28 votes

Inverse square rule for strong forces

Most of the forces induced by a point particle follows the 1/r^2 rule No, it's the forces mediated by point particles with no mass and charge that follow the the 1/r^2 rule. then why does strong ...
Maury Markowitz'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
  • 102k
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
18 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
  • 124k
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
  • 25.5k
17 votes
Accepted

Calculating the Berry curvature in case of degenerate levels (Non abelian Berry curvature): issue

The short answer is that only if the Berry curvature is defined by: (in matrix notation): $$F_{\mu \nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} + [A_{\mu}, A_{\nu}]$$ it becomes gauge covariant, ...
David Bar Moshe's user avatar
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
13 votes

Why hasn't Peter Higgs solved the Yang-Mills prize problem?

The standard model Higgs particle has a weak charge, but no color charge. As a result it generates as mass gap in electroweak theory (masses for the W and Z), but not in QCD (no gluon mass). However, ...
Thomas's user avatar
  • 18.5k
11 votes

Why is a superposition of vacuum states possible in QCD, but not in electroweak theory?

The difference between the two cases is the nature of the vacua. In the case of spontaneous symmetry breaking, you find that the tunneling amplitude between them is proportional to the volume, so ...
ACuriousMind's user avatar
  • 124k
11 votes
Accepted

Gauge invariance is just a redundancy. Why is massive abelian gauge field renormalizable but massive non-abelian gauge field nonrenormalizable?

Certainly I admit that power counting law is violated, but why does violation of power-counting have relation with renormalizablity? As per the Dyson-Weinberg power-counting theorem (see Ref.1, ...
AccidentalFourierTransform's user avatar
11 votes
Accepted

What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

The $\mathrm{SU}(3)$ gauge symmetry is a local symmetry, and therefore it is not Noether's first, but Noether's second theorem that applies to it, which does not yield conserved quantities. For $\...
ACuriousMind's user avatar
  • 124k
10 votes
Accepted

For the Yang-Mills field strength defined as a commutator, why does the $A_\nu\partial_\mu - A_\mu\partial_\nu$ term vanish?

Note that, for example, \begin{align} [A_\mu,\partial_\nu]f&=A_\mu\partial_\nu f-\partial_\nu(A_\mu f)\\ &=A_\mu\partial_\nu f-\partial_\nu(A_\mu)f-A_\mu\partial_\nu f\\ &=-f\partial_\nu ...
Frame's user avatar
  • 606
10 votes

Uniqueness of Yang-Mills theory

If you don't impose power-counting renormalizability, there are a host of other possibilities, since higher order derivatives or higher order interactions can be introduced. For example, terms $(Tr(F^...
Arnold Neumaier's user avatar
10 votes

If the classical Maxwell theory describes E&M fairly, how well-done is the classical Yang-Mills theory for chromodynamics?

This is nicely answered in Jaffe-Witten's "problem description" of the problem of quantization of Yang-Mills theory: By the 1950s, when Yang–Mills theory was discovered, it was already known that ...
Urs Schreiber's user avatar
10 votes

"Hidden" theta-term in Hamiltonian formulation of Yang-Mills theory

It's not particularly strange or unusual for a term to seemingly not appear in the Hamiltonian, but still have a physical effect. For example, the Hamiltonian for a free particle is $$H = \frac{p^2}{...
knzhou's user avatar
  • 102k
10 votes

Is gauge covariant derivative an ordinary covariant derivative?

The gauge covariant derivative is a genuine covariant derivative in the ordinary sense of differential geometry, but in the general sense of a (principal) connection form $A$ inducing covariant ...
ACuriousMind's user avatar
  • 124k
10 votes

Berry phase and Wilson loop

Both the usual gauge fields ($A_\text{QFT}$) and the Berry connection ($A_🍓$) are connections on a principal bundle. In usual gauge theory in QFT you have a spacetime manifold, $S$ and a Lie algebra $...
ɪdɪət strəʊlə's user avatar
10 votes

How does the absence of quadratic terms in the Lagrangian imply massless quanta?

