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It helps to remember that invariant quantities are seen as scalars to the transformation (they have no indices in the target space). In the other hand, covariant quantities are objects that transform in a certain way. Example: Vectors in $R^{2}$, under rotation $R_{ij}$, transform covariantly since $v'_{i}=R_{ij}v_{j}$, but it's length is invariant since ...


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Under the $U(1)$ transform, $A_n \to A_n - \nabla_R \xi_n(R),$ and using Stokes theorem it immediately follows that $\gamma$ is invariant under $U(1).$


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The fact that the theory is not gauge invariant implies that all degrees of freedom of $A_\mu$ must have physical meaning: This is not the theory of photons where only transverse degrees of freedom make sense. This way you must tackle some non-trivial issue like the negative norm associated with temporal modes. This could be avoided by adding a mass to ...


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Your issue is that this interview was not transcribed by a physicist! What he said was "Gauge Symmetry" not "Gate Symmetry". Your googling should work better now, and here is one place to start: https://en.wikipedia.org/wiki/Gauge_theory


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First of all, we are fundamentally interested in E & B. But essentially, it comes down to the fact that only E and B are physically measurable and so $\phi$ and A are mainly only considered as mathematical constructs, but this isn't always true - they can be conceptualised. I apologise that I cannot give you an immediate physical insight, but I can ...


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The vector potential has a divergence of zero; we can obtain some intuition by considering the geometry required by the divergence theorem: the volume integral of the divergence of the vector potential is zero for any volume, hence the total net flux through any surface is zero. So given your specific conditions, you can imagine geometric boundaries and ...


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One can avoid the concept of symmetry breaking in this context, to avoid "non-conservation of the particle number". People have devised way to do that, see for example http://arxiv.org/pdf/cond-mat/0105058v1.pdf. However, all these approaches gives the same results than standard Bogoliubov-like methods in the thermodynamic limit. This is not too ...


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There is no "physical aspect of this fact". The physical variables are the electric and the magnetic field, not the potentials. Introducing the potential is aesthetically and technically pleasing, but it is not necessary. A gauge symmetry is not a physical symmetry. The reason you can have a non-unique potential is that every divergence-free field such as ...



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