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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.
Jul
10
comment Kähler potential vs full effective potential
(cont.) The main theme of the thesis is the exploitation of the Kaehler geometry of coadjoint orbits of semisimple Lie groups. My late doctoral supervisor Dear prof. Michael Marinov managed to get some the mutual work included in the thesis published, but the thesis includes more unpublished work. Again, thank you for the interest and I'll see what I can do.
Jul
10
comment Kähler potential vs full effective potential
@Arnold Neumaier. Thank you very much for the interest. Currently I don't have an electronic copy. Then, I was vey concentrated in my work and I didn't care much of learning Latex or anything else related to computers. So, although the thesis was written in Latex by a friend, I inserted the equations manually. I have a hard copy that I can scan and send. Please give me some time to see what I can do.
Jul
4
comment What is the winding number of a magnetic monopole, and why is it conserved
This is because the commutator of odd valued differential forms comes with a plus sign, and I wasn't strict about normalizations . Please see Yang Zhang monograph for a clear introduction: lepp.cornell.edu/~yz98/notes/…. Please, see also the review article of Eguchi Kilkey Hansen empg.maths.ed.ac.uk/Activities/GT/EGH.pdf, where a good introduction for Lie algebra valued differential forms.
Jun
18
comment Is this a quaternion representation of the equations of motion of General Relativity?
Please notice that the term containing the derivative of the local Lorentz transformation (in both vectorial and quaternionic representations) is only seemingly linear in the velocity components . The derivative of the local Lorentz transformation itself is linear in the velocity components, thus this term is quadratic in the velocity as in the standard form of the geodesic equation.
Jun
6
comment What is spontaneous symmetry breaking in QUANTUM systems?
May be it was not emphasized enough in the answer, but it was supposed to be the main point. Here, spontaneous symmetry breaking can be tested on the finite dimensional (effective) Hilbert space obtained from the quantization (in the sense of geometric quantization) of the trial function manifold. This is the Hilbert space of nonequivalent vaccua. The mere existince of a nontrivial Hilbert space after quantization is the indication of spontaneous symmetry breaking.
May
16
comment Wigner-Eckart theorem of SU(3)
@ramanujan I think that the Wigner-Eckart theorem is not the best method to find the matrix elements in the question. Please tell if you are interested in the matrix elements, I think that I can help you, using other methods
May
16
comment How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
Continued, . If a weight has a a positive component of value $n$ at the $m-$th place, then there are weights in the representation obtained by $1, 2, ., ., ., n$ subtractions of the $m-th$ primitive root. Please, see a less trivial example (which you can use as an exercise if you wish) on page 84 (the representation 16 of SO(9)).
May
16
comment How do I find the tensor components of all weights of a representation of SU(3), e.g. the six dimensional representation (2,0)
The two dimensional vectors are just the weights corresponding to the basis vectors, Sorry for the abuse notation. Please see first equation (5.4), on page 32 of Slansky, where the weight diagrams of the fundamental as well as the (2,0) representations are given. The algorithm for the construction of the weight diagram from the highest weight is described in words in the few lines preceding the equation. The principle is that in the weight basis, the primitive roots correspond to the rows of the Cartan matrix.
May
15
comment Finding wave-fuctions of a Dirac particle for given 4-momentum and spin 4-vector
I suppose that you mean by the spin four vector the Pauli-Lubanski vector $s^{\mu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma}J_{\nu \rho} P_{\sigma}$, and this is the reason that you chose it to be orthogonal to the momentum vector.