• Corpus ID: 208853625

# Classical/Quantum=Commutative/Noncommutative?

@article{Kisil2012ClassicalQuantumCommutativeNoncommut,
title={Classical/Quantum=Commutative/Noncommutative?},
author={V. W. Kisil'},
journal={arXiv: Quantum Physics},
year={2012}
}
• V. Kisil'
• Published 9 April 2012
• Physics
• arXiv: Quantum Physics
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…
4 Citations
Symmetry, Geometry and Quantization with Hypercomplex Numbers
These notes describe some links between the group SL2(R), the Heisenberg group and hypercomplex numbers - complex, dual and double numbers. Relations between quantum and classical mechanics are
Deterministic Actions on Stochastic Ensembles of Particles Can Replicate Wavelike Behaviour of Quantum Mechanics: Does It Matter?
• Physics
• 2018
A synthesis of ideas appeared from the field's founders to modern contextual approaches is proposed, which demonstrates that deterministic physical principles can replicate wavelike probabilistic effects when applied to stochastic ensembles of particles.
An extension of Mobius--Lie geometry with conformal ensembles of cycles and its implementation in a GiNaC library
• V. Kisil
• Mathematics
Proceedings of the International Geometry Center
• 2019
A method is described, which reduces a collection of conformally in\-vari\-ant geometric relations to a system of linear equations, which may be accompanied by one fixed quadratic relation, which operates with numeric and symbolic data of cycles in spaces of arbitrary dimensionality and metrics with any signatures.
Induced Representations and Hypercomplex Numbers
In the search for hypercomplex analytic functions on the halfplane, we review the construction of induced representations of the group $${G = {\rm SL}_2(\mathbb{R})}$$ . Firstly we note that G-action

## References

SHOWING 1-10 OF 32 REFERENCES
On the Theory of quantum mechanics
The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but
DEFORMATION QUANTIZATION: QUANTUM MECHANICS LIVES AND WORKS IN PHASE-SPACE
• C. Zachos
• Physics
International Journal of Modern Physics A
• 2002
Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear
Contextual Approach to Quantum Formalism
The aim of this book is to show that the probabilistic formalisms of classical statistical mechanics and quantum mechanics can be unified on the basis of a general contextual probabilistic model. By
Hypercomplex Representations of the Heisenberg Group and Mechanics
In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework,
p-mechanics as a physical theory: an introduction
This paper provides an introduction to p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. p-mechanics naturally provides a
Quantum Mechanics and a Preliminary Investigation of the Hydrogen Atom
Although the classical electrodynamic theory meets with a considerable amount of success in the description of many atomic phenomena, it fails completely on certain fundamental points. It has long
Quantum Spaces and Their Noncommutative Topology
N oncommutative geometry studies the geometry of " quantum spaces ". Put a little more prosaically, this means the " geometric properties " of noncommu-tative algebras (say, over the field C of
Erlangen Programme at Large 3.2: Ladder Operators in Hypercomplex Mechanics
We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator
Mathematical Foundations of Quantum Mechanics I
We have already used average values of position and momentum of a particle and seen that we can get the average value of an observable F [represented by an operator function \( \hat F\left( {\hat