It is shown that in two-state quantum theory, a generic quantum state can be described by a non-computable real number. In terms of this, the criterion for measurement outcome is simply and deterministically defined. This demonstration is based on a construction of the Riemann sphere whose points represent, not complex numbers, but divergent sequences with bivalent elements. Complex structure arises from self-similar properties of a set of operators which generate these sequences. In general, a rotation of (the coordinates of) the sphere maps a computable real to a non-computable real. This is interpreted physically as a mapping of a physically-measurable state to a counterfactual state. Implications for non-locality, null measurements, many worlds and so on, are discussed. The possible role of the Euler equation as the counterpart of the Schrodinger equation for real-number quantum state evolution is also outlined.
It is shown that in two-state quantum theory, a generic quantum state can be described by a non-computable real number. In terms of this, the criterion for measurement outcome is simply and deterministically defined. This demonstration is based on a construction of the Riemann sphere whose points represent, not complex numbers, but divergent sequences with bivalent elements. Complex structure arises from self-similar properties of a set of operators which generate these sequences. In general, a rotation of (the coordinates of) the sphere maps a computable real to a non-computable real. This is interpreted physically as a mapping of a physically-measurable state to a counterfactual state. Implications for non-locality, null measurements, many worlds and so on, are discussed. The possible role of the Euler equation as the counterpart of the Schrodinger equation for real-number quantum state evolution is also outlined.
The elements of a deterministic quantum theory are developed, which reformulates and extends standard quantum theory. The proposed theory is `realistic' in the sense that in it, a general M-level quantum state is represented by a single real number r. Surprising as it may seem, this real number is shown to contain the same probabilistic information as the standard Hilbert-space state, plus additional information from which measurement outcome is determined. A crucial concept in achieving this is that of Borelian (number-theoretic) normality. The essential role of complex numbers in standard quantum theory is subsumed by the action of a set of self-similar permutation operators on the digits and places of the base-M expansion of a base-M Borelian-normal r; these permutation operators are shown to have complex structure and leave invariant the normality of the underlying real number. The set of real numbers generated by these permutations defines not only the Hilbert space of standard quantum theory, but also, in addition, the sample space from which quantum measurement outcomes can be objectively determined. Dynamical real-number state reduction is precisely described by deterministic number-theoretic operators that reduce the degree of normality of r; from the degree of normality one can infer the standard quantum-theoretic trace rule for measurement probability. All the foundational difficulties of standard quantum theory are described in terms of the proposed theory. It is shown that the real-number states of the proposed theory are precisely its beables.
We present a proposal addressing the foundational question of whether quantum mechanics can be reformulated in a computable way. Using a rational, and deterministic framework based on arbitrary-length integers and algebra. Departing from traditional models that rely on real numbers, complex amplitudes, and probabilistic interpretations, we propose a rational-number-based formulation of quantum mechanics. We demonstrate this approach through the exact eigenvalue formulation of the Schrödinger equation, a harmonization of general relativity with QM, and a reworking of Shor's algorithm using only finite, rational arithmetic (ISA). Our results suggest that quantum theory can be grounded in exact, computable logic that aligns with physical measurement, quantum computing, and existing digital hardware.
We present a proposal addressing the foundational question of whether quantum mechanics can be reformulated in a computable way. Using a rational, and deterministic framework based on arbitrary-length integers and algebra. Departing from traditional models that rely on real numbers, complex amplitudes, and probabilistic interpretations, we propose a rational-number-based formulation of quantum mechanics. We demonstrate this approach through the exact eigenvalue formulation of the Schrödinger equation, a harmonization of general relativity with QM, and a reworking of Shor's algorithm using only finite, rational arithmetic (ISA). Our results suggest that quantum theory can be grounded in exact, computable logic that aligns with physical measurement, quantum computing, and existing digital hardware.
We present a proposal addressing the foundational question of whether quantum mechanics can be reformulated in a computable, rational, and deterministic framework. Departing from traditional models that rely on real numbers, complex amplitudes, and probabilistic interpretations, we propose a rational-number-based formulation of quantum mechanics. We demonstrate this approach through the exact eigenvalue formulation of the Schrödinger equation and a reworking of Shor's algorithm using only finite, rational arithmetic (ISA). Our results suggest that quantum theory can be grounded in exact, computable logic that aligns with both physical measurement, quantum computing, and digital hardware.
It is suggested that the long–standing conceptual difficulties of quantum theory (the measurement problem and associated concept of probability on the one hand and the notions of entanglement and non–locality on the other) arise fundamentally from the role played by the continuum field C of complex numbers in the axioms of standard quantum theory. An alternative representation of the 2Nth roots of unity is developed, based on a self–similar family P of permutation operators, acting on the digits and places of the binary expansion of a real number r0, generically Borel normal. Based on r0 and P, an analytically irregular real–valued function 0≤ r < 1 is constructed on S2, which plays the role of the Riemann sphere. As a result of its irregularity (or, equivalently, granularity), r is only uniquely definable on a countable set of (‘rational’) meridians of S2. P is further generalized to describe permutation–operator representations of the quaternions, from which transformations of r under rotations of the sphere can be defined.Using P, a proposal for a constructive realistic deterministic single–world quantum theory is given. The ‘realistic’ roots–of–unity permutation operator encodes the apparent stochasticity of the complex phase function riEt/ħ in the standard quantum theoretic wave function; similarly, the quaternionic permutations encode ‘realistically’ the spatial entanglement relationships in the wave function. Based on this formalism, finite samples of values of r on the rational meridians are used to define a frequentist–based probability measure consistent with quantum–theoretic probability and correlation. The granularity of r is crucial in accounting for the theory's evasion of the Bell inequalities. Standard quantum phenomena are discussed from the perspective of the proposed theory. Some emphasis is given to the issues of contextuality and non–locality in the proposed theory.
It is argued that any possible definition of a realistic physics theory – i.e., a mathematical model representing the real world – cannot be considered comprehensive unless it is supplemented with requirement of being computationally realistic. That is, the mathematical structure of a realistic model of a physical system must allow the collection of all the system's physical quantities to compute all possible measurement outcomes on some computational device not only in an unambiguous way but also in a reasonable amount of time. In the paper, it is shown that a deterministic quantum model of a microscopic system evolving in isolation should be regarded as realistic since the NP-hard problem of finding the exact solution to the Schrödinger equation for an arbitrary physical system can be surely solved in a reasonable amount of time in the case, in which the system has just a small number of degrees of freedom. In contrast to this, the deterministic quantum model of a truly macroscopic object ought to be considered as non-realistic since in a world of limited computational resources the intractable problem possessing that enormous amount of degrees of freedom would be the same as mere unsolvable.