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S. L. Lyakhovich

Publications and source records attributed to S. L. Lyakhovich.

At least 73 records · Page 4Linked to original sources
arXiv2004

Wick Quantization of Cotangent Bundles over Riemannian Manifolds

A simple geometric procedure is proposed for constructing Wick symbols on cotangent bundles to Riemannian manifolds. The main ingredient of the construction is a method of endowing the cotangent bundle with a formal Kähler structure. The formality means that the metric is lifted from the Riemannian manifold $Q$ to its phase space $T^\ast Q$ in the form of formal power series in momenta with the coefficients being tensor fields on the base. The corresponding Kähler two-form on the total space of $T^\ast Q$ coincides with the canonical symplectic form, while the canonical projection of the Kähler metric on the base manifold reproduces the original metric. Some examples are considered, including constant curvature space and nonlinear sigma models, illustrating the general construction.

hep-th↗
arXiv2002

On the Fedosov Deformation Quantization beyond the Regular Poisson Manifolds

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang-Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.

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arXiv2002

Radiation reaction and renormalization in classical electrodynamics of point particle in any dimension

The effective equations of motion for a point charged particle taking account of radiation reaction are considered in various space-time dimensions. The divergencies steaming from the pointness of the particle are studied and the effective renormalization procedure is proposed encompassing uniformly the cases of all even dimensions. It is shown that in any dimension the classical electrodynamics is a renormalizable theory if not multiplicatively beyond d=4. For the cases of three and six dimensions the covariant analogs of the Lorentz-Dirac equation are explicitly derived.

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arXiv2001

Wick quantisation of a symplectic manifold

The notion of the Wick star-product is covariantly introduced for a general symplectic manifold equipped with two transverse polarisations. Along the lines of Fedosov method, the explicit procedure is given to construct the Wick symbols on the manifold. The cohomological obstruction is identified to the equivalence between the Wick star-product and the Fedosov one. In particular in the Kähler case, the Wick star-product is shown to be equivalent the Weyl one, iff the manifold is a Calabi-Yau one.

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arXiv2001

Wick type deformation quantization of Fedosov manifolds

A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the particular case of Kähler manifold this class is shown to be proportional to the Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed.

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arXiv2000

Wick Type Symbol and Deformed Algebra of Exterior Forms

The covariant descripion is constructed for the Wick-type symbols on symplectic manifolds by means of the Fedosov procedure. The geometry of the manifolds admitting this symbol is explored. The superextended version of the Wick-type star-product is introduced and a possible application of the construction to the noncommutative field theory is discussed.

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arXiv2000

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal M}$ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ${{\mathcal T}^*_ρ}{\mathcal M}$ and the tangent bundle $T {\mathcal M}$. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables.

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arXiv2000

A uniform model of the massive spinning particle in any dimension

The general model of an arbitrary spin massive particle in any dimensional space-time is derived on the basis of Kirillov - Kostant - Souriau approach. It is shown that the model allows consistent coupling to an arbitrary background of electromagnetic and gravitational fields.

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arXiv1999

Algebraic Aspects of Interactions of Massive Spinning Particles in Three Dimensions

The most general 2+1 dimensional spinning particle model is considered. The action functional may involve all the possible first order Poincare invariants of world lines, and the particular class of actions is specified thus the corresponding gauge algebra to be unbroken by inhomogeneous external fields. Nevertheless, the consistency problem reveals itself as a requirement of the global compatibility between first and second class constraints. These compatibility conditions, being unnoticed before in realistic second class theories, can be satisfied for a particle iff the gyromagnetic ratio takes the critical value g=2. The quantization procedure is suggested for a particle in the generic background field by making use of a Darboux co-ordinates, being found by a perturbative expansion in the field multipoles and the general procedure is described for constructing of the respective transformation in any order.

