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S. L. Lyakhovich

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

At least 19 recordsLinked to original sources
arXiv2026

Noether's theorem for the conditional principle of least action

We consider the problem of a conditional extremum of an action in a class of fields constrained by differential equations. For this setup, we propose an extension of Noether's first theorem to connect the symmetries of the action and the imposed equations to the currents conserved at the conditional extrema. The key ingredient of the extension is the gauge symmetry of the differential equations constraining the admissible class of field configurations. We consider a special type of global symmetries of the action which we call conditional symmetries. Such global symmetries must be special cases of gauge transformations of the constraint equations. We construct conservation laws that follow from the conditional symmetries of action. No Lagrange multipliers or other auxiliary fields are introduced and the conserved currents include only the original fields. We also prove the converse theorem which connects the conserved currents to the conditional symmetries of action. The general method is illustrated by several examples.

physics.gen-ph↗
arXiv2024

Topological string as massive spinning particle in three dimensions

A model is proposed for a classical bosonic string in $d=3$ Minkowski space with an action functional that includes Gauss and mean world-sheet curvature. The Lagrangian is invariant under $3d$ Poincaré transformations modulo total divergence. In addition to the diffeomorphism, the action enjoys the extra gauge symmetry with the second derivatives of the scalar gauge parameter. This symmetry gauges out all the local degrees of freedom (DoF's), while some global DoF's survive. The Hamiltonian constrained analysis confirms that the model does not have any local DoF. The world sheet of the string turns out to be a cylinder with time-like axis. The global DoF's of this string describe one single irreducible massive $3d$ particle with spin. The particle momentum is a conserved vector directed along the cylinder axis while the momentum square is a fixed constant determined by the parameters in the action. The total angular momentum is a conserved vector that defines the position of the axis of the cylinder whose specific value is defined by initial data, while the spin, being the product of momentum and angular momentum is fixed by the parameters in the string Lagrangian.

hep-th↗
arXiv2023

Unfree gauge symmetry

The gauge symmetry is said unfree if the gauge transformation leaves the action functional unchanged provided for the gauge parameters are constrained by the system of partial differential equations. The best known example of this phenomenon is the volume preserving diffeomorphism being the gauge symmetry of unimodular gravity (UG). Various extensions are known of the UG, including the higher spin analogs - all with unfree gauge symmetry. Given the distinctions of the unfree gauge symmetry from the symmetry with unrestricted gauge parameters, the algebra of gauge transformations is essentially different. These distinctions have consequences for all the key constituents of general gauge theory, starting from the second Noether theorem, Hamiltonian constrained formalism, BRST complex, and quantization. In this review article, we summarise the modifications of general gauge theory worked out in recent years to cover the case of unfree gauge symmetry.

hep-th↗
arXiv2023

Dualisation of free fields

We consider the general free field theory such that system of equations of motion includes a subsystem with a special property. If the subsystem is considered by itself, it would be a topological field theory having no local degrees of freedom. Various well-known field theories admit such a subsystem, including Einstein's gravity. As the subsystem is a topological theory, the general solution is a pure gauge, modulo global degrees of freedom. Gauge symmetry of the subsystem of linear theory can be explicitly found, providing the general solution to the subsystem. Substituting this solution into entire system, we arrive at an equivalent field theory where the role of the field variables is played by the gauge parameters of the topological subsystem. These fields can be viewed as the potentials for the original ones. A general procedure is proposed for constructing a "parent action" which includes the fields of both dual formulations. At the level of this action, one can switch between dual formulations by imposing appropriate gauge fixing conditions. This general dualisation procedure is exemplified by massless and massive spin two fields. For the massless case, the hook type tensor serves as a potential for the metric, while for the massive case it is the fourth rank tensor with window Young diagram.

hep-th↗
arXiv2022

Gauge symmetry of linearised Nordström gravity and the dual spin two field theory

The field equations are proposed for the third rank tensor field with the hook Young diagram. The equations describe the irreducible spin two massless representation in any $d\geq 3$. The starting point of the construction is the linearised system of Einstein equations which includes the Nordström equation. This equation, being considered irrespectively to the rest of the Einstein system, corresponds to the topological field theory. The general solution is a pure gauge, modulo topological modes which we neglect in this article. We find the sequence of the reducible gauge transformations for the linearised Nordström equation, with the hook tensor being the initial gauge symmetry parameter. By substituting the general solution of the Nordström equation into the rest of the Einstein's system, we arrive at the field equations for the hook tensor. The degree of freedom number count confirms, it is the spin two theory.

