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 Diracs 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 ϵ such that ϵ2 = 0. Thus, the crucial role in quantum theory shall be attributed to the usage of complex numbers.
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 Diracs 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 ϵ such that ϵ2 = 0. Thus, the crucial role in quantum theory shall be attributed to the usage of complex numbers.
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.
The purpose of the article is to show that quantum mechanics is a synthesis of the old classical concepts and the new non-classical concepts (physics and mathematics). To illustrate the relationship of continuity between classical physics and quantum mechanics, the materials of Planck, Broglie, Schrodinger, Bohr and Dirac are considered. The author begins with a comparative study of the fundamental principles of classical and quantum physics, and concludes that, despite their opposition and irreducibility, scientists tried to adopt the principles of classical physics for constructing quantum mechanics. Survey of Planck's papers showed that Planck's formula for the blackbody spectrum containing the quantum of action is obtained as a synthesis of classical concepts and quantum concepts. Survey of Broglie's papers showed that the quantum-mechanical concept of Broglie emerged as a synthesis of the provisions of the classical (variation principles) and quantum physics (Planck's constant). Survey of Schrodinger's papers showed great generality of quantum theory in comparison with classical mechanics. Survey of Bohr's papers showed that Bohr was able to prove the basic concepts of quantum physics with the help of methodological principles. Survey of Dirac's papers showed that Dirac discovered the equivalence of Schrodinger's and Heisenberg's theories, showed that they go back to the same physical and algebraic principles, and that they can be formulated as variants of the quantum-mechanical system. The author considers the methodological features of quantum mechanics and fixes a number of methodological principles by which quantum mechanics is significantly different from classical physics. The category of quantum mechanics, on which an attempt to reconstruct the physical picture of the world based on quantum theory, is briefly discussed. On the basis of this reconstruction, there are the world picture broad categories: mass-field and space-time. The categories are related by the formula: wave-like particle moves in the classical space-time background. The first association the particles and interactions is from quantum mechanics, the second space and time from the special theory of relativity. In conclusion, a list of basic relations of quantum mechanics with classical physics is made. The development of quantum ideas resulted from direct influence of classical physics, so the main heuristic techniques to construct quantum theory are two ideas, namely, the idea of continuity and the idea of conformity which Bohr's methodological principles express. At the same time, quantum theory requires the use of classical concepts for support and verification. This is due to the restrictions imposed by the experimental technique. Thus, the author claims that quantum mechanics is a synthesis of the old classical concepts and the new non-classical concepts (physics and mathematics).
Dirac talked about q-numbers versus c-numbers. Quantum observables are q-number variables that generally do not commute among themselves. He was proposing to have a generalized form of numbers as elements of a noncommutative algebra. That was Dirac's appreciation of the mathematical properties of the physical quantities as presented in Heisenberg's new quantum theory. After all, the familiar real, or complex, number system only came into existence through the history of mathematics. Values of physical quantities having a commutative product is an assumption that is not compatible with quantum physics. The revolutionary idea of Heisenberg and Dirac was pulled back to a much more conservative setting by the work of Schrödinger, followed by Born and Bohr. What Bohr missed is that the real number values we obtained from our measurements are only a consequence of the design of the kind of experiments and our using real numbers to calibrate the output scales of our apparatus. It is only our modeling of the information obtained about the physical quantities rather than what Nature dictates. We have proposed an explicit notion of definite noncommutative values of observables that gives a picture of quantum mechanics as realistic as the classical theory. In this article, we illustrate how matrices can be taken as noncommutative (q-)numbers serving as the values of physical quantities, each to be seen as a piece of quantum information. Our main task is to clarify the subtle issues involved in setting up a conventional scheme assigning matrices as values to the physical quantities.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
This article analyzes the fundamental differences between classical and quantum mechanics from a scientific and theoretical perspective. By comparing the historical development of two physical paradigms, their mathematical frameworks, laws, and areas of application, the study explores their role in explaining natural phenomena. Classical mechanics is based on causal determinism, continuity, and the precise trajectory of motion, whereas quantum mechanics employs concepts of probability, discreteness, and the wave function. To highlight the differences between the theories, key physical laws such as Heisenberg’s uncertainty principle, the Schrödinger equation, and Newton’s laws are compared. The article also discusses the modern interpretation of mechanics in the era of quantum technologies.
The theory of Non-Relativistic Quantum Mechanics was created (or discovered) back in the 1920's mainly by Schrödinger and Heisenberg, but it is fair enough to say that a more modern and unified approach to the subject was introduced by Dirac and Jordan with their (intrinsic) Transformation Theory. In his famous text book on quantum mechanics [1], Dirac introduced his well-known bra and ket notation and a view that even Einstein (who was, as well known, very critical towards the general quantum physical world-view) considered the most elegant presentation of the theory at that time[2]. One characteristic of this formulation is that the observables of position and momentum are truly treated equally so that an intrinsic phase-space approach seems a natural course to be taken. In fact, we may distinguish at least two different quantum mechanical approaches to the structure of the quantum phase space: The Weyl-Wigner (WW) formalism and the advent of the theory of Coherent States (CS). The Weyl-Wigner formalism has had many applications ranging from the discussion of the Classical/Quantum Mechanical transition and quantum chaos to signal analysis[3,4]. The Coherent State formalism had a profound impact on Quantum Optics and during the course of time has found applications in diverse areas such as geometric quantization, wavelet and harmonic analysis [5]. In this chapter we present a compact review of these formalisms (with also a more intrinsic and coordinate independent notation) towards some non-standard and up-to-date applications such as modular variables and weak values.