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Oct
4
comment How to construct the charge conjugation matrix for any given dimension?
I can show you how to construct in this particular case, but can you please check your calculations because in your expressions $(\Gamma^1)^2=(\Gamma^2)^2=1$ while $(\Gamma^3)^2=(\Gamma^4)^2=-1$, i.e., you are working in a signature $(1, 1, -1, -1)$. Is this really the case you need.
Sep
23
comment Killing vectors for SO(3) (rotational) symmetry
cont. For the computation of the Killing vectors according to the given Wikipedia page, one needs in advance to construct the invariant metric. Furthermore, given the invariant metric, the Killing vector components satisfy differential equations which are harder to solve than the algebraic equations in the method described in the answer.
Sep
23
comment Killing vectors for SO(3) (rotational) symmetry
@ramanujan_dirac: Actually, for the construction described above, one does not need to know any property of the Maurer-Cartan form except its evaluation by the insertion of the Euler parameter formula of $g$ into its definition. This form has many interseting properties and applications, please see for further reading Shlomo Sternberg lectures: math.harvard.edu/~shlomo/docs/lie_algebras.pdf.
Sep
6
comment Aharonov-Bohm Effect and Flux Quantization in superconductors
Yes, this is exactly the definition of multivaluedness. Take for example the "funtion" $e^{\frac{i\theta}{2}}$ on the circle, it is multiple valued since it takes two different values at $\theta = 2\pi$ and $\theta = 4\pi$ which are the same physical point. Its modulus is a true function on the circle. Of course, the modulous operation cancels only a single global phase and if the wave function is a superposition, the relative phases will still exist. This is the reason why the wave function "feels" the topology in the Aharonov-Bohm effect.
Sep
5
comment classical dynamics on group manifold SU(2)
I added an update answering the question about the hydrogen atom
Aug
16
comment Why do some anomalies (only) lead to inconsistent quantum field theories
Sorry for the late response, I added an update to the answer addressing your question and relating to Drake's remark. I have edited the account about the Mickelsson’s project which did not reflect my opinion on the importance of this approach which I think that is very valuable.
Aug
13
comment First class and second class constraints
When all the 5 constraints are first class, then one has to choose 5 functions on the phase space (gauge fixing conditions)$\chi_{\alpha}$, such that the matrix $M_{\alpha\beta} = \{\chi_{\alpha}, \phi_{\beta}\}|_{\Sigma^{\prime}}$ is every where nonsingular on the constraint plus gauge fixing surface $\Sigma^{\prime}$
Aug
2
comment Classical and quantum anomalies
Actually, I have a little more information on this subject. I'll try to add an update when I can
Aug
2
comment Classical and quantum anomalies
(cont.) webcache.googleusercontent.com/…
Aug
2
comment Classical and quantum anomalies
(cont.) But, if one uses the Sugawara construction for the Virasoro generators from the current generators, the Virasoro central charge (in the Poisson brackets) will be zero. However, in the Liouville theory one already obtains a nonvanishing Virasoro central charge in the Poisson brackets, please see the following article by Toppan:
Aug
2
comment Classical and quantum anomalies
Here are some examples referring to both your questions. In the WZNW model in two dimensions, one gets the Kac-Moody algebra at the classical level (Poisson brackets) with the correct central charge. In fact this computation was performed by Witten in his original article: projecteuclid.org/…. For a more clear derivation please see: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.8249
Aug
1
comment Classical and quantum anomalies
I have added an update with the elaboration of the spin case
Jul
31
comment Classical and quantum anomalies
When the representation on the wave functions is a ray representation,it is an indication of an anomaly, because the symmetry group is not represented faithfully. Of course group extensions can be studied in general separately from quantum mechanics. I'll update my answer tomorrow and add a few more references.
Jul
31
comment Gauge fixing choice for the gauge field $A_0$
(cont.), please see the following review by: ANTTI SALMELA ethesis.helsinki.fi/julkaisut/mat/fysik/vk/salmela/gausssla.pdf
Jul
31
comment Gauge fixing choice for the gauge field $A_0$
But if one subtitutes $A_0 = 0$ in the Lagrangian before the computation of the equations of motion, then equation of motion of $A_0$, namely, the Gauss law will be lost, because one will not have an $A_0$ in the Lagrangian to take a variation with respect to it. In this case one has to impose the Gauss law "by hand".
Jul
24
comment What does symplecticity imply?
(cont.) There are also advantages in Poisson geometry, since the symplectic leaves of a Poisson manifold may not be manifolds, and also it is more natural to perform stability analysis within Poisson geometry.
Jul
24
comment What does symplecticity imply?
@Arnold (in addition to Jonathan's remark). The same happens for the ideal fluid example. The motion is restricted to a single coadjoint orbit of the volume preserving diffeomorphism group. Actually, the integral curves of any Hamiltonian on any Poisson manifold are restricted to a single symplectic leaf. The importance of Poisson geometry lies in that it includes the solutions of all possible initial conditions in the classical case and all inequivalent quantizations after quantization.
Jul
24
comment What does symplecticity imply?
@Arnold Of course, but symplectic geometry remains important even within Poisson geometry (I think that this is the reason that they don't have an Arxiv subject classification of Poisson geometry). The rigid body example you gave can be formulated on the symplectic manifold $T^*SO(3)$ (Euler angles + angular momenta), then Poisson reduced to $\mathbf{so(3)^*}$ a Poisson non symplectic manifold, but the dynamics actually takes place on a single coadjoint orbit: $S^2$, agian a symplectic manifold.
Jul
23
comment when is the stationary phase approximation exact?
@Arnold. The generalization of the Duistermaat-Heckman theorem to field theories is only to the physics level of rigor. Actually, the field theory examples for which this formalism was applied constitute of finite dimensional reduced phase spaces. For example the Chern-Simons theory whose reduced phase space is the moduli space of flat connections, or the coherent state path integral which is reduced to the lowest landau level by supersymmetry. I think that the first one who in roduced this generalization is Antti Niemi, please see for example Arxiv: hep-th/9301059 (He has also earlier works).
Jul
18
comment Stokes' theorem and alike for fractal surfaces
The main idea in these works is that under some rather mild constraints on the types of functions to be integrated, the integration over fractals can be obtained by a limiting procedure of integrals over polyhedral approximations. Now, Maxwell equations are correct in their integral form: Since quantities like total charges and total currents should be finite, this would require their densities in some cases to have fractal dimensions. The EMF is an integral quantity, thus its calculation from the integral form of the Faraday's should not change.