# Why is gauge symmetry not a true symmetry?

1. Dec 15, 2015

### TimeRip496

A symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation. For example, the speed of light is an example of symmetry and its value will always will always remain the same no matter where and what coordinate system(e.g. cartesian, polar, etc) one use.

But as for gauge symmetry, we can only only use one coordinate though we get to choose what coordinate system we use(gauge fixing?) and we can made the transition between different coordinate system. Is this what gauge symmetry? If it is, is it why gauge symmetry is not a symmetry cause we can't preserve invariance when we move between different coordinate system?

2. Dec 15, 2015

### jfizzix

A symmetry is a transformation that leaves something invariant.

For example, when Newton's equations of motion are invariant by a translation of spatial coordinates (i.e., by changing what you call your zero coordinate point), time coordinate, or a rotation in spatial coordinates, we would say those equations have translational symmetry, time translational symmetry, and rotational symmetry.

Maxwell's equations of electromagnetism have gauge symmetry in that they are invariant under a gauge transformation.

You may be interested to know that there is a profound connection between these continuous symmetries, and conservation laws. The idea, explained by Emmy Noether's theorem is that for each continuous symmetry of the equations of motion, there is a corresponding conserved quantity.

In the ones listed here:
• space translation symmetry leads to conservation of momentum
• time translation symmetry leads to conservation of energy
• rotational symmetry leads to conservation of angular momentum
• gauge symmetry leads to conservation of electric charge (this one's harder to show)
The constancy of the speed of light is not a symmetry, but it is evidence of Lorentz symmetry (that the equations of motion in relativity are invariant under a Lorentz transformation)

3. Dec 15, 2015

### bsmile

The symmetry does not only apply to coordinates. For the gauge symmetry, it applies to a generic physical quantity, which after the transformation, the action is left unchanged.

4. Dec 15, 2015

### TimeRip496

But is there a difference between gauge symmetry and symmetry? Or is gauge symmetry a symmetry, just like translational/rotational symmetry? Because I search online and I see people saying that gauge symmetry is not a true symmetry. This is the part which I don't understand.

5. Dec 15, 2015

### Mentz114

I assume this applies to a Lagrangian or action of some kind. The important symmetries are those that generate conserved quantities and represent physical degrees of freedom. Symmetries that do not generate conserved quantities are 'mopped-up' by adding constraints to the action - i.e. gauge fixing. I think those are the ones referred to as 'not true' symmetries. Dirac wrote an important book on this subject but I can't even recall the title right now.

6. Dec 16, 2015

### atyy

One way to see that gauge symmetry is not a symmetry is that it cannot be "spontaneously" broken.

The laws of physics are invariant under translation by an arbitrary distance. Yet a crystal lattice is not invariant under translation by an arbitrary distance - it is only invariant if one translates it by the lattice spacing. Thus translational symmetry of the underlying laws of physics is "spontaneously" broken by a crystal.

A gauge symmetry is not a symmetry because it is just a way of calling the same thing by more than one name.

7. Dec 16, 2015

### bsmile

Is this the way you define a "spontaneous symmetry broken"? This is quite confusing to me, can anybody explain to me further on it, or you might try to use a metaphor to give one an idea what it is?

8. Dec 17, 2015

### stevendaryl

Staff Emeritus
Hmm. Is there a mathematical or physical criterion for knowing when two descriptions of the universe are just different ways of describing the same situation?

Let's take the paradigm example of a gauge symmetry, which is the electromagnetic field. If you start with Maxwell's equations for the electric field $\vec{E}$ and the magnetic field $\vec{B}$, then you can introduce a pair of nonphysical fields $\Phi$ and $\vec{A}$, and define $\vec{E}$ and $\vec{B}$ in terms of them:

$\vec{E} = - \frac{d}{dt} \vec{A} - \nabla \phi$
$\vec{B} = \nabla \times \vec{A}$

Since $\vec{E}$ and $\vec{B}$ were the physical fields, then you can see that $\vec{A}$ and $\phi$ are not uniquely defined; the physical fields are unchanged by a replacement:

$\vec{A} \rightarrow \vec{A} + \nabla \chi$
$\phi \rightarrow \phi - \frac{d}{dt} \chi$

So the above transformation is nonphysical, since it makes no change to the physical fields.

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?

9. Dec 17, 2015

### atyy

The physical reason is that gauge symmetries that are not recognized by physicists cause mathematicians to buzz annoyingly around the physicists until the physicists cannot stand it anymore :) Then they start to use words like principal bundle :P

If the physicists had started with QED in the path integral formulation, they would need a gauge fixing condition to get the correct results, so they would be able to recognize that it is a redundancy of the description.

How about classically? Let's take the simple case of electrostatics. Maybe take neurobiology. Maybe there would be huge debates about whether the membrane potential of a neuron is really 0 mV (the Hodgkin and Huxley convention) or -70 mV (the modern convention). Maybe it would be never resolved.

