arXiv subjects

Paul Benioff

Publications and source records attributed to Paul Benioff.

At least 19 recordsLinked to original sources

arXiv2022

Local mathematics and scaling field: effects on local physics and on cosmology

The origin of this paper starts with the observation by Yang Mills that what state represents a proton in isospin space at one location does not determine what state represents a proton in isospin space at another location. This is accounted for by the presence of a unitary gauge transformation operator, $U(y,x)$, between vector spaces at different locations. This operator defines the notion of same states for vector spaces at different locations. If $ψ$ is a state in a vector space at $x$ then $U(y,x)ψ$ is the same state in the vector space at $y$. Vector spaces include scalar fields in their axiomatic description. These appear as norms, closure under vector scalar multiplication, etc. This leads to a conflict: local vector spaces and global scalar fields. Here this conflict is removed by replacing global scalar fields with local scalar fields. These are represented by $\bar{S}_{x}$ where $x$ is any location in Euclidean space or space time. Here $S$ represents the different type of numbers, (natural, integers, rational, real, and complex). The association of scalar fields with vector spaces and the Yang Mills observation raises the question, What corresponds to the Yang Mills observation for numbers? The answer is that two different concepts, number and number meaning or value, are conflated in the usual use of mathematics. These two concepts are distinct.

math-ph↗

arXiv2018

Relation between observers and effects of number valuation in science

This paper is a small step towards the goal of constructing a coherent theory of physic and mathematics together. It is based on two ideas, the localization of mathematical systems in space or space time, and the separation of the concepts of number from number value. The separation of number from number value along with the freedom of choice of number values at different points of space or space time enables the introduction of a space or space time dependent number valuation field. The presence of a location dependent number value field affects theoretical descriptions of many physical and geometric quantities. A simple geometric example is worked out in detail, that of the length of a path. The localization of mathematical systems and the separation of number from number value or meaning both emphasize the role of observers. The separation of number from number value shows the role of observers in that value or meaning are conscious observer related concepts. Nothing, including numbers, has value or meaning to an unconscious observer. The localization of mathematical systems also shows the role of observers in that the mathematics that is potentially available to an observer is that at the same position as is the observer. This represents the mathematical knowledge that can reside in an observers brain. As an observer moves in space or space time, the mathematical knowledge potentially available to the observer is the collection of mathematical systems at the same location as the observer. It is hoped that this work, which was begun in 2010, will lead to a better understanding of the relation between the foundations of mathematics and physics, and the role that observers play in this relation.

physics.hist-ph↗

arXiv2018

The no information at a distance principle and local mathematics: some effects on physics and geometry

Local mathematics assumes the existence of number structures of different types, vector spaces, etc. localized at each space time point. Relations between number structures at different locations are based on two aspects: distinction between two so far conflated concepts, number and number value and the "No information at a distance" principle. This principle forbids the choice of the value of a number at one location to determine the value of the same number at another location. Value changing connections, related to a real valued field, $g,$ move numbers between structures at different locations. The effect of the $g$ field, or its exponential equivalent, $g(y)=e^{α(y)},$ on numbers extends to other mathematical structures, vector spaces, etc. The presence of $α$ affects theoretical descriptions of quantities in physics and geometry. Two examples are described, the effect on the Dirac Lagrangian in gauge theory, and the effect on path lengths and distances in geometry. The gradient field of $α$, $\vec{A},$ appears in the Lagrangian as a spin $0$, real scalar field that couples to the fermion field. Any value for the mass of $\vec{A}$ is possible. The lack of direct experimental evidence for the presence of the $g$ or $α$ field means that the field must be essentially constant within a local region of the cosmological universe. Outside the local region there are no restrictions on the field. Possible physical candidates, (inflaton, dark matter, dark energy) for $α$ are noted.

