# Gauge symmetry

1. Nov 16, 2009

### kexue

I would like to hear an original explanation of gauge symmetry. What gauge symmetry really means and why it is needed to describe nature.

I am more or less familiar with the standard treatment of electromagnetism and Yang Mills theories from QFT texts, but feel still unsatisfied since I have not grasped what goes on 'behind the equations', what it all really means.

So could you give me some real and deep insight on what is meant by gauge symmetry?

thank you

2. Nov 16, 2009

### Bob_for_short

"Elementary" particles are often classified according to some symmetry groups (as group representations, multiplets). Requiring the symmetry (invariance) to be local is a way to "introduce" interaction between particles. It is done "by analogy" with QED. There is no deeper meaning in doing so.

IMHO the interactions can be introduced differently. It may help avoid unnecessary "additional" particles and mathematical difficulties.

3. Nov 16, 2009

### humanino

There is a physical meaning to difference of phase between wavefunctions or wavefunction components (as in the phase of a complex number). However, there is no meaning to the absolute phase taken out of context, without reference or comparison to another phase. As a consequence, if you please you may change the origin of phases, or multiply all wavefunctions by the same unitary complex number.

Now you may decide that changing simultaneously all the phases of all the complex numbers in the Universe is not a very physical idea. So instead, you would like to define an origin for phases at each point in spacetime. Monkeying like this with the phase however comes at a cost : now the derivative of the wavefunction has an additional "spurious" (?) contribution coming from your local choice of phase origins. In order to cancel out this derivative term, you have to create a new vector field, and the miracle is that this no else but the photon. We sort-of derived the existence of an interaction from general symmetry principles.

From these elementary consideration, we built what Bob_for_short refers to : gauge symmetry groups and their representations, in that case U(1), the next levels being SU(2) and SU(3) respectively for the weak and strong interactions.

Gauge field theories and their renormalisation is one of the most beautiful and deepest results we know of and have tested successfully about Nature.

4. Nov 18, 2009

### blechman

It all comes down the the longitudinal polarization of spin-1 bosons. These objects are a problem in the real world: they become strongly coupled at some energy scale (roughly $4\pi m$ where m is the gauge boson mass), and you start making funny predictions like "scattering of longitudinally polarized bosons has a probability greater than one!" This is, in essence, the "nonrenormalizable problem" mentioned previously.

The solution to this problem is to somehow GET RID of this polarization. One way to do that is to make the spin-1 boson massless. If you do that, then there IS no longitudinal boson (think photon) and hence there is no problem. But a massless spin 1 particle has a gauge symmetry built in automatically (assuming things like Lorentz invariance), so the two are synonymous.

However, the W,Z bosons have mass, and so that solution won't work. In fact, it was a real problem for a long time, until it was shown that IF the spin-1 particle gets its mass through the "Higgs mechanism" then the longitudinal boson DECOUPLES at high energies where it would otherwise have blown up the world, and hence we are safe! (see Nobel Prize to 't Hooft and Veltmann, 1999).

The key to the "Higgs mechanism" is that the gauge symmetry is actually still there, although realized "nonlinearly". And so once again, the gauge symmetry is what protects you.

This is why gauge symmetry is so important. Hope that helps.

5. Nov 18, 2009

### jambaugh

The crux of gauge theory lies in Noether's theorem: (roughly) (<----- Edit symmetry => theory)

"For every continuous symmetry there corresponds a conserved dynamical quantity et vis versa."

E.g.
Space-time translations <===> Momentum and Energy
Rotations<===> Angular Momentum

Now we can extend this paradigm into "non-physical" i.e. gauge symmetries to explain/model the behavior of conserved quantities we do not originally think of as dynamical variables. We don't normally think of electric charge as a mass times rate of change of some coordinate but we can invoke a U(1) phase coordinate and define charges as "theta momentum". Similarly with the other gauge charges.

Note that the Standard Model is called a "Model" and not a "Theory". We model the dynamics of various charges by appending the gauge degrees of freedom and invoking local symmetries on transformations of these degrees of freedom which are dual to the observed conserved charges (em charge, weak isospin, color, et al). Now we cannot observe the gauge degrees of freedom (which is why they are called "gauge") but we can by modeling them infer how the charges must evolve and interact.

As to why it works... to better understand this you may want to parse though the proofs and variations of Noether's theorem. I see it as the keystone of modern physics...and all the stronger in quantum mechanics where we not only associate, we identify the generators of transformations with observables.

Last edited: Nov 18, 2009
6. Nov 19, 2009

### kexue

Thanks! Some very nice and helpful posts so far. Have to think them through. Keep it coming.

