12,721 reputation
21334
bio website
location
age
visits member for 3 years, 10 months
seen 2 hours ago

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.
Dec
17
comment Collection of histories vs. collection of momentary configurations
@Nick Kidman 1) Yes, it is a quotient space often denoted as $ \mathcal{M}//G$ (with two lines) called the Marsden-Weinstein quotient or the symplectic quotient. $ \mathcal{M}$ is the unreduced phase space and $G$ is the gauge group.In a finite number of dimensions (where we can count), the dimension of the reduced space is $ \mathrm{dim] \mathcal{M} - 2\mathrm{dim] G$. The factor 2 comes from the fact that we need to remove one dimension per symmetry and another dimension for gauge fixing. 2) We can reduce only by the anomaly free subgroup of the gauge group.
Dec
15
comment Separability axiom really necessary?
@moppio89 cont. In quantum mechanics we are interested in computing probabilities. The Hilbert space is an auxiliary tool. The example that I had in mind is the case of bosonization where observable of the same physical system have representations on the Bose and Fermi Hilbert spaces.
Dec
15
comment Separability axiom really necessary?
@moppio89 Yes, I think that their main assumption that what is physically important can be deduced from a scattering process. In addition, the usual notion of vacuum restricts to a subspace of a nonseparable space. Also, may be the mathematical complications constituted a factor due to the need for more advanced measure theoretic analysis on the underlying space than in the case of a countable set.