hep-th↗
arXiv2022

Global conserved quantities and unfree gauge symmetry

We consider a class of theories with unfree gauge symmetry, whose gauge parameters are restricted by differential equations. We demonstrate that such theories admit global conserved quantities, whose on-shell values are defined by asymptotics of the fields rather than Cauchy data. The global conserved quantities can be deduced proceeding from the equations restricting gauge parameters, and they are treated differently by two BRST complexes corresponding to a system with unfree gauge symmetry.

hep-th↗
arXiv2022

Gauge symmetry of unimodular gravity in Hamiltonian formalism

We work out the description of the gauge symmetry of unimodular gravity in the constrained Hamiltonian formalism. In particular, we demonstrate how the transversality conditions restricting the diffeomorphism parameters emerge from the algebra of the Hamiltonian constraints. The alternative form is long known as parametrizing the volume preserving diffeomorphisms by unrestricted two-forms instead of the transverse vector fields. This gauge symmetry is reducible. We work out the Hamiltonian description of this form of unimodular gravity (UG) gauge symmetry. Becchi-Rouet-Stora-Tyutin--Batalin-Fradkin-Vilkovisky (BFV-BRST) Hamiltonian formalism is constructed for both forms of the UG gauge symmetry. These two BRST complexes have a subtle inequivalence: Their BRST cohomology groups are not isomorphic. In particular, for the first complex, which is related to the restricted gauge parameters, the cosmological constant does not correspond to any nontrivial BRST cocycle, while for the alternative complex it does. In the wording of physics, this means $Λ$ is a fixed parameter defined by the field asymptotics rather than the physical observable from the standpoint of the first complex. The second formalism views $Λ$ as the observable with unrestricted initial data.

hep-th↗
arXiv2021

Reducible gauge symmetry versus unfree gauge symmetry in Hamiltonian formalism

The unfree gauge symmetry implies that gauge variation of the action functional vanishes provided for the gauge parameters are restricted by the differential equations. The unfree gauge symmetry is shown to lead to the global conserved quantities whose on shell values are defined by the asymptotics of the fields or data on the lower dimension surface, or even at the point of the space-time, rather than Cauchy hyper-surface. The most known example of such quantity is the cosmological constant of unimodular gravity. More examples are provided in the article for the higher spin gravity analogues of the cosmological constant. Any action enjoying the unfree gauge symmetry is demonstrated to admit the alternative form of gauge symmetry with the higher order derivatives of unrestricted gauge parameters. The higher order gauge symmetry is reducible in general, even if the unfree symmetry is not. The relationship is detailed between these two forms of gauge symmetry in the constrained Hamiltonian formalism. The local map is shown to exist from the unfree gauge algebra to the reducible higher order one, while the inverse map is non-local, in general. The Hamiltonian BFV-BRST formalism is studied for both forms of the gauge symmetry. These two Hamiltonian formalisms are shown connected by canonical transformation involving the ghosts. The generating function is local for the transformation, though the transformation as such is not local, in general. Hence, these two local BRST complexes are not quasi-isomorphic in the sense that their local BRST-cohomology groups can be different. This difference in particular concerns the global conserved quantities. From the standpoint of the BRST complex for unfree gauge symmetry, these quantities are BRST-exact, while for the alternative complex, these quantities are the non-trivial co-cycles.

hep-th↗
arXiv2021

On the world sheet of continuous helicity particle

We consider the class of spinning particle theories, whose quantization corresponds to the continuous helicity representation of the Poincare group. The classical trajectories of the particle are shown to lie on the parabolic cylinder with a lightlike axis irrespectively to any specifics of the model. The space-time position of the cylinder is determined by the values of momentum and total angular momentum. The value of helicity determines the focal distance of parabolic cylinder. Assuming that all the world lines lying on one and the same cylinder are connected by gauge transformations, we derive the geometrical equations of motion for the particle. The timelike world paths are shown to be solutions to a single relation involving the invariants of trajectory up to fourth order in derivatives. Geometrical equation of motion is non-Lagragian, but it admits equivalent variational principle in the extended set of dynamical variables. The lightlike paths are also admissible on the cylinder, but they do not represent the classical trajectories of this spinning particle. The classical trajectories of massless particle (with zero helicity) are shown to lie on hyperplanes, whose spacetime position depends on momentum and total angular momentum.

hep-th↗
arXiv2021

Reducible Stueckelberg symmetry and dualities

We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins.

