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Feynman diagrams are more than just the Lagrangian. They can be acquired by expanding the path integral of the theory into a perturbative series. There is a priori no reason to assume that all quantities needed in order to produce sensible results are consistent with gauge invariance. One possible issue is the problem of regularization: the way your ...


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You may always promote "couplings constants" (charge, mass, etc...) to fields. Now, as a physicist, you need to make some contact with reality. So you have to tell why and which field you are using (for instance the Higgs field (up to a constant), which has a $SU(2)$ charge, is used to replace a constant mass coupling in the interaction $m (\bar e_R e_L + ...


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The Einstein equivalence principle states : The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime. Emphasis added. Note that this principle has done well in explaining quite a few things about gravity. So there is no a priori reason why you ...


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Every Lie group has a set of generators, and typically a group element is found by exponentiate (linear combinations) of these generators. Since the fundamental definition of say $SU(N)$ [similarly $SO(N)$] is something like The group of unitary (orthogonal), $n$ by $n$ matrices with unit determinant then, the fundamental representation is given by ...


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Gravity can be seen as a gauge theory of the Lorentz group (which acts on the tangent space). These was pointed out by Kibble and Sciama during the 50s and 60s. As John said before, it's better seen in terms of differential forms. Another reference you might find interesting is the Lecture notes on Chern-Simons gravity by Jorge Zanelli (available in ...


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Anomalies (not anamolies) are a whole subject whose basics are covered by one or several chapters of almost any good enough quantum field theory textbook so it's counterproductive to retype this whole chapter here. But generally, in quantum field theory, anomalies are quantum mechanical effects breaking symmetries that exist in the classical theory – ...


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As i can remember, for the case of classical electrodynamic the d.o.f counting start after the assumption of the Bianchi Identity, and at the end the desired result came all from the gauge freedom. In fact $\partial_{[\mu}F_{\nu\alpha]}=0$ only choose a suitable form for $F_{\mu\nu}$, for example $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$$ After ...


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You can think of diff as bianchi id. The additional 4 dof is killed by the fact that 4 of the 4 of the EFE are constraints.


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Another take on gauge theories, to add to ACuriousMind's answer: as well as adding degrees of freedom which allow greater wriggle room to bring a broader class of solution techniques to bear, a gauge theory is a way for a theorist to encode experimentally observed symmetries into the a candidate theory. You might, for example, know from the experimental ...


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PhotonicBoom has already provided a nice overview of the basic idea behind gauge theories, let me lay it on a bit thicker with the abstraction: A gauge theory is a theory that has a local gauge symmetry induced by a gauge group $G$, which is required to be a Lie group. Now what do we mean by that? Let $\Sigma$ be our spacetime (of arbitrary dimension and ...


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You are basically right. A gauge theory is a field theory that leaves the equations of motion invariant when you transform the coordinates. It gives physicists the ability to to introduce arbitrary degrees of freedom to play with and simplify problems, as long as the physical quantities remain the same. For example in Electrodynamics you can redefine the ...


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The weak equality $f \approx 0$ means, that we first must evaluate all of the Poisson brackets of the theory (the equations of motion etc.) and only after that we may set $f$ to zero. It's because the hamiltonian doesn't contain info about primary constraints (the good example is electrodynamics), and so it doesn't contain info about the secondary ...


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Alright, let's go on a thrilling tour through the theory of representations. Notation in the following is the same as in the OP, except that we call the trivial representation $\boldsymbol{1}$, as is canon. We start from my expression in the question $$ I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \sum_\rho \sum_{\mu = ...


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I wondered about this one myself a while back. I'm not absolutely positive about this but it is definitely in the ballpark. Here's what I know for the background: I believe the first paper on exotic spheres in physics was by Witten [Commun. Math. Phys., 100, 197–229 (1985)] and centered around the idea that exotic spheres can be interpreted as ...


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The formulation you seek is gauge theory. It is not completely analogous to changing the metric of spacetime, but many similarities can be seen. In this, we take as our starting point a certain gauge group $G$ (In the case of EM, $\mathrm{U}(1)$), which will induce symmetries of our theory, just as the Lorentz group of special relativity is the symmetry of ...


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Not sure to answer properly, but if I remember, using the Dirac bracket allowing you to get rid off the second class constraints and deal at the end only with first class constraints. And still at the end, you consider only weak equality, no =.


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Comment to the question (v2): According to Ref. 1, the weak equality symbol $\approx$ usually means equality modulo all constraints: primary, secondary, tertiary, $\ldots$, constraints. (or in Dirac's classification) first and second class constraints. References: M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994; p. 13.


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Note that "$v_y$" is not velocity. Physically it is the local curvature of the Fermi surface. Also it is possible to treat one amongst the 4 constants (sic) $\eta, v_x, v_y, e$ as an overall scale and drop it from the action. Here I do this with "$v_x$", and write the propagator as $$ G(k)^{-1} = i\eta k_0 + k_x + C k_y^2,$$ where $\eta$ and $C$ are ...


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The names of these creatures are a true mess and there are mainly two independent notation schemes: the mathematical and the physical one. Let $P \to M$ be a $G$-principal bundle. Then $G$ is called the structure group by mathematicians and the gauge group by physicians The (infinite-dimensional) group of automorphism of $P$, or equivalently the group of ...


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In physics one tends to write (for a Yang-Mills field), $A_{\mu}^i$, where $\mu$ is the spacetime index and $i$ is the `group' index. To be more specific, it means that $A_{\mu}$ take values on (i.e., is contracted with the generators of) a Lie algebra, $$A_{\mu} = A_{\mu}^i T^i = A_{\mu}^i (T^i)_{mn}, $$ where in the las equality the explicit matrix indices ...


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This is not quite a complete answer, but more of an overly large comment on terminology. The definitions from nLab don't agree with Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily: Advanced Classical Field Theory, which I'll summarize briefly: The authors call the group $G$ of a (principal) $G$-bundle the structure group. This is standard ...



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