Is gauge theory applicable to finite-dimensional Lie groups?

[Atakor]

Hello,


What about gauging discrete groups ?
(C, P, T (??), Flavour Groups, Fermionic number symmetry...)

 

[xepma]

Exactly what do you mean by "gauging a group"?

If you're talking about discrete gauge theory in the sense that the gauge group is a descrete group, then yes, such theories are around. It's even possible to define it on a lattice -> lattice gauge theory. I think there's an introductory book by J. Smit about this.

 

[timur]

If I understood your question correctly, you cannot "gauge" discrete groups because your want the gauge to be continuous and the only continuous function (whatever that means) having values from a discrete group on a connected space is a constant function, so it is just same as having "global gauge".

 

[Atakor]

Hi,

If you're talking about discrete gauge theory in the sense that the gauge group is a descrete group, then yes, such theories are around. It's even possible to define it on a lattice -> lattice gauge theory. I think there's an introductory book by J. Smit about this.

Yes, that what I mean.. I'm sorry if the my question was ''enigmatic''.
I mean how to gauge a discrete group, in the sens to make its parameters position-dependent.
usually in particle physics.. as far as I know, only continuous symmetries (SU(N),U(1)..) are
gauged.
I tired to do some research in arxiv but all what I found was some work done in the framework of non-commutative geometry.
Can you develop more your answer please.. I don't really understand the physical interpretation of doing this.
Will we still have gauge bosons ? for example..
We can take the simple group Z₂ for example to make things clear.
What does it mean to make permutations of two objects (two particles ?) local ?
what kind of interaction will we generate in that case ? (if any.)

I'll try to find that book.

If I understood your question correctly, you cannot "gauge" discrete groups because your want the gauge to be continuous and the only continuous function (whatever that means) having values from a discrete group on a connected space is a constant function, so it is just same as having "global gauge".

well.. I don't understand what you mean but I think you have in mind a discrete group without spatial dependence (hence the absence of a continuous mapping)..I think.Thanks guys.

 

[timur]

well.. I don't understand what you mean but I think you have in mind a discrete group without spatial dependence (hence the absence of a continuous mapping)..I think.

My point is that you cannot have continuous (nontrivial) spatial dependence with discrete groups. The lattice gauge theory above mentioned is not a theory with discrete gauge group, rather it is a theory with discrete spacetime.

 

[Atakor]

I just browsed the book of J. Smit.. he doesn't gauge discrete groups, rather continuous groups on the lattice.

My point is that you cannot have continuous (nontrivial) spatial dependence with discrete groups.

but why ??
I don't see any reason that forbids it..

 

[timur]

You try to construct a continuous (and nonconstant) function from [0,1] to {-1,1}.

 

[xepma]

Ok, apparently I was wrong about J. Smits' book then. But I do have another source for you: the book by X.G. Wen, Quantum Field Theory of Many-Body Systems.

Back to your question,

As far as I see it we should look at it the following way (and this is to a large extend based on the book by Wen). The presence of a gauge structure implies that we have a redundant description of the physical system. We have multiple ways of describing the same physical picture, which differ only on a mathematical, non-physical level. An example is the vector potential in Electrodynamics, which admits a change by addition of a total derivative which leaves the electromagnetic field in touch.

Note that I avoid the use of the terminology of gauge symmetry (which is not common practice I must add). A symmetry transformation maps states to other states in the Hilbert space, and it does so in a particular way. Gauge structures on the other hand relate different representations of the same state in the Hilbert space. They are not physically distinct, so the word "gauge symmetry" is in some sense a misnomer. There is no "physical symmetry", since we only have one state. The redundancy arises due to the way the state is described.

An example is the Ising model. In the absence of an external field this system has a Z2 symmetry. Flipping all spins leaves the Hamiltonian invariant. We can turn this into a gauge symmetry by simply identifying all states which are mapped under such a global Z2 transformation. The states are now different only on a mathematical, "descriptional" level. The effect is a reduction of the Hilbert space, as the systems is incorporated with a gauge structure.

An example of a local, discrete gauge theory is indeed a Z2 gauge theory defined on a lattice (explained in the book by Wen). The system has spins defined on the nodes of the lattice, and a gauge field defined on the links. The Z2 gauge transformation locally flips a spin, combined with the effect of flipping the sign of the links ending on the node. This seems rather trivial, but the formalism can be extended to account for more less trivial gauge groups as well. As is often the case in a local gauge theory, the physical observables are given by Wilson loops.

Now as for local, discrete gauge theory on a continious coordinate system... My guess is that on some formal level you can get away with defining some weird (local) gauge structure based on a discrete group. However, you bump into problems quite soon since the gauge field is necessarily a discontinous mapping (recall that a gauge field takes on values in its gauge groups, which is a discrete set). If we want to construct any terms in the action we would want them to be gauge invariant and well-defined. I don't see how you could manage that with an everywhere-discontinious field. Derivatives etc are out of the question.

 

[Haelfix]

I don't see any fundamental mathematical issue with gauging a discrete group over a continuous space. I just don't think anyone has found a particularly phenomenologically useful realization of the idea in that context. The sections of the matter fields wouldn't behave much like anything we're used to seeing in the real world.

Quantum mechanically you could also imagine all sorts of really nasty things that would preclude or at least highly constrain such theories from existing. Anomalies and so forth.

 

[Thomas Larsson]

The Ising gauge model is a lattice gauge theory with Z_2 symmetry. It was invented in F.J. Wegner, J. of Math. Phys. 12, 2259 (1971), and is very well-known. On the 3D cubic lattice it is dual to the 3D Ising model and hence of great interest.

 

[Atakor]

thanks all, for your replies..

I'll think about this more..

 

[Coin]

Hm, here's a question that may be a bit offtopic, but I'm just curious. On occasion this board has discussed Garret Lisi's speculative "e8" theory. My understanding was that this theory uses e8 as a "gauge group", and that e8 is a discrete group. Wouldn't this be an example then of what Atakor is asking for? (Or is using something as a "gauge group" different from "gauging a group" somehow?) If so, then how, if at all, how does Lisi/e8 circumvent the problems Timur raises?

 

[xepma]

E8 is an example of a simple Lie group (simple is in this case a mathematical definition). It is therefore not a discrete group, but a continious one (as all Lie groups are).

 

[Thomas Larsson]

Coin, maybe you are thinking of the fact that E8 is a finite-dimensional Lie group. It has infinitely many elements but only finitely many generators.

However, if G is a finite-dimensional Lie group, the corresponding group of gauge transformations is infinite-dimensional; it is the group Map(M,G) of maps from spacetime M to G.