Deformation Quantization Research Articles

Supersymmetric Quesne-Dunkl Quantum Mechanics on Radial Lines

Quantum deformations offer valuable perspectives into quantum mechanics, particularly by advancing our understanding of symmetry and algebraic structures.The Dunkl oscillator, which integrates Dunkl operators into the harmonic oscillator framework, advances the system’s algebraic properties and opens new approaches for exploring quantum phenomena. Supersymmetric quantum mechanics (SSQM) unifies bosonic and fermionic aspects and facilitates the construction of solvable models using generalized Dunkl operators. This paper introduces a new approach to the Dunkl oscillator, employing a complex reflection operator to deepen the understanding of its connection to Hermite polynomials on radial lines. The results offer new perspectives on the Dunkl oscillator’s algebraic structure and its relevance to SSQM and quantum deformation theory, expanding the potential for discovering solvable quantum models.

Superstatistical functions of modified shifted Morse potential for diatomic molecules

The application of the molecular potential model to the study of different thermodynamic systems has been a focus of interest in recent time due to its importance in molecular and chemical physics. Hence, the asymptotic iteration method and the Laguerre polynomials are employed to obtain the eigensolutions of the Schrödinger equation with the modified shifted Morse potential model. Vibrational energy results for different diatomic molecules have been presented numerically, and these results are compared with experimental data and other results from available literature. The variation of normalised wave function and probability density of modified shifted Morse potential for some selected diatomic molecules at the ground- and first-excited states have been considered. Results obtained show that the equally spaced sinusoidal wave curves and varying nodes depend on the level of electron concentrations in the diatomic molecules considered. Partition function and other superstatistical function expressions of modified shifted Morse potential for some diatomic molecules within the modified Dirac delta distribution are obtained, using the Poisson summation formula. The superstatistical plots show a high level of dependence on the temperature parameter, maximum vibrational quantum number and deformation parameter. The superstatistical functions for a deformation parameter of zero correspond to the conventional thermodynamic functions.

Quantisation via Branes and Minimal Resolution

The ‘brane quantisation’ is a quantisation procedure developed by Gukov and Witten (Adv Theor Math Phys 13(5):1445–1518, 2009). We implement this idea by combining it with the tilting theory and the minimal resolutions. This way, we can realistically compute the deformation quantisation on the space of observables acting on the Hilbert space. We apply this procedure to certain quantisation problems in the context of generalised Kähler structure on P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document}. Our approach differs from and complements that of Bischoff and Gualtieri (Commun Math Phys 391(2):357–400, 2022). We also benefitted from an important technical tool: a combinatorial criterion for the Maurer–Cartan equation, developed by Barmeier and Wang (Deformations of path algebras of quivers with relations, 2020. arXiv:2002.10001).

Deformation quantization of projective schemes and differential operators

Deformation quantization of projective schemes and differential operators

Asymptotic safe nonassociative quantum gravity with star R-flux products, Goroff–Sagnotti counter-terms, and geometric flows

Nonassociative modifications of general relativity, GR, defined by star products with R-flux deformations in string gravity consist an important subclass of modified gravity theories, MGTs. A longstanding criticism for elaborating quantum gravity, QG, argue that the asymptotic safety does not survive once certain perturbative terms (in general, nonassociative and noncommutative) are included in the projection space. The goal of this work is to prove that a generalized asymptotic safety scenario allows us to formulate physically viable nonassociative QG theories using effective models defined by generic off-diagonal solutions and nonlinear symmetries in nonassociative geometric flow and gravity theories. We elaborate on a new nonholonomic functional renormalization techniques with parametric renormalization group, RG, flow equations for effective actions supplemented by certain canonical two-loop counter-terms. The geometric constructions and quantum deformations are performed for nonassociative phase spaces modelled as R-flux deformed cotangent Lorentz bundles. Our results prove that theories involving nonassociative modifications of GR can be well defined both as classical nonassociative MGTs and QG models. Such theories are characterized by generalized G. Perelman thermodynamic variables which are computed for certain examples of nonassociative geometric and RG flows.

Physical Motivation On Deformation Quantisation

We study the Poisson structures associated with deformation quantisation and its non-degradable factor on the Casimir function. We also describe a filtered associative algebra in a quotient space as a Poisson algebra and the automorphism of the Poisson bracket is discussed.

Quot scheme and deformation quantization

Quot scheme and deformation quantization

Star exponentials from propagators and path integrals

In this paper we address the relation between the star exponentials emerging within the Deformation Quantization formalism and Feynman’s path integrals associated with propagators in quantum dynamics. In order to obtain such a relation, we start by visualizing the quantum propagator as an integral transform of the star exponential by means of the symbol corresponding to the time evolution operator and, thus, we introduce Feynman’s path integral representation of the propagator as a sum over all the classical histories. The star exponential thus constructed has the advantage that it does not depend on the convergence of formal series, as commonly understood within the context of Deformation Quantization. We include some basic examples to illustrate our findings, recovering standard results reported in the literature. Further, for an arbitrary finite dimensional system, we use the star exponential introduced here in order to find a particular representation of the star product which may be recognized as the one encountered in the context of the quantum field theory for a Poisson sigma model.

A deformation quantization for non-flat spacetimes and applications to QFT

We provide a deformation quantization, in the sense of Rieffel, for all globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e. a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e. is Hadamard, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).

Quantum deformation of cubic string field theory

In this paper, we will analyze a quantum deformation of cubic string field theory. This will be done by first constructing a quantum deformation of string theory, in a covariant gauge, and then using the quantum deformed stringy theory to construct a quantum deformation of string field theory. This quantum-deformed string field will then be used to contract a quantum-deformed version of cubic string field theory. We will explicitly demonstrate that the axioms of cubic string field theory hold even after quantum deformation. Finally, we will analyze the effect of the quantum deformation of string field theory on the string vertices.

Supersymmetric Quantum Mechanics on a noncommutative plane through the lens of deformation quantization

A gauge invariant mathematical formalism based on deformation quantization is outlined to model an N=2 supersymmetric system of a spin 1/2 charged particle placed in a nocommutative plane under the influence of a vertical uniform magnetic field. The noncommutative involutive algebra (C∞(R2)[[ϑ]],∗r) of formal power series in ϑ with coefficients in the commutative ring C∞(R2) was employed to construct the relevant observables, viz., SUSY Hamiltonian H, supercharge operator Q and its adjoint Q† all belonging to the 2 × 2 matrix algebra M2(C∞(R2)[[ϑ]],∗r) with the help of a family of gauge-equivalent star products ∗r. The energy eigenvalues of the SUSY Hamiltonian all turned out to be independent of not only the gauge parameter r but also the noncommutativity parameter ϑ. The nontrivial Fermionic ground state was subsequently computed associated with the zero energy which indicates that supersymmetry remains unbroken in all orders of ϑ. The Witten index for the noncommutative SUSY Landau problem turns out to be −1 corroborating the fact that there is no broken supersymmetry for the model we are considering.

Influence of quantum correction on the Schwarzschild black hole polarized image

Using a model of an accretion disk around a Schwarzschild black hole, the analytic estimates for image polarization were derived by Narayan et al. (Astrophys J 102:912, 2021). Recently, the EHT team also obtained polarization images of the Sgr A∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{*}$$\\end{document} and measured both linear and circular polarization (Astrophys J Lett 964:L25, 2024). We find that quantum correction effects can also influence polarization information. Considering the quantum corrected Schwarzschild black hole (Kazakov–Solodukhin black hole), we derive the polarization intensity of the target black hole and investigate polarization images under different parameters. It is found that a larger quantum deformation leads to an expansion of the polarization region, while the polarization intensity value decrease. Under different observation angles, magnetic fields, fluid direction angles, and fluid velocity conditions, we also derive polarization images of corrected black holes. These key indicators not only affect the intensity of polarization but also the direction of polarization. We establish the relationship between polarization intensity and quantum correction deformation parameters, revealing a gradual decline in polarization intensity with reduced radius and an anti-polarization behavior induced by the progressive increase in deformation parameters at a constant radius. Our analysis may provide observational evidence for quantum effect of general relativity.

Classical limits of Hilbert bimodules as symplectic dual pairs

Hilbert bimodules are morphisms between C*-algebraic models of quantum systems, while symplectic dual pairs are morphisms between Poisson geometric models of classical systems. Both of these morphisms preserve representation-theoretic structures of the relevant types of models. Previously, it has been shown that one can functorially associate certain symplectic dual pairs to Hilbert bimodules through strict deformation quantization. We show that, in the inverse direction, strict deformation quantization also allows one to functorially take the classical limit of a Hilbert bimodule to reconstruct a symplectic dual pair.

Function theory off the complexified unit circle: Fréchet space structure and automorphisms

Motivated by recent work on strict deformation quantization of the unit disk and the Riemann sphere, we study the Fréchet space structure of the set of holomorphic functions on the complement \(\Omega:=\{(z,w)\in \hat{\mathbb{C}}^2\colon z\cdot w\not=1\}\) of the complexified unit circle \(\{(z,w) \in \hat{\mathbb{C}}^2 \colon z\cdot w=1\}\). We also characterize the subgroup of all biholomorphic automorphisms of \(\Omega\) which leave the canonical Laplacian on \(\Omega\) invariant.

Wick-type deformation quantization of contact metric manifolds

Wick-type deformation quantization of contact metric manifolds

Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei

There is a surge of research devoted to the formalism and physical manifestations of non-Lorentzian kinematical symmetries, which focuses especially on the ones associated with the Galilei and Carroll relativistic limits (the speed of light taken to infinity or to zero, respectively). The investigations have also been extended to quantum deformations of the Carrollian and Galilean symmetries, in the sense of (quantum) Hopf algebras. The case of 2+1 dimensions is particularly worth to study due to both the mathematical nature of the corresponding (classical) theory of gravity, and the recently finalized classification of all quantum-deformed algebras of spacetime isometries. Consequently, the list of all quantum deformations of (anti-)de Sitter-Carroll algebra is immediately provided by its well-known isomorphism with either Poincaré or Euclidean algebra. Quantum contractions from the (anti-)de Sitter to (anti-)de Sitter-Carroll classification allow to almost completely recover the latter. One may therefore conjecture that the analogous contractions from the (anti-)de Sitter to (anti-)de Sitter-Galilei r-matrices provide (almost) all coboundary deformations of (anti-)de Sitter-Galilei algebra. This scheme is complemented by deriving (Carrollian and Galilean) quantum contractions of deformations of Poincaré algebra, leading to coboundary deformations of Carroll and Galilei algebras.

Higher Deformation Quantization for Kapustin–Witten Theories

Higher Deformation Quantization for Kapustin–Witten Theories

Computation of soliton structure and analysis of chaotic behaviour in quantum deformed Sinh-Gordon model.

Soliton dynamics and nonlinear phenomena in quantum deformation has been investigated through conformal time differential generalized form of q deformed Sinh-Gordon equation. The underlying equation has recently undergone substantial amount of research. In Phase 1, we employed modified auxiliary and new direct extended algebraic methods. Trigonometric, hyperbolic, exponential and rational solutions are successfully extracted using these techniques, coupled with the best possible constraint requirements implemented on parameters to ensure the existence of solutions. The findings, then, are represented by 2D, 3D and contour plots to highlight the various solitons' propagation patterns such as kink-bright, bright, dark, bright-dark, kink, and kink-peakon solitons and solitary wave solutions. It is worth emphasizing that kink dark, dark peakon, dark and dark bright solitons have not been found earlier in literature. In phase 2, the underlying model is examined under various chaos detecting tools for example lyapunov exponents, multistability and time series analysis and bifurcation diagram. Chaotic behavior is investigated using various initial condition and novel results are obtained.

Strict Deformation Quantization and Local Spin Interactions

We define a strict deformation quantization which is compatible with any Hamiltonian with local spin interaction (e.g. the Heisenberg Hamiltonian) for a spin chain. This is a generalization of previous results known for mean-field theories. The main idea is to study the asymptotic properties of a suitably defined algebra of sequences invariant under the group generated by a cyclic permutation. Our point of view is similar to the one adopted by Landsman, Moretti and van de Ven (Rev Math Phys 32(10):2050031, 2020, https://doi.org/10.1142/S0129055X20500312), who considered a strict deformation quantization for the case of mean-field theories. However, the methods for a local spin interaction are considerably more involved, due to the presence of a strictly smaller symmetry group.

F-algebroids and deformation quantization via pre-Lie algebroids

F-algebroids and deformation quantization via pre-Lie algebroids