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.

Universe equivalence means here that for any system type, S,S y is the same system in y asS x is in x .
For the purposes of this work, it is useful to have a specific definition of mathematical systems. Here the mathematical logical definition of a system of a given type as a structure [12,13] is used. A structure consists of a base set, a few basic operations, none or a few basic relations, and a few constants. The structure must satisfy a set of axioms appropriate for the type of system being considered. For example,N satisfies a set of axioms for the natural numbers as the nonnegative elements of a discrete ordered commutative ring with identity [14], is a real number structure that satisfies the axioms for a complete ordered field [15], andC is a complex number structure that satisfies the axioms for an algebraically closed field of characteristic 0 [16].
is a structure that satisfies the axioms for a Hilbert space [17]. Here ψ x is a state variable inH x . There are no constants inH x . The subscript, x, indicates that these structures are contained in x . The other idea introduced here is the use of scaling factors for structures for the different number types. These scale structures are based on the observation [18,19,20] that it is possible to define, for each number type, structures in which number values are scaled relative to those in the structures shown above. The scaling of number values must be compensated for by scaling of the basic operations and constants in such a way that the scaled structure satisfies the relevant set of axioms if and only if the original structure does.
Scaling of number structures introduces scaling into other mathematical systems that are based on numbers as scalars for the system. Hilbert spaces are examples as they are based on the complex numbers as scalars.
The fact that number structures can be scaled allows one to introduce scaling factors that depend on space time or space and time. If y = x+μdx is a neighbor point of x, then the real scaling factor from x to y is defined by Here A is a real valued gauge field that determines the amount of scaling, and µ and dx are, respectively, a unit vector and length of the vector from x to y. Also · denotes the scalar product. For y distant from x, r y,x is obtained by a suitable path integral from x to y.
Space time scaling of numbers would seem to be problematic since it appears to imply that comparison of theoretical and experimental numbers obtained at different space time points have to be scaled to be compared. This is not the case. As will be seen, number scaling plays no role in such comparisons. More generally, it plays no role in what might be called, "the commerce of mathematics and physics".
Space time dependent number scaling is limited to expressions in theoretical physics that require the mathematical comparison of mathematical entities at different space time points. Typical examples are space time derivatives or integrals. Local availability of mathematics makes such a comparison problematic. If f is a space time functional that takes values in some structureS, then "mathematics is local" requires that for each point, y, f (y) is an element of S y . In this case space time integrals or derivatives of f make no sense as they require addition or subtraction of values of f in different structures. Addition and subtraction are defined only within structures, not between structures.
This problem is solved by choosing some point x, such as an observers location, and transforming eachS y into a local representation ofS y onS x . Two methods are available for doing this: parallel transformations for which the local representation ofS y onS x isS x itself, and correspondence transformations. These give a local, scaled representation ofS y onS x in that each element ofS y corresponds to the same element ofS x , multiplied by the factor r y,x .
The rest of this paper explains, in more detail, these ideas and some consequences for physics. The next section describes representations of number types that differ by scaling factors. Sections 3 and 4 describe space time fields of complex and real number structures and the representation of r y,x in terms of a gauge field, as in Eq. 5. This is followed by a discussion of the local availability of mathematics and the assignment of separate mathematical universes to each space time point. Section 6 describes correspondence and parallel transforms. It is shown that A plays no role in the commerce of mathematics and physics. This involves the comparison and movement of the outcomes of theoretical predictions and experiments and the general use of numbers. Section 7 applies these ideas to quantum theory, both with and without the presence of A. Parallel and correspondence transformations are used to describe the wave packet representation of a quantum system. It is seen that there is a wave packet description that closely follows what what is actually done in measuring the position distribution and position expectation value. The coherence is unchanged in such a description.
The next to last section uses "mathematics is local" and the scaling of numbers to insert A into gauge theories. The discussion is brief as it has already been covered elsewhere [18,20]. A appears in the Lagrangians as a boson for which a mass term is not forbidden. The last section concludes the paper.
The origin of this work is based on aspects of mathematical locality that are already used in gauge theories [21,22] and their use in the standard model [23]. In these theories, an n dimensional vector space,V x , is associated with each point, x, in space time. A matter field ψ(x) takes values inV x . Ordinary derivatives are replaced by covariant derivatives, D µ,x , because of the problem of comparing values of ψ(x) with ψ(y) and to introduce the freedom of choice of bases. These derivatives use elements of the gauge group, U (n), and their representations in terms of generators of the Lie algebra, u(n), to introduce gauge bosons into the theories.

