Hamilton-Jacobi Theory in k-Symplectic Field Theories

In this paper we extend the geometric formalism of Hamilton-Jacobi theory for Mechanics to the case of classical field theories in the k-symplectic framework

Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds

A description of time-dependent Mechanics in terms of Lagrangian submanifolds of Dirac manifolds (in particular, presymplectic and Poisson manifolds) is presented. Two new Tulczyjew triples are discussed. The first one is adapted to the restricted Hamiltonian formalism and the second one is adapted to the extended Hamiltonian formalism

Time-of-arrival formalism for the relativistic particle

A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in 3+1 dimensions. For each detector position, there exists a subspace of detected states in the Hilbert space of solutions to the Klein Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.Comment: 16 pages, no figures, uses LaTeX. The section "Interpretation" has been completely rewritten and some errors correcte

Hamiltonian Dynamics of Linearly Polarized Gowdy Models Coupled to Massless Scalar Fields

The purpose of this paper is to analyze in detail the Hamiltonian formulation for the compact Gowdy models coupled to massless scalar fields as a necessary first step towards their quantization. We will pay special attention to the coupling of matter and those features that arise for the three-handle and three-sphere topologies that are not present in the well studied three torus case -in particular the polar constraints that come from the regularity conditions on the metric. As a byproduct of our analysis we will get an alternative understanding, within the Hamiltonian framework, of the appearance of initial and final singularities for these models.Comment: Final version to appear in Classical and Quantum Gravit

Nonholonomic constraints in kk-symplectic Classical Field Theories

A $k$-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.Comment: 27 page

PT-symmetry from Lindblad dynamics in a linearized optomechanical system

We analyze a lossy linearized optomechanical system in the red-detuned regime under the rotating wave approximation. This so-called optomechanical state transfer protocol provides effective lossy frequency converter (quantum beam-splitter-like) dynamics where the strength of the coupling between the electromagnetic and mechanical modes is controlled by the optical steady-state amplitude. By restricting to a subspace with no losses, we argue that the transition from mode-hybridization in the strong coupling regime to the damped-dynamics in the weak coupling regime, is a signature of the passive parity-time (PT) symmetry breaking transition in the underlying non-Hermitian quantum dimer. We compare the dynamics generated by the quantum open system (Langevin or Lindblad) approach to that of the PT-symmetric Hamiltonian, to characterize the cases where the two are identical. Additionally, we numerically explore the evolution of separable and correlated number states at zero temperature as well as thermal initial state evolution at room temperature. Our results provide a pathway for realizing non-Hermitian Hamiltonians in optomechanical systems at a quantum level

Unitarity violation in non-abelian Pauli-Villars regularization

We regularize QCD using the combination of higher covariant derivatives and Pauli-Villars determinants proposed by Slavnov. It is known that for pure Yang-Mills theory the Pauli-Villars determinants generate unphysical logarithmic radiative corrections at one loop that modify the beta function. Here we prove that when the gauge fields are coupled to fermions so that one has QCD, these unphysical corrections translate into a violation of unitarity. We provide an understanding of this by seeing that Slavnov's choice for the Pauli-Villars determinants introduces extra propagating degrees of freedom that are responsible for the unitarity breaking. This shows that Slavnov's regularization violates unitarity, hence that it should be rejected

On the Hamilton-Jacobi Theory for Singular Lagrangian Systems

We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.Comment: 36 page