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Paradox Integral - Generalized Version of Feynman’s Integral

Pavel Cherkashin

Published Jun 21, 2023• 0 citations
Publisher

Abstract

The Paradox Integral is a mathematical tool that can be used to calculate the probability of different paths that a system could take. It is a generalization of Feynman's path integral, which is used in quantum mechanics to calculate the probability of different outcomes of an experiment.The Paradox Integral is based on the principle of cross-dimensional dependence, which states that the largest exponent is the smallest coefficient. This means that the destination can be derived from the initial conditions, and the answer can be derived from the question.The Paradox Integral can be used to identify voids of knowledge and build sufficient pathways with their associated probabilities. This can be used to solve problems and make predictions.

DOI
10.31219/osf.io/qvxw7

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The Paradox Integral is a mathematical tool that can be used to calculate the probability of different paths that a system could take. It is a generalization of Feynman's path integral, which is used in quantum mechanics to calculate the probability of different outcomes of an experiment.The Paradox Integral is based on the principle of cross-dimensional dependence, which states that the largest exponent is the smallest coefficient. This means that the destination can be derived from the initial conditions, and the answer can be derived from the question.The Paradox Integral can be used to identify voids of knowledge and build sufficient pathways with their associated probabilities. This can be used to solve problems and make predictions.
A path , in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable . In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)]. For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e). Probability amplitude and time evolution Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.
The definition of Feynman path integrals (Feynman functional integrals) as integrals with respect to a generalized measure, called the Lebesgue–Feynman measure in the paper and being an infinite-dimensional analogue of the classical Lebesgue measure on finite-dimensional Euclidean space, is discussed. This definition, which is a formalization of Feynman’s original definition, is different from those used previously in the mathematical literature. It makes it possible to give a description of the origin of quantum anomaly which is a mathematically correct version of the description given in the book Path Integrals and Quantum Anomalies by K. Fujikawa and H. Suzuki (Oxford, 2004) (and erroneously qualified as wrong in the book Functional Integration: Action and Symmetries by P. Cartier and C. DeWitt-Morette (Cambridge Univ. Press, Cambridge, 2006)).
The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism. It is a generalization of the classical least action principle. The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state. Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral. We look in to the subject of two kinds of slit-experiments: The square slit, the single- and the double-Gaussian slit. Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics. It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral. Further applications of the path integral are discussed.
The Feynman path integral is defined over the space RT\mathbb{R}^TRT of all possible paths; it has been a powerful tool to develop Quantum Mechanics. The absolute value of Feynman's integrand is not integrable, then Lebesgue integration theory could not be used by Feynman. However, it exists formally as a Henstock integral (which does not require the measure concept) and is a suitable alternative to the ordinary integrals that normally appear in path integrals. Feynman proved the equivalence of his theory with the traditional formulation of Quantum Mechanics, since his path integral satisfies Schrödinger's equation. On the other hand, Feynman's path integral is related to the diagrams of Feynman. For the application of this integral in Feynman's diagrams it is necessary to exchange the integral ∫RT\int_{\mathbb{R}^T}∫RT​ and the series. We discuss the impossibility to exchange the integral and the sum, considering integral of Henstock and the version of Dominated Convergence Theorem. Even it has not been proved through the several mathematical formalisms that have been used.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.