10. Dec 17, 2015

### atyy

You are at a Chinese dinner, so the table is round. Is your pair of chopsticks on your right or left? You will eat fine either way, but in any particular dinner, someone will choose one, forcing everyone else to make the same choice.

[Apparently this metaphor goes back in some version to Abdus Salam http://philsci-archive.pitt.edu/563/1/SSB.pittarchive.mss.pdf and http://www.nytimes.com/1996/11/23/world/abdus-salam-is-dead-at-70-physicist-shared-nobel-prize.html?_r=0]

11. Dec 17, 2015

### stevendaryl

Staff Emeritus
This is worth understanding. Without gauge-fixing, what goes wrong in QED? Presumably, there are too many degrees of freedom for the Lagrange equations of motion to give unique equations of motion for the fields?

12. Dec 17, 2015

### atyy

The way I have always heard it explained is that the path integral sums over all paths, so it is essentially a sort of counting. Without the gauge fixing, one overcounts.

A quick google suggests the discussion on p277 of http://eduardo.physics.illinois.edu/phys582/582-chapter9.pdf.

13. Dec 17, 2015

### atyy

The rest of your post is right, but this point isn't. It is the global U(1) symmetry that leads to conservation of electric charge, not the gauge "symmetry".

*Sometimes people call the global symmetry a gauge symmetry. However in the context of this thread, the OP is distinguishing the gauge redundancy from the "physical" symmetries, so in the language of the OP, it is not the gauge symmetry that leads to charge conservation.

Last edited: Dec 17, 2015
14. Dec 17, 2015

Symmetries transform one state into another state related by some operation. A continuous symmetry comes with a conserved charge/current. You can also have discrete symmetries with no such conserved charge like parity, time reversal, etc.

A gauge transformation is a local transformation that depends on the space time position. In the standard model, it is an element of some Lie algebra depending on the theory (QED is U(1) QCD SU(3), etc). You can also have "discrete" lattice gauge theories, the simplest being a Z2 "Ising" gauge theory. This implies you have some nontrivial topological order and you get corresponding nontrivial topological excitations.

Gauge symmetry is not a real symmetry since a gauge transformation does not relate different states. It shows two states are actually the same. So if you have two states and you can access one from the other, they are actually the same state. To include both would make the Hilbert space over complete. So fixing a gauge projects onto the physical degrees of freedom. In the path integral, this chooses a gauge orbit so you are not over counting states.

Another way to see this is that initially, when you try to solve for the photon propagator, you need to add some xi term, serving as a Lagrange multiplier (some constraint) for the propagator to be invertible. That is what gauge fixing is.

In the abelian U(1) case, gauge fixing is quite easy. However, for non abelian theories if you want to pick a gauge which makes Lorentz invariance manifest, you run into problems since the path integral will give you the inverse of a determinant depending on the gauge field. That's where ghosts come in, etc.

15. Dec 17, 2015

### ShayanJ

I'm OK with your argument. It really makes sense to think about gauge symmetries that way, only as redundancies in our descriptions. But the strange thing here is that these non-physical symmetries are actually determining how the physics should be. The gauge fields that we add to our classical Lagrangians are the result of our demand that those Lagrangians should be gauge-invariant. This is really strange that non-physical properties are determining how physics should be!

16. Dec 18, 2015

### atyy

The traditional "gauge principle" used to determine "derive" coupling between electric charge and electric potential is also misleading terminology, although it is what everyone uses. In fact, the gauge principle taken at face value cannot determine the coupling uniquely. Rather the "gauge principle" is more appropriately called a "minimal coupling" principle, similar to the equivalence principle of general relativity.

17. Dec 18, 2015

### ShayanJ

So we're just pretending that the demand of gauge invariance is determining the interaction term?

18. Dec 18, 2015

### stevendaryl

Staff Emeritus
The way I've heard it argued, local gauge invariance implies the existence of gauge fields of some kind, whether or not it determines the interaction term. So rather than starting with an electromagnetic field $A^\mu$ and trying to deduce its interaction with some matter field $\phi$, you start with the free-field $\phi$. Note that the equations of motion are invariant under a global phase change: $\phi \rightarrow e^{i \chi}\phi$. Now, you insist (and I'm not sure why you feel that you have the right to insist this) that the equations should also be invariant under a local phase change, where $\chi$ varies from point to point. But if $\chi$ varies, then you end up with extra terms in your Lagrangian of the form $(\phi^* \partial_\mu \phi) \partial^\mu \chi$ (and its complex-conjugate). So the invariance is spoiled unless you have another field that also changes with $\chi$. Adding a gauge field $A^\mu$ that changes with $\chi$ according to $\delta A^\mu \propto \partial^\mu \chi$ is the simplest choice, but the local gauge symmetry forces you to have some kind of interaction term.

19. Dec 18, 2015

### atyy

20. Dec 18, 2015

### ShayanJ

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.