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arXiv2016

Effect of number scaling on entangled states in quantum mechanics

A summary of number structure scaling is followed by a description of the effects of number scaling in nonrelativistic quantum mechanics. The description extends earlier work to include the effects on the states of two or more interacting particles. Emphasis is placed on the effects on entangled states. The resulting scaling field is generalized to describe the effects on these states. It is also seen that one can use fiber bundles with fibers associated with single locations of the underlying space to describe the effects of scaling on arbitrary numbers of particles.

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arXiv2015

Effects of a scalar scaling field on quantum mechanics

This paper describes the effects of a complex scalar scaling field on quantum mechanics. The field origin is an extension of the gauge freedom for basis choice in gauge theories to the underlying scalar field. The extension is based on the idea that the value of a number at one space time point does not determine the value at another point. This, combined with the description of mathematical systems as structures of different types, results in the presence of separate number fields and vector spaces, both as structures, at different space time locations. Complex number structures and vector spaces at each location, are scaled by a complex space time dependent scaling factor. The effect of this scaling factor on several physical and geometric quantities has been described in other work. Here the emphasis is on quantum mechanics of one and two particles, their states and properties. Multiparticle states are also briefly described. The effect shows as a complex, nonunitary, scalar field connection on a fiber bundle description of nonrelativistic quantum mechanics. The lack of physical evidence for the presence of this field so far means that the coupling constant of this field to fermions is very small. It also means that the gradient of the field must be very small in a local region of cosmological space and time. Outside this region there are no restrictions on the field gradient.

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arXiv2015

Space and time dependent scaling of numbers in mathematical structures: Effects on physical and geometric quantities

The relationship between the foundations of mathematics and physics is a topic of of much interest. This paper continues this exploration by examination of the effect of space and time dependent number scaling on theoretical descriptions of some physical and geometric quantities. Fiber bundles provide a good framework to introduce a space and time or space time dependent number scaling field. The effect of the scaling field on a few nonlocal physical and geometric quantities is described. The effect on gauge theories is to introduce a new complex scalar field into the derivatives appearing in Lagrangians. U(1) invariance of Lagrangian terms does not affect the real part of the scaling field. For this field, any mass is possible. The scaling field is also shown to affect quantum wave packets and path lengths, and geodesic equations even on flat space. Scalar fields described so far in physics, are possible candidates for the scaling field. The lack of direct evidence for the field in physics restricts the scaling field in that the gradient of the field must be close to zero in a local region of cosmological space and time. There are no restrictions outside the region. It is also seen that the scaling field does not affect comparisons of computation or measurements outputs with one another. However it does affect the assignment of numerical values to the outputs of computations or measurements. These are needed because theory predictions are in terms of numerical values.

math-ph↗

arXiv2015

Fiber bundle description of number scaling in gauge theory and geometry

This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, $C^{c}_{x}\times V^{c}_{x} $ for each c in GL(1,C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structures that contain the values of a vector valued matter field at x. The vertical components of connections between neighboring fibers are taken to be the gradient field A(x)+iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless and no mass restrictions for A. Addition of an electromagnetic field dies not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with a fiber at point x containing scaled pairs, $R^{r}_{x}\times T^{r}_{x}$ of real number and tangent space structures for each x and and nonnegative real r. B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient field, A(x)+iB(x) means that it either couples very weakly to matter fields or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.

math-ph↗

arXiv2015

Principal fiber bundle description of number scaling for scalars and vectors: Application to gauge theory

