Papers by Simon Lyakhovich
Physical Review D, 2002
The effective equations of motion for a point charged particle taking account of radiation reacti... more 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.
Nuclear Physics B, 2004
We define and study invariants which can be uniformly constructed for any gauge system. By a gaug... more We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti-)Poisson supermanifold endowed with an odd Hamiltonian self-commuting vector field called a homological vector field. This definition encompasses all the cases usually included into the notion of a gauge theory in physics as well as some other similar (but different) structures like Lie or Courant algebroids. For Lagrangian gauge theories or Hamiltonian first class constrained systems, the homological vector field is identified with the classical BRST transformation operator. We define characteristic classes of a gauge system as universal cohomology classes of the homological vector field, which are uniformly constructed in terms of this vector field itself. Not striving to exhaustively classify all the characteristic classes in this work, we compute those invariants which are built up in terms of the first derivatives of the homological vector field. We also consider the cohomological operations in the space of all the characteristic classes. In particular, we show that the (anti-)Poisson bracket becomes trivial when applied to the space of all the characteristic classes, instead the latter space can be endowed with another Lie bracket operation. Making use of this Lie bracket one can generate new characteristic classes involving higher derivatives of the homological vector field. The simplest characteristic classes are illustrated by the examples relating them to anomalies in the traditional BV or BFV-BRST theory and to characteristic classes of (singular) foliations.
To describe a massive particle with fixed, but arbitrary, spin on $d=4$ anti-de Sitter space $M^4... more To describe a massive particle with fixed, but arbitrary, spin on $d=4$ anti-de Sitter space $M^4$, we propose the point-particle model with configuration space ${\cal M}^6 = M^{4}\times S^{2}$, where the sphere $S^2$ corresponds to the spin degrees of freedom. The model possesses two gauge symmetries expressing strong conservation of the phase-space counterparts of the second- and fourth-order Casimir operators
Journal of High Energy Physics, 2005
We consider a generic gauge system, whose physical degrees of freedom are obtained by restriction... more We consider a generic gauge system, whose physical degrees of freedom are obtained by restriction on a constraint surface followed by factorization with respect to the action of gauge transformations; in so doing, no Hamiltonian structure or action principle is supposed to exist. For such a generic gauge system we construct a consistent BRST formulation, which includes the conventional BV Lagrangian and BFV Hamiltonian schemes as particular cases. If the original manifold carries a weak Poisson structure (a bivector field giving rise to a Poisson bracket on the space of physical observables) the generic gauge system is shown to admit deformation quantization by means of the Kontsevich formality theorem. A sigma-model interpretation of this quantization algorithm is briefly discussed. * sll@phys.tsu.ru † sharapov@phys.tsu.ru.
Nuclear Physics B, 2001
A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of ... more 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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Physics Letters B, 2002
We consider a general N=(2,2) non-linear sigma model with a torsion. We show that the consistency... more We consider a general N=(2,2) non-linear sigma model with a torsion. We show that the consistency of N=(2,2) supersymmetry implies that the target manifold is necessary equipped with two (in general, different) Poisson structures. Finally we argue that the Poisson geometry of the target space is a characteristic feature of the sigma models with extended supersymmetry.
Arbitrary superspin massive superparticles
Physics Letters B, 1995
ABSTRACT We propose an exactly solvable model for a massive N-extended superparticle with pure (h... more ABSTRACT We propose an exactly solvable model for a massive N-extended superparticle with pure (half-)integer superspin , …. Regardless of the superspinvalue, the configuration space is 4|4N × S2, where S2 corresponds to spinning degrees of freedom. Being explicitly super-Poincaré invariant, the model possesses two gauge symmetries implying strong conservation of the squared momentum and superspin. Hamilton constrained dynamics is developed and canomical quantization is studied. For N = 1 we show that the physical super wave-functions are to be on-shell massive chiral superfields. Central-charge and higher-dimensional generalizations of the model are given.
Physics Letters B, 2002
We propose a new BRST-like quantization procedure which is applicable to dynamical systems contai... more We propose a new BRST-like quantization procedure which is applicable to dynamical systems containing both first and second class constraints. It requires no explicit separation into first and second class constraints and therefore no conversion of second class constraints is needed. The basic ingredient is instead an invariant projection operator which projects out the maximal subset of constraints in involution. The hope is that the method will enable a covariant quantization of models for which there is no covariant separation into first and second class constraints. An example of this type is given.