Representations of different number types
Here the mathematical logical definition [12,13] of mathematical systems as structures is used. A structure consists of a base set, basic operations, relations, and constants that satisfy a set of axioms relevant to the system being considered. As each type of number is a mathematical system, this description leads to structure representations of each number type.
The types of numbers usually considered are the natural numbers,N , the integers,Ī, the rational numbers, Ra, the real numbers,R, and the complex numbers,C. Structures for the real and complex numbers can be defined bȳ A letter with an over line, such asR, denotes a structure. A letter without an over line, as R in the definition ofR, denotes a base set of a structure. The main point of this section is to show, for each type of number, the existence of many structures that differ from one another by scale factors. To see how this works it is useful to consider a simple case for the natural numbers, 0, 1, 2, · · · . LetN be represented bȳ whereN satisfies the axioms of arithmetic [14]. The structureN is a representation of the fact that 0, 1, 2 · · · with appropriate basic operations and relations are natural numbers. However, subsets of 0, 1, 2, · · · , along with appropriate definitions of the basic operations, relations, and constants are also natural number structures.
As an example, consider the even numbers, 0, 2, 4, · · · inN LetN 2 be a structure for these numbers wherē Here N 2 consists of the elements of N with even number values inN . The structureN 2 shows that the elements of N that have value 2n inN have value n inN 2 . Thus the element that has value 2 inN has value 1 inN 2 , etc. The subscript 2 on the constants, basic operations, and relations inN 2 denotes the relation of these structure elements to those inN . The definition ofN 2 floats in the sense that the specific relations of the basic operations, relation, and constants to those inN must be specified. These are chosen so thatN 2 satisfies the axioms of arithmetic if and only ifN does. A suitable choice that satisfies this requirement is another representation ofN 2 defined byN This structure is called the representation ofN 2 onN . N 2 2 shows explicitly the relations between the basic operations, relations, and number values inN 2 and and those inN . For example, 1 2 ↔ 2, + 2 ↔ +, × 2 ↔ ×/2, < 2 ↔< . These relations are such thatN 2 2 , and therebyN 2 , satisfies the axioms of arithmetic if and only ifN does. N 2 2 also shows the presence of 2 as a scaling factor. Elements of the base set N 2 that have value n inN 2 have value 2n inN . Note that, by themselves, the elements of the base set have no intrinsic number values. The values are determined by the axiomatic properties of the basic operations, relations, and constants in the structure containing them.
This description of scaled representations applies to the other types of numbers as well. For real numbers let r be a positive real number inR, Eq. 6. LetR be another real number structure. Define the representation ofR r onR by the structure,R R r r shows that number values inR r are related to those inR by a scaling factor r.R r r gives the definitions of the basic operations, relation, and constants in R r in terms of those inR. These definitions must satisfy the requirement that R r satisfies the real number axioms if and only ifR r r does if and only ifR does. 1 Note that the base set R is the same for all three structures. Also the elements of R do not have intrinsic number values independent of the structure containing R. They attain number values only inside a structure where the values depend on the structure containing R.
The relationships between number values inR r r ,R r , andR can be represented by a new term, correspondence. One says that the number value a r in R r corresponds to the number value ra inR. This is different from the notion of sameness. InR, ra is different from the value a. However, a is the same value inR as a r is inR r as ra is inR r r . The distinctions between the concepts of correspondence and sameness does not arise in the usual treatments of numbers. The reason is that sameness and correspondence coincide when r = 1.
For complex numbers, the structures, in addition toC, Eq. 6, arē and the representation ofC r onC as Here r is a real number value inC. a is the same number value inC as a r is in C r . Otherwise the description is similar to that for the natural and real numbers. More details on these and other number type representations are given in [19].

Fields of mathematical structures
As was noted in the introduction, the local availability of mathematics results in the assignment of separate structures,S x , to each point, x, of space time. Here S denotes a type of mathematical structure. The discussion is limited to the main system types of concern. These are the real numbers, the complex numbers, and Hilbert spaces. Hilbert spaces are included here because the freedom of choice of scaling factors for number types affects Hilbert spaces as they are based on complex numbers as scalars.