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arXiv1998

BRST-anti-BRST covariant theory for the second class constrained systems. A general method and examples

The BRST-anti-BRST covariant extension is suggested for the split involution quantization scheme for the second class constrained theories. The constraint algebra generating equations involve on equal footing a pair of BRST charges for second class constraints and a pair of the respective anti-BRST charges. Formalism displays explicit Sp(2) \times Sp(2) symmetry property. Surprisingly, the the BRST-anti-BRST algebra must involve a central element, related to the nonvanishing part of the constraint commutator and having no direct analogue in a first class theory. The unitarizing Hamiltonian is fixed by the requirement of the explicit BRST-anti-BRST symmetry with a much more restricted ambiguity if compare to a first class theory or split involution second class case in the nonsymmetric formulation. The general method construction is supplemented by the explicit derivation of the extended BRST symmetry generators for several examples of the second class theories, including self--dual nonabelian model and massive Yang Mills theory.

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arXiv1998

Hidden supersymmetry and Berezin quantization of N=2, D=3 spinning superparticles

The first quantized theory of N=2, D=3 massive superparticles with arbitrary fixed central charge and (half)integer or fractional superspin is constructed. The quantum states are realized on the fields carrying a finite dimensional, or a unitary infinite dimensional representation of the supergroups OSp(2|2) or SU(1,1|2). The construction originates from quantization of a classical model of the superparticle we suggest. The physical phase space of the classical superparticle is embedded in a symplectic superspace $T^\ast({R}^{1,2})\times{L}^{1|2}$, where the inner Kähler supermanifold ${L}^{1|2}=OSp(2|2)/[U(1)\times U(1)]=SU(1,1|2)/[U(2|2)\times U(1)]$ provides the particle with superspin degrees of freedom. We find the relationship between Hamiltonian generators of the global Poincaré supersymmetry and the ``internal'' SU(1,1|2) one. Quantization of the superparticle combines the Berezin quantization on ${L}^{1|2}$ and the conventional Dirac quantization with respect to space-time degrees of freedom. Surprisingly, to retain the supersymmetry, quantum corrections are required for the classical N=2 supercharges as compared to the conventional Berezin method. These corrections are derived and the Berezin correspondence principle for ${L}^{1|2}$ underlying their origin is verified. The model admits a smooth contraction to the N=1 supersymmetry in the BPS limit.

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arXiv1998

Existence Theorem for Split Involution Constraint Algebra

Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described.

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arXiv1998

Geometric quantization of N=2, D=3 superanyon

A classical model of N=2, D=3 fractional spin superparticle (superanyon) is presented, whose first-quantization procedure combines the Berezin quantization for the superspin degrees of freedom and the canonical quantization for the space-time ones. To provide the supersymmetry for the quantised theory, certain quantum corrections are required to the N=2 supersymmetry generators as compared to the Berezin procedure. The renormalized generators are found and the first quantised theory of N=2 superanyon is constructed.

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arXiv1997

N=1, D=3 Superanyons, osp(2|2) and the Deformed Heisenberg Algebra

We introduce N=1 supersymmetric generalization of the mechanical system describing a particle with fractional spin in D=1+2 dimensions and being classically equivalent to the formulation based on the Dirac monopole two-form. The model introduced possesses hidden invariance under N=2 Poincaré supergroup with a central charge saturating the BPS bound. At the classical level the model admits a Hamiltonian formulation with two first class constraints on the phase space $T^*(R^{1,2})\times {\cal L}^{1|1}$, where the Kähler supermanifold ${\cal L}^{1|1}\cong OSp(2|2)/U(1|1)$ is a minimal superextension of the Lobachevsky plane. The model is quantized by combining the geometric quantization on ${\cal L}^{1|1}$ and the Dirac quantization with respect to the first class constraints. The constructed quantum theory describes a supersymmetric doublet of fractional spin particles. The space of quantum superparticle states with a fixed momentum is embedded into the Fock space of a deformed harmonic oscillator.

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