The article shows that quantum phenomena can be explained by the algebraic structure of space of the fundamental physical value of action. The eigenvalue operator problem is obtained as a consequence of an algebraic action vector multiplication law, existence of which we conceded. Thus we come to the explanation of quantum phenomena by the algebraic structure of the space of action vectors. Also the authors planned the path for explanations of types of quantum operators in relation to classical differentiation operators: multipliers included into the structure constants can be transferred to the operators of differentiation. Such multipliers include imaginary unit. In addition, it was established understanding of wave function as a partial differential of an action vector.
We develop a new concept of quantum mechanics which is based on a generalized space-time and on an action vector space similar to it. Both spaces are provided by algebraic properties. This allows to calculate the Dirac matrixes and to derive quantum mechanics equations from structure equations of the specified algebras. A new interpretation of the wave function is given as differential of the action vector. A generalization of the Dirac equation for 8-component wave function is derived. It is interpreted as the equation for two leptons of the same generation. A procedure of the approximate description of free leptons is formulated. The generalized equation of quantum mechanics is reduced to the Dirac, Pauli and Schrödinger equations by the sequential use of this procedure. We explain the existence of three lepton generations.
This work reformulates quantum mechanics on the real plane
(R2), establishing an algebraic isomorphism to the complex field
(C) and demonstrating that the complex structure inherent to the Schrödinger equation can be fully captured by coupled real-valued dynamics. By constructing a field
(R2,+,⋅) isomorphic to
C, the real counterpart of the Schrödinger equation is derived, yielding a system of coupled real partial differential equations. This system reveals a symplectic structure where the symplectic matrix
J geometrically represents the imaginary unit
i, acting as a phase-space rotation operator. Time-independent solutions decouple into identical real Helmholtz equations, preserving spectral equivalence. The reformulation provides geometric insights into foundational aspects: the geometric representation of
i, the emergence of probability conservation from real-component interactions, and the encoding of quantum phase relationships through the multiplicative structure of the
R2 field. This perspective reinforces connections to classical Hamiltonian mechanics and geometric quantization, suggesting that complex numbers are structural, emerging from field operations on the real plane, rather than fundamental. Furthermore, quantum computation elements like qubits and gates are shown to have equivalent representations as real symplectic and special orthogonal matrices, maintaining unitary equivalence. We introduce a framework for quantum computation within a real vector space and provide a generalized protocol for generating entangled Bell states. These results offer a new perspective on the necessity of complex numbers in quantum theory, providing a framework with potential computational advantages for hybrid quantum-classical systems and enhanced geometric interpretability.
This book is a comprehensive text in the field of quantum mechanics, covering fundamental concepts including the state of a quantum mechanical system, operators, superposition principle and measurement postulate. The notion of an operator and the algebra of operators are introduced with the help of elementary concepts of mathematical analysis. Mathematical tools developed will help readers in understanding the difficulties encountered in classical physics while trying to explain the experimental results involving atomic spectra and other phenomena. The differential equations that arise while solving eigenvalue problems are solved rigorously, to make the text self-sufficient. The solutions are then physically interpreted and explained. The text offers solved examples, analogous and homework problems to help students in solving practical problems of physics requiring quantum mechanical treatment.
The fundamental role of an operator is to effect a transformation. Differential operators and matrices are two important examples of how operators can be represented in a concrete way that makes quantitative calculations possible on systems that they represent. Both types of operator play important roles in quantum mechanics and the people shall meet them again. In a sense an operator interrogates a system represented by its eigenfunction, which results in a definite answer given by the eigenvalue. Operators with real eigenvalues also play a key role in quantum theory. Operators with real eigenvalues are conventionally represented by what are termed Hermitian or self-adjoint operators. Non-commutation has important consequences when it comes to information about two different parameters associated with a system. The connection between differentiation of a product, the Leibniz rule, and the commutator bracket, is indicative of the underlying algebraic structure of differential calculus.
Abstract We investigate operator algebraic origins of the classical Koopman–von Neumann wave function
ψKvN ψ KvN as well as the quantum-mechanical one
ψQM ψ QM . We introduce a formalism of Operator Mechanics (OM) based on a noncommutative Poisson, symplectic, and noncommutative differential structures. OM serves as a pre-quantum algebra from which algebraic structures relevant to real-world classical and quantum mechanics follow. In particular,
ψKvN ψ KvN and
ψQM ψ QM are both consequences of this pre-quantum formalism. No a priori Hilbert space is needed. OM admits an algebraic notion of operator expectation values without invoking states. A phase space bundle
E E follows from this.
ψKvN ψ KvN and
ψQM ψ QM are shown to be sections in
E E . The difference between
ψKvN ψ KvN and
ψQM ψ QM originates from a quantization map interpreted as “twisting” of sections over
E E . We also show that the Schrödinger equation is obtained from the Koopman–von Neumann equation. What this suggests is that neither the Schrödinger equation nor the quantum wave function are fundamental structures. Rather, they both originate from a pre-quantum operator algebra. Finally, we comment on how entanglement between these operators suggests emergence of space; and possible extensions of this formalism to field theories.