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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 $T$.
Jan
7
comment Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed
@Hamurabi I have corrected the answer
Jan
6
comment Vector potential $A$ on a 2-sphere $S^2$ of radius $R$ with some points removed
@Hamurabi Sorry that it is taking more time than I thought. I'll try to finish as soon as possible.
Dec
30
comment Braiding statistics of anyons from a Non-Abelian Chern-Simon theory
@Hamurabi There exist generalizations of the Knizhnik-Zamolodchikov, for example arXiv:arXiv:hep-th/9510143, hep-th/9410091, for elliptic curves and Riemann surfaces.
Dec
19
comment Braiding statistics of anyons from a Non-Abelian Chern-Simon theory
@Ideat cont. Witten's approach is more "thermodynamical" and he prefers to see the traces of the non-Abelian statistics in the partition function. To be more precise,(and this fact was also written in not so much words in Witten's paper): A non-Abelian Wilson loop can be thought of as a particle moving on a group or a flag manifold stuck to the boundary in the limit where its mass vanishes. Then its dynamics restricts upon quantization to the lowest Landau level producing the correct Wilson loop insertion.
Dec
19
comment Braiding statistics of anyons from a Non-Abelian Chern-Simon theory
@Idear These two approaches are very similar (but not exactly equivalent). Actually, Witten in his Jones polynomial paper (on page 365) refers to this similarity and asserts that the Wilson loop can be "thought of" as the trajectory of a particle in 2+1 dimensions. Witten refers to a famous paper by Polyakov adopting the strategy that Lee and Oh used later. This is only one of the numerous issues that Witten only talked about in his Jones polynomial paper (even without giving a single formula), which proved to be very fruitful for subsequent research.