Complex numbers
Parallel transformations betweenC x andC y for two points, x, y, define the notion of same number values between the structures. Let F y,x be an isomorphism fromC x ontoC y . WithC and F y,x defines the notion of same number value and same operation inC y as that inC x . This is expressed by Here a y is the same (or F y,x -same) number value inC y as a x is inC x . Op y is the same operation inC y as Op x is inC x . Op denotes any one of the operations, +, −, ×, ÷. Note that F y,x is independent of paths between x and y. This follows from the requirement that for a path P from x to y and a path Q from y to z, Here Q * P is the concatenation of Q to P . If z = x then the path is cyclic and the final structure is identical to the initial one. This gives the result that This shows that F P y,x is path independent so that a path label is not needed. Note that The subscript order in F y,x gives the path direction, from x to y. At this point the freedom to choose complex number structures at each space time point is introduced. This is an extension, to number structures, of the freedom to choose basis sets in vector spaces as is used in gauge theories [22,21] This can be accounted for by factoring F y,x into a product of two isomorphisms as in Here y = x +νdx is taken to be a neighbor point of x.
The action of W y r and W r y is given bȳ Here r y,x is a real number inC x that is associated with the link from x to y.
As was the case for F y,x the order of the subscripts determines the direction of the link. Thus r x,y is a number inC y for the same link but in the opposite direction and (r x,y ) x r y,x = 1.
Here (r x,y ) x is the same number value inC x as r x,y is inC y . In the following, the subscripts y, x are often suppressed on r y,x to simplify the notation. The structureC r x is defined to be the representation ofC y onC x . As is the case forC r r , Eq. 13, the number values and operations inC r x are defined in terms of the corresponding number values and operations inC x : The multiplication and division by r, shown in × x /r, r÷ x , are operations inC x . Note that the number value r inC x is the multiplicative identity inC r x . Alsō C r x has the same base set, C x , as doesC x . The corresponding definition of W r x is given by W r x is an isomorphism in that Here a x and b x are number values inC x and O x denotes the basic operations inC x . W y r has a similar definition as it is an isomorphism fromC r x toC y . Since the definition is similar it will not be given here.
C r x can also be represented in a form similar to that of Eq. 12 as This structure can be described as the representation ofC y at x. The relation between the number values and operations inC r,x and those inC x is provided byC r x which defines the number values and operations ofC r,x in terms of those inC x . In this sense bothC r,x andC r x are different representations of the same structure. From now onC r,x andC r x will be referred to as the representation of C y at x and onC x respectively.
The relations between the basic operations and constants ofC r x and those of C x lead to an interesting property. Let f r x (a r x ) be any analytic function onC r x . It follows that Here f x is the same function onC x as f r x is onC r x . Also a x and b x are the same number values inC x as a r x and b r x are inC r x . This result follows from the observation that any term (a r The n factors and n-1 multiplications in the numerator contribute a factor of r. This is canceled by a factor of r in the denominator. The one r factor arises from the relation of division inC r x to that inC x . Eq. 27 follows from the fact that Eq. 28 holds for each term in any convergent power series. As a result it holds for the power series itself.

Real numbers
Since the treatment for real numbers is similar to that for complex numbers, it will be summarized here. The representations ofR y at x and onR x are given by Eqs. 10 and 11 as Here r = r x,y is a positive real number. The definition of parallel transforms for complex number structures applies here also. Let F y,x transformR x toR y . F y,x defines the notion of same real number in that a y = F y,x (a x ) is the same real umber inR y as a x is inR x . As was shown in Eqs. 20 and 21, F y,x can be factored into two operators as in W r x defines the scaled representation ofR y onR x . It is given explicitly by Eq. 24. W y r maps the scaled representation ontoR y . Eqs. 27 and 28 also hold for the relations between any real valued analytic function onR r x and its correspondent onR x in that f r x (a r x ) = rf x (a x ).

