Skip to main content
32 votes

What is $U(1)$ symmetry?

Let us first refer to symmetry generically. When we say a theory is symmetric under $G$ ($G$ some group) we mean that the elements of $G$ transform the states, and the operators of a theory, (in the ...
ohneVal's user avatar
  • 4,438
29 votes
Accepted

Physical difference between gauge symmetries and global symmetries

The first answer to such a question must always be: A gauge symmetry has no "physical" meaning, it is an artifact of our choice for the coordinates/fields with which we describe the system (cf. Gauge ...
ACuriousMind's user avatar
  • 126k
29 votes
Accepted

Covariance in gauge theories: why should the Lagrangian be gauge invariant?

We do not start from the assumption that the Lagrangian "should" be invariant under gauge transformations. This assumption is often made because global symmetries are seen as more natural than local ...
ACuriousMind's user avatar
  • 126k
28 votes
Accepted

Why are gauge theories called so?

Because Weyl's original gauge theory (1918-1920), which was also the very first unified field theory of electromagnetism and gravity (Kaluza only published in 1921), had a field of scales/gauges ...
Conifold's user avatar
  • 5,343
27 votes
Accepted

Hamiltonian for relativistic free particle is zero

...what I would like to know is why we get a zero Hamiltonian. I suspect that this is due to the reparametrization invariance... Will this always happen? Why? Yes, it is due to reparameterization ...
Chiral Anomaly's user avatar
22 votes

Does the structure constant of Yang-Mills field change sign under time reversal?

Rant on notation I hate the physics convention of defining transformations by writing something like "$S \to S$, $\phi \to 2 \phi$, $x \to x + 1$, apples $\to$ oranges, $1 \to -1$, $\pi \to e$, ...
knzhou's user avatar
  • 103k
21 votes
Accepted

Is the magnetic vector potential "real" in classical electromagnetism?

The vector potential is gauge-dependent and unobservable in both classical and quantum mechanics. Only gauge-invariant quantities — including the electric and magnetic fields — are observable. Even in ...
d_b's user avatar
  • 8,309
20 votes
Accepted

How do symmetries “define” physical laws?

A theory is typically described by a Lagrangian, and varying this gives us the equations of motion of the system. The symmetries you describe are symmetries of the Lagrangian i.e. they are ...
John Rennie's user avatar
20 votes
Accepted

What is the analog of the Aharonov-Bohm effect for general gauge fields and for gravity?

Review of the electromagnetic case In the EM case, the Aharonov-Bohm effect can be deduced like this. The lagrangian for a non-relativistic charged particle is $$ L\sim \dot{\mathbf{x}}^2/2+\dot{\...
Chiral Anomaly's user avatar
19 votes

What defines a large gauge transformation, really?

Bundles and compactified spacetime A gauge theory cannot be looked at purely locally, it has inherently global features one cannot see locally. The proper mathematical formalization of a Yang-Mills ...
ACuriousMind's user avatar
  • 126k
19 votes

Why do we use potential for quantizing the electromagnetic field?

The $\bf E$ and $\bf B$ fields viewed as independent quantum oscillators contain too many DOFs, if that's what you mean. But I'm getting ahead of myself. Here is one line of reasoning: It is ...
Qmechanic's user avatar
  • 205k
18 votes
Accepted

Examples of "gauging a global symmetry"

Here is a simple example, one of the first you should try to understand. The theory has a free $U(1)$ scalar field $\phi$ in $d+1$ spacetime dimensions, discussed in the modern notation of ...
Ryan Thorngren's user avatar
18 votes

Haag's comment on the relation between fields and particles

Good question. Some preliminary remarks. The map "one particle" $\leftrightarrow$ "one field" holds, at best, in the weakly coupled regime, where fields are (by construction, cf. ...
AccidentalFourierTransform's user avatar
18 votes
Accepted

How many colors really are there in QCD?

Color charge is a general term that describes how a particle transforms under $SU(3)$ transformations, i.e. what is its $SU(3)$ representation. The terms red, green and blue refer to the fundamental ...
Prahar's user avatar
  • 26.4k
18 votes
Accepted

Trouble reconciling these two views on gauge theory

Gauge theory resembles general relativity applied to extra dimensions beyond the big four. Whether that's the true nature of gauge fields or just a convenient mental picture isn't clear, but it's at ...
benrg's user avatar
  • 27.4k
17 votes
Accepted

Yang-Mills vs Einstein-Hilbert Action

In Yang-Mills, the gauge connection plays the role of a potential and the curvature form plays the role of a "field strength". In GR, the metric tensor plays the role of a potential, and the ...
Bence Racskó's user avatar
17 votes

Which global symmetry of Minkowski space (if any) gets gauged to the diffeomorphism invariance of general relativity?