I will expand a bit on one of the answers to be clearer why a quadratic term usually leads to the 'mass' term. Because the coefficient in front of the $A_\mu A^\mu$ term has dimension $2$ doesn't by ...
Avantgarde's user avatar
  • 3,832
9 votes
Accepted

Yang-Mills potential and principal bundles

The term gauge transformation refers to two related notions in this context. Let $P$ be a principal $G$-bundle over a manifold $M$, and let $\cup_i U_i$ be a cover of $M$. A connection on $P$ is ...
Elliot Schneider's user avatar
9 votes
Accepted

Why can't compact symplectic groups $Sp(n)\equiv USp(2n)\equiv U(2n)\cap Sp(2n,\mathbb{C})$ be gauge groups in Yang-Mills theory?

The structure of standard model $SU(3)\times SU(2)\times U(1)$ is chiral which basically tells you the necessity of chiral fermions. If left-handed fermions transform under a representation $R$ of the ...
ved's user avatar
  • 787
9 votes

Geometry of Yang-Mills theory

Hands down, the best reference book to learn about gauge theories is DeWitt's The Global Approach to Quantum Field Theory. In this book you can find the most general formulation of both classical and ...
9 votes

Physical content of $[D_\mu, D_\nu]=ieQF_{\mu\nu}$ and $[D_\mu, D_\nu]=igT^a G^a_{\mu\nu}$

It may be worth looking at this formula as the infinitesimal limit of an exponentiated relation. We have $$ \exp(- a^\mu D_\mu) \exp(- b^\nu D_\nu) \exp( a^\mu D_\mu) \exp( b^\nu D_\nu) \approx 1 + ...
Adam Latosiński's user avatar
9 votes
Accepted

Why is the standard model gauge group $SU(3) \times SU(2) \times U(1)$ and not $U(3) \times U(2) \times U(1)$?

Background. Indeed, the quark bilinears have an extra exact symmetry $U(1)_B$ (baryon number) and the lepton ones an extra symmetry $U(1)_L$ (lepton number). These two commute with each other and the ...
Cosmas Zachos's user avatar
9 votes

2 coinciding D-branes leads to a $U(2)$ gauge theory

The answer to your second question is the easier one, because it is purely about the structure of the Lie algebra $\mathfrak{u}(N)$: The group $\mathrm{U}(N)$ has dimension $N^2$ and its maximal torus ...
ACuriousMind's user avatar
  • 124k
9 votes
Accepted

How do I understand the Hodge $⋆$ operator in Yang-Mills Lagrangian?

The Hodge dual of a $p$-form (antisymmetric $(0,p)$ tensor) in $d$ dimensions is a $(d-p)$-form whose components are defined by $$ (\star A)_{\mu_1 \cdots \mu_{d-p}} \equiv \frac{1}{p!} \epsilon_{\...
Prahar's user avatar
  • 25.5k
8 votes
Accepted

Interpretation of the field strength tensor in Yang-Mills Theory

Classically, the gauge field strength is a curvature of a connection, the same way that the Riemann tensor is. Since $F^a_{\mu\nu}$ lives in the adjoint representation of the gauge group, you can ...
Ben Niehoff's user avatar
  • 1,051
8 votes
Accepted

Is color charge quantized?

Naively, color can vary continuously between the colors according to a gauge transformation $\psi\mapsto \mathrm{e}^{\mathrm{i}\epsilon^a T^a}$ for some $\mathfrak{su}(2)$-valued object $\epsilon$, ...
ACuriousMind's user avatar
  • 124k
8 votes
Accepted

Why do we solve the Wess-Zumino consistency condition using the method of descent?

The case of perturbative anomalies is straightforward: the only such anomalies are of the ABJ type, i.e., they appear in even dimensions and are associated to chiral fermions. These are completely ...
AccidentalFourierTransform's user avatar
8 votes

How are sources described in gauge theory?

Because gauge invariance requires covariant "conservation" $$ 0=\nabla_\mu J^\mu\equiv \partial_\mu J^\mu +[A_\mu, J^\mu], $$ the sources $J^\mu$ in non-abelian theory cannot be $c$-...
mike stone's user avatar
  • 52.4k
8 votes
Accepted

Precise definition of "simple gauge group" in the Yang-Mills existence/mass gap problem

In high-energy physics, "simple" always means "any group whose Lie algebra is any of the algebras in the Cartan classification $\{A,B,C,D,E,F,G\}$" where $A_n=\mathfrak{su}_{n-1}$, ...
AccidentalFourierTransform's user avatar

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