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arXiv:1110.1388

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

Paul A. Benioff

Published Oct 6, 2011• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1110.1388

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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, Vx,V_{x},Vx​, to each point, x,x,x, of space time. VxV_{x}Vx​ contains all the mathematics that an observer, Ox,O_{x},Ox​, at x,x,x, can know. Each VxV_{x}Vx​ 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 VxV_{x}Vx​, of real and complex number systems in VyV_{y}Vy​. The representations are scaled by a factor ry,xr_{y,x}ry,x​ relative to the systems in Vx.V_{x}.Vx​. For yyy a neighbor point of x,x,x, ry,xr_{y,x}ry,x​ is the exponential of the scalar product of a gauge field, A⃗(x),\vec{A}(x),A(x), and the vector from xxx to y.y.y. For yyy distant from x,x,x, ry,xr_{y,x}ry,x​ is a path integral from xxx to y.y.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 A⃗\vec{A}A on wave packets and canonical momenta in quantum theory, and some properties of A⃗\vec{A}A in gauge theories, is described.
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, x, to each space time point, x.. The mathematics available to an observer, Ox, at x is contained in 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 x, of mathematical systems in y . If ay or ψy is a number or a quantum state in y, then the corresponding number or state in x is ry,xax or ry,xψx. Here ax and ψx are the same number and state in x as ay and ψy are in y . If y = x+ μdx is a neighbor point of x, then the scaling factor is ry,x = exp( A(x) • μdx) where A is 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 ay and ψy. Internal scaling occurs in expressions with integrals or derivatives over space time. An example is the replacement of the position expectation value, ∫ψ*(y)yψ(y)dy, by ∫x ry,ψx* x(yx)yxψx(yx)dyx. This is an integral in 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. For observers in L and events outside L, at cosmological distances, scaling is not limited by theory experiment agreement requirements.
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,g,g, move numbers between structures at different locations. The effect of the ggg field, or its exponential equivalent, g(y)=eα(y),g(y)=e^{α(y)},g(y)=eα(y), on numbers extends to other mathematical structures, vector spaces, etc. The presence of ααα 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 ααα, A⃗,\vec{A},A, appears in the Lagrangian as a spin 000, real scalar field that couples to the fermion field. Any value for the mass of A⃗\vec{A}A is possible. The lack of direct experimental evidence for the presence of the ggg or ααα 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 ααα are noted.
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.
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.