Hilbert spaces
As noted in the introduction, Hilbert space structures have the form shown in Eq. 4 asH Complex numbers are included implicitly in that Hilbert spaces are closed under multiplication of vectors by complex numbers. Also scalar products are bilinear maps with complex values. As was the case for numbers, parallel transformation ofH y to x mapsH y ontoH x . If scaling of the numbers is included, then the local representation of This equation gives explicitly the relations of operations and vectors of the local representation ofH y to those inH x . The relations are defined by the requirement thatH r x satisfy the Hilbert space axioms [17] if and only ifH x does. 2 Here rψ x is the same vector inH r x as ψ x is inH x . The description ofH r x given here is suitable for use in section 7 where wave packets for quantum systems are discussed. For gauge theories, the Hilbert spaces contain vectors for the internal variables of matter fields. In this case one has to include a gauge field to account for the freedom to choose bases [22,21]. The local representation ofH y onH x is then given by [18,20] If theH x are n dimensional, then V is an element of the gauge group, U (n).

Gauge fields
As was noted in the introduction, for y = x +νdx, r y,x can be represented as the exponential of a vector field: 2 Support for the inclusion of r as a vector multiplier, as inH r x , Eq. 32, is based on the equivalence between finite dimensional vector spaces and products of complex number fields [17]. IfHy andHx are n dimensional spaces, thenHy ≃C n y andHx ≃C n x . These equivalences extend to the local representation ofHy onHx. As the local representation ofCy onCx,C r x is the scalar field base for the local Hilbert space representation. It follows thatH r x is equivalent to (C r x ) n . A vector in (C r x ) n corresponds to an n-tuple, {a r x,j : j = 1, · · · , n} of number values inC r x . Use of the fact that the value a r x,j inC r x , corresponds to the number value ra j,x inCx shows that the n-tuple in (C r x ) n corresponds to the n-tuple, r{a j,x : j = 1, · · · , n} inC n x . These equivalences should extend to the case wherē Hy andHx are separable, which is the case here.
(sum over repeated indices implied). A(x) is also referred to as a gauge field as it gives the relations between neighboring complex number structures at different space time points. To first order in small quantities, The use of r y,x makes clear the fact that the setup described here is a generalization of the usual one. To see this, set A(x) = 0 everywhere. Then r y,x = 1 for all y, x and the local representations ofC y andR y onC x andR x areC x and R x . Since theC x andR x are then independent of x, one can replaceC x andR x with just one complex and real number structure,C andR.

Scale factors for distant points
The description of r y,x can be extended to points y distant from x. Let P be a path from x to y parameterized by a real number, s, such that P (0) = x and P (1) = y. Let r P y,x be the scale factor associated with the path P . If a y is a number value inC y , then a y corresponds to the number value, r P y,x a x , inC x where a x = F x,y a y is the same number value inC x as a y is inC y .
One would like to express r P y,x as an exponential of a line integral along P of the field A(x). However this is problematic because the integral corresponds to a sum over s of complex number values inC P (s) . Such a sum is not defined because addition is defined only within a number structure. It is not defined between different structures. This can be remedied by referring all terms in the sum to one number structure such asC x . To see how this works, consider a two step path from x to y = x +ν 1 ∆ x and from y to z = y +ν 2 ∆ y . ∆ y is the same number inC y as ∆ x is inC x .
Let a z be a number value inC z . a z corresponds to the number value r z,y × y a y inC y . Here a y = F y,z a z is the same number value inC y as a z is inC z . InC x , r z,y × y a y corresponds to the number value given by Here (r z,y ) x = F x,y (r z,y ) is the same number inC x as r z,y is inC y and × x /r y,x is the representation of × y onC x . TheC x multiplications are implied in the righthand term and a x = F x,y a y . The factor (r z,y ) x r y,x can be expressed in terms of the field A. It is ∆ y is replaced here by its same value ∆ x inC x . Let P be an n step path where P (0) = x 0 = x, P (j) = x j , P (n−1) = x n−1 = y and x j+1 = x j +ν j ∆ xj . Then r P y,x a x is given by The subscript x denotes the fact that all terms in the product and in the exponential, are values inC x . For example (r xj+1,xj ) x = F x,xj r xj+1,xj is the same value inC x as r xj+1,xj is inC xj . An ordering of terms in the product of Eq, 38 is not needed because the different r factors commute with one another. This can be extended to a line integral along P . The result is [20] r P y,x = exp{ The subscript x on the factors in the integral mean that the terms are all evaluated inC x . It is unknown if the field A and thereby r P y,x is or is not independent of the path P from x to y. If A is not integrable, then the path dependence introduces complications. In particular it means that for y distant from x, there is no path independent way to describe the local representation ofC y onC x . The local representation would have to include a path variable as inC In this work, this complication will be avoided by assuming that A is integrable. Then r p y,z = r y,x , independent of P . Let P be a path from x to y and Q be another path from y to x. Then integrability gives Here Q * P is the concatenation of Q to P . This result gives

