
Analysis in Higher Gauge Theory - espeed
https://golem.ph.utexas.edu/category/2018/10/analysis_in_higher_gauge_theor.html
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ur-whale
Would anyone on HN care to explain to physics laymen what Gauge theory is
about in the first place?

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danbruc
A field theory is a theory that describes a physical phenomenon by fields and
how they evolve and interact over time. A field is just a function that takes
a point in space and time and gives you a value. You can model temperature as
a function that given a point in space and instant in time gives you the
temperature at that specific point in space and time. Or the wind velocity
which will assign a vector indicating the direction and speed of the wind to
each point in space and time instead of a single number as in the case of
temperature - this is the difference between a scalar field and a vector field
and there are other types of fields like tensor fields. A field theory then
provides you with the equations that explain how fields change over space and
time and how they interact with each other in case the theory has multiple
interacting fields.

A gauge theory then is a special kind of field theory, one which is invariant
under some symmetry transformation. You could for example come up with the
idea to describe temperature with complex numbers instead of real numbers. The
equations wouldn't really change, they would essentially just ignore the
imaginary part of the temperature making 20°C, 20 + 42i°C, and 20 - 7i°C three
different representations of the same temperature. In consequence you are free
to adjust the imaginary parts of all the temperatures without this affecting
your predictions.

As it turns out if you want to use field theory to describe quantum physics
and if you want your field theory to have certain nice properties, for example
making it obvious that the theory respects locality, then you are forced to
use gauge theories, i.e. you are unable to write a field theory without those
redundancies where several different values correspond to the same real
physical state. In the example from above, the real temperature is of course
20°C but because of the way you would like to formulate your theory, you are
forced to use complex numbers for the temperature and because the resulting
theory is so nice to work with you accept this and just declare that all
temperatures with the same real component are the same regardless of the
imaginary part.

And while you often hear that the world is fundamentally made out of fields,
that is not true, or at least it does not have to be true. Quantum field
theory is one of our best tools to describe our world but the fields are just
a nice and convenient mathematical tools, they are not in one to one
correspondence with actually existing things.

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sideshowb
Came back here to see this question answered. Was not disappointed. +10 for
clarity :)

So, the phase of a wavefunction in QM makes it a gauge theory, then? It
doesn't matter what the actual phase is at any point, but it does matter how
it varies between one point and the next.

(For the physics layman - you might enjoy my game
[https://omnisplore.wordpress.com/2016/04/25/learning-
quantum...](https://omnisplore.wordpress.com/2016/04/25/learning-quantum-
mechanics-the-easy-way/) )

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danbruc
_So, the phase of a wavefunction in QM makes it a gauge theory, then?_

There are two kinds of symmetries, global symmetries and local symmetries. In
case of a global symmetry the same transformation is applied everywhere, for
example moving all particles of a system to the right by one meter. In case of
a local symmetry the transformation can vary over space and time, so you could
move different particles by different distances, for example double the x
coordinate.

Classical mechanics is invariant under global translations, it does not matter
where the system is, here or one meter to the left, what matters are just
differences between positions, the distances between particles. On the other
hand classical mechanics is not invariant under local translations because it
will alter the distances between particles. Local symmetries are a much
stronger constraints than global symmetries which are essentially just a
special case of local symmetries where the transformation parameters are fixed
across space and time.

The wave function has a global symmetry, you can rotate the phase, but phase
differences are important and you can therefore not apply different phase
rotations to different parts of the wave function. On the other hand the
relevant symmetries in gauge theories are local symmetries, so the global
symmetry of the wave function is not what makes a theory of quantum physics a
gauge theory. In case of electrodynamics it is for example the electromagnetic
four-potential that has a gauge freedom, i.e. there are many different
electromagnetic four-potentials that give rise to identical physically
observable magnetic and electric fields.

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sideshowb
Thanks - look forward to learning more when I have time. Are gauge theories
something you encounter on the path to QFT?

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personlurking
Dang, this link was greyed out for me when I had not previously clicked on it.
FYI.

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posterboy
That couldn't have anything to do with HN but all with your browser. Most
likely you do have clicked through to the URI another time before when the
link was up and you just forgot because it is not anything you would or could
remember, higher gauge theory I mean. That happened to me with a pdf as I will
save too many pdfs to read and open too many pages in tabs to fit memory.

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personlurking
Yeah, the only thing is I've never clicked on that link and I'm not
knowledgeable enough, in any way, to understand "higher gauge theory".

It also seems I got several downvotes for my comment, trying to reach
moderator dang, which I've seen other HNers do in the comments section, for
example, when a title should be altered, etc.

