Lagrangian formalism and Lie group approach for commutative semigroup of differential equations

非局域拉格朗日场与第二 Noether 定理：非对易 $U(1)$ 规范理论

Nonlocal Lagrangian fields and the second Noether theorem. Non-commutative $U(1)$ gauge theory

本文系统梳理非局域拉格朗日形式体系，扩展适用于非局域情形的第二 Noether 定理，并将其用于非对易 $U(1)$ 规范理论的哈密顿量与量子化分析。

Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations

A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of $n$-fold iterated integrals of some elementary functions as $n\to\infty$. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman--Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae gi…

Non-commutative Nash inequalities

A set of functional inequalities - called Nash inequalities - are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative Lp spaces, where their relationship to Poincare and log-Sobolev inequalities are fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.

Feynman's Path Integrals as Evolutionary Semigroups

We show that, for a class of systems described by a Lagrangian L(x,\dot{x},t) = 1/2\dot{x}^{2} - V(x,t) the propagator can be reduced via Noether's Theorem to a standard path integral multiplied by a phase factor. Using Henstock's integration technique, this path integral is given a firm mathematical basis. Finally, we recast the propagator as an evolutionary semigroup.

Transgression Forms and Abelian Semigroups in Supergravity

Two main themes populate this Thesis's pages: transgression forms as Lagrangians for gauge theories and the Abelian semigroup expansion of Lie algebras. A transgression form is a function of two gauge connections whose main property is its full invariance under gauge transformations. From this form a Lagrangian is built, and equations of motion, boundary conditions and associated Noether currents are derived. A subspace separation method, based on the extended Cartan homotopy formula, is proposed, which allows to (i) split the Lagrangian in 'bulk' and 'boundary' contributions and (ii) separate…

Quantum extensions of dynamical systems and of Markov semigroups

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in…