New answers tagged

1

The solution that you wrote in your last (not numbered) equation is not a basis of a Hilbert space of sections because the phase factor: $(-1)^n e^{2i\pi A}$ depends on $n$. The phase factor should not depend on $n$. Please see your (correct) equation (2) defining the boundary conditions, in which the phase factor does not depend on $n$. Thus there is no ...


0

If the vacuum of the theory is supersymmetric - i.e. SUSY is not broken - then it is annihilated by the SUSY generators. On the other hand, using the SUSY algebra one can show that the hamiltonian can be written in terms of the SUSY generators. This implies that the vacuum $|0\rangle$ is supersymmetric if and only if $\langle 0|H|0\rangle=0$, i.e. the vev ...


0

Maxwell's equations, in their microscopic form as formulated by Lorentz, are the standard postulates of electrodynamics. From them all electromagnetic formulas and properties can be derived. The symmetry of these equations relative to spatial translation implies, as follows from Noether's theorem, the law of conservation of linear momentum of the charges ...


2

The field stregth tensor of a Yang-Mills theory is defined as $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+ie\left[A_\mu,A_\nu\right].$$ In general, the gauge field is in the adjoint representation of the gauge group (we normally say it takes value in the algebra) so it is written as $$A_\mu=A_\mu^aT_a,\quad a=1,2,\ldots dim(G),$$ where $dim(G)$ ...


1

In order for a theory to present stable monopole solutions it has to satisfy three requirements: i) It has to have the topological conditions, generally showed as non trivial second homotopy group of the vacuum manifold. ii) It has to satisfy a quantization condition $$e^{ieQ_m}=\mathbb 1,$$ where $Q_m$ is the (non-Abelian) magnetic charge. This is a ...


9

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 make sense. This way you must tackle some non-trivial issue like the negative norm associated with temporal modes. This could be avoided by adding a mass to ...


0

To answer your second question, group theory has a lot of application in physics, especially in particle physics. I guess you are unfamiliar with Special unitary group. Basically, if we consider $SU(2) \times U(1)|$ Yang-Mills system along with the Higgs field, we'd describe electromagnetism along with weak force. Also, let us consider $SU(3)|$ Yang-Mills ...


0

First of all, what's the motivation for the Yang-Mills action and how should I understand the coupling constants $\theta$ and $g$? I would say motivation comes from experiments. For instance it is an experimental fact that the electric charge is conserved. The associated current is also conserved, in the sense of $$\partial_\mu J^\mu=0.$$ Therefore we ...


-4

Do Weyl fermions carry electric charge? That depends on whether Weyl fermions exist, and whether they are what people say they are. See this from the article mentioned by John Rennie: 'Whereas electrons and all the other known fermions have mass, in 1929, mathematician and physicist Hermann Weyl theorized that massless fermions that carry electric ...


2

We should probably start by pointing out that no Weyl fermion has ever been observed. The recent observations are of quasiparticles that behave like Weyl fermions. Speaking rather loosely (and at the risk of upsetting the QFT experts hereabouts) a Dirac fermion can be viewed as a sum of two Weyl fermions, and the observations are of paired quasiparticles ...


1

Of course an anomalous global symmetry destroys the associated Ward identity, but...we don't care so much about that. The Ward identity of global symmetries is not needed for consistency of the theory. However, a broken local Ward identity completely destroys the associated gauge theory, in particular since the decoupling of the unphysical degrees of freedom ...


0

In fact there are two different meaning for the term "global anomaly", which is a pity: global anomaly as opposed to gauge anomaly; global gauge anomaly as opposed to local gauge anomaly. An anomaly can arise from global and gauge symmetries. So here global refers to the fact that the symmetry group is not gauged: these symmetries have a physical ...


0

First of all, we are fundamentally interested in E & B. But essentially, it comes down to the fact that only E and B are physically measurable and so $\phi$ and A are mainly only considered as mathematical constructs, but this isn't always true - they can be conceptualised. I apologise that I cannot give you an immediate physical insight, but I can ...


0

The vector potential has a divergence of zero; we can obtain some intuition by considering the geometry required by the divergence theorem: the volume integral of the divergence of the vector potential is zero for any volume, hence the total net flux through any surface is zero. So given your specific conditions, you can imagine geometric boundaries and ...


0

Yes, any group can be a gauge group. To each oriented edge of the lattice you assign a group element, with the opposite orientations of an edge associated to inverse group elements. The observables are conjugacy classes of products of group elements around a loop of oriented edges. The most basic such observable is the product of group elements around an ...


4

It is the celebrated spin connection on the tangent space, gauging Lorentz rotations so you can take Lorentz covariant derivatives on spinors---you would not be able to do Supergravity without it. As you see, however, $\omega_\mu^{ab}$ is a composite gauge field, that is, it is is an elaborate function of Vierbeine (or Vielbeine) and their derivatives, ...


2

I cannot quite vouch for exhaustive panoramas, but the crucial point is that GL(N), SU(N) matrices are representable in a nonhermitean basis discovered by Sylvester in 1882, the clock and shift matrices which he called nonions for N=3 (long before the Gell-Mann basis!), sedenions, etc. Their braiding relations, and maximal grading, and hence commutators, ...


3

There most definitely is, and your text should have used it in defining the unitary gauge more conventionally: the SU(2) group element parameterization of physics, that is the rotation matrix for spinors R. Absorb v into the definition of σ, where it belongs and from where it can re-emerge at will. $$R=\exp (i\theta ~\hat{n}\cdot\vec{\sigma})=I \cos \theta ...


3

There is no "physical aspect of this fact". The physical variables are the electric and the magnetic field, not the potentials. Introducing the potential is aesthetically and technically pleasing, but it is not necessary. A gauge symmetry is not a physical symmetry. The reason you can have a non-unique potential is that every divergence-free field such as ...



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