Physical Review D, 1996
A universal model for D=4 spinning particle is constructed with the configuration space chosen as... more A universal model for D=4 spinning particle is constructed with the configuration space chosen as R 3,1 ×S 2 , where the sphere corresponds to the spinning degrees of freedom. The Lagrangian includes all the possible world-line first order invariants of the manifold. Each combination of the four constant parameters entering the Lagrangian gives the model, which describes the proper irreducible Poincaré dynamics both at the classical and quantum levels, and thereby the construction uniformly embodies the massive, massless and continuous helicity cases depending upon the special choice of the parameters. For the massive case, the connection with the Souriau approach to elementary systems is established. Constrained Hamiltonian formulation is built and Dirac canonical quantization is performed for the model in the covariant form. An explicit realization is given for the physical wave functions in terms of smooth tensor fields on R 3,1 × S 2 . One-parametric family of consistent interactions with general electromagnetic and gravitational fields is introduced in the massive case. The spin tensor is shown to satisfy the Frenkel-Nyborg equation with arbitrary fixed giromagnetic ratio in a limit of weakly varying electromagnetic
Nuclear Physics B, 1996
The general procedure of constructing a consistent covariant Dirac-type bracket for models with m... more The general procedure of constructing a consistent covariant Dirac-type bracket for models with mixed first and second class constraints is presented. The proposed scheme essentially relies upon explicit separation of the initial constraints into infinitely reducible first and second class ones (by making use of some appropriately constructed covariant projectors). Reducibility of the second class constraints involved manifests itself in weakening some properties of the bracket as compared to the standard Dirac one. In particular, a commutation of any quantity with the second class constraints and the Jacobi identity take place on the second class constraints surface only. The developed procedure is realized for N = 1 Brink-Schwarz superparticle in arbitrary dimension and for N = 1, D = 9 massive superparticle with Wess-Zumino term. A possibility to apply the bracket for quantizing the superparticles within the framework of the recent unified algebra approach by is examined. In particular, it is shown that for D = 9 massive superparticle it is impossible to construct Dirac-type bracket possessing (strong) Jacobi identity in a full phase space.
Dynamical systems defining Jacobi's ϑ-constants
Journal of Mathematical Physics, 2011
ABSTRACT We propose a system of equations that defines Weierstrass-Jacobi's eta- and thet... more ABSTRACT We propose a system of equations that defines Weierstrass-Jacobi's eta- and theta-constant series in a differentially closed way. This system is shown to have a direct relationship to a little-known dynamical system obtained by Jacobi. The classically known differential equations by Darboux-Halphen, Chazy, and Ramanujan are the differential consequences or reductions of these systems. The proposed system is shown to admit the Lagrangian, Hamiltonian, and Nambu formulations. We explicitly construct a pencil of nonlinear Poisson brackets and complete set of involutive conserved quantities. As byproducts of the theory, we exemplify conserved quantities for the Ramamani dynamical system and quadratic system of Halphen-Brioschi.
International Journal of Modern Physics A, 2001
In a classical Hamiltonian theory with second class constraints the phase space functions on the ... more In a classical Hamiltonian theory with second class constraints the phase space functions on the constraint surface are observables. We give general formulas for extended observables, which are expressions representing the observables in the enveloping unconstrained phase space. These expressions satisfy in the unconstrained phase space a Poisson algebra of the same form as the Dirac bracket algebra of the observables on the constraint surface. The general formulas involve new differential operators that differentiate the Dirac bracket. Similar extended observables are also constructed for theories with first class constraints which, however, are gauge dependent. For such theories one may also construct gauge invariant extensions with similar properties. Whenever extended observables exist the theory is expected to allow for a covariant quantization. A mapping procedure is proposed for covariant quantization of theories with second class constraints.
Theoretical and Mathematical …, 2001
We propose an explicit construction of the deformation quantization of the general second-class c... more We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.
Nuclear Physics B, 2014
In their simplest form, metric-like Lagrangians for higher-spin massless fields display constrain... more In their simplest form, metric-like Lagrangians for higher-spin massless fields display constrained gauge symmetries, unless auxiliary fields are introduced or locality is foregone. Specifically, in its standard incarnation, gauge invariance of Maxwell-like Lagrangians relies on parameters with vanishing divergence. We propose an alternative form of the corresponding local symmetry involving unconstrained parameters of mixed-symmetry type, described by rectangular two-row Young diagrams and entering high-derivative gauge transformations. The resulting gauge algebra appears to be reducible and we display the full pattern of gauge-for-gauge parameters, testing its correctness via the corresponding counting of degrees of freedom. Incidentally, this shows that massless higher spins admit a local unconstrained formulation with no need for auxiliary fields.
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Papers by Simon Lyakhovich