QFT and local gauge invariance

[losang]

Why is local gauge invariance needed in qft? I read that is allows interactions whereas global gauge invariance does not but was given no reason.

 

[Maien]

I'm not sure but as far as I know QFT lagrangians are invariant under global gauge transformation which is not the case for local gauge transformation. Requiring local gauge invariance leads to introducing a gauge field which leads to the interaction term. So, interaction terms are only introduced when we require local gauge invariance!

 

[haushofer]

Why is local gauge invariance needed in qft? I read that is allows interactions whereas global gauge invariance does not but was given no reason.

Why it is needed we don't know :)

Take the case of QED. If we take a look at the Lagrangian for a free fermion and a free electromagnetic field, we see two symmetries: a gauge symmetry of the vector potential, and a phase symmetry of the fermion; we can shift the fermion field by a constant phase without altering the equation. This is a global symmetry. However, the gauge symmetry of the electromagnetic field is local; it's just the derivative of some scalar function.

So that makes you wonder: what happens if I promote the global symmetry of the fermion field to a local symmetry? In doing this, we can easily see that the derivative term in the Lagrangian spoils the invariance. This is not hard to understand; in a derivative we take the difference of fields at two different coordinates, and these two quantities transform differently. So we introduce some sort of connection; it's a correction term with which we make the Lagrangian invariant under local fermionic phase transformations (U(1) transformations) and see what happens. If we work this out, this correction term appears to transform exactly like the vector potential!

So, we introduced the interaction between electromagnetic fields and fermions by promoting a global symmetry to a local one. In general relativity we do the same thing: we need a directional derivative which transforms properly under coordinate transformations. For this we need a connection, and with some variational arguments we see that this connection is constructed out of the metric and its first derivative. The metric is just the gravitational potential in general relativity, so we could say that by demanding the equations of motion to be covariant with a dynamical space-time, we introduced the force field of gravity.

Ofcourse, in the QED case the transformations take place in some abstract inner space, while in general relativity we work in space-time. But the idea is the same.

Why this description works? I don't have a clue. But it appears that this gives a very nice way of describing interactions, also in the standard model. Because this idea can easily be extended to more complicated (Non-Abelian) transformations like SU(2) or SU(3), and these result in the weak and strong forces.

 

[Haelfix]

You can look at local gauge invariance in a number of ways (try Wiki for some detail), but I like to think of it as a postulate that happens to provide a class of field theories that satisfy experiment (the standard model).

One of the reasons its nice, is that it guarentees renormalizability of the ensuing quantum theory. It also allows you to vastly constrain the number of possible models you can write down into a small finite subclass (each of which has been studied to death and turn out to be important).

 

[nrqed]

You can look at local gauge invariance in a number of ways (try Wiki for some detail), but I like to think of it as a postulate that happens to provide a class of field theories that satisfy experiment (the standard model).

One of the reasons its nice, is that it guarentees renormalizability of the ensuing quantum theory. It also allows you to vastly constrain the number of possible models you can write down into a small finite subclass (each of which has been studied to death and turn out to be important).

By itself gauge invariance does not guarantee renormalizability, right? I mean that there is no problem in writing down gauge invariant terms of arbitrarily high dimension that are therefore non-renormalizable.

But it's true that if we keep only dimension four operators, gauge invariance build in constraints that are useful to prove renormalizability. But is it a sufficient criterion, Haelfix? I mean, if we would start with a lagrangian of a scalar coupled to a U(1) gauge boson and impose renormalizability could we reconstruct just by this argument scalar electrodynamics?

One neat thing about gauge invariance is that, from an effective field theory point of view, it "protects" the mass of the gauge boson to be driven to the cutoff scale. So if the gauge boson is massless, it will remain massless even if interactions are turned on. If the gaug boson has a mass through spontaneous symmetry breaking like in the Standard Model, loop corrections won't drive the renormalized mass to the Planck scale.

In the end, a fundamental theory should "explain" why the SM has those gauge invariances in a more satisfactory way than what we can say now.

I have always thought that ta a deeper level, gauge invariance woudl arise from some sort of deeper conenction between boson and fermion fields. People always say that SUSY is the only symmetry unifying bosons and fermions. And it's true that a SUSY transformation truly changes one type of field into the other type. But the gauge invariances of the SM also connect the two types of fields in a different way since a transformation of the gauge field must be correlated with a transformation of the other field. It's interesting that making the SUSY transformations local yields the force of gravity while making the other invariances local yield the other three forces.

Sorry for the rambling post :-)

 

[Haelfix]

Yes its true, its not a sufficient criteria. You definitely need to add some extra conditions to guarantee renormalizability.

For instance lorentz invariance, minimality of operators to dimension four terms and so forth. (See THooft/Veltmans proof of the renormalization of gauge theories).

Unfortunately its not enough to uniquely pick out the SM either, but one is getting close. I think you'd probably need some unitarity relations on the Smatrix as well and you can then minimal bootstrap the rest starting from a few given canonical interactions.. This was done in a neat little book I read back in grad school (who's name I have promptly forgotten)

As for the gauge principle itself. I completely agree, its a mystery. Why should a redundancy of description and the ensuing nonabelian representation theory be so powerful a concept and so important to modern quantum physics (or in other words, the geometrization of some internal space). People have noticed the similarity with gravity and written much about it as well, as well as the importance of topology for spin bundles as a possible clue, but of course it still remains somewhat adhoc and in need of enlightening from a more fundamental theory.