David Bar Moshe

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1924

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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 Michael Marinov managed to get some the mutual work included in the thesis published, but the thesis includes more unpublished work.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.
May 14	comment	Uncertainty principle in infinite potential well Yes, in this case, it is true. Gieres reference emphasizes that care must be taken in handling the Dirac bra and ket formalism in the case of infinite dimensional Hilbert spaces. The operator domains must be taken into account in deciding self adjontness.
May 14	comment	Uncertainty principle in infinite potential well The Hilbert space $D_{\theta}$ is not stable under the action of the position operator. The solution is allow for operators to act on a much larger space than this Hilbert space, please see page 18 in the reference given in the main text. Now, the operators are just multiplication and derivative,the uncertainty relation is verifiable by elementary calculus. What is wrong to to write is the following: $(p_{\theta}\Psi_1, \Psi_2) = (\Psi_1, p_{\theta}\Psi_2) $ in the case that $\Psi_1 = \psi_n$ and $\Psi_2 = x \psi_n$. You can verify that equality is wrong again by using elementary calculus.
May 6	comment	Fermi statistics and Berry phase @VijayMurthy In fact, the standard quantization of anyons is performed along similar lines. The key point is that Wess Zumino Witten terms are deform the symplectic structure, thus modify the angular momentum relations. Please see the following paper by Nair citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.148.8249, in which the relativistic anyons fractional statistics are introduced throug a magnetic monopole term in the momentum space.
Feb 26	comment	Feynman rules with helicity states. I have added an update to the answer, in which I have corrected the index error and added an explanation on the relation of the helicity to the polarization vectors (not based on Wigner's classification). Also, I have added an explanation about the photon spin.
Feb 18	comment	Constructing a Hamiltonian (as a polynomial of $q_i$ and $p_i$) from its spectrum Surely, the solution can be not unique, for example, the adjoint action by a unitary transformation would not change the spectrum. Will you be satisfied with any Hamiltonian having the given spectrum?
Jan 27	comment	Why is there no theta-angle (topological term) for the weak interactions? @Thomas: To the right handed fermions. They are not coupled to the gauge fields so their transformation does not change the path integral measure
Jan 26	comment	Why is there no theta-angle (topological term) for the weak interactions? @Luboš I am not an expert, from reading only, I think that your suggestion is quite close to one solution to the strong CP problem assuming the mass of the u-quark is exactly zero,though not widely accepted.
Jan 15	comment	Torsion and gauge invariant EM kinetic term @Pavel, Sorry for the late response, I am afraid I didn't understand the question. Nevertheless, here are some more details , which I hope will be helpful. When formulated by means of the Vector potential, the Maxwell Lagrangian has the same form as on a Riemannian manifold. However, when written in terms of the tetrad components of the gauge field (which are locally $u(1)$ valued sections of the frame bundle, it becomes dependent on the spin connection and as a consequence one gets the correct Maxwell equations $\nabla *F = 0$ from its variation with respect to the tetrad components $A_a$.