In Hamiltonian mechanics the equations of motion may be considered as a condition on the tangent vectors to the solution; they should be null-vectors of the symplictic structure. Usually the formalism for the field case is done by replacing the finite dimensional configuration space by an infinite dimensional one. In the present paper we work in worldsheet-targetspace formalism. The null-vectors of symplectic 2-form are replaced by null-polyvectors of a higher rank form on a finite dimensional manifold. The action in this case is an integral of a differential form over a surface in phase space. The method to obtain such a description from the Lagrange formalism generalizes the Legendre transformation. The requirement for this transformation to preserve the value of the action and its extremality leads to a natural definition of this procedure.
We study Lagrangian systems with a finite number of degrees of freedom that are non-local in time. We obtain an extension of Noether theorem and Noether identities to this kind of Lagrangians. A Hamiltonian formalism is then set up for this systems. $n$-order local Lagrangians can be treated as a particular case and the standard results for them are recovered. The method is then applied to several other cases, namely two examples of non-local oscillators and the p-adic particle.
This article aims to study non-local Lagrangians with an infinite number of degrees of freedom. We obtain an extension of Noether's theorem and Noether's identities for such Lagrangians. We then set up a Hamiltonian formalism for them. In addition, we show that $n$-order local Lagrangians can be treated as a particular case and the standard results can be recovered. Finally, this formalism is applied to the case of $p$-adic open string field.
The Dirac's formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant and we calculate the Feynman propagator using the extended phase description. We show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space.
In recent years, the non-relativistic quantum dynamics derived from three assumptions; i) probability current conservation, ii) average energy conservation, and iii) an epistemic momentum uncertainty [A. Budiyono and D. Rohrlich, \href{https://doi.org/10.1038/s41467-017-01375-w}{Nat. Commun 8, 1306 (2017)}]. Here we show that, these assumptions can be derived from a natural extension of classical statistical mechanics.
