arXiv subjects

V. G. Kupriyanov

Publications and source records attributed to V. G. Kupriyanov.

At least 19 recordsLinked to original sources

arXiv2026

Canonical representation of the Snyder-de Sitter algebra with correct flat and commutative limits

The Snyder--de Sitter algebra provides a Lorentz-covariant deformation of phase-space geometry characterized by a curvature parameter $α$ and a noncommutativity parameter $β$. We construct an explicit canonical (Darboux) representation of this algebra that is regular in both parameters. Starting from the symplectic structure associated with the Snyder--de Sitter Poisson brackets, we derive the canonical transformation between the physical phase-space variables and Darboux coordinates and obtain its inverse in closed form to all orders in $α$ and $β$. The resulting representation has well-defined flat ($α\to0$) and commutative ($β\to0$) limits, reducing respectively to the Snyder and de Sitter phase-space algebras, while the simultaneous limit $(α,β)\to(0,0)$ yields the standard canonical phase-space coordinates. We then apply this representation to the construction of Poisson gauge transformations on Snyder--de Sitter phase space. In particular, we derive the corresponding gauge transformation matrix and Poisson field strength, both of which are regular in $α$ and $β$. Our construction provides a convenient framework for investigating gauge theories and other physical systems formulated on Snyder--de Sitter phase space.

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arXiv2025

Symplectic realization of generalized Snyder-Poisson algebra

We investigate Snyder space-time and its generalizations, including Yang and Snyder-de-Sitter spaces, which constitute manifestly Lorenz invariant noncommutative geometries. This work initiates a systematic study of gauge theory on such spaces in the semi-classical regime, formulated as Poisson gauge theory. As a first step, we construct the symplectic realization of the relevant noncommutative spaces, a prerequisite for defining Poisson gauge transformations and field strengths. We present a general method for representing the Snyder algebra and its extensions in terms of canonical phase space variables, enabling both the reproduction of known representations and the derivation of novel ones. These canonical constructions are employed to obtain explicit symplectic realizations for the Snyder-de-Sitter space and construct the deformed partial derivative which differentiates the underlying Poisson structure. Furthermore, we analyze the motion of freely falling particles in these backgrounds and comment on the geometry of the associated spaces.

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arXiv2024

Charged Particle in Lie-Poisson Electrodynamics

Lie-Poisson electrodynamics describes the semi-classical limit of non-commutative $U(1)$ gauge theory, characterized by Lie-algebra-type non-commutativity. We focus on the mechanics of a charged point-like particle moving in a given gauge background. First, we derive explicit expressions for gauge-invariant variables representing the particle's position. Second, we provide a detailed formulation of the classical action and the corresponding equations of motion, which recover standard relativistic dynamics in the commutative limit. We illustrate our findings by exploring the exactly solvable Kepler problem in the context of the $λ$-Minkowski (or the angular) non-commutativity, along with other examples.

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arXiv2023

Lie-Poisson gauge theories and $κ$-Minkowski electrodynamics

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly gauge-covariant expressions for the Euler-Lagrange equations of motion, the Bianchi and the Noether identities. We discuss the non-Lagrangian equations of motion, and apply our findings to the $κ$-Minkowski case. We construct a family of exact solutions of the deformed Maxwell equations in the vacuum. In the classical limit, these solutions recover plane waves with left-handed and right-handed circular polarization, being classical counterparts of photons. The deformed dispersion relation appears to be nontrivial.

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arXiv2022

On the L$_\infty$ structure of Poisson gauge theory

The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L$_\infty^{full}$ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory. We derive the minimal set of non-vanishing $\ell$-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L$_\infty$ algebras. On the other hand, it can be used as a starting point for bootstrapping the full non-commutative gauge theory. The first few brackets of such a theory are constructed explicitly in the text. In addition we show that the derivation properties of $\ell$-brackets on L$_\infty^{full}$ with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, L$_\infty^{full}$ does not have a structure of P$_\infty$ algebra.

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arXiv2022

Poisson gauge models and Seiberg-Witten map

The semiclassical limit of full non-commutative gauge theory is known as Poisson gauge theory. In this work we revise the construction of Poisson gauge theory paying attention to the geometric meaning of the structures involved and advance in the direction of a further development of the proposed formalism, including the derivation of Noether identities and conservation of currents. For any linear non-commutativity, $Θ^{ab}(x)=f^{ab}_c\,x^c$, with $f^{ab}_c$ being structure constants of a Lie algebra, an explicit form of the gauge Lagrangian is proposed. In particular a universal solution for the matrix $ρ$ defining the field strength and the covariant derivative is found. The previously known examples of $κ$-Minkowski, $λ$-Minkowski and rotationally invariant non-commutativity are recovered from the general formula. The arbitrariness in the construction of Poisson gauge models is addressed in terms of Seiberg-Witten maps, i.e., invertible field redefinitions mapping gauge orbits onto gauge orbits.

