[Paper Review] Symmetry in noncommutative quantum mechanics

Olivier Espinosa, Patricio Gaete|ArXiv.org|Jun 7, 2002

Noncommutative and Quantum Gravity Theories1 references3 citations

TL;DR

This paper proposes a symmetry-preserving Hamiltonian formulation in noncommutative quantum mechanics (NCQM) by modifying the standard approach that naively sets $ H_\theta = H $. It introduces a minimal deformation of the Hamiltonian using a generalized star product and canonical transformations, ensuring that rotational and gauge symmetries of the commutative theory are preserved. The key result is that the energy spectrum of the noncommutative theory exactly matches the standard quantum mechanical spectrum, with experimental differences manifesting only in the structure of eigenstates.

ABSTRACT

We reconsider the generalization of standard quantum mechanics in which the position operators do not commute. We argue that the standard formalism found in the literature leads to theories that do not share the symmetries present in the corresponding commutative system. We propose a general prescription to specify a Hamiltonian in the noncommutative theory that preserves the existing symmetries. We show that it is always possible to choose this Hamiltonian in such a way that the energy spectrum of the standard and non-commuting theories are identical, so that experimental differences between the predictions of both theories are to be found only at the level of the detailed structure of the energy eigenstates.

Motivation & Objective

• To address the fundamental flaw in existing NCQM formulations that break symmetries like rotational and gauge invariance present in the commutative theory.
• To develop a minimal, physically consistent prescription for constructing the noncommutative Hamiltonian $ H_\theta $ from a given standard Hamiltonian $ H $.
• To ensure that the energy spectrum of the noncommutative theory matches that of the standard theory, so that observable differences are restricted to the structure of eigenstates.

Proposed method

• Propose a new prescription for constructing $ H_\theta $ using a generalized star product and canonical transformations that preserve the symmetries of the original system.
• Use the formalism of harmonic oscillator operators to express position and momentum operators in terms of $ a, a^\dagger, b, b^\dagger $, enabling explicit computation of physical observables.
• Derive the deformed position operators $ \rho_x, \rho_y $ relative to the guiding center in the Landau problem, incorporating noncommutativity through $ \theta $-dependent corrections.
• Construct a unitary transformation $ T_{\theta\chi} $ that implements gauge transformations in the noncommutative theory, ensuring the Hamiltonian transforms covariantly.
• Show that the standard gauge transformation law cannot be preserved in NCQM due to nontrivial commutators involving $ X_i $ and $ \partial_j \chi({\bf R}) $, necessitating a modified transformation rule.
• Demonstrate that the energy spectrum remains identical to the commutative case by computing $ \langle \rho_x^2 + \rho_y^2 \rangle $ in Fock states, revealing $ \theta $-dependent corrections only in the eigenstate structure.

Experimental results

Research questions

• RQ1 Can a noncommutative quantum mechanical Hamiltonian be constructed such that it preserves the rotational and gauge symmetries of the corresponding commutative theory?
• RQ2 Why do standard approaches to NCQM—such as setting $ H_\theta = H $ or using the Moyal star product—lead to theories that break fundamental symmetries?
• RQ3 Is it possible to construct a noncommutative Hamiltonian with the same energy spectrum as the standard theory, so that experimental differences arise only in the detailed structure of eigenstates?
• RQ4 How does noncommutativity affect the expectation value of $ \rho_x^2 + \rho_y^2 $, a measure of the electron's orbital radius in the Landau problem?
• RQ5 What is the correct form of the gauge transformation in noncommutative quantum mechanics that preserves the canonical commutation relations and the Hamiltonian's transformation law?

Key findings

• The proposed Hamiltonian $ H_\theta $ preserves rotational and gauge symmetries, unlike standard NCQM approaches that break these symmetries.
• The energy spectrum of the noncommutative theory is identical to that of the standard theory, with no $ \theta $-dependent shifts in energy levels.
• The expectation value $ \langle \rho_x^2 + \rho_y^2 \rangle_{n,l} $ is corrected by terms proportional to $ \theta $ and $ \theta^2 $, specifically $ \ell_B^2(2n+1)\left[1 - \frac{\theta}{2\ell_B^2} + \frac{\theta^2}{8\ell_B^4}\left(1 + \frac{l}{2n+1}\right)\right] $, showing $ \theta $-dependence in eigenstate structure.
• The gauge transformation in NCQM must be modified: $ T_{\theta\chi} = \exp\left[i\frac{e}{\hbar}\chi(\mathbf{R} + \frac{\theta}{2\hbar}\tilde{\mathbf{P}})\right] $, which reduces to the standard form only in the $ \theta \to 0 $ limit.
• The standard gauge transformation law cannot be preserved in NCQM because it would violate the canonical commutator $ [X_i, P_j] = i\hbar\delta_{ij} $, necessitating a deformed transformation rule.
• The formalism ensures that $ H_\theta $ is invariant under the modified gauge transformation $ T_{\theta\chi} $, confirming gauge invariance in the noncommutative framework.