The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value are combined in the usual representations of number structures. This is valid as long as just one structure of each number type is being considered. It is not valid when different structures of each number type are being considered. Elements of base sets of number structures, considered by themselves, have no meaning. They acquire meaning or value as elements of a number structure. Fiber bundles over a space or space time manifold, M, are described. The fiber consists of a collection of many real or complex number structures and vector space structures. The structures are parameterized by a real or complex scaling factor, s. A vector space at a fiber level, s, has, as scalars, real or complex number structures at the same level. Connections are described that relate scalar and vector space structures at both neighbor M locations and at neighbor scaling levels. Scalar and vector structure valued fields are described and covariant derivatives of these fields are obtained. Two complex vector fields, each with one real and one imaginary field, appear, with one complex field associated with positions in $M$ and the other with position dependent scaling factors. A derivation of the covariant derivative for scalar and vector valued fields gives the same vector fields. The derivation shows that the complex vector field associated with scaling fiber levels is the gradient of a complex scalar field. Use of these results in gauge theory shows that the imaginary part of the vector field associated with M positions acts like the electromagnetic field. The physical relevance of the other three fields, if any, is not known.

math-ph↗

arXiv2014

Effects of mathematical locality and number scaling on coordinate chart use

A stronger foundation for earlier work on the effects of number scaling, and local mathematics is described. Emphasis is placed on the effects of scaling on coordinate systems. Effects of scaling are represented by a scalar field, $θ,$ that appears in gauge theories as a spin zero boson. Gauge theory considerations led to the concept of local mathematics, as expressed through the use of universes, as collections of local mathematical systems at each point, x, of a space time manifold, M. Both local and global coordinate charts are described. These map M into either local or global coordinate systems within a universe or between universes, respectively. The lifting of global expressions of nonlocal physical quantities, expressed by space and or time integrals or derivatives on M, to integrals or derivatives on coordinate systems, is described. The assumption of local mathematics and universes makes integrals and derivatives, on M or on global charts, meaningless. They acquire meaning only when mapped into a local universe. The effect of scaling, by including the effect of $θ$ into the local maps, is described. The lack of experimental evidence for $θ$ so far shows that the coupling constant of $θ$ to matter fields must be very small compared to the fine structure constant. Also the gradient of $θ$ must be very small in the local region of cosmological space and time occupied by us as observers. So far, there are no known restrictions on $θ$ or its gradient in regions of space and/or time that are far away from our local region.

math-ph↗

arXiv2013

Effects of gauge theory based number scaling on geometry

Effects of local availability of mathematics (LAM) and space time dependent number scaling on physics and, especially, geometry are described. LAM assumes separate mathematical systems as structures at each space time point. Extension of gauge theories to include freedom of choice of scaling for number structures, and other structures based on numbers, results in a space time dependent scaling factor based on a scalar boson field. Scaling has no effect on comparison of experimental results with one another or with theory computations. With LAM all theory expressions are elements of mathematics at some reference point. Changing the reference point introduces (external) scaling. Theory expressions with integrals or derivatives over space or time include scaling factors (internal scaling) that cannot be removed by reference point change. Line elements and path lengths, as integrals over space and/or time, show the effect of scaling on geometry. In one example, the scaling factor goes to 0 as the time goes to 0, the big bang time. All path lengths, and values of physical quantities, are crushed to 0 as $t$ goes to 0. Other examples have spherically symmetric scaling factors about some point, $x.$ In one type, a black scaling hole, the scaling factor goes to infinity as the distance, $d$, between any point $y$ and $x$ goes to 0. For scaling white holes, the scaling factor goes to 0 as $d$ goes to 0. For black scaling holes, path lengths from a reference point, $z$, to $y$ become infinite as $y$ approaches $x.$ For white holes, path lengths approach a value much less than the unscaled distance from $z$ to $x.$

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arXiv2012

Gauge theory extension to include number scaling by boson field: Effects on some aspects of physics and geometry

