arXiv subjects

Simon Lyakhovich

Publications and source records attributed to Simon Lyakhovich.

6 recordsLinked to original sources

arXiv2025

Gauge symmetry and partially Lagrangian systems

We consider a classical field theory whose equations of motion follow from the least action principle, but the class of admissible trajectories is restricted by differential equations. The key element of the proposed construction is the complete gauge symmetry of these additional equations. The unfree variation of the trajectories reduces to the infinitesimal gauge symmetry transformation of the equations, restricting the trajectories. We explicitly derive the equations that follow from the requirement that this gauge variation of the action vanishes. The system of equations for conditional extrema is not a Lagrangian system as such, but it admits an equivalent Hamiltonian formulation with a non-canonical Poisson bracket. The bracket is degenerate, in general. Alternatively, the equations restricting the dynamics could be added to the action with Lagrange multipliers with unrestricted variation of the original variables. In this case, we would arrive at the Lagrangian equations for the original variables involving Lagrange multipliers and for Lagrange multipliers themselves. In general, these two methods are not equivalent because the multipliers can bring extra degrees of freedom compared to the case of equations derived by unfree variation of the action. We illustrate the general method with two examples. The first example is a particle in a central field with varying trajectories restricted by the equation of conservation of angular momentum. The phase space acquires one more dimension, and there is an extra conserved quantity $K$ which is responsible for the precession of trajectories. The second example is linearized gravity with the Einstein-Hilbert action, and the class of varying fields is restricted by the linearized Nordström equation. This conditional extrema problem is shown to lead to the linearized Cotton gravity equations.

math-ph↗

arXiv2025

Degree of freedom count in linear gauge invariant PDE systems

We consider not necessarily Lagrangian partial linear differential equations (PDE) with constant coefficients. Einstein proposed a definition of the"strength" of such a field theory that defines its degree of freedom (DoF). Einsteinian strength is based on the asymptotic number of free Taylor series coefficients of bounded degree in the general solution of the PDE system. The direct count of DoF in this way is a complex and technically demanding process. Proceeding from Einsteinian strength of equations and making use of commutative algebra tools, we deduce another DoF count recipe which is formulated in terms of orders of the field equations, their gauge symmetries and gauge identities. This DoF count recipe covers the case of reducible gauge field theories, and it is easy to use. We begin with interpreting the matrix of the system as a linear map between polynomial modules. First, proceeding from Einsteinian definition, we derive an explicit formula for DoF as the multiplicity of a certain extension module. Second, we prove another explicit formula for DoF in terms of orders of equations and gauge generators. A notable consequence of this formula is that two Hermitian conjugate systems have identical DoF. Every classical field theory defines the BRST complex which has the natural grading known as the ghost number. We equip this complex with another grading, which we call differential order. This grading is 1 for every space time-derivative, while all thefields, ghosts, and antifields are assigned with this degree in a certain way, depending on their ghost number and the orders of equations and gauge generators. We compute the Euler characteristic of the BRST complex with respect to this new grading. This provides homological interpretation of DoF for linear gauge system as minus the residue at infinity of the logarithmic derivative of the Euler characteristics for its BRST complex.

math-ph↗

arXiv2005

Non-Abelian Conversion and Quantization of Non-scalar Second-Class Constraints

We propose a general method for deformation quantization of any second-class constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a non-trivial vector bundle over the phase-space manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the ``constraint bundle'', which becomes a key ingredient of the conversion procedure for the non-scalar constraints. Unlike in the case of scalar second-class constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting non-scalar second-class constraints into non-Abelian first-class ones. The BRST-extended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of second-class systems with non-scalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative star-product on the phase space.

hep-th↗

arXiv2002

Poisson geometry of sigma models with extended supersymmetry

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.

hep-th↗

arXiv2002

Projection operator approach to general constrained systems

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.

hep-th↗

arXiv2001

Extended observables in theories with constraints

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.

hep-th↗