Why is gauge symmetry not a true symmetry?

[vanhees71]

If you use this as your definition of symmetry, then gauge transformations count, but so do anomalous symmetries. So it's not a good definition for quantum systems. E.g. pure Yang Mills is classically scale invariant but it isn't a symmetry for the quantized theory.

I believe saying that gauge transformations aren't symmetries or are "redundancies" makes sense because they do not transform distinct states into each other. If you take an arbitrary state and apply a gauge transformation to it, you end up with the same physical state. Whereas global symmetries do transform between states, for example it transforms between states in an irreducible multiplet. Or as another example, maybe every energy eigenstate is in a 1D irrep of a symmetry, but a linear combination of two states in different irreps will transform to a different linear combination.

Ok, to have anomalies included, just substitute "effective action" instead of action, and everything is fine. In the classical theory it's by the way sufficient that the variation of the action functional obeys the symmetry. I'm not so sure about the quantum case. For the validity of the Ward-Takahashi identities of proper vertex functions, excluding vacuum diagrams, it should be sufficient, but I'm not sure concerning the effective action itself, which gets a physical observable in the equilibrium case, where it is a thermodynamical potential.

 

[DrDu]

However, you could imagine an alternate history of science in which Maxwell first discovered the laws of electromagnetism in terms of $\vec{A}$ and $\phi$, and then later discovered that his laws were invariant under the above transformations. What reason would physicists have for concluding that this was a gauge symmetry, and not a true physical symmetry?

A and phi would still not be observables as we can't measure them unambiguously.

I asked here not a long time ago how to recognize a gauge theory if one only looks at the observables. I won't claim I understood the answers, but A. Neumaier cited some interesting papers who showed how to eliminate non-observables from some simple model-gauge theories and quantize the system:

https://www.physicsforums.com/threa...-gauge-theory-in-terms-of-observables.839208/

 

[DrDu]

Anyway, my original comment is still applicable. As gauge invariance is not a physical symmetry and is only a redundancy in our description, this seems really strange that a non-physical thing is determining the physics. For me, either gauge invariance is somehow physical or there is something really physical that can be used instead of gauge invariance.

I agree 100% with you. However, we face this almost everywhere in physics. For example, enumerating the positions of identical particles is also such a kind of redundancy. It is more physical, to use only a set of relative position vectors pointing from e.g. one position to the other and restricting the position vector to lie in one half space, identifying points on the boundary. If you do this, you get a multiply connected space and it turns out that the usual classification of particles in terms of permutation symmetry is related to the topological different paths in that space. I would love to see something similar for e.g. the gauge group U(1) of electromagnetism.

 

[DrDu]

It would be so nice if this is true. Unfortunately, the interaction Lagrangian $A_{\mu}J^{\mu}$ cannot be expressed in terms of the $F_{\mu\nu}$.

I don't take this to be god given: E.g. you could use for the interaction $F_{\mu\nu}P^{\mu\nu}$ with the polarisation P obeying $\partial_\mu P^{\mu\nu}=J^\nu$. The two expressions can be shown to be equivalent up to a total derivative.

 

[RUTA]

You can also view gauge invariance as the result of relationalism. Rovelli writes, "Gauge is ubiquitous. It is not unphysical redundancy of our mathematics. It reveals the relational structure of our world." Rovelli, C.: Why Gauge? (2013) http://arxiv.org/pdf/1308.5599v1.pdf, p 7. Mathematically speaking in terms of a path integral for lattice gauge theory, that the difference matrix K (derivatives are differences in discrete form) for the action on the spacetime lattice has a non-trivial null space (at least one eigenvalue = 0) means its rows are not linearly independent. Fadeev-Popov gauge fixing is just the restriction of the path integral to the row space of K (subspace with non-zero eigenvalues). When you restrict the source vector J to the row space of K you get conserved J. All that follows from simple relationalism, e.g., stating that I'm behind you in a line at the grocery store entails that you're in front of me (you can see the "redundancy" here). In the same way, constructing the rows for the difference matrix K entails a redundancy that leads to the non-trivial null space.

 

[samalkhaiat]

I don't take this to be god given

No, it is forced on you by local gauge symmetry, with very accurate experimental predictions.

E.g. you could use for the interaction $F_{\mu\nu}P^{\mu\nu}$ with the polarisation P obeying $\partial_\mu P^{\mu\nu}=J^\nu$.

Who orderd this P for you? Plus, if both tensors P and F satisfy the same field equation $\partial P = \partial F = J$, then $F \propto P$ So, your "interaction term" is proportional to the free electromagnetic Lagrangian $F^{2}$.

 

[DrDu]

This form of the Lagrangian involving the polarisation is quite common in solid state and molecular physics.

 

[DrDu]

Thinking about it, I forgot an epsilon tensor in the definition of P.

 

[radium]

I think the most natural way to think of symmetries in a quantum system is to think about the transformation of the state. Symmetries and gauge invariance can have profound effects in many quantum mechanical systems which are not immediately obvious in the classical theory, so you cannot look at just the action since you have things like dimensional transmutation from loop effects.

Going back to the subject of gauge invariance, the way to think about it is in terms of the Hilbert space which relates to gauge orbits etc. This is actually pretty clear in a Z2 lattice gauge theory. If you formulate the theory in terms of a superposition a of loops, you will see just by counting arguments that your Hilbert space is overcomplete. This means that many states you have are actually equivalent by some gauge transformation and must be projected out using some gauge fixing condition to get the physical state of the systems. In this case you actually have a richer phenomena of projective symmetry. This means that a combination of symmetry operations equal to the identity in the symmetry group are actually only the same up to some gauge transformation.

I think the best way to phrase all this is that while the choice of gauge is not observable, gauge invariance most definitely has physical consequences.

Some more examples: boundary conditions become incredibly important.
Some examples are Chern-Simons theory and 2+1D QED. In the first case, you have a term which is a total derivative. If you add a boundary to this system, the bulk theory is no longer gauge invariant, you must consider the whole system together to preserve gauge invariance. This can explain the edge states in quantum hall systems/topological insulators. In compact 2+1D QED, you have additional "large" gauge transformations allowed which are topologically nontrivial. This causes the charge to be quantized. By contrast, charge is not quantized for non compact 2+1D QED. Overall, gauge invariance is very important for systems with topological/symmetry protected topological order.

 

[king vitamin]

Ok, to have anomalies included, just substitute "effective action" instead of action, and everything is fine. In the classical theory it's by the way sufficient that the variation of the action functional obeys the symmetry. I'm not so sure about the quantum case. For the validity of the Ward-Takahashi identities of proper vertex functions, excluding vacuum diagrams, it should be sufficient, but I'm not sure concerning the effective action itself, which gets a physical observable in the equilibrium case, where it is a thermodynamical potential.

It's been known for a long time that the effective action is gauge-dependent (I believe Jackiw is the original reference: http://journals.aps.org/prd/abstract/10.1103/PhysRevD.9.1686 ). Perhaps only demanding that the extrema are gauge invariant would work? I think this is still a major topic of research, see e.g. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.113.241801 from last year (which already has several references).

 

[vanhees71]

Sure, it's gauge-dependent, because you have to fix the gauge, and the proper vertex functions, whose generating functional the effective (quantum) action is, are gauge dependent. What's gauge independent are observable quantities like S-matrix elements.

 

[LarryS]

The book https://www.amazon.com/dp/080187971X/?tag=pfamazon01-20 contains a very well written, non-mathematical explanation of gauge symmetry, especially in relation to EM.