Gauge Theory Explained: Fundamentals, Scope and Postulates

[DiracPool]

I've been reading up on gauge theory and it isn't easy. Can someone give me an easy summary of its fundamental scope and postulates without too much math. It seems really important insofar as it defines itself as something of a "parent" theory to most of the leading cosmological models of the day, such as string theory, standard model, loop quantum gravity, etc. I just want to get a feel of what it is and WHY it is, how did this thing that seems to have been founded by a search for symmetry become the parent of all theoretical physics models?

 

[DimReg]

Gauge theory is as best as I can put it, the study of physical systems whose mathematical models have more degrees of freedom than are physical. However, that isn't quite the whole story. If you study spin 1 fields in QFT, you'll find that you can't avoid negative norm states (which are bad because they imply negative probabilities). You can make them "go away" by adding some principle that makes these negative norm states unphysical, which is typically done using a symmetry (that is, you assert that if two states are related by a symmetry transformation they are the same physically).

So attempting to talk about spin 1 particles forces you to have a symmetry in your theory. This symmetry forces you to write a lagrangian of a certain form (one that produces the same physics when acted on by that symmetry). Once you have that lagrangian, you "simply" proceed with the rest of your theory. That's how you get gauge theory.

It's scope is all (or at least most) of physics. For example, the standard model has the symmetry group of SU(3) X SU(2) X U(1). As to why it's so important, it's because for whatever reason, there are spin 1 particles in the world. If there was only spin 0 and spin 1/2, such symmetry wouldn't be needed.

 

[friend]

It's scope is all (or at least most) of physics. For example, the standard model has the symmetry group of SU(3) X SU(2) X U(1). As to why it's so important, it's because for whatever reason, there are spin 1 particles in the world. If there was only spin 0 and spin 1/2, such symmetry wouldn't be needed.

I wonder how we discovered these symmetries U(1)XSU(2)XSU(3)? Were they necessary in order to solve the partial differential equations involving the wavefunctions of QED and QCD? Or is it more the case that these symmetries were simply recognized in the pattern of data discovered in experiments?

 

[marcus]

I wonder how we discovered these symmetries U(1)XSU(2)XSU(3)? Were they necessary in order to solve the partial differential equations involving the wavefunctions of QED and QCD? Or is it more the case that these symmetries were simply recognized in the pattern of data discovered in experiments?

Wikipedia has quite a substantial article on gauge theory, including it's history from the first dim glimmerings of the idea. It covers a lot in a few paragraphs--it might interest you (and others!) if you haven't seen it.

http://en.wikipedia.org/wiki/Gauge_theory

 

[fzero]

I wonder how we discovered these symmetries U(1)XSU(2)XSU(3)? Were they necessary in order to solve the partial differential equations involving the wavefunctions of QED and QCD? Or is it more the case that these symmetries were simply recognized in the pattern of data discovered in experiments?

The U(1) gauge invariance of electromagnetism seems to have been understood by Maxwell after arriving at his equations (http://arxiv.org/abs/hep-ph/0012061 would probably explain the history). The SU(2) and SU(3) symmetries were deduced from the data and later understood in terms of the gauge theory framework. In the case of the SU(2) weak interactions, the couplings between the up and down-type quarks, as well as the electron to the neutrino, led to specific charge assignments of weak isospin and hypercharge to the elementary particles. Later, Glashow, Weinberg and Salam discovered how to understand this in terms of a spontaneously broken gauge symmetry.

In the case of SU(3) color, it was recognized that certain baryons were composed of 3 of the same flavor of quark. Since quarks have to be fermions, and the Pauli exclusion principle requires a wavefunction to be antisymmetric under fermion exchange, it was understood that quarks must have a new quantum number (see http://en.wikipedia.org/wiki/Quantum_chromodynamics#History). It took another decade and additional data on bound states of the charm quark before the gauge theory of the strong interactions could be rigorously tested.

 

[friend]

The U(1) gauge invariance of electromagnetism seems to have been understood by Maxwell after arriving at his equations (http://arxiv.org/abs/hep-ph/0012061 would probably explain the history). The SU(2) and SU(3) symmetries were deduced from the data and later understood in terms of the gauge theory framework.

