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IEEE International Conference on Artificial Intelligence and Knowledge Engineering (AIKE)

The Quantum Mechanics Propagator as the Machine Learning Performance in Space-Time Displacements

Huber Nieto–Chaupis

Published Dec 1, 2021• 1 citation
Publisher

Abstract

The role of evolution operator is to provide the time displacement of wave function through the Hamiltonian of the system. The usage of coordinates representation gives the well-known propagator that is the Green’s function. In this paper it is emphasized that once the propagator is projected onto a scenario of machine learning it would acquire the role of performance in according to the criteria of Tom Mitchell. In this manner from the resulting wave function the probability is simulated presenting noteworthy morphologies in the which the system displays high values of probability for the measurement of distances.

DOI
10.1109/aike52691.2021.00029

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The role of evolution operator is to provide the time displacement of wave function through the Hamiltonian of the system. The usage of coordinates representation gives the well-known propagator that is the Green’s function. In this paper it is emphasized that once the propagator is projected onto a scenario of machine learning it would acquire the role of performance in according to the criteria of Tom Mitchell. In this manner from the resulting wave function the probability is simulated presenting noteworthy morphologies in the which the system displays high values of probability for the measurement of distances.
The goal of this paper is the presentation of the elementary procedures that normally are done in nonrelativistic Quantum Mechanics in terms of the principles of Machine Learning. In essence, this paper discusses Mitchell’s criteria, whose block fundamental dictates that the universal evolution of any system is composed by three fundamental steps: (i) Task, (ii) Performance and (iii) Experience. In this paper, the quantum mechanics formalism reflected on the usage of evolution operator and Green’s function are assumed to be part of mechanisms that are inherently engaged to the Machine Learning philosophy. The action for measuring observables through experiments and the intrinsic apparition of statistical or systematic errors are discussed in terms of “quantum learning”.
We present a theory of Machine Learning based entirely on the formalism of Quantum Mechanics from the fact that the diverse instances on the application of the algorithms would contain certain concepts linked to stochastic. In this manner, the probabilistic formalism of the Quantum Mechanics might be well applied. Thus, we implement the Mitchell's criteria with mathematical methodologies based on the Hilbert's space as well as the employment of quantum operators to describe the behavior of the experience in terms of probabilities. We illustrate the application of this theory through a quantitative analysis of the time evolution of the experience.
A theory consisting in quantum mechanics and theorem of Bayes, is presented. In essence, the Bayes probability has been built from two subspaces. While in one some quantum measurements are done, in the another it is seen that the probabilities acquire their highest values. Thus, the validity of a prior probability makes sense is there is a clear difference between the done measurement of probability amplitude. Thus, the principles of machine learning compacted in the criteria of Tom Mitchell have been employed. The simulations have shown that the size of space has direct impact on the prior probability that presumably would get low values of probability in a limited subspace. These values have turned out to be strongly correlated to the times in which measurements are done in a big space. Therefore, it is evident the prospective applicability of this novel approach in all those scenarios that require of a quantum measurement in separated times.
In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schr\"odinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
This paper proposes a hybrid model of machine learning, based at the principles of quantum mechanics in conjunction to Bayes theorem. With such construction, it is tested onto a scenario dictated by the Mitchell criteria. By assuming stochastic system whose main dependence is through time, the proposed model exhibits a high experience probability just at the beginning of processes. The results of this paper would corroborate the hypothesis that stochastic system might be strongly dependent at the very beginning or initial conditions.