David Bar Moshe
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 Jun 5 comment The $U(1)$ charge of a representation @user41746 $SU(5)$ is $A_4$, so the metric tensor should be $4\times 4$ matrix, please see the first matrix in table 7 page 82. Its third row for n=4 is just the given vector. I can’t see this exercise on page 84 (This page belongs to the appendix and it contains tables 10 and 11a. However, on page 16 equation (3.3), Slansky obtained the same hypercharges as in the question. May 22 comment The derivation of the Belinfante-Rosenfeld tensor @ramanujan_dirac: The energy momentum tensor constitutes of the source term in Einstein equations: $R_{\mu\nu}-g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$, and both the Ricci curvature and the metric tensor are symmetric. May 1 comment The curvature of the space of commuting hermitian matrices @vesofilev: Sorry for not understanding you. The following reference arxiv.org/abs/0801.2913 contains a parameterization of the matrix $U$ for the flag manifold $\frac{SU(3)}{U(1)\times U(1)}$ on page 9, the third paragraph. (The authors call it the dressing matrix). Apr 30 comment The curvature of the space of commuting hermitian matrices @vesofilev: The $SU(2)$ case is a special case. Take for example the space of one commuting matrix in $SU(3)$, the spaces obtained from the adjoint action on $diag(0,0,1)$ and $diag(0,1,1)$ are disconnected. Apr 30 comment The curvature of the space of commuting hermitian matrices The space of $p$ commuting $n\times n$ matrices is not connected, since a unitary transformation does not change the rank of a matrix. Each connected component corresponds to a specific choice of the ranks of the commuting matrices. The connected components are generalized flag manifolds. Since they are homogeneous spaces, they are of constant curvature and there are many ways to calculate their curvatures.The following lecture notes may be of help to you uregina.ca/~mareal/flag-coh.pdf Mar 25 comment Soliton Moduli Spaces and Homotopy Theory @jamalS cont. There are more applications besides anomalies such as spinc quantization, charge quantization and fractionalization, Brane charges, supersymmetric quantum mechanics, Berezin-Toeplitz quantization and may be more. I don’t know of a single review comprising all these applications. Mar 25 comment Soliton Moduli Spaces and Homotopy Theory @jamalS There are analogous computations for the monopole moduli space dimension in the following review arxiv.org/abs/hep-th/0609055v2 by Weinberg and Yi, and for nonAbelian vortices in arxiv.org/abs/hep-th/0306150v1 by Hannay and Tong. There is a more mathematical review of the instanton and Seiberg-Witten moduli spaces dimension computation by David Bleecker milne.ruc.dk/~Booss/A-S-Index-Book/BlckBss_2012_04_25.pdf. Mar 20 comment Lie group of Schrodinger Wave equation @user35952 I'll try to answer your new question soon, I'll try to clarify the previous "Poincare group vs Galilean group" answer Mar 13 comment Which symmetric pure qudit states can be reached within local operations? @Piotr Migdal Thank you very much Mar 9 comment Lie algebra of axial charges @gian_25 The matrix representation is just the 4-dimensional fundamental representation of $SO(4) = SU(2) \times SU(2)$, denoting the indices of the 4-space corresponding to $(\sigma, \pi_1, \pi_2, \pi_3)$ by $0,1,2,3$. Then$(T^{V}_i)_{jk} = i \epsilon_{ijk}, (T^{(A)}_i)_{jk} = i (\delta_{ji}\delta_{k0}+ \delta_{ki}\delta_{i0})$. Please observe that the commutator of two axial generators is a vector generator. Mar 6 comment How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative? @MBN there are partial results, for example Ellis meghnad.iucaa.ernet.in/~tarun/pprnt/topology/… mentions a result by Clarke that every 3+1 noncompact space-time can be isometrically embedded into a flat $\mathbb{R}^{87,3}$. Mar 6 comment How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative? @linuxfreebird. In the case when the transformation is invertible, then your analysis is correct, but this special case is not really interesting in general relativity. I updated my answer with a detailed explanation. Mar 6 comment Quantization of electrostatic $\vec E$ field? @webb corrected and cited. But this does not change the argument: a nonvanishing commutation relation between any component of the electric and magnetic fields imply nonzero uncertainty. Feb 27 comment Conservation of phase space volume in Rindler space-time @V. Moretti Thank you very much for the information Feb 26 comment Conservation of phase space volume in Rindler space-time @V. Moretti I am referring to the canonical quantization of the Hamiltonian (Last equation in the text) $H = gx \sqrt{p^2+m^2}$ on $L^2(\mathbb{R}, dx)$. Operator ordering will be needed to reach a self adjoint operator. I think that no solution is available to this problem because of the geodesic incompleteness of the Rindler space. Feb 25 comment Conservation of phase space volume in Rindler space-time @Nathaniel cont. In the field case, you are talking about an infinite dimensional Hamiltonian system. It is not straightforward to define a volume in infinite dimensions. (It must be regularized). Although, there are some results on regularized volumes in general, I think that there is only little work, if any, on Liouville's theorem in infinite dimensions. Feb 25 comment Conservation of phase space volume in Rindler space-time @Nathaniel In general, the phase space of a system of several particles, even in interaction, is the Cartesian product of their individual phase spaces. The phase space is the space of initial data and one needs two initial conditions per particle. Thus, the same result should be valid. Feb 18 comment Physical intuition for deformation quantization of Poisson manifolds @user40276 - 1)I have added a few examples of explicitly known star products. 2) Consider for example vector fields which can be realized as differential operators on each chart, but not every set of differential operators defined on each chart is a vector field. Only the cases in which these differential operators satisfy the correct transformation properties on the overlaps make them vector fields, i.e., when they are global sections of $TM$. Feb 18 comment Physical intuition for deformation quantization of Poisson manifolds @Alex Nelson - Corrected. Thank you. Jan 9 comment Topological ground state degeneracy of SU(N), SO(N), Sp(N) Chern-Simons theory @Idear For the case of $SO(N)$ You can use equation (1b) together with the coefficients given in sections 3.B and 3.D of Gannon's article arxiv.org/abs/hep-th/0106123 to compute the integrable highest weights. I'll correct and update my answer soon. I used the symbol # for the cardinality of the set. Yes, the rank is the dimension of the Cartan matrix, it is also the dimension of the maximal torus $T$.