Star Product Formalism Research Articles

Correction to: Star-Product Formalism for the Probability and Mean-Value Representations of Qudits

In section Acknowledgments, the Russian Science Foundation Grant No. 1871-10091 should read No. 1971-10091.

Star-Product Formalism for the Probability and Mean-Value Representations of Qudits

We develop the quantizer–dequantizer formalism for the probability and mean-value representations of qubits and qutrits. In these representations, the elements of symbols of operators are measurable physical quantities. We derive the star-product kernels, which lead to explicit expressions of the associative product of the symbols of the density operators and quantum observables for qubits. The extension of the quantizer–dequantizer formalism associated with the probability and mean-value descriptions of qudit states is discussed.

Probability representation of quantum mechanics and star product quantization

This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.

Discovering the manifold facets of a square integrable representation: from coherent states to open systems

Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary — or, in general, projective unitary — representations implement the action of an abstract symmetry group on physical states and observables. More specifically, a major role is played by the so-called square integrable representations. Indeed, the properties of these representations are fundamental in the definition of certain families of generalized coherent states, in the phase-space formulation of quantum mechanics and the associated star product formalism, in the definition of an interesting notion of function of quantum positive type, and in some recent applications to the theory of open quantum systems and to quantum information.

Quadratic Tomography Star Product Algebra and its Classical Limit

We consider quadratic tomography in star product formalism. The contraction and the behavior of the associative algebra of quadratic tomographic symbols in $\hbar\rightarrow 0$ limit are discussed. A simple $k$-deformation example is illustrated.

Wigner functions for angle and orbital angular momentum: Operators and dynamics

Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces S^1 x R, i.e. for the canonical pair angle and angular momentum, was presented (arXiv:1601.02520), main properties of those functions derived, discussed and their usefulness illustrated by examples. The present paper is a continuation which compares properties of the new Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions on planar ones in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase space functions is discussed. The star product formalism is shown to be completely implementable. In addition basic dynamical laws for the new Wigner and Moyal functions are derived as generalized quantum Liouville and energy equations. They are very similar to those of the planar case, but also show characteristic differences.

Star product, discrete Wigner functions, and spin-system tomograms

We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.

Generalized tomographic maps and star-product formalism

We elaborate on the notion of generalized tomograms, both in the classical and quantum domains. We construct a scheme of star-products of thick tomographic symbols and obtain in explicit form the kernels of classical and quantum generalized tomograms. Some of the new tomograms may have interesting applications in quantum optical tomography.

FROM QUANTUM DEFORMATIONS OF RELATIVISTIC SYMMETRIES TO MODIFIED KINEMATICS AND DYNAMICS

Starting from noncommutative generalization of Minkowski space we consider quantum deformed relativistic symmetries which lead to the modification of kinematics of special relativity. The noncommutative field theory framework described by means of the star product formalism is briefly described. We briefly present the quantum modifications of Einstein gravity.

An introduction to the tomographic picture of quantum mechanics

Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.

Spin description in the star product and path integral formalisms

Spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. One can then establish the relation to other approaches that describe spin with fermionic variables. The fermionic star product formalism and the fermionic path integral formalism are related in analogy to their bosonic counterparts.

Einstein-Riemann Gravity on Deformed Spaces

A differential calculus, differential geometry and the E-R Gravity theory are studied on noncommutative spaces. Noncommutativity is formulated in the star product formalism. The basis for the gravity theory is the infinitesimal algebra of diffeomorphisms. Considering the corresponding Hopf algebra we find that the deformed gravity is based on a deformation of the Hopf algebra.

Deformed geometric algebra and supersymmetric quantum mechanics

Deforming the algebraic structure of geometric algebra on the phase space with a Moyal product leads naturally to supersymmetric quantum mechanics in the star product formalism.

Geometric algebra and star products on the phase space

Superanalysis can be deformed with a fermionic star product into a Clifford calculus that is equivalent to geometric algebra. With this multivector formalism it is then possible to formulate Riemannian geometry and an inhomogeneous generalization of exterior calculus. Moreover, it is shown here how symplectic and Poisson geometry fit in this context. The application of this formalism together with the bosonic star product formalism of deformation quantization leads then on space and space-time to a natural appearance of spin structures and on phase space to BRST structures that were found in the path integral formulation of classical mechanics. Furthermore it will be shown that Poincaré and Lie–Poisson reduction can be formulated in this formalism.

Superfield theory and supermatrix model

AbstractWe show that any superfield theory can be reformulated as a supermatrix model. The claim is based on a formal argument, turning on a non‐commutativity on the superspace and taking a fine tuned commutative limit. Once the non‐commutativity is introduced on the superspace, superfield theories can be formulated in two equivalent ways, through the star product formalism and in terms of the supermatrices. We elaborate the duality between them by constructing the isomorphism explicitly and relating the super space integral of the star product Lagrangian to the supertrace of the supermatrix model Lagrangian. We show there exists an interesting fine tuned commutative limit where the duality can be still maintained. More detailed analysis can be found in [1].

Center of mass tomography for reconstructing quantum states of multipartite systems

We investigate a one random variable tomography map, describing quantum state of a multipartite system. The random variable is analogous to center of mass considered in rotated and scaled reference frames in the phase space. Starting from the star product formalism, we construct the map, investigate its properties (including the symmetry properties in respect to identical particles permutations), derive the evolution equation, and consider a multimode oscillator as an example. The physical meaning of the map is analyzed in detail.

Superfield theory and supermatrix model

We study the noncommutative superspace of arbitrary dimensions in a systematic way. Superfield theories on a noncommutative superspace can be formulated in two folds, through the star product formalism and in terms of the supermatrices. We elaborate the duality between them by constructing the isomorphism explicitly and relating the superspace integrations of the star product Lagrangian or the superpotential to the traces of the supermatrices. We show there exists an interesting fine tuned commutative limit where the duality can be still maintained. Namely on the commutative superspace too, there exists a supermatrix model description for the superfield theory. We interpret the result in the context of the wave particle duality. The dual particles for the superfields in even and odd spacetime dimensions are D-instantons and D0-branes respectively to be consistent with the T-duality.

Kernel of star-product for spin tomograms

The spin-state tomogram is presented as the modulus squared of the matrix element of the SU( 2)-group irreducible representation. For the spin system, we established explicitly a realization of the set of operators defining for spin tomograms the star-product formalism in terms of irreducible tensors. On the set of spin tomograms, the delta-function and ‘standard’ and Moyal-like kernels of the tomogram star-product are calculated explicitly in terms of Clebsch– Gordan and Racah coefficients and matrix elements of the SU( 2) irreducible representation. In the limit of infinite spin, the spin tomogram is shown to become the tomogram of the harmonic-oscillator state.

Tomography of nonclassical states

A review of the symplectic tomography method is presented. Superpositions of different types of photon states are considered within the framework of the tomography approach. Such nonclassical photon states as even and odd coherent states, crystallized Schrodinger cat states, and other superposition states are studied using the construction of symplectic tomograms (tomographic symbols) and the star-product formalism for tomograms.

Tomography for Several Particles with One Random Variable

The tomographic map of the quantum state of a system with several degrees of freedom, which depends on one random variable, analogous to the rotated and scaled center-of-mass of the system, is constructed. The time-evolution equation of the tomogram for this map is given in the explicit form. The properties of the map such as the transition probabilities between different states and relation to the star-product formalism are elucidated. An example of a multimode oscillator is considered in detail. Identical particles are discussed within the framework of the proposed tomography scheme.