The Mathematical Description of a Generic Physical System

Tal Hendel

Abstract Panmicropsychism is the hypothesis that the physical ultimates of our universe (i.e., its fundamental physical ingredients) are also its phenomenal ultimates (i.e., its fundamental phenomenal ingredients). In this paper, I argue that, just as the mathematical description of the physical ultimates of our universe displays simplicity, symmetry, and beauty, panmicropsychists should similarly expect the mathematical description of the phenomenal ultimates to reflect these same qualities. I exemplify this claim using what is arguably the simplest of all types of phenomenal qualities – color. Specifically, I utilize phenomenological data on color to construct the maximally symmetric mathematical description of phenomenal color space. I then show that this mathematical description is isomorphic to the mathematical description of two-state quantum systems (qubits) in a mixed state. Based on this isomorphism, I suggest that color may be the phenomenal dual aspect of two-state quantum systems.

K. Kong Wan

A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.

Netherland, Albert V. ‎ Herrebrugh

A mathematical model for the slit experiments in the heart of quantum mechanics is developed to gain insight into quantum ‎theory. The proposed system-theoretical model is entirely based on commutative mathematics, i.e. convolution, and integral ‎transformations, and starts with spacetime functions with inherent energy-based cause and effect relations of the state-‎function Ѱ in the complex Hilbert space. The benefits of his approach are as 1-Invariance in time reversal. 2-Deterministic ‎result functions in the model in line with the outcome of slit experiments. 3- Separation of causality and cross-correlations of ‎attained states. 4- Disappearance of a posteriori probability of quantum states. 5- Quantum a priori fixed states after ‎causality interactions have ended, (even) when quanta are (light-years) separated. The model predicts the patterns in the ‎experiments with mathematical functions of the energy distributions. The quantum mechanical counterpart description of the ‎physical reality of slit experiments thus may be considered complete in A. Einstein’s definition. The patterns in double slit ‎experiments are found to be an effect of energy (amplitude-) modulation. An equivalent double-slit pattern can be retrieved ‎from an input modulated 1-slit experiment excluding interference interpretations. The system-theoretical model uses generic ‎properties of quanta and evolves into determinism in ‎quantum mechanics slit experiments. The mathematics in the model ‎handles beables by treatment ‎of momentum p in system theoretical I/O relations of the transformed functions and allows the ‎proposed description by the avoidance of a direct addressing of the individual quanta through variables. The following ‎method yields exact, non-probabilistic results.‎