[Paper Review] Classical/Quantum=Commutative/Noncommutative?

VV Kisil|arXiv (Cornell University)|Apr 9, 2012

Quantum Mechanics and Applications23 references3 citations

TL;DR

This paper challenges the foundational idea that quantum mechanics fundamentally arises from noncommutativity, arguing instead that classical mechanics can be formulated using noncommutative structures via the nilpotent unit ε (where ε² = 0) and a non-zero Planck constant. It demonstrates that noncommutativity is not essential to quantum theory, as shown by the path integral formulation's independence from commutator relations, and repositions complex numbers as the central algebraic feature of quantum mechanics.

ABSTRACT

In 1926, Dirac stated that quantum mechanics can be obtained from classical theory through a change in the only rule. In his view, classical mechanics is formulated through commutative quantities (c-numbers) while quantum mechanics requires noncommutative one (q-numbers). The rest of theory can be unchanged. In this paper we critically review Dirac's proposition. We provide a natural formulation of classical mechanics through noncommutative quantities with a non-zero Planck constant. This is done with the help of the nilpotent unit, which squares to zero. Thus, the crucial rôle in quantum theory shall be attributed to the usage of complex numbers. The paper provides English and Russian versions.

Motivation & Objective

• To critically re-examine Dirac's claim that quantum mechanics arises from replacing commutative c-numbers with noncommutative q-numbers.
• To demonstrate that classical mechanics can be consistently formulated using noncommutative structures, even with a non-zero Planck constant.
• To argue that noncommutativity is not essential to quantum theory, using the path integral formulation as a counterexample.
• To reframe the foundational role in quantum mechanics from noncommutativity to the use of complex numbers.
• To provide an alternative mathematical foundation for quantum theory that avoids dimensional inconsistencies in observable algebra.

Proposed method

• Introduces the nilpotent unit ε (ε² = 0) to construct a noncommutative algebra for classical mechanics with a non-zero Planck constant.
• Uses the algebraic framework of rings and algebras to formalize observables, avoiding dimensional inconsistencies in addition.
• Analyzes the path integral formulation (Feynman, 1965) as a noncommutativity-free route to quantum mechanics.
• Highlights that key quantum texts like Feynman’s QED and path integral lectures do not rely on commutator relations.
• Reinterprets quantum theory through complex numbers as the central algebraic structure, rather than noncommutativity.
• Constructs a classical theory using ε-algebras that mimics quantum behavior without invoking noncommutativity.

Experimental results

Research questions

• RQ1 Can classical mechanics be formulated using noncommutative structures without relying on the standard c-number framework?
• RQ2 Is noncommutativity truly essential to quantum mechanics, or can quantum theory be constructed without it?
• RQ3 What is the role of complex numbers in quantum theory compared to noncommutativity?
• RQ4 How does the path integral formulation challenge the idea that noncommutativity defines quantum mechanics?
• RQ5 Can dimensional consistency in observable algebra be preserved in a noncommutative classical framework?

Key findings

• Classical mechanics can be consistently formulated using noncommutative structures via the nilpotent unit ε with ε² = 0, even with a non-zero Planck constant.
• The path integral formulation of quantum mechanics does not require noncommutativity, as it is not mentioned in key texts like Feynman’s QED or his path integral lectures.
• The assumption that all observables can be added is physically inconsistent due to dimensional incompatibility, undermining the algebraic foundation of Dirac’s c-number/q-number dichotomy.
• Noncommutativity is not a necessary feature of quantum theory, as demonstrated by formulations that bypass commutator relations entirely.
• The central algebraic feature of quantum mechanics is not noncommutativity but the use of complex numbers, which should be re-evaluated as the true foundation.
• The paper provides a mathematically consistent classical theory using ε-algebras that exhibits quantum-like behavior without noncommutativity.