Deformation Quantization Research Articles

On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.

Chern classes of quantizable coisotropic bundles

Let M be a smooth algebraic variety of dimension 2(p+q) with an algebraic symplectic form and a compatible deformation quantization of the structure sheaf. Consider a smooth coisotropic subvariety Y of codimension q and a vector bundle E on Y . We show that if the pushforward of E admits a deformation quantization (as a module), then its “trace density” characteristic class lifts to a cohomology group associated to the null foliation of Y . Moreover, it can only be nonzero in degrees 2q,\dotsc,2(p+q) . For Lagrangian Y , this reduces to a single degree 2q . Similar results hold in the holomorphic category.

Identities for deformation quantizations of almost Poisson algebras

Identities for deformation quantizations of almost Poisson algebras

Phase of Topological Lattice with Leaky Guided Mode Resonance.

Topological nature in different areas of physics and electronics has often been characterized and controlled through topological invariants depending on the global properties of the material. The validity of bulk-edge correspondence and symmetry-related topological invariants has been extended to non-Hermitian systems. Correspondingly, the value of geometric phases, such as the Pancharatnam-Berry or Zak phases, under the adiabatic quantum deformation process in the presence of non-Hermitian conditions, are now of significant interest. Here, we explicitly calculate the Zak phases of one-dimensional topological nanobeams that sustain guided-mode resonances, which lead to energy leakage to a continuum state. The retrieved Zak phases show as zero for trivial and as π for nontrivial photonic crystals, respectively, which ensures bulk-edge correspondence is still valid for certain non-Hermitian conditions.

Generalized time-dependent SIS Hamiltonian models: Exact solutions and quantum deformations

The theory of Lie–Hamilton systems is used to construct generalized time-dependent SIS epidemic Hamiltonians with a variable infection rate from the ‘book’ Lie algebra. Although these are characterized by a set of non-autonomous nonlinear and coupled differential equations, their corresponding exact solution is explicitly found. Moreover, the quantum deformation of the book algebra is also considered, from which the corresponding deformed SIS Hamiltonians are obtained and interpreted as perturbations in terms of the quantum deformation parameter of previously known SIS systems. The exact solutions for these deformed systems are also obtained.

Noncommutative geometry and deformation quantization in the quantum Hall fluids with inhomogeneous magnetic fields

It is well known that noncommutative geometry naturally emerges in the quantum Hall states due to the presence of strong and constant magnetic fields. Here, we discuss the underlying noncommutative geometry of quantum Hall fluids in which the magnetic fields are spatially inhomogenoeus. We analyze these cases by employing symplectic geometry and Fedosov’s deformation quantization, which rely on symplectic connections and Fedosov’s star-product. Through this formalism, we unveil some new features concerning the static limit of the Haldane’s unimodular metric and the Girvin–MacDonald–Platzman algebra of the density operators, which plays a central role in the fractional quantum Hall effect.

The Enhanced Period Map and Equivariant Deformation Quantizations of Nilpotent Orbits

The Enhanced Period Map and Equivariant Deformation Quantizations of Nilpotent Orbits

On Possible Minimal Length Deformation of Metric Tensor, Levi-Civita Connection, and the Riemann Curvature Tensor

The minimal length conjecture is merged with a generalized quantum uncertainty formula, where we identify the minimal uncertainty in a particle’s position as the minimal measurable length scale. Thus, we obtain a quantum-induced deformation parameter that directly depends on the chosen minimal length scale. This quantum-induced deformation is conjectured to require the generalization of Riemannian spacetime geometry underlying the classical theory of general relativity to an eight-dimensional spacetime fiber bundle, which dictates the deformation of the line element, metric tensor, Levi-Civita connection, Riemann curvature tensor, etc. We calculate the deformation thus produced in the Levi-Civita connection and find it to explicitly and exclusively depend on the product of the minimum measurable length and the particle’s spacelike four-acceleration vector, L2x¨2. We find that the deformed Levi-Civita connection preserves all properties of its undeformed counterpart, such as torsion freedom and metric compatibility. Accordingly, we have constructed a deformed version of the Riemann curvature tensor whose expression can be factorized in all its terms with different functions of L2x¨2. We also show that the classical four-manifold status of being Riemannian is preserved when the quantum-induced deformation is negligible. We study the dependence of a parallel-transported tangent vector on the spacelike four-acceleration. We illustrate the impact of the minimal-length-induced quantum deformation on the classical geometrical objects of the general theory of relativity using the unit radius two-sphere example. We conclude that the minimal length deformation implies a correction to the spacetime curvature and its contractions, which is manifest in the additional curvature terms of the corrected Riemann tensor. Accordingly, quantum-induced effects endow an additional spacetime curvature and geometrical structure.

Note on the Corrected Hawking Temperature of a Parametric Deformed Black Hole with Influence of Quantum Gravity

In this paper, using the Hamilton-Jacobi method, we discuss the tunnelling of fermions when the dual influence of quantum gravity and the deformation of a parameterized black hole are taken into account. With the influence of the generalized uncertainty principle, there exists an offset around the standard Hawking temperature. We investigate a parametric deformed black hole and find that the corrected temperature is lower than the standard one, so there exists a remnant of the black hole, and the correction is not only determined by the mass and the energy of the emitted fermion but also determined by the mass of the black hole and the deformation parameter. Under the dual influence of quantum gravity and deformation, the correction effect of quantum gravity is the main influencing factor, while the correction effect of the deformation parameter is secondary. For both the massive and massless cases, the quantum gravity correction factor is only determined by the energy of the emitted fermion, while the deformation correction factor is only determined by the mass of the black hole.

Quantizable functions on Kähler manifolds and non-formal quantization

Applying the Fedosov connections constructed in [7], we find a (dense) subsheaf of smooth functions on a Kähler manifold X which admits a non-formal deformation quantization. When X is prequantizable and the Fedosov connection satisfies an integrality condition, we prove that this subsheaf of functions can be quantized to a sheaf of twisted differential operators (TDO), which is isomorphic to that associated to the prequantum line bundle. We also show that examples of such quantizable functions are given by images of quantum moment maps.

Covariant action for conformal higher spin gravity

Conformal higher spin (HS) gravity is a HS extension of Weyl gravity and is a family of local HS theories, which was put forward by Segal and Tseytlin. We propose a manifestly covariant and coordinate-independent action for these theories. The result is based on an interplay between HS symmetries and deformation quantization: a locally equivalent but manifestly background-independent reformulation, known as the parent system, of the off-shell multiplet of conformal HS fields (Fradkin–Tseytlin fields) can be interpreted in terms of Fedosov deformation quantization of the underlying cotangent bundle. This brings into the game the invariant quantum trace, induced by the Feigin–Felder–Shoikhet cocycle of Weyl algebra, which extends Segal’s action into a gauge invariant and globally well-defined action functional on the space of configurations of the parent system. The same action can be understood within the worldline approach as a correlation function in the topological quantum mechanics on the circle.

Influence of accretion disk on the optical appearance of the Kazakov-Solodukhin black hole

The optical characteristics of the Kazakov-Solodukhin black hole (BH) surrounded by a thin disk is investigated. By utilizing the ray-tracing method, we derive the direct and secondary images to illustrate the influence of observational inclination angle (${\ensuremath{\theta}}_{0}$) and quantum correction parameter ($a$) on the observable properties. Our calculation involves determining the effective potential, light deflection, and azimuthal angle of the BH as well as the radiation flux of the accretion disk. By assuming three function models, we determine that the exponential function is the most suitable for describing the relationship between the observed flux and the quantum deformation parameter. Our research reveals that the observable characteristics of a BH are affected not only by the shape of the accretion disk but also by its spatiotemporal properties and the observer's inclination angle. We observe that a BH encircled by an optically thick accretion disk yields different results from those of an optically thin disk.

