Generalized geometric commutator theory and quantum geometric bracket and its uses

• URL: http://arxiv.org/abs/2001.08566v4
• Date: Mon, 26 Dec 2022 08:20:45 GMT
• Title: Generalized geometric commutator theory and quantum geometric bracket and its uses
• Authors: Gen Wang
• Abstract summary: Inspired by the geometric bracket for the generalized covariant Hamilton system, we abstractly define a generalized geometric commutator $$left[ a,b right]=left[ a,b right]_cr+Gleft(s, a,b right)$$ formally equipped with geomutator $Gleft(s, a,b right)=aleft[ s,b right]_cr-bleft[ s,a right]_cr$ defined in terms
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Inspired by the geometric bracket for the generalized covariant Hamilton system, we abstractly define a generalized geometric commutator $$\left[ a,b \right]={{\left[ a,b \right]}_{cr}}+G\left(s,a,b \right)$$ formally equipped with geomutator $G\left(s, a,b \right)=a{{\left[ s,b \right]}_{cr}}-b{{\left[ s,a \right]}_{cr}}$ defined in terms of structural function $s$ related to the structure of spacetime or manifolds itself for revising the classical representation ${{\left[ a,b \right]}_{cr}}=ab-ba$ for any elements $a$ and $b$ of any algebra. Then we use the generalized geometric commutator to define quantum covariant Poisson bracket that is related to the quantum geometric bracket defined by geomutator as a generalization of quantum Poisson bracket. The covariant dynamics includes the generalized Heisenberg equation as a natural extension of Heisenberg equation and G-dynamics based on the quantum geometric bracket, meanwhile, the geometric canonical commutation relation is induced. As an application, we reconsider the canonical commutation relation and the quantization of field to be more complete.

Related papers

• Quantum geometric Wigner construction for $D(G)$ and braided racks [0.0]
  A quantum double $D(G)=Bbb C(G)rtimes Bbb C G$ of a finite group plays an important role in the Kitaev model for quantum computing. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincar'e group of $Bbb R1,3$.
  arXiv  Detail & Related papers  (2024-07-16T15:21:28Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Quantum charges of harmonic oscillators [55.2480439325792]
  We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
  arXiv  Detail & Related papers  (2024-04-02T09:16:18Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Beyond the Berry Phase: Extrinsic Geometry of Quantum States [77.34726150561087]
  We show how all properties of a quantum manifold of states are fully described by a gauge-invariant Bargmann. We show how our results have immediate applications to the modern theory of polarization.
  arXiv  Detail & Related papers  (2022-05-30T18:01:34Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Quantum scrambling of observable algebras [0.0]
  quantum scrambling is defined by how the associated physical degrees of freedom get mixed up with others by the dynamics. This is accomplished by introducing a measure, the geometric algebra anti-correlator (GAAC) of the self-orthogonalization of the commutant of $cal A$ induced by the dynamics. For generic energy spectrum we find explicit expressions for the infinite-time average of the GAAC which encode the relation between $cal A$ and the full system of Hamiltonian eigenstates.
  arXiv  Detail & Related papers  (2021-07-02T14:30:58Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version [0.0]
  We propose a new mathematical approach to Bona's non-linear generalization of quantum mechanics. It highlights the central role of self-consistency. Some new mathematical concepts are introduced, which are possibly interesting by themselves.
  arXiv  Detail & Related papers  (2020-09-10T16:20:25Z)
• Getting to the Bottom of Noether's Theorem [0.0]
  We show that Noether's theorem holds whenever we can map observables to generators in such a way that each observable generates a one- parameter group that preserves itself. We show this expresses a relation between quantum and statistical mechanics, closely connected to the principle that "inverse temperature is imaginary time"
  arXiv  Detail & Related papers  (2020-06-26T00:56:17Z)
• Generalized Extended Momentum Operator [0.0]
  We study and generalize the momentum operator satisfying the extended uncertainty principle relation (EUP) This generalized extended momentum operator (GEMO) consists of an arbitrary auxiliary function of position operator, $mu left( xright) $, in such a combination that not only GEMO satisfies the EUP relation but also it is Hermitian.
  arXiv  Detail & Related papers  (2020-02-24T13:19:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.