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arXiv:1407.6272

The expectation value of the field operator

Dan Solomon

Published Jul 14, 2014• 0 citations
arXivHTMLPDF

Abstract

Much of the mathematical development of quantum field theory has been in support of determining the S-matrix in order to calculate scattering cross sections. However there is also an interest in determining how expectation values of field operators evolve in time from an initial state. In this paper I will examine some aspects of this problem.

DOI
10.48550/arxiv.1407.6272

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Much of the mathematical development of quantum field theory has been in support of determining the S-matrix in order to calculate scattering cross sections. However there is also an interest in determining how expectation values of field operators evolve in time from an initial state. In this paper I will examine some aspects of this problem.
We present our study on complete measurements of two-photon ionization in atomic rubidium. In this study we developed an effective technique to measure the angular distributions of photoelectrons. Through the measurements with this technique and using elliptically-polarized light we have successfully determined the atomic parameters—the relative cross sections for various ionization channels as well as the phase difference between s- and d-continuum waves. The measured phase differences are in excellent agreement with expected values over the entire range of photoelectron kinetic energy from 0 to 0.5 eV. The measured relative cross sections for ionization into S and D channels are also in good agreement with theoretical data. Both the relative cross sections and phase differences show a slow variation as a function of photoelectron energy. The ratios of cross sections of the two D channels are not in agreement with the expected value, suggesting fine structure effects. We also studied a quantum mechanical interference using two-color laser fields and observed the extremely asymmetric photoelectron angular distribution produced by interfering even- and odd-order photoionization processes in atomic rubidium as the optical phase was varied. We observed these effects with both perpendicularly and parallel-polarized fields components. It is worthy to note that the quantum interference observed with perpendicularly-polarized two-color fields is unique. Based on the analysis of measured angular distributions we derived the phase difference between even and odd continuum waves for the first time. The measured value of the phase difference is in good agreement with the calculated value from quantum defect theory.
This paper describes a computer model of Quantum Field Theory (QFT), referred to in this paper as QTModel. After specifying the initial configuration for a QFT process (e.g. scattering) the model generates the possible applicable processes in terms of Feynman diagrams, the equations for the scattering matrix, and evaluates probability amplitudes for the scattering matrix and cross sections. The computations of probability amplitudes are performed numerically. The equations generated by QTModel are provided for demonstration purposes only. They are not directly used as the base for the computations of probability amplitudes. The computer model supports two modes for the computation of the probability amplitudes: (1) computation according to standard QFT, and (2) computation according to a proposed functional interpretation of quantum theory.
Abstract In this paper are introduced several novel techniques having as their objects, first, the gain of a new understanding of the divergence difficulties of orthodox quantum theory, and secondly, the systematic development of the quantum mechanics of fields in terms of the density matrix. There is included a new presentation of a form of perturbation theory which has the outstanding advantages over the usual one that it is much more quickly convergent, and leaves the density matrix normalized at every stage so that the diagonal elements representing occupational probabilities cannot diverge. Exact general formulae are derived for self-energies and cross-sections for the purpose of examining their convergence properties. The general theory of fields is developed abinitio, and it is shown that the use of the density matrix in place of the wave vector illuminates and simplifies the customary theory. The second quantization of the density matrix which follows throws unexpected light on the existence of particles with negative expectation values. Finally, the whole theory is applied to the crucial example of the electron or positron in an electromagnetic field. It is confirmed that the self-energy of the electron in the orthodox theory cannot be made finite without the introduction of negative energy photons.