hep-th↗
arXiv2021

Extended Chern-Simons model for a vector multiplet

We consider a gauge theory of vector fields in $3d$ Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern-Simons (ECS) equations with higher derivatives. If the color index takes $n$ values, the third-order model admits a $2n$-parameter series of second-rank conserved tensors, which includes the canonical energy-momentum. Even though the canonical energy is unbounded, the other representatives in the series can have bounded from below $00$-component. The theory admits consistent self-interactions with the Yang-Mills gauge symmetry. The Lagrangian couplings preserve the unbounded from below energy-momentum tensor, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of conserved tensor with a bounded from below $00$-component. These models are stable at the non-linear level. The dynamics of interacting theory admits a constraint Hamiltonian form. The Hamiltonian density is given by the $00$-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. The particular attention is paid to the "triply massless" ECS theory, which demonstrates instability already at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize dynamics in the vicinity of the local minimum of energy. The equations of motion of stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a bounded from below Hamiltonian.

hep-th↗
arXiv2021

General method for including Stueckelberg fields

A systematic procedure is proposed for inclusion of Stueckelberg fields. The procedure begins with the involutive closure when the original Lagrangian equations are complemented by all the lower order consequences. The involutive closure can be viewed as Lagrangian analogue of complementing constrained Hamiltonian system with secondary constraints. The involutively closed form of the field equations allows for explicitly covariant degree of freedom number count, which is stable with respect to deformations. If the original Lagrangian equations are not involutive, the involutive closure will be a non-Lagrangian system. The Stueckelberg fields are assigned to all the consequences included into the involutive closure of the Lagrangian system. The iterative procedure is proposed for constructing the gauge invariant action functional involving Stueckelberg fields such that Lagrangian equations are equivalent to the involutive closure of the original theory. The generators of the Stueckelberg gauge symmetry begin with the operators generating the closure of original Lagrangian system. These operators are not assumed to be a generators of gauge symmetry of any part of the original action, nor are they supposed to form an on shell integrable distribution. With the most general closure generators, the consistent Stueckelberg gauge invariant theory is iteratively constructed, without obstructions at any stage. The Batalin-Vilkovisky form of inclusion the Stueckelberg fields is worked out and existence theorem for the Stueckelberg action is proven.

hep-th↗
arXiv2020

Hamiltonian constraints and unfree gauge symmetry

We study Hamiltonian form of unfree gauge symmetry where the gauge parameters have to obey differential equations. We consider the general case such that the Dirac-Bergmann algorithm does not necessarily terminate at secondary constraints, and tertiary and higher order constraints may arise. Given the involution relations for the first-class constraints of all generations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form, including the differential equations constraining gauge parameters. All the field theories with unfree gauge symmetry share the common feature: they admit sort of "global constants of motion" such that do not depend on the local degrees of freedom. The simplest example is the cosmological constant in the unimodular gravity. We consider these constants as modular parameters rather than conserved quantities. We provide a systematic way of identifying all the modular parameters. We demonstrate that the modular parameters contribute to the Hamiltonian constraints, while they are not explicitly involved in the action. The Hamiltonian analysis of the unfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, including explicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRST Hamiltonian quantization method for the case of unfree gauge symmetry. The main distinction is in the content of the non-minimal sector and gauge fixing procedure. The general formalism is exemplified by traceless tensor fields of irreducible spin $s$ with the gauge symmetry parameters obeying transversality equations.

hep-th↗
arXiv2020

Resonance and stability of higher derivative theories of derived type

We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives.

hep-th↗
arXiv2019

Unfree gauge symmetry in the Hamiltonian formalism

The constrained Hamiltonian formalism is worked out for the theories where the gauge symmetry parameters are unfree, being restricted by differential equations. The Hamiltonian BFV-BRST embedding is elaborated for this class of gauge theories. The general formalism is exemplified by the linearized unimodular gravity.

hep-th↗
arXiv2019

Stable interactions between higher derivative extended Chern-Simons and charged scalar field

We consider inclusion of interactions between the higher derivative extended Chern-Simons and charged scalar field. We demonstrate that the order $N$ extended Chern-Simons and order $2n$ charged scalar admit the $(N+n)$-parameter series of interaction vertices. The interactions are in general non-Lagrangian, but they preserve a certain conserved second-rank tensor, whose parameters are determined by the coupling. The $00$-component of this tensor can be bounded even if the canonical energy of the model is unbounded before the inclusion of interaction. If the $00$-component of conserved tensor is bounded, the theory is stable.

hep-th↗