# Tag Info

6

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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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 ...

3

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 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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The terms "Landau gauge" and "Feynman gauge" (among others) were introduced by Bruno Zumino. I accidentally learned about it an hour ago from David Derbes http://motls.blogspot.com/2014/06/bruno-zumino-1923-2014.html?m=1 in this blog post about a sad event, Bruno Zumino's death a week ago. David Derbes wrote: I met Bruno Zumino at the Scottish ...

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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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Hints: The left-hand side $$-\frac{1}{2}g^2f^{abc}f^{cde}\left(A_{\mu}^{b}c^{d}c^{e}+{\rm cycl}(b,d,e)\right)$$ of eq. (16.47) can be relabelled as $$-\frac{1}{2}g^2\left( f^{abc}f^{cde}+{\rm cycl}(b,d,e)\right)A_{\mu}^{b}c^{d}c^{e}.$$ P&S assume that the structure constants $f^{abc}$ are totally antisymmetric, cf. text below eq. (15.79).

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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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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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I admit I am a bit confused by your terminology, but here is how I learned it: Let $P$ be a $G$-principal bundle and $\Sigma$ a spacetime. gauge group: The fibers of the $G$-principal bundle over the spacetime, i.e. the group $G$. (Local) group of gauge transformations: The group of diffeomorphisms $t : P \rightarrow P$, which are fiber-preserving and ...

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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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