Structure of unital 3-fields

We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring $mathcal{R}$ having $mathbb{Z}diagup 2mathbb{Z}$ as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of $mathcal{R}$. We include a number of illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.

Conformal inversion and Maxwell field invariants in four- and six-dimensional spacetimes

Conformally compactified (3+1)-dimensional Minkowski spacetime may be identified with the projective light cone in (4+2)-dimensional spacetime. In the latter spacetime the special conformal group acts via rotations and boosts, and conformal inversion acts via reflection in a single coordinate. Hexaspherical coordinates facilitate dimensional reduction of Maxwell electromagnetic field strength tensors to (3+1) from (4 + 2) dimensions. Here we focus on the operation of conformal inversion in different coordinatizations, and write some useful equations. We then write a conformal invariant and a pseudo-invariant in terms of field strengths; the pseudo-invariant in (4+2) dimensions takes a new form. Our results advance the study of general nonlinear conformal-invariant electrodynamics based on nonlinear constitutive equations.

Formulation of singular theories in a partial Hamiltonian formalism using a new bracket and multi-time dynamics

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space. In this case the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.

Partial Hamiltonian formalism, multi-time dynamics and singular theories

We formulate singular classical theories without involving constraints. Applying the action principle for the action (27) we develop a partial (in the sense that not all velocities are transformed to momenta) Hamiltonian formalism in the initially reduced phase space (with the canonical coordinates $q_{i},p_{i}$, where the number $n_{p}$ of momenta $p_{i}$, $i=1,...,n_{p}$ (17) is arbitrary $n_{p}leq n$, where $n$ is the dimension of the configuration space), in terms of the partial Hamiltonian $H_{0}$ (18) and $(n-n_{p})$ additional Hamiltonians $H_{alpha}$, $alpha=n_{p}+1,...,n$ (20). We obtain $(n-n_{p}+1)$ Hamilton-Jacobi equations (25)-(26). The equations of motion are first order differential equations (33)-(34) with respect to $q_{i},p_{i}$ and second order differential equations (35) for $q_{alpha}$. If $H_{0}$, $H_{alpha}$ do not depend on $dot{q}_{alpha}$ (42), then the second order differential equations (35) become algebraic equations (43) with respect to $dot{q}_{alpha}$. We interpret $q_{alpha}$ as additional times by (45), and arrive at a multi-time dynamics. The above independence is satisfied in singular theories and $r_{W}leq n_{p}$ (58), where $r_{W}$ is the Hessian rank. If $n_{p}=r_{W}$, then there are no constraints. A classification of the singular theories is given by analyzing system (62) in terms of $F_{alphabeta}$ (63). If its rank is full, then we can solve the system (62); if not, some of $dot{q}_{alpha}$ remain arbitrary (sign of a gauge theory). We define new antisymmetric brackets (69) and (80) and present the equations of motion in the Hamilton-like form, (67)-(68) and (81)-(82) respectively. The origin of the Dirac constraints in our framework is shown: if we define extra momenta $p_{alpha}$ by (86), then we obtain the standard primary constraints (87), and the new brackets transform to the Dirac bracket. Quantization is discussed.

Generalized interaction in multigravity

A general approach to description of multigravity models in D-dimensional space-time is presented. Different possibilities of generalization of the invariant volume are given. Then a most general form of the interaction potential is constructed, which for bigravity coincides with the Pauli-Fierz model. A thorough analysis of the model along the 3+1 expansion formalism is done. It is shown that the absence of ghosts the considered bigravity model is equivalent in the weak field limit to the massive gravity (the Pauli-Fierz model). Thus, on the concrete example it is shown, that the interaction between metrics leads to nonvanishing mass of graviton.

A new Hamiltonian formalism for singular Lagrangian theories

We introduce a version of the Hamiltonian formalism based on the Clairaut equation theory, which allows us a self-consistent description of systems with degenerate (or singular) Lagrangian. A generalization of the Legendre transform to the case, when the Hessian is zero is done using the mixed (envelope/general) solutions of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the initial Lagrange equations, but contains nondynamical momenta and unresolved velocities. This system is reduced to the physical phase space and presented in the Hamiltonian form by introducing a new (non-Lie) bracket.

Constrained systems and the Clairaut equation

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to systems with constraints, the procedure of finding a Hamiltonian for a degenerate Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairaut transformation. In this way the unconstrained version of Hamiltonian equations is obtained. The Legendre-Clairaut transformation presented is involutive. We demonstrate the origin of the Dirac primary constraints, along with their explicit form, and this is done without using the Lagrange multiplier method.

Quantum enveloping algebras with von Neumann regular Cartan-like generators and the Pierce decomposition

Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.

Generalizations of Nonlinear and Supersymmetric Classical Electrodynamics

We first write down a very general description of nonlinear classical electrodynamics, making use of generalized constitutive equations and constitutive tensors. Our approach includes non-Lagrangian as well as Lagrangian theories, allows for electromagnetic fields in the widest possible variety of media (anisotropic, piroelectric, chiral and ferromagnetic), and accommodates the incorporation of nonlocal effects. We formulate electric-magnetic duality in terms of the constitutive tensors. We then propose a supersymmetric version of the general constitutive equations, in a superfield approach.