Noncommutative Harmonic Oscillator Research Articles

SPECIAL VALUES OF THE SPECTRAL ZETA FUNCTION OF THE NON-COMMUTATIVE HARMONIC OSCILLATOR AND CONFLUENT HEUN EQUATIONS

We study the special values at s = 2 and 3 of the spectral zeta function ζQ(s) of the non-commutative harmonic oscillator Q(x, Dx) introduced in A. Parmeggiani and M. Wakayama (Proc. Natl Acad. Sci. USA 98 (2001), 26-31; Forum Math. 14 (2002), 539-604). It is shown that the series defining ζQ(s) converges absolutely for Re s > 1 and further the respective values ζQ(2) and ζQ(3) are represented essentially by contour integrals of the solutions, respectively, of a singly confluent Heun ordinary differential equation and of exactly the same but an inhomogeneous equation. As a by-product of these results, we obtain integral representations of the solutions of these equations by rational functions.

Non-commutative Harmonic Oscillators and the Connection Problem for the Heun Differential Equation

We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.

Heisenberg Evolution in a Quantum Theory of Noncommutative Fields

A quantum theory of noncommutative fields was recently proposed by Carmona, Cortez, Gamboa and Mendez (hep-th/0301248). The implications of the noncommutativity of the fields, intended as the requirements $[\phi,\phi^{+}]=\theta\delta^{3}(x-x^{\prime}), [\pi,\pi^{+}]=B\delta^{3}(x-x^{\prime}) $, were analyzed on the basis of an analogy with previous results on the so-called ``noncommutative harmonic oscillator construction". Some departures from Lorentz symmetry turned out to play a key role in the emerging framework. We first consider the same hamiltonian proposed in hep-th/0301248, and we show that the theory can be analyzed straightforwardly within the framework of Heisenberg evolution equation without any need of making reference to the ``noncommutative harmonic oscillator construction". We then consider a rather general class of alternative hamiltonians, and we observe that violations of Lorentz invariance are inevitably encountered. These violations must therefore be viewed as intrinsically associated with the proposed type of noncommutativity of fields, rather than as a consequence of a specific choice of Hamiltonian.

Extended space duality in the non-commutative plane

Non-commutative (NC) effects in planar quantum mechanics are investigated. We have constructed a Master model for a non-commutative harmonic oscillator by embedding it in an extended space, following the Batalin–Tyutin [Int. J. Mod. Phys. A 6 (1992) 3255] prescription. Different gauge choices lead to distinct NC structures, such as NC coordinates, NC momenta or non-commutativity of a more general kind. In the present framework, all of these can be studied in a unified and systematic manner. Thus the dual nature of theories having different forms of non-commutativity is also revealed.

Noncommutative quantum mechanics

We discuss the main features of noncommutative quantum mechanics, a version of nonrelativistic quantum mechanics that involves noncommuting coordinates. After finding a representation for the algebra obeyed by the coordinates and momenta, we analyze the changes due to the noncommutative nature of the coordinates. The noncommutative two-dimensional harmonic oscillator is discussed in detail. Under certain restrictions, the effect of the noncommutativity is found to be equivalent to a Landau interaction. The modifications produced by the noncommutativity on the thermodynamic functions of the oscillator also are studied.

Noncommutative quantum mechanics and Bohm’s ontological interpretation

We carry out an investigation into the possibility of developing a Bohmian interpretation based on the continuous motion of point particles for noncommutative quantum mechanics. The conditions for such an interpretation to be consistent are determined, and the implications of its adoption for noncommutativity are discussed. A Bohmian analysis of the noncommutative harmonic oscillator is carried out in detail. By studying the particle motion in the oscillator orbits, we show that small-scale physics can have influence at large scales, something similar to the IR-UV mixing.

ON THE SPECTRUM AND THE LOWEST EIGENVALUE OF CERTAIN NON-COMMUTATIVE HARMONIC OSCILLATORS

We study here various expansions and approximations of the spectrum of non-commutative harmonic oscillators of the typeQ (x, Dx) = A (−∂x2/2 + x2/2) + B (x∂x + 1/2), x ∈ R, where A, B are constant 2 × 2 matrices such that A is real symmetric positive (or negative) definite and B ≠ 0 is real skew-symmetric, when the Hermitian matrix A + iB is positive (or negative) definite. Special emphasis is put on the lowest eigenvalue. These expansions are written in terms of det(A )/pf(B )2 in the limit det(A ) → +∞ for constant pf(B ) and constant Tr(A )/√det(A ).

