arXiv subjects

Paul Benioff

Publications and source records attributed to Paul Benioff.

At least 37 records · Page 2Linked to original sources

arXiv2007

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

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.

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arXiv2007

A Representation of Real and Complex Numbers in Quantum Theory

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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arXiv2006

Fields of quantum reference frames based on different representations of rational numbers as states of qubit strings

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 representations of 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.

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arXiv2006

Fields of Iterated Quantum Reference Frames based on Gauge Transformations of Rational String States

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.

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arXiv2005

Complex Rational Numbers in Quantum Mechanics

A binary representation of complex rational numbers and their arithmetic is described that is not based on qubits. It takes account of the fact that $0s$ in a qubit string do not contribute to the value of a number. They serve only as place holders. The representation is based on the distribution of four types of systems, corresponding to $+1,-1,+i,-i,$ along an integer lattice. Complex rational numbers correspond to arbitrary products of four types of creation operators acting on the vacuum state. An occupation number representation is given for both bosons and fermions.

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arXiv2005

A representation of complex rational numbers in quantum mechanics

A representation of complex rational numbers in quantum mechanics is described that is not based on logical or physical qubits. It stems from noting that the zeros in a product qubit state do not contribute to the number. They serve only as place holders. The representation is based on the distribution of four types of systems on an integer lattice. The four types, labelled as positive real, negative real, positive imaginary, and negative imaginary, are represented by creation and annihilation operators acting on the system vacuum state. Complex rational string number states correspond to strings of creation operators acting on the vacuum. Various operators, including those for the basic arithmetic operations, are described. The representation used here is based on occupation number states and is given for bosons and fermions.

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arXiv2004

The Theory Experiment Connection: Rn Space and Inflationary Cosmology

Based on a discussion of the theory experiment connection, it is proposed to tighten the connection by replacing the real and complex number basis of physical theories by sets R_{n},C_{n} of length 2n finite binary string numbers. The form of the numbers in R_{n} is based on the infinite hierarchy of 2n figure outputs from measurements of any physical quantity with an infinite range (distance, energy, etc.). A space and time based on these numbers is described. It corresponds to an infinite sequence of spherical scale sections R_{n,e} (e an integer). Each section has the same number of points but the size increases exponentially with increasing e. The sections converge towards an origin which is a space singularity. Iteration of an order preserving transformation, F_{<} or its inverse shows exponential expansion or contraction of the space with the origin as a source or sink of space points. The suitability of R_{n} space as a framework for inflationary cosmology is based on a constant iteration rate for F_{<} and time dependent e and n=n(t). At t=0 all space is restricted to a region of scale sections R_{n_{0},e} with n_{0} (small) and e\leq e_{0} (negative). Inflation, which occurs naturally, is stopped automatically at time t_{I} by increasing n from n_{0} to n_{I}>>n_{0}. Here n_{I} is such that the outermost Δscale sections of R_{n_{0}} space, which are moving at velocities >c at time t_{I}, are contained in the scale section R_{n_{I},0} of R_{n_{I}} space. This is needed if R_{n_{I},0} space is to be similar to the usual R space. Hubble expansion and the redshift are accounted for by a continuing slow increase in n. Comparison with experimental data suggests that the rate of increase must be at least one n unit every 30-60 million years.

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arXiv2004

Tightening the Theory-Experiment Connection in Physics: R_{n} based Space and Time

Some aspects of replacing C based physics by C_{n} based physics are discussed. Here C_{n} ={R_{n},I_{n}} where R_{n} and I_{n} are the real and imaginary components of the numbers in C_{n}, and both R_{n} and I_{n}) are sets of length 2n string numbers in some basis. The discussion here is limited to describing the experimental basis and choice for the numbers in R_{n}, and a few basic but interesting properties of R_{n} based space and time.

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arXiv2004

Towards a Coherent Theory of Physics and Mathematics: The Theory-Experiment Connection

The problem of how mathematics and physics are related at a foundational level is of much interest. One approach is to work towards a coherent theory of physics and mathematics together. Here steps are taken in this direction by first examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C_{n} which is the set of pairs, R_{n},I_{n}, of n figure rational numbers in some basis. The properties of these numbers are based on the type of numbers that represent measurement outcomes for continuous variables. A model of space and time based on R_{n} is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n to infinity to obtain locally flat continuum based space and time is outlined. Possibly the most interesting result is that R_{n} based space is invariant under scale transformations which correspond to expansion and contraction of space relative to a flat background. Also the location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics based on C_{n} and on R_{n} space are briefly investigated.

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arXiv2003

Resource Limited Theories and their Extensions

This work is based on the idea that extension of physical and mathematical theories to include the amount of space, time, momentum, and energy resources required to determine properties of systems may influence what is true in physics and mathematics at a foundational level. Background material, on the dependence of region or system sizes on both the resources required to study the regions or systems and the indirectness of the reality status of the systems, suggests that one associate to each amount, r, of resources a domain, D_{r}, a theory, T_{r}, and a language, L_{r}. D_{r} is limited in that all statements in D_{r} require at most r resources to verify or refute. T_{r} is limited in that any theorem of T_{r} must be provable using at most r resources. Also any theorem of T_{r} must be true in D_{r}. L_{r} is limited in that all expressions in L_{r} require at most r resources to create, display, and maintain. A partial ordering of the resources is used to describe minimal use of resources, a partial ordering of the T_{r}, and motion of an observer using resources to acquire knowledge. Reflection principles are used to push the effect of Godel's incompleteness theorem on consistency up in the partial ordering. It is suggested that a coherent theory of physics and mathematics, or theory of everything, is a common extension of all the T_{r}.

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arXiv2002

Cyclic networks of quantum gates

In this article initial steps in an analysis of cyclic networks of quantum logic gates is given. Cyclic networks are those in which the qubit lines are loops. Here we have studied one and two qubit systems plus two qubit cyclic systems connected to another qubit on an acyclic line. The analysis includes the group classification of networks and studies of the dynamics of the qubits in the cyclic network and of the perturbation effects of an acyclic qubit acting on a cyclic network. This is followed by a discussion of quantum algorithms and quantum information processing with cyclic networks of quantum gates, and a novel implementation of a cyclic network quantum memory. Quantum sensors via cyclic networks are also discussed.

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arXiv2002

Language is Physical

Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication or thinking would be possible. Efficient implementability for creating and reading language expressions is discussed and illustrated with a quantum mechanical model. The reason for interest in language is that language expressions can have meaning, either as an informal language or as a formal language associated with a mathematical or physical theory. It is noted that any universally applicable physical theory, or coherent theory of physics and mathematics together, includes in its domain physical models of expressons for both the informal language used to discuss the theory and expressions of the theory itself. It follows that there must be formulas in the formal theory that express some of their own physical properties. Inclusion of intelligent system in the theory domain means that the theory, e.g., quantum mechanics, must describe in some sense its own validation. Maps of language expressions into physical states are discussed as are conditions under which such a map is a Godel map. The possibility that language is also mathematical is very briefly discussed.

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arXiv2002

Towards a Coherent Theory of Physics and Mathematics

As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently strong, and it must maximally describe its own validity and sufficient strength. The mathematical logical definition of validity is used, and sufficient strength is seen to be a necessary and useful concept. The requirement of maximal description of its own validity and sufficient strength may be useful to reject candidate coherent theories for which the description is less than maximal. Other aspects of a coherent theory discussed include universal applicability, the relation to the anthropic principle, and possible uniqueness. It is suggested that the basic properties of the physical and mathematical universes are entwined with and emerge with a coherent theory. Support for this includes the indirect reality status of properties of very small or very large far away systems compared to moderate sized nearby systems. Discussion of the necessary physical nature of language includes physical models of language and a proof that the meaning content of expressions of any axiomatizable theory seems to be independent of the algorithmic complexity of the theory. Gödel maps seem to be less useful for a coherent theory than for purely mathematical theories because all symbols and words of any language musthave representations as states of physical systems already in the domain of a coherent theory.

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arXiv2002

Use of Mathematical Logical Concepts in Quantum Mechanics: An Example

The representation of numbers by product states in quantum mechanics can be extended to the representation of words and word sequences in languages by product states. This can be used to study quantum systems that generate text that has meaning. A simple example of such a system, based on an example described by Smullyan, is studied here. Based on a path interpretation for some word states, definitions of truth, validity, consistency and completeness are given and their properties studied. It is also shown that the relation between the meaning, if any, of word states and the algorithmic complexity of the process generating the states must be quite complex or nonexistent.

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arXiv2001

The Representation of Numbers in Quantum Mechanics

Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to addition of k^{j-1} if j>0 and k^{j} if j<0 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on the successor for j=1 only. This is the only successor defined in the usual axioms of arithmetic.

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arXiv2001

Efficient Implementation and the Product State Representation of Numbers

The relation between the requirement of efficient implementability and the product state representation of numbers is examined. Numbers are defined to be any model of the axioms of number theory or arithmetic. Efficient implementability (EI) means that the basic arithmetic operations are physically implementable and the space-time and thermodynamic resources needed to carry out the implementations are polynomial in the range of numbers considered. Different models of numbers are described to show the independence of both EI and the product state representation from the axioms. The relation between EI and the product state representation is examined. It is seen that the condition of a product state representation does not imply EI. Arguments used to refute the converse implication, EI implies a product state representation, seem reasonable; but they are not conclusive. Thus this implication remains an open question.

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arXiv2001

Space Searches with a Quantum Robot

Quantum robots are described as mobile quantum computers and ancillary systems that move in and interact with arbitrary environments. Their dynamics is given as tasks which consist of sequences of alternating computation and action phases. A task example is considered in which a quantum robot searches a space region to find the location of a system. The possibility that the search can be more efficient than a classical search is examined by considering use of Grover's Algorithm to process the search results. For reversible searches this is problematic for two reasons. One is the removal of entanglements generated by the search process. The other is that even if the entanglement can be avoided, the search process in 2-D space regions is no more efficient than a classical search. However, quantum searches of space regions with 3 or more dimensions are more efficient than classical searches. Reasons why quantum robots are interesting independent of these results are briefly summarized.

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arXiv2000

The Representation of Natural Numbers in Quantum Mechanics

This paper represents one approach to making explicit some of the assumptions and conditions implied in the widespread representation of numbers by composite quantum systems. Any nonempty set and associated operations is a set of natural numbers or a model of arithmetic if the set and operations satisfy the axioms of number theory or arithmetic. This work is limited to k-ary representations of length L and to the axioms for arithmetic modulo k^{L}. A model of the axioms is described based on states in and operators on an abstract L fold tensor product Hilbert space H^{arith}. Unitary maps of this space onto a physical parameter based product space H^{phy} are then described. Each of these maps makes states in H^{phy}, and the induced operators, a model of the axioms. Consequences of the existence of many of these maps are discussed along with the dependence of Grover's and Shor's Algorithms on these maps. The importance of the main physical requirement, that the basic arithmetic operations are efficiently implementable, is discussed. This conditions states that there exist physically realizable Hamiltonians that can implement the basic arithmetic operations and that the space-time and thermodynamic resources required are polynomial in L.

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