arXiv subjects

Wataru Ichinose

Publications and source records attributed to Wataru Ichinose.

6 recordsLinked to original sources

arXiv2023

From each of Feynman's and von Neumann's postulates to the restricted Feynman path integrals: a mathematical theory of temporally continuous quantum measurements

Feynman proposed a postulate or a method of quantization in his celebrated paper in 1948. Applying Feynman's postulate to temporally continuous quantum measurements of the positions of particles, Mensky proposed the restricted Feynman path integrals for continuous quantum measurements after phenomenological considerations. Our aim in the present paper is to give a rigorous proof that Mensky's restricted Feynman path integrals emerge out of the Feynman's postulate under a simple approximation. In addition, it is proved that the restricted Feynman path integrals emerge out of von Neumann's postulate on instantaneous measurements as well as Feynman's postulate. The quantum systems that we study include spin systems. These results are applied to formulations of the multi-split experiments, the quantum Zeno and the Aharanov-Bohm effects.

math-ph↗

arXiv2021

On the mathematical formulation of the restricted Feynman path integrals through broken line paths

The restricted Feynman path integrals (RFPIs) have been proposed to study continuous quantum measurements in physics. The RFPIs are heuristically determined in terms of the usual probability amplitude multiplied by weight for each path, which contains information about the results and the resolution of the measuring device. In the present paper we will consider the RFPIs particularly for the position measurements and will prove rigorously that these RFPIs are well defined in the $L^{2}$ space and are the solutions to the non-self-adjoint Schroedinger equations. Our results in the present paper give a generalization of the results on the usual Feynman path integrals for the Schroedinger equations.Furthermore, our results are extended to quantum spin systems.

math-ph↗

arXiv2019

On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential

The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials $(V,A_{1},A_{2},...,A_{d})$ such that $V(t,x) = |x|^{2(l+1)} + $`` a polynomial of degree $(2l + 1)$ in $x$ " ($l = 0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^2$-valued continuous functions with respect to the time variable.

math-ph↗

arXiv2017

On the Schrödinger equations with time-dependent potentials growing polynomially in the spatial direction

The Cauchy problem for the Schrödinger equations is studied with time-dependent potentials growing polynomially in the spatial direction. First the existence and the uniqueness of solutions are shown in the weighted Sobolev spaces. In addition, we suppose that our potentials are depending on a parameter. Secondly it is shown that if potentials depend continuously and differentiably on the parameter, the solutions to the Schrödinger equations respectively become continuous and differentiable with respect to its parameter.

math.AP↗

arXiv2017

Notes on the Feynman path integral for the Dirac equation

This paper is a continuation of the author's preceding one. In the preceding paper the author has rigorously constructed the Feynman path integral for the Dirac equation in the form of the sum-over-histories, satisfying the superposition principle, over all paths of one electron in space-time that goes in any direction at any speed, forward and backward in time with a finite number of turns. In the present paper, first we will generalize the results in the preceding paper and secondly prove in a direct way that our Feynman path integral satisfies the unitarity principle and the causality one.

math-ph↗

arXiv2008

Mathematical Remarks on the Feynman Path Integral for Nonrelativistic Quantum Electrodynamics

The Feynman path integral for nonrelativistic quantum electrodynamics is studied mathematically of a standard model in physics, where the electromagnetic potential is assumed to be periodic with respect to a large box and quantized thorough its Fourier coefficients. In physics, the Feynman path integral for nonrelativistic quantum electrodynamics is defined very formally. For example, as is often seen, even independent variables are not so clear. First, the Feynman path integral is defined rigorously under the constraints familiar in physics. Secondly, the Feynman path integral is also defined rigorously without the constraints, which is stated in Feynman and Hibbs' book without any comments. So, our definition may be completely new. Thirdly, the vacuum and the state of photons of momentums and polarization states are expressed by means of concrete functions of variables consisting of the Fourier coefficients of the electromagnetic potential. Our results above have many applications as is seen in Feynman and Hibbs' book, though the applications are not rigorous so far. It is also proved rigorously by means of the distribution theory that the Coulomb potentials between charged particles naturally appear in the Feynman path integral above. As is well known, this shows that photons give the Coulomb forth.

math-ph↗