The h Bar

Silly Goose

00:36

does anyone know of a (shorter) introduction to gauge theories? as in, more like an introductory note of a length like 20 pages or so

ACuriousMind

@ClaudioMenchinelli yes

@SillyGoose what do you expect to learn in 20 pages :P

Silly Goose

a very broad picture of the structures involved :P at least a naming of what mathematical things correspond to what physical things

ACuriousMind

also: What kind of "gauge theories"? Arbitrary ones? Yang-Mills theories? Literally just electromagnetism? Lagrangian formulation? Hamiltonian formulation?

Silly Goose

(but not necessarily definitions of the mathematical things)

maybe just electromagnetism or the simplest case that still shows off the characteristics of a general gauge theory

ACuriousMind

I mean, the core characteristic is just that the theory is symmetric under a space-time dependent symmetry $A\mapsto A + \mathrm{d}f$ for any function $f$

you don't even need a single page for that :P

Silly Goose

00:44

what is $A$?

ACuriousMind

the 4-potential?

Sir Cumference

@SillyGoose I mean you could start here

Introduction to gauge theory

A gauge theory is a type of theory in physics. The word gauge means a measurement, a thickness, an in-between distance (as in railroad tracks), or a resulting number of units per certain parameter (a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition). Modern theories describe physical forces in terms of fields, e.g., the electromagnetic field, the gravitational field, and fields that describe forces between the elementary particles. A general feature of these field theories is that the fundamental fields cannot be directly measured; however, some associated...

Silly Goose

@ACuriousMind what is meant by the theory is symmetric?

Sir Cumference

At least as far as I've been introduced to them, it's largely about finding potentials that yield equivalent forces/fields/etc.

ACuriousMind

@SillyGoose just the ordinary sense of symmetry? Like, the action is invariant under this transformation

Sir Cumference

00:46

Think about how shifting your potential by a constant doesn't have any effect on the force

There are generally other ways you could change a potential and still end up with the same physical results

Different potentials that yield identical physical results (regardless of which potential you pick) are said to be gauge invariant among each other

Typically the choice of potential is just a matter of mathematical convenience. E.g. sometimes if we have a given electric field, it'll be useful to choose a potential that satisfies certain conditions (e.g. regarding its derivative)

Silly Goose

@ACuriousMind should i have in mind quasi-symmetries as described by QM here: physics.stackexchange.com/questions/11313/…? Instead of leaving the action entirely invariant: $\delta S = 0$?

ACuriousMind

@SillyGoose you should always be prepared for "symmetry" to actually mean that notion of quasi-symmetry

but that's not special to gauge theories at all

Silly Goose

@SirCumference hm i think i get this notion, i think i am wanting to gain some familiarity with the terminology/structures of gauge theory

ACuriousMind

the problem is that there really isn't a lot of elementary examples of gauge theories

Sir Cumference

@SillyGoose As usual I would start with a simpler intuition/familiarity and build up from there

ACuriousMind

00:51

or, rather, there isn't a lot of non-artificial elementary examples

really usually the only gauge theory you'll see is electromagnetism until we get to Yang-Mills theories to describe QCD

Sir Cumference

In e.g. electromagnetism, there are plenty of potentials that all yield the same electromagnetic fields, so it's often a matter of using the most mathematically convenient one (i.e. the most preferable "gauge")

ACuriousMind

and electromagnetism is a really poor example of the general "structures" of gauge theories for various reasons

Silly Goose

cripes

Sir Cumference

Tbh for now just keep it simple and know that it's a matter of swapping between potentials and still getting the same end result

There's a much deeper discussion on the relationships between gauge invariant potentials, but at the moment I'd say go through the Introduction wikipedia page

e.g. you'll find out about how you can identify "convenient" potentials by making some mathematical restrictions (for instance, requiring $\nabla \cdot A = 0$)

as well as whether certain restrictions uniquely identify a potential or not

> In field theories, different configurations of the unobservable fields can result in identical observable quantities. A transformation from one such field configuration to another is called a gauge transformation; the lack of change in the measurable quantities, despite the field being transformed, is a property called gauge invariance.

This line is pretty much the general idea

ACuriousMind

there's already one problem: Because physics often only encounters gauge theories in the context of EM and Yang-Mills theories, people tend to talk about it as field theories, but there are definitions of gauge theories that apply to finite-dimensional (i.e. non-field-theoretic) systems just as well

yet understanding the formulation of gauge theories in this general sense will take a lot of effort to connect to the way physics usually talks about them

thesmartwaterbear

03:49

You guys know anything about the standard model?

naturallyInconsistent

03:59

What are we supposed to do with a question like this?

nickbros123

04:20

I rly like EnM cos there are no particles in the fundamental equations of Maxwell, you just need $\rho(\vec r, t)$ $\vec J(\vec r, t)$ and you're basically done in principle. Everything's a fluid !1!1!1

Silly Goose

consider the irreps of $sl(2; \mathbb{C})$ labeled by positive integers $m \in \mathbb{N}$.

is the fact that irreps labeled by some fixed $m$ are isomorphic representations a consequence of Shur's lemma?

I am not seeing how this proof I am following (from B.C. Hall) proves that same $m$ irreps of $sl(2;\mathbb{C})$ are isomorphic

it seems like i could use shur's lemma given that i construct an intertwining map between the irreps $\pi_m$ and $\pi_{m'}$ where $m = m'$ that is nonzero

oh wait

nvm I don't get it

naturallyInconsistent