Local availability of mathematics
The local availability of mathematics means that for an observer, O x , at point x, the mathematics that O x can use, or is aware of, is locally available at x. Since mathematical systems are represented by structures, [12,13], one can use x to denote the collection of all these structures.
x includes real and complex number and Hilbert space structuresR x ,C x ,H x , structures for operator algebras as well as many other structure types. All the mathematics that O x uses to make physical predictions and physical theories use the systems in x . Similarly, all the mathematics available to an observer O y at point y is contained in y .
An important requirement is that the mathematics available, in principle at least, to an observer must be independent of the observers location. This means that y must be equivalent to x . For each system structure in y , there must be a corresponding structure in x . Conversely, for each system structure in x there must be a corresponding structure in y . Furthermore the corresponding structures in y and x must be related by parallel transforms that map one structure to another. These parallel transforms define what is meant by the same structure and the same structure elements and operations in x as in y , and conversely. This use of parallel transforms is an extension to other types of mathematical systems, of the definitions and use of parallel transforms, Section 3, to relate complex and real number structures at different space time points. It is based on the description of each type of mathematical system as structures, each consisting of a base set, basic operations, relations and constants, that satisfy a set of axioms relevant to the system type. [12,13].
The association of an observer to a point, as in O x , is an idealization, mainly because observers, e.g. humans, have a finite size. Because of this, an observer's location is a region and not a point. This is the case if one notes that the observer's brain is the seat of all mathematical knowledge and limits consideration to the brain. In addition, quantum mechanics introduces an inherent uncertainty to the location of any system. In spite of these caveats, the association of an observer to a point will be used here.
An important aspect of x is that O x must be able to use the systems in x to describe the systems in y . This can be done by means of parallel transform maps or correspondence maps from systems in y to those in x . Parallel transforms map elements and operations of system structuresS y to the same elements and operations ofS x . In this case O x can use the mathematics ofS x as a stand in for the mathematics ofS y .
Correspondence maps take account of scaling of real and complex numbers in relating systems at y to those at x. In this case O x describes the mathematics ofS y in terms of the local representation,S r x , ofS y onS x . If S = R or S = C then O x would describe the properties ofR y orC y in terms of the local scaled systemsR r x andC r x . The existence of correspondence maps means that for each system type, S, x contains all the scaled systems,S r x , for each point y, in addition toS x . (Recall r = r y,x .) They include scaled real numbers,R r x , complex numbers,C r x , and scaled Hilbert spaces,H r x , as well as many other system types. All these scaled systems are available to an observer, O x at x. Since they are locally available, O x takes account of scaling by using them to make theoretical calculations that require inclusion of numbers or vectors at different space or space time points. If O x does not use these correspondence maps and restricts use to parallel transform maps only, then the setup becomes simpler in that eachS r x is identical toS x . This raises the question of when correspondence maps can be used instead of parallel transform maps. This will be discussed in the next sections. Here the use of correspondence maps follows from the inclusion into physics of the freedom to choose number systems at different space time points. In this sense it extends the freedom to choose bases in vector spaces in gauge field theory [21,22] to the underlying scalars.

