The discrete phase space representation of quantum mechanics involving a characteristic length is investigated. The continuous (1 + 1)-dimensional phase space is first discussed for the sake of simplicity. It is discretized into denumerable infinite number of concentric circles. These circles, endowed with “unit area”, are degenerate phase cells resembling closed strings.Next, Schrödinger wave equation for one particle in the three dimensional space under the influence of a static potential is studied in the discrete phase space representation of wave mechanics. The Schrödinger equation in the arena of discrete phase space is a partial difference equation. The energy eigenvalue problem for a three dimensional oscillator is exactly solved.Next, relativistic wave equations in the scenario of three dimensional discrete phase space and continuous time are explored. Specially, the partial finite difference-differential equation for a scalar field is investigated for the sake of simplicity. The exact relativistic invariance of the partial finite difference-differential version of the Klein-Gordon equation is rigorously proved. Moreover, it is shown that all nine important Green’s functions of the partial finite difference-differential wave equation for the scalar field are non-singular.In the appendix, a possible physical interpretation for the discrete orbits in the phase space as degenerate, string-like phase cells is provided in a mathematically rigorous way.
Continuous degrees of freedom The quantum systems we have discussed so far have been described by finite-dimensional Hilbert spaces. Basic measurements on such systems have a finite number of possible outcomes, and a quantum state predicts a discrete probability distribution over these. Now we wish to extend our theory to handle systems with one or more continuous degrees of freedom, such as the position of a particle that can move in one dimension. This will require an extension of our theory to Hilbert spaces of infinite dimension, and to systems with continuous observables. There is a philosophical issue here. How do we know that there really are infinitely many distinct locations for a particle? The short answer is, we don't. It might be that space itself is both discrete (at the tiniest scales) and bounded (at the largest), so that the number of possible locations of a particle is some very large but finite number. If this is the case, then the continuum model for space is nothing more than a convenient approximation. Infinity is just a simplified way of describing a quantity that is immense, but still finite. In this section, we will adopt this view of infinity. We will imagine that any continuous variable is really an approximation of a “true” discrete variable. This idea will motivate the continuous quantities and operations that we need.
Recently we have presented a hidden-variable model of measurements for a qubit where the hidden-variable state-space dimension is one-half the quantum-state manifold dimension. The absence of a short memory (Markov) dynamics is the price paid for this dimensional reduction. The conflict between having the Markov property and achieving the dimensional reduction was proved by Montina [A. Montina, Phys. Rev. A 77, 022104 (2008)] using an additional hypothesis of trajectory relaxation. Here we analyze in more detail this hypothesis introducing the concept of invertible process and report a proof that makes clearer the role played by the topology of the hidden-variable space. This is accomplished by requiring suitable properties of regularity of the conditional probability governing the dynamics. In the case of minimal dimension the set of continuous hidden variables is identified with an object living an
N-dimensional Hilbert space whose dynamics is described by the Schr\"odinger equation. A method for generating the economical non-Markovian model for the qubit is also presented.
Abstract—This paper completely deconstructs the calculus-centered mathematical redundancy and probabilistic uncertainty faced by modern physics, and re-establishes the fundamental mechanism of the universe into a clear system of elementary algebraic equations. Conventional academia has treated "time (t)" as a physical variable, interpreting the behavior of the microscopic world through complex partial differential equations and probability clouds. This Foundation establishes a unique grand premise: "Time (t) is not a physical reality but merely a unit for establishing a reference, and a Field is not an applied Force but the geometric structure of space (the thickness of a water pipe)." Based on this premise, the three pillars of quantum mechanics—Louis de Broglie's matter wave theory, Werner Heisenberg's uncertainty principle, and Erwin Schrödinger's wave equation—are substituted into closed algebraic structures. Cross-verification through six empirical cases from both macroscopic and microscopic realms—including multidimensional orbits of hydrogen atoms, particle constraints in quantum dots, actual nuclear proton behavior, and deep learning data-driven energy levels—proves that this algebraic formula system precisely converges with the high-level calculus predictions of conventional academia down to the decimal points with 100% accuracy.
Transport equations describe how physical quantities -- such as mass, energy, momentum, concentration, probability, or fields -- are carried, propagated, or redistributed through space and time, forming a foundational class of partial differential equations across science and engineering. However, high-dimensional partial differential equations are difficult to represent on digital grids because the number of degrees of freedom grows exponentially with dimension. Continuous-variable quantum photonics on the other hand can represent and evolve these large-scale fields without first discretizing space into a discrete grid. We demonstrate a large-scale analog photonic simulator for the constant-coefficient advection equation, a transport equation that is a fundamental benchmark for scientific computing. The solution of a
d-variable advection equation is encoded into
d optical modes, so that the partial differential equation evolution maps directly to programmable phase-space displacements generated by optical quadrature momenta. Using a time-domain continuous-variable quantum photonic platform, we validate programmable control with
20,000 single-mode squeezed states and
20,000 two-mode squeezed states, and implement transport dynamics on a
20,000-mode cluster-state resource. Homodyne measurements then verifies mode-resolved displacement control, which can provide first and second-order moment information of the solution to the advection equation, with final achievable relative error as low as
0.8% and
0.92% for first and second-order moment observables respectively. Our results establish continuous-variable photonics as a suitable programmable analog platform for large-scale advection equations.