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arXiv2020

$k$-Minkowski-deformation of $U(1)$ gauge theory

We construct a non-commutative kappa-Minkowski deformation of U(1) gauge theory, following a general approach, recently proposed in JHEP 2008 (2020) 041. We obtain an exact (all orders in the non-commutativity parameter) expression for both the deformed gauge transformations and the deformed field strength, which is covariant under these transformations. The corresponding Yang-Mills Lagrangian is gauge covariant and reproduces the Maxwell Lagrangian in the commutative limit. Gauge invariance of the action functional requires a non-trivial integration measure which, in the commutative limit, does not reduce to the trivial one. We discuss the physical meaning of such a nontrivial commutative limit, relating it to a nontrivial space-time curvature of the undeformed theory. Moreover, we propose a rescaled kappa-Minkowski non-commutative structure, which exhibits a standard flat commutative limit.

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arXiv2016

Weak associativity and deformation quantization

Non-commutativity and non-associativity are quite natural in string theory. For open strings it appear due to the presence of non-vanishing background two-form in the world volume of Dirichlet brane, while in closed string theory the flux compactifications with non-vanishing three-form also lead to non-geometric backgrounds. In this paper, working in the framework of deformation quantization, we study the violation of associativity imposing the condition that the associator of three elements should vanish whenever each two of them are equal. The corresponding star products are called alternative and satisfy an important for physical applications properties like the Moufang identities, alternative identities, Artin's theorem, etc. The condition of alternativity is invariant under the gauge transformations, just like it happens in the associative case. The price to pay is the restriction on the non-associative algebra which can be represented by the alternative star product, it should satisfy the Malcev identity. The example of nontrivial Malcev algebra is the algebra of imaginary octonions. For this case we construct an explicit expression of the non-associative and alternative star product. We also discuss the quantization of Malcev-Poisson algebras of general form, study its properties and provide the lower order expression for the alternative star product. To conclude we define the integration on the algebra of the alternative star products and show that the integrated associator vanishes.

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arXiv2016

Alternative multiplications and non-associativity in physics

Some physical systems like the quantum mechanics with magnetic charges or field theoretical models appearing in the context of string theory are formulated in terms of non-associative algebras. Hence, demand non-associative star products for its description. However, unlike its associative counterpart the situation with non-associative star products is quite unclear. At the moment it is not evident which condition should be used instead of the associativity in the definition of the non-associative star multiplication. In mathematics, the natural generalization of the associativity is the requirement that the multiplication should be alternative, the associator of three elements should vanish if each two of them are equal. We show that for the alternative closed star product the integrated associator vanishes. In particular, it means that on-shell there will be no violation of associativity in string scattering amplitudes. We discuss the examples of the alternative and closed star products. In particular, it is constructed the star product corresponding to the algebra of octonions.

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arXiv2015

Nonassociative Weyl star products

Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires (weak) hermiticity and the Weyl condition. We also propose an iterative procedure that allows to compute the star product up to any desired order.

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arXiv2015

Noncommutative $R^d$ via closed star product

We consider linear star products on $R^d$ of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, $Tr( f\star g)= Tr( f\cdot g)$. We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibnitz rule holds true up to a total derivative. As a particular example we study the space $R^3_θ$ with $\mathfrak{su}(2)$ type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibnitz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.

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arXiv2014

Dirac equation on coordinate dependent noncommutative space-time

We consider the consistent deformation of the relativistic quantum mechanics introducing the noncommutativity of the space-time and preserving the Lorentz symmetry. The relativistic wave equation describing the spinning particle on coordinate dependent noncommutative space-time (noncommutative Dirac equation) is proposed. The fundamental properties of this equation, like the Lorentz covariance and the continuity equation for the probability density are verified. To this end using the properties of the star product we derive the corresponding probability current density and prove its conservation. The energy-momentum tensor for the free noncommutative spinor field is calculated. We solve the free noncommutative Dirac equation and show that the standard energy-momentum dispersion relation remains valid in the noncommutative case.