Thank you. Your reference was interesting reading.

What does, "and later understood in terms of gauge theory framework" mean? This seems to imply a gauge invariance of some formulation (partial differential equations). That's the part of the story that I'm missing. I could have those details for QED. But I don't know about QCD.

 

[haushofer]

Well, SU(2)XU(1) has four generators, which gives upon gauging four gauge fields. This corresponds after spontaneous symmetry breaking to one massless photon and three massive vector bosons. SU(3) has eight generators, and thus gives upon gauging eight gauge fields. This corresponds to eight massless gluons.

That structure has to be measured. We can't calculate how many fundamental forces we have and how many gauge bosons. Gauge theory gives the structure, but the specific choice of groups has to be found by measurement.

 

[friend]

Well, SU(2)XU(1) has four generators, which gives upon gauging four gauge fields. This corresponds after spontaneous symmetry breaking to one massless photon and three massive vector bosons. SU(3) has eight generators, and thus gives upon gauging eight gauge fields. This corresponds to eight massless gluons.

That structure has to be measured. We can't calculate how many fundamental forces we have and how many gauge bosons. Gauge theory gives the structure, but the specific choice of groups has to be found by measurement.

I remember, way back in school, the professor showing us how to come up with quantized angular momentum in terms of j(j+1), or something like that, by solving some partial differential equations using special function, Bessel functions or Hankel functions or something that. But recently I've seen how they come up with the j(j+1) for angular momentum simply on the grounds of SU(2). So that makes me wonder where the SU(2) symmetry comes from, from solving PDEs of from just symmetry fitting the data.

Although, I could be conflating subjects here. My memory is a bit rusty on those special functions.

 

[fzero]

I remember, way back in school, the professor showing us how to come up with quantized angular momentum in terms of j(j+1), or something like that, by solving some partial differential equations using special function, Bessel functions or Hankel functions or something that. But recently I've seen how they come up with the j(j+1) for angular momentum simply on the grounds of SU(2). So that makes me wonder where the SU(2) symmetry comes from, from solving PDEs of from just symmetry fitting the data.

Although, I could be conflating subjects here. My memory is a bit rusty on those special functions.

As a mathematical object, SU(2) simply exists. We can define it just by giving the elements and their multiplication table (for the group) or commutation relations (for the algebra). But the connection with physics is in systems for which a given group is a symmetry of the system. So, for instance, for a spherically symmetric potential (the "central-force problem"), some set of differential equations (Maxwell, Einstein, Schrodinger...) will have a rotational symmetry. Solutions will then be given in terms of spherical harmonics, which can be viewed as a particular representation of the rotation group in 3d.

Now, the SU(2) of the weak interactions is completely independent from rotations in 3d space (it is an internal symmetry). Indeed it was necessary to deduce the symmetry in an indirect way from data on particle interactions. It was noted that, in order to explain how elementary particles were coupled by the weak interaction to other particles, we should group the left-handed component of the electron and neutrino into an SU(2) doublet

$$\begin{pmatrix} e_L \\ \nu_{eL} \end{pmatrix}$$

and similarly for the up and down quarks

$$\begin{pmatrix} u_L \\ d_{L} \end{pmatrix} .$$

Because this is an external, rather than a spacetime symmetry, it is not a symmetry of a differential equation in quite the same way as the rotation group in a central-force problem.

It might be worthwhile to review how SU(2) appears in QM, both for spin and orbital angular momentum, before you tackle gauge theories.

 

[Naty1]

Hey DiracPool,
I was just reading in Wikipedia about gauge theory, something I know virtually nothing about...did not get very far yet...

but, found some insights here: http://en.wikipedia.org/wiki/Coordinate_condition

and here they even have an illustration/picture of it's application

http://en.wikipedia.org/wiki/Gauge_fixing

In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom...

and at the bottom of this article:

http://en.wikipedia.org/wiki/Gauge_theory

are dozens of related links...haven't gotten to those yet...

good luck.