Generalized Bopp shift, Darboux Canonicalization, and the Kinematical Inequivalence of NCQM and QM

• URL: http://arxiv.org/abs/2603.00524v1
• Date: Sat, 28 Feb 2026 07:49:53 GMT
• Title: Generalized Bopp shift, Darboux Canonicalization, and the Kinematical Inequivalence of NCQM and QM
• Authors: S. Hasibul Hassan Chowdhury,
• Abstract summary: Two-dimensional noncommutative quantum mechanics is often formulated through linear transformations of represented position and momentum operators.<n>We clarify the representation-theoretic meaning of such constructions at the level of kinematical symmetry groups and irreducible unitary representations.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Two-dimensional noncommutative quantum mechanics (NCQM) is often formulated through linear transformations of represented position and momentum operators and through Darboux-type canonicalizations. We clarify the representation-theoretic meaning of such constructions at the level of kinematical symmetry groups and irreducible unitary representations. The standard NCQM commutators are naturally encoded by a step-two nilpotent Lie group $G_{\hbox{\tiny{NC}}}$ with three-dimensional center, whose irreducible sectors are labeled by central characters (equivalently, coadjoint-orbit labels), parametrized on the regular stratum by $(\hbar,\vartheta,B_{\mathrm{in}})$. In this language, ordinary two-dimensional quantum mechanics (QM) is the quotient (equivalently, inflation) sector $(\hbar,0,0)\subset \widehat{G_{\hbox{\tiny{NC}}}}$, the unitary dual of $G_{\hbox{\tiny{NC}}}$; i.e., it consists of those representations that factor through the central quotient $G_{\hbox{\tiny{NC}}}\twoheadrightarrow G_{\hbox{\tiny{WH}}}$, where $G_{\hbox{\tiny{WH}}}$ denotes the Weyl--Heisenberg group. We show that generalized Bopp-shift and Seiberg--Witten-type linear recombinations of represented operators, and the existence of an auxiliary quadruple satisfying the canonical commutation relations obtained by Darboux canonicalization, do not imply unitary equivalence between a fixed generic NCQM sector $(\hbar_{0},\vartheta_{0},B_{0})$ and the ordinary-quantum-mechanics sector $(\hbar_{0},0,0)$ of $\widehat{G_{\hbox{\tiny{NC}}}}$.

Related papers

• Transmutation based Quantum Simulation for Non-unitary Dynamics [35.35971148847751]
  We present a quantum algorithm for simulating dissipative diffusion dynamics generated by positive semidefinite operators of the form $A=Ldagger L$.<n>Our main tool is the Kannai transform, which represents the diffusion semigroup $e-TA$ as a Gaussian-weighted superposition of unitary wave propagators.
  arXiv  Detail & Related papers  (2026-01-07T05:47:22Z)
• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids [21.876059213677966]
  We employ the $mathrmSU(n)_k$ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions.<n>The spins on all lattice sites are chosen to transform under the $mathrmSU(n)$ irreducible representation with a single row and $k$ boxes in the Young tableau.
  arXiv  Detail & Related papers  (2025-01-16T14:42:00Z)
• Gaussian quantum Markov semigroups on finitely many modes admitting a normal invariant state [0.0]
  Gaussian quantum Markov semigroups (GQMSs) are of fundamental importance in modelling the evolution of several quantum systems.<n>We completely characterize those GQMSs that admit a normal invariant state and we provide a description of the set of normal invariant states.<n>We study the behavior of such semigroups for long times: firstly, we clarify the relationship between the decoherence-free subalgebra and the spectrum of $mathbfZ$.
  arXiv  Detail & Related papers  (2024-12-13T10:01:18Z)
• Quantum Cellular Automata on Symmetric Subalgebras [6.158725838873227]
  We investigate quantum cellular automata on one-dimensional spin systems defined over a subalgebra of the full local operator algebra.<n>For systems where each site carries a regular representation of $G$, we establish a complete classification of such subalgebra QCAs.
  arXiv  Detail & Related papers  (2024-11-28T17:22:50Z)
• Representation theory of Gaussian unitary transformations for bosonic and fermionic systems [0.0]
  We analyze the behavior of the sign ambiguity that one needs to deal with when moving between the groups of the symplectic and special annihilation group. We show how we can efficiently describe group multiplications in the double cover without the need of going to a faithful representation on an exponentially large or even infinite-dimensional space.
  arXiv  Detail & Related papers  (2024-09-18T01:22:38Z)
• 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)
• DHR bimodules of quasi-local algebras and symmetric quantum cellular automata [0.0]
  We show that for the double spin flip action $mathbbZ/2mathbbZtimes mathbbZ/2mathbbZZcurvearrowright mathbbC2otimes mathbbC2$, the group of symmetric QCA modulo symmetric finite depth circuits in 1D contains a copy of $S_3$, hence is non-abelian.
  arXiv  Detail & Related papers  (2023-03-31T18:33:07Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension [0.0]
  We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_omega,lambda:=beginbmatrix-fraclambda+omegax&-partial_x.
  arXiv  Detail & Related papers  (2021-07-08T11:48:57Z)
• Sub-bosonic (deformed) ladder operators [62.997667081978825]
  We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness. This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states. In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
  arXiv  Detail & Related papers  (2020-09-10T20:53:58Z)

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.