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.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.
For EM gauge field, yes. For non-Abelian Yang-Mills gauge field, no.Did you mean what is preserved is the gauge field (which came from the fact laws of physics be invariant under "local" gauge transformations)?
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##.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?
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##.
What do you mean by non-local?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?
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.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.
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 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 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.
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 want to 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.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 want to 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.
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
can you gauge the more elementary ingredients so the theory is no longer sensitive to angles, coordinate or time?