This is a bit of a philosophical/conceptual question. I've done tons of reading on it, of course, but haven't found anything that makes me go 'ah ha'!(adsbygoogle = window.adsbygoogle || []).push({});

I am working steadily through the mathematical formalism of differential geometry, but am struggling to grasp how the things we say in this language actually explain the physics we observe.

Everywhere I go, 'gauge symmetry' and 'gauge invariance' seem to be at the heart of things. I often read statements along the lines of 'gauge symmetries give rise to forces' or 'fundamental interactions are mediated by gauge fields' etc. And a lot of hand-wavey stuff about how the fundamental forces serve to 'reconcile' the different gauge (phase) settings at different points.

I'm at the edge of my understanding with this stuff, and the unsatisfactory nature of talk like this is probably mostly an artifact of my reading level. I am rapidly learning the mathematical formalism of differential geometry. I'm fairly comfortable with group theory, Lie derivative, covariant derivative, and I am now learning about fiber bundles. I have not internalized anything about connections yet, which I understand to be important.

Anyhow, my pointed question is:

1. What does 'gauge theory' actually explain? In other words, how would nature be different if we couldn't use gauge theory to describe it? Would the fundamental forces just not exist? I've read that these local symmetries 'constrain' or 'determine' the laws of nature. In what sense? What limitations, specifically, does the demand of local symmetry put on our formulation of natural laws? I haven't been able to find an answer to this that makes sense to me. (Just because I'm still learning.)

My dim understanding is that there is a symmetry group parameter that we can shift by a different amount for each point in the space. But global invariance needs to be maintained, so we need a way to reconcile or compensate for the fact that the parameter was shifted by different amounts at two points. Somehow (?) the forces of nature are the manifestation, to our experience, of nature doing this reconciliation.

But unpacking this, 1) what is the motivation for going from global to local symmetry? Is it theoretical, experimental, both? 2) what is the motivation for demanding global invariance? Why, physically, do the varying shifts in the parameter (applying different group elements to different points in the space) have to be reconciled? What would the big disaster be if we didn't?

Another, related but even more philosophical, question:

2. Do gauge theories/symmetry principles actually explain anything? Or are they just more elegant and satisfying ways of labeling the same phenomena? In other words, I suspect that the forces of nature don't 'arise from' symmetry principles causally but that they are just equivalent to symmetry principles. So you could work from symmetry to forces of nature or forces of nature to symmetry.

So is saying the forces happen 'because' of symmetry pretty much the same as saying that you flipped heads 'because' you didn't flip tails? That isn't an explanation as much as a reformulation. Is it the same with gauge principles? Or is there substantive explanation to be found there which satisfies the yearning for explanation, in the way that the kinetic theory satisfyingly explained the Second Law of Thermodynamics or GR satisfyingly explains gravity as spacetime curvature?

I'm really just trying to fill in gaps in my understanding and thought asking specific questions might accelerate that. Thanks.

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# A What does gauge theory explain?

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