Mathematical Studies on Quantum Systems and Locally Quantum Systems

Paul A. Benioff

Effects on Quantum Physics of the Local Availability of Mathematics and Space Time Dependent Scaling Factors for Number Systems

Konstantin A. Makarov, Eduard Tsekanovskiı̆

Paul A. Benioff

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.

G. Fano, S. M. Blinder

Paul Benioff

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.

Vikke Vuorenpää