A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.     

Quantum decision theory as quantum theory of measurement

We present a general theory of quantum information processing devices, that can be applied to human decision makers, to atomic multimode registers, or to molecular high-spin registers. Our quantum decision theory is a generalization of the quantum theory of measurement, endowed with an action ring, a prospect lattice and a probability operator measure. The algebra of probability operators plays the role of the algebra of local observables. Because of the composite nature of prospects and of the entangling properties of the probability operators, quantum interference terms appear, which make actions noncommutative and the prospect probabilities nonadditive. The theory provides the basis for explaining a variety of paradoxes typical of the application of classical utility theory to real human decision making. The principal advantage of our approach is that it is formulated as a self-consistent mathematical theory, which allows us to explain not just one effect but actually all known paradoxes in human decision making. Being general, the approach can serve as a tool for characterizing quantum information processing by means of atomic, molecular, and condensed-matter systems.     

Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding ＊-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.     

Exact master equation for a noncommutative Brownian particle

We derive the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators on the plane with spatial noncommutativity. The results obtained are exact to all orders in the noncommutative parameter. As a by-product we derive some miscellaneous results such as the equilibrium Wigner distribution for the reservoir of noncommutative oscillators, the weak coupling limit of the master equation and a set of sufficient conditions for strict purity decrease of the Brownian particle. Finally, we consider a high-temperature Ohmic model and obtain an estimate for the time scale of the transition from noncommutative to ordinary quantum mechanics. This scale is considerably smaller than the decoherence scale.     

Spin Hall effect in noncommutative coordinates

A semiclassical constrained Hamiltonian system which was established to study dynamical systems of matrix valued non-Abelian gauge fields is employed to formulate spin Hall effect in noncommuting coordinates at the first order in the constant noncommutativity parameter θ. The method is first illustrated by studying the Hall effect on the noncommutative plane in a gauge independent fashion. Then, the Drude model type and the Hall effect type formulations of spin Hall effect are considered in noncommuting coordinates and θ deformed spin Hall conductivities which they provide are acquired. It is shown that by adjusting θ different formulations of spin Hall conductivity are accomplished. Hence, the noncommutative theory can be envisaged as an effective theory which unifies different approaches to similar physical phenomena.     

'Wick Rotations': The Noncommutative Hyperboloids and Other Surfaces of Rotations

A 'Wick rotation' is applied to a noncommutative sphere to produce a noncommutative version of the hyperboloids. A harmonic basis of the associated algebra is given. A method of constructing noncommutative analogues of surfaces of rotation, examples of which include the paraboloid and the q-deformed sphere, is given. Also given are mappings between noncommutative surfaces, stereographic projections to the complex plane and unitary representations. A relationship with one-dimensional crystals is highlighted.     

Star products and geometric algebra

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.     

Van der Waals interactions and Photoelectric Effect in Noncommutative Quantum Mechanics

We calculate the long-range Van der Waals force and the photoelectric cross section in a noncommutative setup. It is argued that non-commutativity effects could not be discerned for the Van der Waals interactions. The result for the photoelectric effect shows deviation from the usual commutative one, which in principle can be used to put bounds on the space-space non-commutativity parameter.     

Quantum Overloading Cryptography Using Single-Photon Nonlocality

Using the single-photon nonlocality, we propose a quantum novel overloading cryptography scheme, in which a single photon carries two bits information in one-way quantum channel. Two commutative modes of the single photon, the polarization mode and the spatial mode, are used to encode secret information. Strict time windows are set to detect the impersonation attack. The spatial mode which denotes the existence of photons is noncommutative with the phase of the photon, so that our scheme is secure against photon-number-splitting attack. Our protocol may be secure against individual attack.     

Quantum diffusions and the noncommutative torus

Quantum diffusions driven by the creation and annihilation processes on the noncommutative torus algebra are considered. A cohomological obstruction to the existence of such a diffusion is overcome by constructing a diffusion on a larger algebra. There are implications for models in solid-state physics based on the noncommutative torus.     

Schlesinger Transformations for Linearisable Equations

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every Itô algebra with a quotient identity has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy–Khinchin Theorem in the classical case. Two basic examples of noncommutative Itô finite group algebras are considered.     

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space.Based on the deformed boson algebra,we construct coherent state representations.We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations.It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.     

Algebraic Treatment of the MIC-Kepler System in Spherical Coordinates

The MIC-Kepler system is studied via the Milshtein Strakhovenko variant of the so（2,1） Lie algebra. Green＇s function is constructed in spherical coordinates, with the help of the Kustaanheimo Stiefel variables and the generators of the S0（2,1） group. The energy spectrum and the normalized wavefunctions of the bound states are obtained.     

Finite-Dimensional Simple Leibniz Pairs and Simple Poisson Modules

Simple modules over the Leibniz pairs are studied. Simple Poisson modules over Poisson algebras of the semisimple associative algebra structure are determined and they are nothing but simple bimodules over simple associative algebras with standard noncommutative Poisson algebra structure.     

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.     

Noncommutative lattices and their continuum limits

We consider finite approximations of a topological space M by noncommutative lattices of points. These lattices are structure spaces of noncommutative C^*-algebras which in turn approximate the algebra C(M) of continuous functions on M. We show how to recover the space M and the algebra C(M) from a projective system of noncommutative lattices and an inductive system of noncommutative C^*-algebras, respectively.     

Noncommutative two-tori with real multiplication as noncommutative projective varieties

We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of [Categories of holomorphic bundles on noncommutative two-tori. math.AG/0211262] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

Reflection equation- and FRT-type algebras

We introduce two types of algebras which include respectively the well known reflection equation (RE) and Faddeev-Reshetikhin-Takhtayan algebras associated with a quasitriangular Hopf algebraH. We show that these two types of algebras are twist-equivalent. It follows that a RE algebra is a module algebra over a twisted tensor square ofH. We present some applications to the equivariant quantization. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.