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arXiv2014

Conserved symmetries in noncommutative quantum mechanics

We consider a problem of the consistent deformation of physical system introducing a new features, but preserving its fundamental properties. In particular, we study how to implement the noncommutativity of space-time without violation of the rotational symmetry in quantum mechanics or the Lorentz symmetry in f{i}eld theory. Since the canonical (Moyal) noncommutativity breaks the above symmetries one should work with more general case of coordinate-dependent noncommutative spaces, when the commutator between coordinates is a function of these coordinates. F{i}rst we describe in general lines how to construct the quantum mechanics on coordinate-dependent noncommutative spaces. Then we consider the particular examples: the Hydrogen atom on rotationally invariant noncommutative space and the Dirac equation on covariant noncommutative space-time.

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arXiv2013

Hydrogen atom on curved noncommutative space

We have calculated the hydrogen atom spectrum on curved noncommutative space defined by the commutation relations $\left[ \hat {x}^{i},\hat{x}^{j}\right] =iθ\hatω^{ij}\left( \hat {x}\right) $, where $θ$ is the parameter of noncommutativity. The external antisymmetric field which determines the noncommutativity is chosen as $ω^{ij}(x) =\varepsilon^{ijk}{x}_{k}f\left( {x_i}x^{i}\right) $. In this case the rotational symmetry of the system is conserved, preserving the degeneracy of the energy spectrum. The contribution of the noncommutativity appears as a correction to the fine structure. The corresponding nonlocality is calculated: $ΔxΔy \geq \frac{θ^2}{4} |m\langle f^2\rangle| $, where $m$ is a magnetic quantum number.

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arXiv2013

Quantum mechanics with coordinate dependent noncommutativity

Noncommutative quantum mechanics can be considered as a first step in the construction of quantum field theory on noncommutative spaces of generic form, when the commutator between coordinates is a function of these coordinates. In this paper we discuss the mathematical framework of such a theory. The noncommutativity is treated as an external antisymmetric field satisfying the Jacoby identity. First, we propose a symplectic realization of a given Poisson manifold and construct the Darboux coordinates on the obtained symplectic manifold. Then we define the star product on a Poisson manifold and obtain the expression for the trace functional. The above ingredients are used to formulate a nonrelativistic quantum mechanics on noncommutative spaces of general form. All considered constructions are obtained as a formal series in the parameter of noncommutativity. In particular, the complete algebra of commutation relations between coordinates and conjugated momenta is a deformation of the standard Heisenberg algebra. As examples we consider a free particle and an isotropic harmonic oscillator on the rotational invariant noncommutative space.

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arXiv2012

Dynamics of a Dirac Fermion in the presence of spin noncommutativity

Recently, it has been proposed a spacetime noncommutativity that involves spin degrees of freedom, here called "spin noncommutativity". One of the motivations for such a construction is that it preserves Lorentz invariance, which is deformed or simply broken in other approaches to spacetime noncommutativity. In this work, we gain further insight in the physical aspects of the spin noncommutativity. The noncommutative Dirac equation is derived from an action principle, and it is found to lead to the conservation of a modified current, which involves the background electromagnetic field. Finally, we study the Landau problem in the presence of spin noncommutativity. For this scenario of a constant magnetic field, we are able to derive a simple Hermitean non-commutative correction to the Hamiltonian operator, and show that the degeneracy of the excited states is lifted by the noncommutativity at the second order or perturbation theory.

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arXiv2010

Dynamical noncommutativity

The model of dynamical noncommutativity is proposed. The system consists of two interrelated parts. The first of them describes the physical degrees of freedom with coordinates q^1, q^2, the second one corresponds to the noncommutativity r which has a proper dynamics. After quantization the commutator of two physical coordinates is proportional to the function of r. The interesting feature of our model is the dependence of nonlocality on the energy of the system. The more the energy, the more the nonlocality. The lidding contribution is due to the mode of noncommutativity, however, the physical degrees of freedom also contribute in nonlocality in higher orders in θ.

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arXiv2010

Noncommutativity due to spin

Using the Berezin-Marinov pseudoclassical formulation of spin particle we propose a classical model of spin noncommutativity. In the nonrelativistic case, the Poisson brackets between the coordinates are proportional to the spin angular momentum. The quantization of the model leads to the noncommutativity with mixed spacial and spin degrees of freedom. A modified Pauli equation, describing a spin half particle in an external e.m. field is obtained. We show that nonlocality caused by the spin noncommutativity depends on the spin of the particle; for spin zero, nonlocality does not appear, for spin half, $ΔxΔy\geqθ^{2}/2$, etc. In the relativistic case the noncommutative Dirac equation was derived. For that we introduce a new star product. The advantage of our model is that in spite of the presence of noncommutativity and nonlocality, it is Lorentz invariant. Also, in the quasiclassical approximation it gives noncommutativity with a nilpotent parameter.

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