I think A. Zee has a nice explanation in his nutshell book, too. It goes like this: a massless spin-1 particles has two degress of freedom, but it is being described by a Lorentz invariant theory with four degrees of freedom, hence we have a redundant description. Gauge symmetry is no symmetry at all, which would map one physical state in another. Instead it always describes the same state.

7. Nov 19, 2009

### Bob_for_short

In my opinion, the Noether's theorem is useful if there exists physically meaningful and mathematically well-defined solutions of the equations. It is not the case in Classical Electrodynamics (CED) where self-action gives run-away exact solutions. Thus it is not clear in QED and QFT if the Noether's theorem says anything in favour of self-action terms and local gauge invariance.

Last edited: Nov 19, 2009
8. Nov 19, 2009

### blechman

Zee's explanation is more in line with mine about how you have "unphysical" polarizations (longitudinal and scalar) and these screw things up. When you have a massive spin-1 particle, one of these unphysical modes becomes physical and you have to be careful.

The fiber bundles approach to gauge theory is very beautiful mathematics, but (and I do NOT mean this disrespectfully, so don't get upset everyone!!) I never found it useful in developing a physical (practical) picture of the meaning of gauge invariance. But that's just me. Roger Penrose has a very nice description of this in his book "The Road to Reality" if you would like to see it in more detail, explained relatively straightforwardly.

9. Nov 19, 2009

### jambaugh

If you want to get into some more depth on the definition of gauge you can look in Henneaux and Teitelboim's Quantization of Gauge Systems

Here is what I could gather (with my own interpretation) from reading it. We work classically in phase space with it's symplectic geometry and Poisson bracket. Ultimately the Poisson bracket is a type of Lie bracket and what is happening is that we are embedding the Lie algebra of the physical system (observables) in this symplectic algebra.

By Lie algebra of the observables I mean the Lie algebra of the generators of the transformations associated with the observables via Noether.

The classical state manifold of a physical system needn't be a symplectic manifold however it can always be embedded in one and we do this by extending phase space. And the commutator of generators associated with observables is represented by the Poisson brackets. This extension of the state manifold to phase space adds dimensions which are the gauge degrees of freedom.

So instead of points in state space being the states, you have a fibration of the extended phase-space in which it is embedded. States correspond to curves (gauge orbits).

For example consider abstractly a state space in the shape of a circle embedded in an x,p phase-space. The physical states are now rays corresponding to a given angle and the gauge degree of freedom is radial scaling which comes about from embedding the 1dim circle in 2-dim (x,p) space.

We have a fibration of the phase-space with base (circle) and fiber (radial rays). ( One could also define a different fibration with base circle and fibers spirals. These are but one of an infinite set of fibrations defining a fiber bundle from a given space.)

We then can "fix the gauge" picking representative points in each fiber (preferably in a smooth way) which gets us back to defining states as points on a manifold. When we do this we introduce geometric artifacts which must be incorporated in the bracket structure so it still closes and respects our choice of gauge. This is where we get the Dirac Bracket from the Poisson Bracket.

OK. So in summary I see gauge degrees of freedom as extended dimensions we add to physical systems in order to fit them into a specific canonical structure. Then when we canonically quantize we retain the gauge degrees of freedom having a larger algebra than the operator algebra generated by observables. In standard quantum mechanics we needn't worry about this too much we simply pick the Hilbert space corresponding to the physical modes. But in QFT we need to retain the canonical position-momentum structure so we retain also the gauge extension structure.

Something similar also occurs when we work in the relativistic setting of QM since we must pick infinite dimensional representations (to retain unitarity) of non-compact groups such as the Lorentz and Poincare group. The physical system may not be infinite dimensional or may be but the representation my yet have non-physical degrees of freedom. We again get gauge degrees of freedom which we must constrain out. I think there must be some way to bypass all this extension but that would require a non-unitary QM which is problematic when conserving probabilities.

I suspect that to quantize gravity someone (hopefully me) must address this issue. The non-renormalizability of canonical quantum gravity is (I think) exactly the explosion of these gauge degrees of freedom in embedding the theory in the standard canonical structure. As beautiful as the whole Lagrangian and Hamiltonian formulation is, it is also, I fear, trapping us within this problem. (I also see string-brane theory as further compounding the problem by picking a bigger more esoteric replacement for the canonical structure.)

10. Nov 19, 2009

### Bob_for_short

I see. You reason as a mathematician.

11. Nov 20, 2009

### bcrowell

Staff Emeritus