arXiv subjects

Dikanaina Harrivel

Publications and source records attributed to Dikanaina Harrivel.

5 recordsLinked to original sources

arXiv2008

First integrals for non linear hyperbolic equations

Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct formally these series by mean of generating functions based on an algebraic framework inspired by the construction of Fock spaces in quantum field theory. Then we build an analytic setting in which all these constructions really make sense and give rise to convergent series.

math.AP↗

arXiv2006

Perturbative classical and quantum field theory

In a first part we study the phi^{p+1}--field theory from the classical point of view. Using Butcher series we compute explicitly the perturbative expansion of the solutions and we prove that this expansion converges if the coupling constant is small enough. Then we show that we can formally recover the Heisenberg's interacting quantum field directly from this expansion. In other words we write the Heisenberg interacting field as a Butcher series.

math-ph↗

arXiv2006

Covariant Deformation Quantization of Free Fields

We define covariantly a deformation of a given algebra, then we will see how it can be related to a deformation quantization of a class of observables in Quantum Field Theory. Then we will investigate the operator order related to this deformation quantization.

math-ph↗

arXiv2006

Butcher series and control theory

We show how solutions of a non--linear differential equation can be written as sum indexed by planar trees: the Butcher series. Then we use that property in order to control non--linear differential equation. We show that if the linearized system is controllable then the system itself is controllable if the nonlinear term is small enough and we express explicitly the control as a sum indexed by planar tree which each terms is obtained by minimization of a functional.

math.OC↗

arXiv2005

Planar Binary Trees and Perturbative Calculus of Observables in Classical Field Theory

Consider the Klein-Gordon equation coupled with an interaction term $(\Box+m^2)ϕ+λψ^p$. For the linear Klein-Gordon equation, a kind of generalized Noether's theorem gives us a conserved quantity. The purpose of this paper is to find an analogue of this conserved quantity in the interacting case. We see that it can be done perturbatively, and we define explicitely a conserved quantity using a perturbative expansion based on Planar Binary Tree and a kind of Feynman rules. Only the case $p=2$ is treated but our approach can be generalized to any $ϕ^p$-theory.

math.AP↗