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Mathematics

Canonical Commutation Relation Derived from Witt Algebra

Huber Nieto–Chaupis

Published Jun 7, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.3390/math13121910

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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.
Despite the fact that it has been known since the time of Heisenberg that quantum operators obey a quantum version of Newton's laws, students are often told that derivations of quantum mechanics must necessarily follow from the Hamiltonian or Lagrangian formulations of mechanics. Here, we first derive the existing Heisenberg equations of motion from Newton's laws and the uncertainty principle using only the equations F=dPdtF=\frac{dP}{dt}F=dtdP​, P=mdVdtP=m\frac{dV}{dt}P=mdtdV​, and [X,P]=i\left[X,P\right]=i[X,P]=i. Then, a new expression for the propagator is derived that makes a connection between time evolution in quantum mechanics and the motion of a classical particle under Newton's laws. The propagator is solved for three cases where an exact solution is possible 1) the free particle 2) the harmonic oscillator 3) a constant force, or linear potential in the standard interpretation. Such a picture may be useful for students as they make the transition from classical to quantum mechanics and help solidify the equivalence of the Hamiltonian, Lagrangian, and Newtonian formulations of physics in their minds.
Abstract In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.
The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
The complete mathematical structure of quantum mechanics is derived from first principles, tracing all fundamental results to the constraint axiom via octonionic algebra. Without postulates, empirical fitting, or measurement assumptions, the following results are established: (1) Quantum states must reside in a complex Hilbert space over the complex numbers. (2) The canonical commutation relation between position and momentum follows from octonionic non-commutativity together with holographic information bounds. (3) The Heisenberg uncertainty principle arises as a mathematical theorem from the Cauchy–Schwarz inequality. (4) The Schrödinger equation emerges from time translation symmetry. (5) Spin one-half arises from the structure of the relevant Clifford algebra. (6) The Pauli matrices originate from the quaternion subalgebra contained within the octonions. (7) The Born rule appears as the unique probability measure on complex projective space. Related resourcesAdditional preprints, theoretical frameworks, and ongoing work by the author are available at:https://murad-ahmadov.github.io/
We show that quantum theory (QT) is a substructure of classical probabilistic physics. The central quantity of the classical theory is Hamilton’s function, which determines canonical equations, a corresponding flow, and a Liouville equation for a probability density. We extend this theory in two respects: (1) The same structure is defined for arbitrary observables. Thus, we have all of the above entities generated not only by Hamilton’s function but also by every observable. (2) We introduce for each observable a phase space function representing the classical action. This is a redundant quantity in a classical context but indispensable for the transition to QT. The basic equations of the resulting theory take a “quantum-like” form, which allows for a simple derivation of QT by means of a projection to configuration space reported previously [Quantum Stud Math Found 5:219–227, 2018]. We obtain the most important relations of QT, namely the form of operators, Schrödinger’s equation, eigenvalue equations, commutation relations, expectation values, and Born’s rule. Implications for the interpretation of QT are discussed, as well as an alternative projection method allowing for a derivation of spin.