The distinguishability between pairs of quantum states, as measured by quantum fidelity, is formulated on phase space. The fidelity is physically interpreted as the probability that the pair are mistaken for each other upon an measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of states. The symplectic capacity for a given state, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalize the Gibbs-Liouville theorem in classical mechanics, which state that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schroedinger equation is the mathematical representation for the conservation of fidelity.
The distinguishability between pairs of quantum blobs, as measured by quantum fidelity, is defined on complex phase space. Fidelity is physically interpreted as the probability that the pair are mistaken for each other upon a measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of states. The symplectic capacity for a given state, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalizes the Liouville theorem in classical mechanics, which states that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schrödinger equation is the mathematical representation for the conservation of fidelity.
Quantum blobs are the smallest units of phase space that are compatible with the Robertson-Schrödinger indeterminacy relation and invariant under general symplectic transformations [1]. In this article, the distinguishability between pairs of quantum blobs, as measured by quantum fidelity, is defined on complex phase space. Fidelity is physically interpreted as the probability that the pair are mistaken for each other upon a measurement. The mathematical representation is based on the concept of symplectic capacity in symplectic topology. The fidelity is the absolute square of the complex-valued overlap between the symplectic capacities of the pair of quantum blobs. The symplectic capacity for a given quantum blob, onto any conjugate plane of degrees of freedom, is postulated to be bounded from below by the Gromov width h/2. This generalizes the Liouville theorem in classical mechanics, which states that the volume of a region of phase space is invariant under the Hamiltonian flow of the system, by constraining the shape of the flow. It is shown that for closed Hamiltonian systems, the Schrödinger equation is the mathematical representation for the conservation of fidelity.
We show that in the opposition to a rather common opinion quantum mechanics is not complete. It is possible to introduce so called hidden variable — in our model classical fields — and combine the statistical predictions of quantum mechanics with deterministic dynamics of those hidden variables. Quantum mechanics can be considered as an approximative description of physical processes based on neglecting by quantities of the magnitude o(α), where α is the dispersion of fluctuations of the Gaussian background field. In this paper we present the detailed presentation of theory of infinite‐dimensional phase space and derive main equations of quantum mechanics (e.g., Schrödinger’s equation, Heisenberg’s equation and von Neumann equation) from the Hamilton equation on the infinite‐dimensional symplectic space. We emphasize (to escape misunderstanding) that our paper is not about quantization of systems with infinite number of degrees of freedom, but about representation of quantum systems as classical systems with infinite number of degrees of freedom. We also investigate the purely mathematical problem of preserving of the dispersion of Gaussian fluctuations by Hamiltonian flows.
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.
In this article, it is argued that the Gibbs-Liouville theorem is a mathematical representation of the statement that closed classical systems evolve deterministically. From the perspective of an observer of the system, whose knowledge about the degrees of freedom of the system is complete, the statement of deterministic evolution is equivalent to the notion that the physical distinctions between the possible states of the system, or, in other words, the information possessed by the observer about the system, is never lost. Thus, it is proposed that the Gibbs-Liouville theorem is a statement about the dynamical evolution of a closed classical system valid in such situations where information about the system is conserved in time. Furthermore, in this article it is shown that the Hamilton equations and the Hamilton principle on phase space follow directly from the differential representation of the Gibbs-Liouville theorem, i.e. that the divergence of the Hamiltonian phase flow velocity vanish. Thus, considering that the Lagrangian and Hamiltonian formulations of classical mechanics are related via the Legendre transformation, it is obtained that these two standard formulations are both logical consequences of the statement of deterministic evolution, or, equivalently, information conservation.