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Fundamental theories are gauge theories 
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#1
Jan614, 02:12 PM

P: 119

Hi folks  I was reading some (nontechnical) work by Frank Wilczek, in which he stated that any fundamental theory  that is, well behaved in the E →∞ limit  must be a local gauge theory. Does anyone know of the reasons for why this is thought to be the case?
Even sketchy remarks appreciated! Thanks! 


#2
Jan614, 04:56 PM

P: 82

I believe this is a comment about the re normalization. How parameters of the theory change as the scale (energy) changes.
Though im not sure what is meant by a fundamental theory. For example I believe one can have a field theory with global symmetries describing condensed matter systems. And the SM is known to be effective. So is there then no known correct fundamental theorem? 


#3
Jan1014, 02:15 AM

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#4
Jan1014, 03:11 AM

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P: 303

Fundamental theories are gauge theories
The reason for this is that essentially matter is described by Fermions/Spinors (spin 1/2), roughly speaking and the interactions between matter are mediated by Spin0 or Spin1 particles.
In four dimensions, spin0 particles are trivial on their own (don't interact) and so the theory needs spin1 particles. However if these spin1 particles are fundamentally massive (that is, if their mass is not generated by their interactions with another field), then the theory will contain irremovable infinities. Only if the spin1 particles are massless will there be no infinities. So we need massless spin1 particles. Finally massless spin1 particles are not well suited to being described by fields (due to them have too few degress of freedom) however. Whenever we try the field has artificial unphysical extra degrees of freedom. The fact that these degrees of freedom can be altered without affecting the physics is known gauge symmetry. So basically in four dimensions an interacting theory must be a gauge theory, because only the interaction of massless spin1 particles is infinity/divergence free. 


#5
Jan1014, 03:41 AM

P: 82

That was a good explanation DarMM, thanks.
Regarding the SM being effective, the evidence is neutrino masses and dark matter. Clearly an extension is required, even if we do not care about naturalness. There is a nice talk on this by Strumia, http://workshops.ift.uamcsic.es/WMH126/strumia.pdf. Where towards the end he discusses running the SM to Mplanck and above. 


#6
Jan1214, 03:45 AM

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#7
Jan1214, 03:55 AM

P: 82

Okay,
then, > "And the *minimal* SM is known to be effective" ;) 


#8
Jan1214, 04:51 AM

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I still don't agree, but perhaps this is due to my understanding of "effective theory".
For me an effective theory is a theory formulated in terms of nonfundamental d.o.f. which are sufficient to explain physics within a certain energy range, but which are to be replaced by new d.o.f. beyond a certain energy scale. Examples are solid stage physics (with phonons and other effective d.o.f.), chiral effective theories (with pions and nucleons), etc. What about asymptotic safety approach to quantum gravity, Connes noncommutative geometry? We know for sure that we need an UV completion of SM + gravity, but it can very well be that there are no new fundamental d.o.f. from which the known particles of the SM do emerge (as bound states or whatever) but simply a new mechanism for UV completion whereas the d.o.f. known from the SM remain "fundamental". I would not call such a theory "effective". All what I am saying is that currently we really do not know how such an UV completion will look like. 


#9
Jan1214, 08:15 PM

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#10
Jan1314, 04:07 AM

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P: 303

In the field formalism, these extra polarisations manifest as gauge degrees of freedom, so the electron also picks up a guage variance. Since the photons form the electrons Coloumb field or electric charge, which is transformed by global phase changes, these nonphysical photons induce a gauged phase. 


#11
Jan1314, 09:28 PM

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Oh,.... perhaps you're alluding to the schemes for dressing the electron field with a particular coherent photon state which causes (some of) the gauge degrees of freedom to disappear, while also giving the bare electron a Coulomb field, and banishing the QED IR divergences? 


#12
Jan1414, 04:06 AM

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I'm talking about if we do QED with ##A_{\mu}## an unphysical object, containing two extra, unitarity breaking degrees of freedom. If we do this, instead of using a constrained field, or even better loop like operators that actually represent the photon, then these unphysical photons will contribute to the Coloumb field of the electron in the theory. In other words, the theory contains electron states with unphysical Coloumb fields, unitarity breaking Coloumb fields. This Coloumb field is unphysical. Only on a restricted subspace ##\mathcal{H}_{phys}## of the space of states of the theory will the Coloumb field not contain these extra polarisation, however since the electron field is an operator over the whole space it picks up the extra gauge phase regardless. 


#13
Jan1414, 04:48 AM

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#14
Jan1414, 12:07 PM

P: 82

Hi sorry I come back into this a bit late.
What I meant was that, we must introduce at least new neutrino states. Thus, extending the SM beyond its current form without right handed guys. 


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