What Does Gauge Invariance Tell Us About Reality?

[Ghost Repeater]

This is not a technical question. I'd like to have a more conceptual discussion about what - if anything - gauge invariance tells us about reality. If we could, please try to keep the discussion at the level of undergrad or beginning grad.

To focus my questions and keep things elementary, I'd like to take the example of the coupling between quantum phase and electromagnetic potential. There are many ways of formulating the so-called 'gauge argument'. Perhaps the most succinct way is just to say that without the electromagnetic potential term appended onto the derivative operator (e.g. in the Hamiltonian or Lagrangian), the quantum phase of the particle/matter field would not be arbitrary.

It is in this sense that we so often hear it said that phase (or gauge) symmetry constrains interactions, in the sense that insisting that quantum phase be arbitrary requires the existence of an electromagnetic potential.

I want to know people's thoughts on the nature of this causal link - if, in fact, it is a causal link. There seems to be a great deal of conceptual confusion about it and different takes on whether and why it's important.

This is especially true around the question of why we require the quantum phase to be arbitrary in the first place, i.e. why would we postulate local phase symmetry to begin with? This seems to leave a huge gap in the physical motivation of the theory.

I've heard quite a few things. For example

1. David Griffiths, in his 'Introduction to Particle Physics', baldly states that he knows of no reason why local phase symmetry 'should' exist.

2. John Barrow, in a popular science book, seems to think that local gauge symmetry is desirable because of simple locality concerns. E.g. if we never promote global gauge transformation to local, then we are essentially relying on 'spooky action at a distance'.

3. Phase transformations are often treated as essentially just coordinate transformations, i.e. resetting our zero point in the coordinatization of internal spaces, and the argument is made that the patterns that encode physical results should never depend on coordinates we choose to describe those results. On this reading, the coupling between phase transformation and electromagnetic potential just arises from the fact that our descriptive conventions should be irrelevant to our physical laws, i.e. an extension of the kind of thinking that brought Einstein to relativity.

So is there any 'underlying logic' to local gauge symmetry, or is it just a brute fact at this point? Albeit a brute fact that, once given, implies (and possibly explains?) electromagnetic interaction via photons.

To approach the question another way, I often think of gauge symmetry giving rise to forces by analogy with conservation laws in elementary mechanics. In that realm, it seems it would be valid to reason as follows: "Things change according to Newton's laws because otherwise, energy and momentum would not be conserved. Energy and momentum must be conserved because otherwise, the time or place at which experiments are done would not be arbitrary, which means you could get different laws of physics just by using a different coordinate system."

This last point is where intuition clicks, I think. At least for me. It makes deep, intuitive sense that we should not get different laws or patterns just from formulating those relationships using a set of numbers that have been shifted over. This doesn't strike me as a brute fact but as an intuition that could have explanatory power.

So is there a logical chain that grounds electromagnetic force in local phase symmetry and local phase symmetry in this idea that physics must be independent of the numerical conventions/coordinate systems we use? Or am I being misled by analogy with the thinking that led to relativity?

EDIT: There's also a philosophical question as to conflating 'the reason a thing exists' with 'the reason that we can know a thing exists.' So phase symmetry may not cause electromagnetic force but may just be the way that we (could in principle) deduce that it exists. But a deduction of this kind is not necessarily an explanation.

 

[cosmik debris]

1. David Griffiths, in his 'Introduction to Particle Physics', baldly states that he knows of no reason why local phase symmetry 'should' exist.

I can't answer your question but this made me laugh. I don't know David but he might be amused too.

Cheers

 

[king vitamin]

There are many ways of formulating the so-called 'gauge argument'. Perhaps the most succinct way is just to say that without the electromagnetic potential term appended onto the derivative operator (e.g. in the Hamiltonian or Lagrangian), the quantum phase of the particle/matter field would not be arbitrary.

It is in this sense that we so often hear it said that phase (or gauge) symmetry constrains interactions, in the sense that insisting that quantum phase be arbitrary requires the existence of an electromagnetic potential.

I want to know people's thoughts on the nature of this causal link - if, in fact, it is a causal link. There seems to be a great deal of conceptual confusion about it and different takes on whether and why it's important.

If you want my opinion, this reasoning is entirely aesthetic and is not a physical reasoning at all.

A more physical line of reasoning - which is, however, a bit more technical - comes from Weinberg's field theory books. One begins with the irreducible representations of the Lorentz group (the Wigner classification), and then you find that there is a big difference between fields with and without mass. For massless fields with spin (really helicity), there are only two physical degrees of freedom (=the representations are two-dimensional), whereas massive spin-$\ell$ particles have $(2 \ell+1)$ degrees of freedom. But if you try to imbed a helicity-$\ell$ particle into a rank $\ell$ Lorentz tensor, as you would try to do if you want to write down a Lorentz-invariant Lagrangian, however, you do not get the correct number of degrees of freedom. You're forced to use a redundant description to eliminate the rest.

So to summarize (and I apologize that the above was a bit more technical than you asked for): it appears you need gauge invariance to describe massless particles with spin greater than 1/2 in a Lagrangian formalism. You may have encountered the fact that there is no fixed gauge which is manifestly Lorentz invariant and unitary, which is a manifestation of this.