Correspondence maps and parallel transform maps
It is proposed here that correspondence maps be used in any theoretical physics expression that requires These involve what is referred to here as the commerce of mathematics and physics. 3 To see this suppose O x wants to compare the numerical output of either an experiment or a theoretical computation, done at x, with the numerical output of either an experiment or computation done at y. Let b x and d y be the real valued numerical outcomes obtained. Use of the correspondence maps means that O x would compare b x with the local representation of d y at x, that is, with the number r y,x d x .
Here d x is the same number inR x as d y is inR y . This is contradicted by experience. There is no hint of a factor r y,x in comparing outcomes of repeated experiments, or comparing experimental outcomes with theoretical predictions, or in any other use of numbers in commerce. If one ignores statistical and quantum uncertainties, numerical outcomes of repeated experiments or repeated computations are the same irrespective of when and where they are done. 4 The reason for this is a consequence of a basic fact. This is that no experiment and no computation ever directly yields a numerical value as an outcome. Instead the outcome of any experiment is a physical system in a physical state that is interpreted as a numerical value. Similarly the outcome of a computation is a physical system in a state that is interpreted as a numerical value.
The crucial word here is interpreted. If ψ y is the output state of a measurement apparatus for an experiment at point y, and φ x is the output state of a computation at point x, then the numerical values of these output states are given by a y = I y (ψ y ) and b x = I x (φ x ). Here I y and I x are interpretive maps from the output states of the measurement system intoR y and from the computation output states intoR x respectively. The space time dependence of the maps is indicated by the x, y subscripts.
The "Naheinformationsprinzip", no information at a distance principle [24,22], forbids direct comparison of the information in ψ y with that in φ x . This means that I y (ψ y ) and I x (φ x ) cannot be directly compared. Instead the in-formation contained in ψ y and that contained in φ x must be transported, by physical means, to a common point for comparison.
There are many different methods of physical information transmission. Included are optical and electronic methods as well as older slower methods. All methods involve motion of an information carrying physical system from one point to another, "information is physical" [25]. The physical system used should be such that the state of the information carrying degrees of freedom does not change during transmission from one point to another. Figure 1 illustrates schematically, in one dimensional space and time, the nonrelativistic transmission of a theory computation output state obtained at x ′ , u and an experimental output state obtained at y, v to a common point, x, t, for comparison. Here x, x ′ , y are space locations and u, v, t are times. Figure 1: A simple example of comparing theory with experiment. The ovals denote the output computation and experiment systems in states φ c and φ e at space and times, x ′ , u and y, v. P and Q denote the paths followed by these systems. One has P (u) = x ′ and Q(v) = y. The double oval in the center denotes the the two systems at the point, x, of path intersection where P (t) = Q(t) = x. The interpretation maps are denoted by I P (s),s and I Q(s),s for different times s. The real number structuresR P (s),s andR Q(s),s are associated with each point in the paths P and Q. F x,t;x ′ ,u and F x,t:y,v are parallel transform operators that map the real number structures at the points of theory and experiment completion to the point of path intersection.
The figure, and the discussion, illustrate a general principle. All activities in the commerce of mathematics and physics consist of physical procedures and operations that generate physical output systems in states that are interpreted as numerical values. The "no information at a distance" principle forbids direct comparison of the associated number values at different points. Instead the systems or suitable information carriers must be brought to a common point where the numerical information, as number values in just one real number structure, can be locally compared.
Similar considerations apply to storage of outcomes of experiments or computations either in external systems or in the observers brain. As physical dynamic systems, observers move in space time. If P is a path taken by an observer, with P (τ ) the observers location at proper time τ, the mathematics available to O P (τ ) is that in P (τ ) . If φ(P (τ )) denotes the state of a real number memory trace in an observers brain, then the number value represented by φ(P (τ )) is given by I P (τ ) (φ(P (τ ))). This is a number value inR P (τ ) . At a later proper time τ ′ , the number value represented by the memory trace is I P (τ ′ ) (φ(P (τ ′ )). If there is no degradation of the memory trace, then I P (τ ) (φ(P (τ ))) is the same number value inR P (τ ) as I P (τ ′ ) (φ(P (τ ′ )) is inR P (τ ′ ) . Correspondence maps play no role here either.

Quantum theory
As might be expected, the local availability of mathematics and the freedom of choice of number scaling factors, have an effect on quantum theory. This is a consequence of the use of space time integrals and derivatives in the theory. For example, one would expect to see the gauge field A appear in quantum descriptions of physical systems. To see how this effect arises, it is useful to limit the treatment to nonrelativistic quantum mechanics on three dimensional Euclidean space, R 3 .

