Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

S. Hasibul Hassan Chowdhury

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 $\widehat{G_{\hbox{\tiny{NC}}}}$.

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

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 with three-dimensional center, whose irreducible sectors are labeled by central characters (equivalently, coadjoint-orbit labels), parametrized on the regular stratum by . In this language, ordinary two-dimensional quantum mechanics (QM) is the quotient (equivalently, inflation) sector , the unitary dual of ; i.e., it consists of those representations that factor through the central quotient , where 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 and the ordinary-quantum-mechanics sector of .
Paper Structure (15 sections, 3 theorems, 48 equations)

This paper contains 15 sections, 3 theorems, 48 equations.

Table of Contents

  1. Motivation and scope
  2. On basis change and kinematical equivalence
  3. The kinematical group $G_{\hbox{\tiny{NC}}}$ and its unitary irreducible representation classes
  4. Ordinary quantum mechanics as an inflation family inside $\widehat{G_{\hbox{\tiny{NC}}}}$
  5. Central quotient at the Lie algebra level
  6. The induced surjection $\pi:G_{\hbox{\tiny{NC}}}\twoheadrightarrow G_{\hbox{\tiny{WH}}}$
  7. Inflation and the class $(\hbar_{0},0,0)\in\widehat{G_{\hbox{\tiny{NC}}}}$
  8. Coadjoint orbits and dependence on the central character
  9. Bopp shifts and Darboux canonicalization: several layers inside a fixed class in $\widehat{G_{\hbox{\tiny{NC}}}}$
  10. The $(r,s)$--family as an explicit linear recombination of represented operators
  11. Darboux canonicalization and the computational CCR layer
  12. An intrinsic canonicalization example (generic locus).
  13. CCR normal form and kinematical nonreducibility to the $(\hbar,0,0)$ sector
  14. A remark on star-product level identifications
  15. Conclusion and Outlook

Key Result

Proposition II.4

If $\pi,\pi'\in\widehat{G_{\hbox{\tiny{NC}}}}$ are kinematically equivalent, then they have the same central character. In particular, they have the same unitary-dual label (in any fixed parametrization of $\widehat{G_{\hbox{\tiny{NC}}}}$).

Theorems & Definitions (13)

  • Definition II.1: Hilbert-space basis change
  • Definition II.2: NCQM operator quadruple associated with a $G_{\hbox{\tiny{NC}}}$-irrep
  • Definition II.3: Kinematical equivalence
  • Proposition II.4: Unitary equivalence preserves the central character
  • proof
  • Theorem II.5: Main kinematical inequivalence theorem
  • proof
  • Remark VI.1: Unitary equivalence inside the $(r,s)$-family versus non-equivalence with the CCR operator quadruple
  • Remark VI.2: Interpretation of the generalized Bopp shift map $S(r,s)$
  • Remark VI.3: Relation between $T$ and the inverse generalized Bopp shift map
  • ...and 3 more