On Two Non-Ergodic Reversible Cellular Automata, One Classical, the Other Quantum.

We propose and discuss two variants of kinetic particle models-cellular automata in 1 + 1 dimensions-that have some appeal due to their simplicity and intriguing properties, which could warrant further research and applications. The first model is a deterministic and reversible automaton describing two species of quasiparticles: stable massless matter particles moving with velocity ±1 and unstable standing (zero velocity) field particles. We discuss two distinct continuity equations for three conserved charges of the model. While the first two charges and the corresponding currents have support of three lattice sites and represent a lattice analogue of the conserved energy-momentum tensor, we find an additional conserved charge and current with support of nine sites, implying non-ergodic behaviour and potentially signalling integrability of the model with a highly nested R-matrix structure. The second model represents a quantum (or stochastic) deformation of a recently introduced and studied charged hardpoint lattice gas, where particles of different binary charge (±1) and binary velocity (±1) can nontrivially mix upon elastic collisional scattering. We show that while the unitary evolution rule of this model does not satisfy the full Yang-Baxter equation, it still satisfies an intriguing related identity which gives birth to an infinite set of local conserved operators, the so-called glider operators.

Cohomology and deformation quantization of Poisson conformal algebras

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and give some constructions of them. Then we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.

One-loop corrections to the celestial chiral algebra from Koszul Duality

We consider self-dual Yang-Mills theory (SDYM) in four dimensions and its lift to holomorphic BF theory on twistor space. Following the work of Costello and Paquette, we couple SDYM to a quartic axion field, which guarantees associativity of the (extended) celestial chiral algebra at the quantum level. We demonstrate how to reproduce their one-loop quantum deformation to the chiral algebra using Koszul duality.

Pseudoclassical system with gauge and time-reparametrization invariance

We present a pseudoclassical mechanics model which exhibits gauge symmetry and time-reparametrization invariance. As such, first- and second-class constraints restrict the phase space, and the Hamiltonian weakly vanishes. We show that the Dirac conjecture does not hold---the secondary first-class constraint is not a symmetry generator---and only the gauge fixing condition associated with the primary first-class constraint is needed to remove the gauge ambiguities. The gauge fixed theory is equivalent to the Fermi harmonic oscillator extended by a boundary term. We quantize in the deformation quantization and in the Schr\"odinger representation approaches and observe that the boundary term prepares the system in the state of positive energy.

Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetry

We study the “twisted” Poincaré duality of smooth Poisson manifolds, and show that, if the modular vector field is diagonalizable, then there is a mixed complex associated to the Poisson complex, which, combining with the twisted Poincaré duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology. This generalizes the previous results obtained by Xu for unimodular Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure is preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras it is also preserved by Koszul duality.

Kontsevich Graphons

The article applies graph functions to extend the Kontsevich differential graded Lie algebraic formalism (in Deformation Quantization) to infinite Kontsevich graphs on the basis of the Connes-Kreimer Hopf algebraic renormalization and the theory of noncommutative differential geometry.

Asymptotic Quantization of a Particle on a Sphere

Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.

All basic quantizations of D=3, N=1 Lorentz supersymmetry

By the supersymmetrization of a simple algebraic technique proposed in Lukierski and Tolstoy (Eur Phys J C 77:226, 2017) we obtain the complete classification of all basic (nonisomorphic) quantum deformations for the orthosymplectic Lie superalgebra mathfrak {osp}(1|2;{mathbb {C}}) and its pseudoreal and real forms in terms of the classical r-matrices. In particular, we prove that pseudoreal compact form has only one quantum deformation (standart q-analog), and the D=3, N=1 Lorentz supersymmetry, which is the non-compact real form of mathfrak {osp}(1|2;{mathbb {C}}), has four different Hopf-algebraic quantum deformations: two standard q-analogs, and two (Jordanian and super-Jordanian) twist deformations. All basic Hopf-algebraic quantum deformations are presented in the explicit form.