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arXiv:0704.3574

Space of Quantum Theory Representations of Natural Numbers, Integers, and Rational Numbers

Paul A. Benioff

Published Apr 26, 2007• 2 citations
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Abstract

This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' and k. This suggests that one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the base k q's as composites of the prime number q's.

DOI
10.48550/arxiv.0704.3574

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This paper extends earlier work on quantum theory representations of natural numbers N, integers I, and rational numbers Ra to describe a space of these representations and transformations on the space. The space is parameterized by 4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific representation of X = N, I, Ra as a Fock space F^{X}{k,m,h} of states of finite length strings of qukits q and a string state basis B^{X}{k,m,h,g}. The pair (m,h) locates the q string in a square integer lattice I \times I, k is the q base, and the function g fixes the gauge or basis states for each q. Maps on the parameter set induce transformations on on the representation space. There are two shifts, a base change operator W_{k',k}, and a basis or gauge transformation function U_{k}. The invariance of the axioms and theorems for N, I, and Ra under any transformation is discussed along with the dependence of the properties of W_{k',k} on the prime factors of k' and k. This suggests that one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the base k q's as composites of the prime number q's.
In this paper fields of quantum reference frames based on gauge transformations of rational string states are described in a way that, hopefully, makes them more understandable than their description in an earlier paper. The approach taken here is based on three main points: (1) There are a large number of different quantum theory representations of natural numbers, integers, and rational numbers as states of qubit strings. (2) For each representation, Cauchy sequences of rational string states give a representation of the real (and complex) numbers. A reference frame is associated to each representation. (3) Each frame contains a representation of all mathematical and physical theories that have the real and complex numbers as a scalar base for the theories. These points and other aspects of the resulting fields are then discussed and justified in some detail. Also two different methods of relating the frame field to physics are discussed. 1
In this paper fields of quantum reference frames based on gauge transformations of rational string states are described in a way that, hopefully, makes them more understandable than their description in an earlier paper. The approach taken here is based on three main points: (1) There are a large number of different quantum theory representations of natural numbers, integers, and rational numbers as states of qubit strings. (2) For each representation, Cauchy sequences of rational string states give a representation of the real (and complex) numbers. A reference frame is associated to each representation. (3) Each frame contains a representation of all mathematical and physical theories that have the real and complex numbers as a scalar base for the theories. These points and other aspects of the resulting fields are then discussed and justified in some detail. Also two different methods of relating the frame field to physics are discussed.
A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.
A quantum theoretic representation of real and complex numbers is described here as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are 4 types of qubits each with associated single qubit annihilation creation (a-c) operators that give the state and location of each qubit type on a 2 dimensional integer lattice. The string states, defined as finite products of creation operators acting on the qubit vacuum state correspond to complex rational numbers with real and imaginary components. These states span a Fock space. Arithmetic relations and operations are defined for the string states. Cauchy sequences of these states are defined, and the arithmetic relations and operations lifted to apply to these sequences. Based on these, equivalence classes of these sequences are seen to have the requisite properties of real and complex numbers. The representations have some interesting aspects. Quantum equivalence classes are larger than their corresponding classical classes, but no new classes are created. There exist Cauchy sequences such that each state in the sequence is an entangled superposition of the real and imaginary components, yet the sequence is a real number. Also, except for coefficients of superposition states, the construction is done with no reference to the real and complex number base, R and C, of the Fock space.
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This work is based on a description of quantum reference frames that seems more basic than others in the literature. Here a frame is based on a set of real and of complex numbers and a space time as a 4-tuple of the real numbers. There are many isomorphic frames as there are many isomorphic sets of real numbers. Each frame is suitable for construction of all physical theories as mathematical structures over the real and complex numbers. The organization of the frames into a field of frames is based on the representations of real and complex numbers as Cauchy operators defined on complex rational states of finite qubit strings. The structure of the field is based on noting that the construction of real and complex numbers as Cauchy operators in a frame can be iterated to create new frames coming from a frame. Gauge transformations on the rational string states greatly expand the number of quantum frames as, for each gauge U, there is one frame coming from the original frame. Forward and backward iteration of the construction yields a two way infinite frame field with satisfying properties. There is no background space time and there are no real or complex numbers for the field as a whole. Instead these are relative concepts associated with each frame in the field. Extension to include qukit strings for different k bases, is described as is the problem of reconciling the frame field to the existence of just one frame with one background space time for the observable physical universe.