Harmonic oscillator with minimal length uncertainty relations and ladder operators

We construct creation and annihilation operators for harmonic oscillators with minimal length uncertainty relations. We discuss a possible generalization to a large class of deformations of cannonical commutation relations. We also discuss dynamical symmetry of noncommutative harmonic oscillator.

Quantum theory of noncommutative fields

Generalizing the noncommutative harmonic oscillator construction, we propose a new extension of quantum field theory based on the concept of "noncommutative fields". Our description permits to break the usual particle-antiparticle degeneracy at the dispersion relation level and introduces naturally an ultraviolet and an infrared cutoff. Phenomenological bounds for these new energy scales are given.

A remark on systems of differential equations sssociated with representations of $\SPE(\r)$ and their perturbations

We give here a number of examples of non-commutative harmonic oscillators ``in disguise'' that can be exactly solved by using the tensor product of the oscillator representation and the finite dimensional representation of $\SPE(\r)$, and its perturbations.

Noncommutative quantization in 2D conformal field theory

The simplest possible noncommutative harmonic oscillator in two dimensions is used to quantize the free closed bosonic string in two flat dimensions compactified on the torus. The partition function is not deformed by the introduction of noncommutativity, if we rescale the time by a factor that depends on the noncommutativity parameter and change the compactification radius appropriately. The four-point function and certain 2n-point functions are deformed, preserving, nevertheless, the sl(2,C) covariance.

Noncommutative 3D harmonic oscillator

We find transformation matrices allowing us to express a noncommutative three-dimensional harmonic oscillator in terms of an isotropic commutative oscillator, following the 'philosophy of simplicity' approach. Noncommutative parameters have a physical interpretation in terms of an external magnetic field. Furthermore, we show that for a particular choice of noncommutative parameters there is an equivalent anisotropic representation, whose transformation matrices are far more complicated. We indicate a way to obtain the more complex solutions from the simple ones.

Isotropic representation of the noncommutative 2D harmonic oscillator

We show that a 2D noncommutative harmonic oscillator has an isotropic representation in terms of commutative coordinates. The noncommutativity in the new mode induces energy level splitting and is equivalent to an external magnetic field effect. The equivalence of the spectra of the isotropic and anisotropic representation is traced back to the existence of the SU(2) invariance of the noncommutative model.

The noncommutative harmonic oscillator in more than one dimension

The noncommutative harmonic oscillator, with noncommutativity not only in position space but also in phase space, in arbitrary dimension is examined. It is shown that the ⋆-genvalue problem, which replaces the Schrödinger problem in this case, can be decomposed into separate harmonic oscillator equations for each dimension. The two-dimensional noncommutative harmonic oscillator (four noncommutative phase-space dimensions) is investigated in greater detail. The requirement of the existence of rotationally symmetric solutions leads to a two parameter harmonic oscillator which is completely solved in this case. The angular momentum operator is derived and its ⋆-genvalue problem is shown to be equivalent to the usual eigenvalue problem of the ⋆-genfunction related wave function. The ⋆-genvalues of the angular momentum are found to depend on the energy difference of the oscillations in the two dimensions. Furthermore two examples of a symmetric noncommutative harmonic oscillators are analyzed. The first is the noncommutative two-dimensional Landau problem with harmonic oscillator potential, which shows degeneracy in the energy levels for certain critical values of the noncommutativity parameters, and the second is the three-dimensional harmonic oscillator with noncommuting coordinates and momenta.

Path integral formulation of noncommutative quantum mechanics

We propose a phase-space path integral formulation of noncommutative quantum mechanics, and prove its equivalence to the operatorial formulation. As an illustration, the partition function of a noncommutative two-dimensional harmonic oscillator is calculated.

Non-Commutative Harmonic Oscillators¶and Fuchsian Ordinary Differential Operators

The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.