In gauge theories, separate vector spaces, Vx, are assigned to each space time point x with unitary operators as maps between basis vectors in neighboring Vx. Here gauge theories are extended by replacing the single underlying set of complex scalars, C, with separate sets, Cx, at each x, and including scaling between the Cx. In gauge theory Lagrangians, number scaling shows as a scalar boson field, B, with small coupling to matter fields. Freedom of number scaling is extended to a model with separate number structures assigned to each point x. Separate collections, Ux, of all mathematical systems based on numbers, are assigned to each x. Mathematics available to an observer, Ox, at x is that in Ux. The B field induces scaling between structures in the different Ux. Effects of B scaling on some aspects of physics and geometry are described. The lack of experimentally observed scaling means that B(z) is essentially constant for all points, z, in a region, Z, that can be occupied by us as observers. This restriction does not apply outside Z. The effects of B scaling on line elements, curve lengths, and distances between points, are examined. Oz's description, using Uz in Z, of elements at points, x, outside Z, includes scaling from x to z. Integrals over curves include scaling factors inside the integrals. Two examples are discussed. One shows that B(t) can be such that mathematical, physical, and geometric quantities approach zero as the time t approaches zero. This mimics the big bang in that distances approach zero. Examples of black and white scaling holes are described in which B(x) is plus or minus infinity at a point x.

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arXiv2012

Local Availability of mathematics and number scaling: Effects on quantum physics

Local availability of mathematics and number scaling provide an approach to a coherent theory of physics and mathematics. Local availability of mathematics assigns separate mathematical universes, U_{x}, to each space time point, x. The mathematics available to an observer, O_{x}, at x is contained in U_{x}. Number scaling is based on extending the choice freedom of vector space bases in gauge theories to choice freedom of underlying number systems. Scaling arises in the description, in U_{x}, of mathematical systems in U_{y}. If a_{y} or ψ_{y} is a number or a quantum state in U_{y}, then the corresponding number or state in U_{x} is r_{y,x}a_{x} or r_{y,x}ψ_{x}. Here a_{x} and ψ_{x} are the same number and state in U_{x} as a_{y} and ψ_{y} are in U_{y}. If y=x+\hatμdx is a neighbor point of x, then the scaling factor is r_{y,x}=\exp(\vec{A}(x)\cdot\hatμdx) where \vec{A} is a vector field, assumed here to be the gradient of a scalar field. The effects of scaling and local availability of mathematics on quantum theory show that scaling has two components, external and internal. External scaling is shown above for a_{y} and ψ_{y}. Internal scaling occurs in expressions with integrals or derivatives over space or space time. An example is the replacement of the position expectation value, \intψ^{*}(y)yψ(y)dy, by \int_{x}r_{y,x}ψ^{*}_{x}(y_{x})y_{x}ψ_{x}(y_{x})dy_{x}. This is an integral in U_{x}. The good agreement between quantum theory and experiment shows that scaling is negligible in a space region, L, in which experiments and calculations can be done, and results compared. L includes the solar system, but the speed of light limits the size of L to a few light years. Outside of $L$, at cosmological distances, the limits on scaling are not present.

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arXiv2011

Effects on quantum physics of the local availability of mathematics and space time dependent scaling factors for number systems

The work is based on two premises: local availability of mathematics to an observer at any space time location, and the observation that number systems, as structures satisfying axioms for the number type being considered, can be scaled by arbitrary, positive real numbers. Local availability leads to the assignment of mathematical universes, $V_{x},$ to each point, $x,$ of space time. $V_{x}$ contains all the mathematics that an observer, $O_{x},$ at $x,$ can know. Each $V_{x}$ contains many types of mathematical systems. These include the different types of numbers (natural numbers, integers, rationals, and real and complex numbers), Hilbert spaces, algebras, and many other types of systems. Space time dependent scaling of number systems is used to define representations, in $V_{x}$, of real and complex number systems in $V_{y}$. The representations are scaled by a factor $r_{y,x}$ relative to the systems in $V_{x}.$ For $y$ a neighbor point of $x,$ $r_{y,x}$ is the exponential of the scalar product of a gauge field, $\vec{A}(x),$ and the vector from $x$ to $y.$ For $y$ distant from $x,$ $r_{y,x}$ is a path integral from $x$ to $y.$ Some consequences of the two premises will be examined. Number scaling has no effect on general comparisons of numbers obtained as computations or as experimental outputs. The effect is limited to mathematical expressions that include space or space time integrals or derivatives. The effect of $\vec{A}$ on wave packets and canonical momenta in quantum theory, and some properties of $\vec{A}$ in gauge theories, is described.