First, general relativity is not a gauge theory in the narrow sense (of having a gauge field) if you consider the second-order formalism in which only the metric is dynamical. The Einstein-Hilbert ...
ACuriousMind's user avatar
  • 126k
17 votes

Can we understand from basic QED, why is the photon electrically neutral?

Actually I could give a similar answer as given by "Silly Goose" that the gauge transformations of QED form an abelian group, therefore the structure constants of its Lie-algebra are zero, ...
Frederic Thomas's user avatar
16 votes
Accepted

Why do negative norm states break unitarity?

I asked Mark Srednicki about this, and he told me that it's not really correct to say that negative-norm states break unitarity, because negative-norm states don't exist by the definition of the inner ...
tparker's user avatar
  • 48.1k
16 votes
Accepted

Why do (can) we impose local gauge invariance?

I'm with you. I don't want to be unprofessional, but I find the whole "breaking causality" thing to be complete bogus. I see absolutely no way that the humble Klein Gordon field "breaks ...
user1379857's user avatar
  • 11.5k
16 votes
Accepted

Small confusion about the Aharonov-Bohm effect

This "derivation" hits a pet peeve of mine, which is that mathematical treatments of topological phases persistently confuse the phase shift resulting from a physical process with abstract, ...
knzhou's user avatar
  • 103k
16 votes

What is the analog of the Aharonov-Bohm effect for general gauge fields and for gravity?

Since you've already received a good answer to the full question, I'll just add a bit more detail on the analogue of the Aharanov-Bohm effect for gravity. As noted in the existing answer and elsewhere,...
knzhou's user avatar
  • 103k
16 votes
Accepted

What are some of the benefits of understanding differential geometry?

This is a classic tradeoff in research-level physics. A more sophisticated approach can often (a) explain otherwise mysterious results, (b) provide an invariant formulation that does not depend on ...
Andrew's user avatar
  • 50.2k
16 votes
Accepted

Gauge Theory determined by Gauge Group and Representation: What about specifying the bundle?

You have correctly diagnosed at the end of your question why many texts never bother with bundles: As long as we're only doing physics on $\mathbb{R}^4$ or are only interested in local phenomena that ...
ACuriousMind's user avatar
  • 126k
15 votes
Accepted

Understanding gauge fields as connections on a principal G-bundle

A gauge field is a connection on a principal bundle. I'll try to show here roughly how this formulation is related to the common construction in physics of the gauge fields in the process of making ...
coconut's user avatar
  • 4,693
15 votes

Can we use the term "U(1) gauge invariance" for the free electromagnetic field?

OP has a point. The field $A^\mu$ is a connection, and therefore it lives in the algebra of the gauge group, not in the group itself. In this case, $\mathfrak u(1)=\mathbb R$. At first sight, this is ...
AccidentalFourierTransform's user avatar
15 votes
Accepted

Are Maxwell's equations "physical"?

Just a quick complaint about naming first: Maxwell's equations are written in terms of the electric and magnetic fields, which are physical degrees of freedom. You're instead talking about the Euler-...
knzhou's user avatar
  • 103k
15 votes
Accepted

Justification of the $U(1)$ gauge for electromagnetism?

Ultimately, the physical reason for doing this is that it works. There’s a fairly natural line of reasoning which leads to this procedure - that’s not a proof, because proofs don’t exist in physics, ...
J. Murray's user avatar
  • 70k
14 votes

To which extent is general relativity a gauge theory?

The precise sense in which general relativity is a gauge theory has been known (but apparently largely overlooked) for decades. The original sources for what I'm summarizing in the following are a ...
ACuriousMind's user avatar
  • 126k
14 votes
Accepted

Why can't a real scalar couple to the electromagnetic field?

The fact that the theory is not gauge invariant implies that all degrees of freedom of $A_\mu$ must have physical meaning: This is not the theory of photons where only transverse degrees of freedom ...
Valter Moretti's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible