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arXiv:2407.14965

Commutation relations for functions of canonical conjugate operators

Conrado Badenas

Published Jul 20, 2024• 0 citations
arXivHTMLPDF

Abstract

In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.

DOI
10.48550/arxiv.2407.14965

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In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.
We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.
The position operator in the Heisenberg representation obeys the usual classical equation of motion. The similarity between the Heisenberg equations of motion and the classical Hamiltonian equations of motion arises from the fact that the commutator of the operators for the physical quantities plays a similar role in quantum mechanics, to that of the Poisson bracket of the classical quantities A and B, in classical mechanics. Generally, the Heisenberg equations of motion are more difficult to solve than the corresponding classical equations because of the lack of commutation of quantum mechanical operators. The chapter explains the notion of a time-ordered product of operators, and then can be written much more compactly.
From an arbitrary definition of operators inspired by oscillators of Virasoro, an algebra is derived. It fits the structure of Virasoro algebra with null central charge or Witt algebra. The resulting formalism has yielded commutators with a dependence on integer numbers, and it follows the Witt-like algebra. Also, the quantum mechanics evolution operator for the case of the quantum harmonic oscillator was identified. Furthermore, the Schrödinger equation was systematically derived under the present framework. When operators are expressed in the framework of Hilbert space states, the resulting Witt algebra seems to be proportional to the well-known canonical commutation relation. This has demanded the development of a formalism based on arbitrary and physical operators as well as well-defined rules of commutation. The Witt-like was also redefined through the direct usage of the uncertainty principle. The results of the paper might suggest that Witt algebra encloses not only quantum mechanics’ fundamental commutator but also other unexplored relations among quantum mechanics observables and Witt algebra.