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arXiv2011

New Gauge Fields from Extension of Parallel Transport of Vector Spaces to Underlying Scalar Fields

Gauge theories can be described by assigning a vector space V(x) to each space time point x. A common set of complex numbers, C, is usually assumed to be the set of scalars for all the V{x}. This is expanded here to assign a separate set of scalars, C{x}, to V{x} for each x. The freedom of choice of bases, expressed by the action of a gauge group operator on the V{x}, is expanded here to include the freedom of choice of complex scale factors, c_{y,x}, as elements of GL(1,C) that relate C{y} to C{x}. A gauge field representation of c_{y,x} gives two gauge fields, A(x) and iB(x). Inclusion of these fields in the covariant derivatives of Lagrangians results in A(x) appearing as a gauge boson for which mass is optional and B(x) as a massless gauge boson. B(x) appears to be the photon field. The nature of A(x) is not known at present. One does know that the coupling constant of A(x) to matter fields is very small compared to the fine structure constant.

math-ph↗

arXiv2011

Representations of Each Number Type that Differ by Scale Factors

For each type of number, structures that differ by arbitrary scaling factors and are isomorphic to one another are described. The scaling of number values in one structure, relative to the values in another structure, must be compensated for by scaling of the basic operations and relations (if any) in the structure. The scaling must be such that one structure satisfies the relevant number type axioms if and only if the other structure does.

math.RA↗

arXiv2011

New Gauge Field from Extension of Space Time Parallel Transport of Vector Spaces to the Underlying Number Systems

One way of describing gauge theories in physics is to assign a vector space $V_{x}$ to each space time point $x.$ For each $x$ the field $ψ$ takes values $ψ(x)$ in $V_{x}.$ The freedom to choose a basis in each $V_{x}$ introduces gauge group operators and their Lie algebra representations to define parallel transformations between vector spaces. This paper is an exploration of the extension of these ideas to include the underlying scalar complex number fields. Here a Hilbert space, $\bar{H}_{x},$ as an example of $V_{x},$ and a complex number field, $C_{x},$ are associated with each space time point. The freedom to choose a basis in $H_{x}$ is expanded to include the freedom to choose complex number fields. This expansion is based on the discovery that there exist representations of complex (and other) number systems that differ by arbitrary scale factors. Compensating changes must be made in the basic field operations so that the relevant axioms are satisfied. This results in the presence of a new real valued gauge field $\vec{A}(x).$ Inclusion of $\vec{A}(x)$ into covariant derivatives in Lagrangians results in the description of $\vec{A}(x)$ as a gauge boson that can have mass. The great accuracy of QED suggests that the coupling constant of $\vec{A}(x)$ to matter fields is very small compared to the fine structure constant. Other physical properties of $\vec{A}(x)$ are not known at present.

quant-ph↗

arXiv2009

A Possible Approach to Inclusion of Space and Time in Frame Fields of Quantum Representations of Real and Complex Numbers

This work is based on the field of reference frames based on quantum representations of real and complex numbers described in other work. Here frame domains are expanded to include space and time lattices. Strings of qukits are described as hybrid systems as they are both mathematical and physical systems. As mathematical systems they represent numbers. As physical systems in each frame the strings have a discrete Schrodinger dynamics on the lattices. The frame field has an iterative structure such that the contents of a stage j frame have images in a stage j-1 (parent) frame. A discussion of parent frame images includes the proposal that points of stage j frame lattices have images as hybrid systems in parent frames. The resulting association of energy with images of lattice point locations, as hybrid systems states, is discussed. Representations and images of other physical systems in the different frames are also described.

quant-ph↗

arXiv2008

Reference Frame Fields based on Quantum Theory Representations of Real and Complex Numbers

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.

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