Classical/Quantum=Commutative/Noncommutative?

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 r^ole in quantum theory shall be attributed to the usage of complex numbers. The paper provides English and Russian versions.

💡 Research Summary

This paper presents a critical re-examination of Paul Dirac’s famous proposition that quantum mechanics can be derived from classical mechanics by a single change: replacing commutative quantities (c-numbers) with noncommutative ones (q-numbers). The author, Vladimir V. Kisil, argues that this “quantum = noncommutative” dogma is an oversimplification and offers a refined perspective on the mathematical foundations of both theories.

The analysis begins by highlighting conceptual problems in Dirac’s algebraic approach. The idea that all observables form an algebra where addition is always possible is physically unsound, as it ignores dimensional analysis (e.g., adding position and momentum is meaningless). The author then notes that noncommutativity is not an indispensable axiom for quantum theory, citing Feynman’s path integral formulation, which successfully derives quantum phenomena without initially invoking operator noncommutativity. This suggests the core ingredients are instead the non-zero Planck constant (ħ) and the imaginary unit (i).

The paper then reviews the standard group-theoretic derivation of quantum mechanics from the unitary irreducible representations (UIRs) of the Heisenberg group. The Stone-von Neumann theorem guarantees that all such representations with a given ħ are equivalent, leading to the Heisenberg commutation relation and the Weyl quantization scheme.

The central and novel contribution of the paper is the construction of a classical counterpart using the same Heisenberg group. The author defines a non-unitary representation, ρ_εh, of the Heisenberg group acting on a space of functions valued in the algebra of dual numbers. This algebra is generated by a nilpotent unit ε, where ε² = 0, in contrast to the complex unit i (i² = -1) used in the quantum case. This representation is induced by a dual-number-valued character of the group’s center.

Remarkably, the infinitesimal generators of this classical representation are also noncommutative, satisfying a modified Heisenberg relation: