teleparallel gravity) by 'gauging' the generators of the translation group. See for instance the attached.
If space time is emergent and there are more elementary ingredients such as perhaps the spin networks in loop quantum gravity.. can you gauge the more elementary ingredients so the theory is no longer sensitive to angles, coordinate or time? Is there existing model for this?
 
  • #17
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If space time is emergent
This sounds like another topic.

can you gauge the more elementary ingredients so the theory is no longer sensitive to angles, coordinate or time?
I don't understand this. "Gauge the ingredients"?
 
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I am reading "Deep Down Things" also. I had this impression that the local invariance the gauge terms are meant to repair in the Wave Equation included Lorentz Invariance.

I was then blown away by the intro to Gauge Fixing because of how EM (and the other gauge symmetry groups) seemed to emerge from a case where some poor unit of space-time is having a dilemma about it's metric i.e. which observer is observing it's observables - the regular one or some boosted one.

So I am still struggling with the section later in the book on Higgs later because I was thinking the (many-body) GR consistent wave equation was the problem gauge-fixing kind-of solved to begin with.

How could a unit of space-time be having such a dilemma? Well, I don't seem to be the only one with that confusion.
I haven't tried to Grok the second paper yet and I need to find out what the "Gribov" problem is but it sounds relevant. First paper is wild.


https://arxiv.org/abs/1808.05842
On the possibility of laboratory evidence for quantum superposition of geometries
Marios Christodoulou, Carlo Rovelli
(Submitted on 17 Aug 2018)
We analyze the recent proposal of measuring a quantum gravity phenomenon in the lab by entangling two particles gravitationally. We give a generally covariant description of this phenomenon, where the relevant effect turns out to be a quantum superposition of proper times. We point out that measurement of this effect would count as evidence for quantum superposition of spacetime geometries. This interpretation addresses objections appeared in the literature. We observe that the effect sheds light on the Planck mass, and argue that it is very plausibly a real effect.


https://arxiv.org/abs/1809.05093
Switching quantum reference frames in the N-body problem and the absence of global relational perspectives
Augustin Vanrietvelde, Philipp A Hoehn, Flaminia Giacomini
(Submitted on 13 Sep 2018)
Given the importance of quantum reference systems to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of quantum reference systems, which is valid in both fields. Here, we continue with such a unifying approach, begun in arxiv:1809.00556, whose key ingredients is a gravity-inspired symmetry principle, which enforces physics to be relational and leads, thanks to gauge related redundancies, to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure turns out to be the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. Quantum reference frame switches thereby amount to a symmetry transformation. In the quantum theory, they require a transformation that takes one from the Dirac to a reduced quantum theory and we show that it amounts to a trivialization of the constraints and a subsequent projection onto the classical gauge fixing conditions. We illustrate this method in the relational N-body problem with rotational and translational symmetry. This model is particularly interesting because it features the Gribov problem so that globally valid gauge fixing conditions are impossible which, in turn, implies also that globally valid relational frame perspectives are absent in both the classical and quantum theory. These challenges notwithstanding, we exhibit how one can systematically construct the quantum reference frame transformations for the three-body problem.
 
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