Effect of the local availability of mathematics on quantum theory
The local availability of mathematics requires that the usual setup of just onē C,R,H is replaced by separate number structures,R x ,C x and separate Hilbert spaces,H x , associated with each x in R 3 . 5 It follows that mathematical operations, such as space or time derivatives or integrals, which involve nonlocal mathematical operations on numbers or vectors at different points, cannot be done. The reason is that these operations violate mathematical locality.
To preserve locality, one must use either parallel transformations or correspondence transformations. These two methods are well illustrated by considering a single particle wave packets. The usual representation has the form ψ = ψ(y)|y dy where the integral is over all space points in R 3 . One result of the local availability of mathematics is that, for each y, the vector. ψ(y)|y . is inH y , just as ψ(y) is a number value inC y . It follows that the space integral over y makes no sense. It describes a suitable limit of adding vectors that belong to different Hilbert spaces. Addition is not defined between different spaces; it is defined only within one Hilbert space and complex number structure.
The use of parallel transformations replaces Eq. 42 by Here ψ(y) x = F x,y ψ(y) is the same number value inC x as ψ(y) is inC y , the number triple, y x , in |y x x is the same triple inR 3 x as y is inR 3 y , and |y x x is the same state inH x as |y is inH y . The differentials dy x = dy 1 x dy 2 x dy 3 x refer toR 3 x . The subscript x in indicates that the integral is based onH x ,C x . The representations of sameness given above are shown explicitly by Also |F x,y (y) x is the same basis vector inH x as |y is inH y .
Note that the point x on which the integral is based is arbitrary. Eq. 43 holds if the subscript x is replaced by another point z. Then the integral is based onC z ,H z .
The use of parallel transforms is applicable to other aspects of quantum mechanics. For each y in R 3 , the momentum operator, p y , for vectors inH y is given by Here, i y , y are numbers inC y . The action of p y on a vector ψ at point y gives for the jth component p y,j ψ = i y y ∂ j,y ψ = ψ(y + dy j ) − ψ(y) dy j .
As was the case for the space integral, this expression makes no sense because ψ(y + dy j ) is inC y+d j y and ψ(y) is inC y . This can be remedied by replacing ∂ j,y by ∂ ′ j,y where Here ψ(y + dy j ) y = F y,y+d j y ψ(y + dy j ). It follows from this that the expression for the momentum becomes The Hamiltonian for a single quantum system in an external potential, acting on a state, ψ y at point y, is given by Here, y and m y are Planck's constant and the particle mass. They have values inR y . The values of the external potential, V (y), are also inR y .
The main difference between this and the usual expression for a Hamiltonian is the replacement of ∂ j,y with ∂ ′ j,y . Otherwise, the expressions are the same. For a single particle state, ψ, the momentum representation is ψ = ψ(p)|p dp.
Here dp = dp 1 dp 2 dp 3 . Since the amplitude ψ(p) is a complex number value and no location for the value is specified, one may choose any location, x, such as that of an observer, O x , to assign ψ(p) as a number value inC x and the integral as an element of x .
The relation between ψ(p) and ψ(x) is given by the Fourier transform. The components of the space integral in must all be mapped to a common point, x, for the integral to make sense. This gives Here is the same number inC x as e iz pzz is inC z . The treatment described can be extended to multiparticle entangled states. It is sufficient to consider two particle states. For example a two particle state ψ 1,2 where the total momentum of the two particles is 0 can be expressed by Use of Fourier transforms gives The integral must be transformed to a Hilbert space with just one scalar field. For a point x withC x ,H x , the integrand factors are parallel transformed to obtain Here Also p x is the same value inC x as p is inC z1 in the z 1 integral, Eq. 54, and −p x is the same value inC x as −p is inC z2 in the z 2 integral.

Inclusion of number system scale factors
The above shows that the imposition of "mathematics is local" on quantum theory is more complex than the usual treatment with just one scalar field and one Hilbert space for all space points. Since the description with parallel transforms is equivalent to the usual one, the added complexity is not needed if one goes no further with it. This is not the case if one extends the treatment to include space dependent scaling factors for the differentC x ,R x . For a given x, the local representations ofC y onC x are given by scaled representations,C ry,x x , ofC y onC x . Also the local representation ofH y onH x with effects of the number scaling included, is given byH For y = x +νdx, a neighbor point of x, the scaling factor, r y,x is given by, r y,x = e A(x)·νdx , Eq. 34. If y is distant from x and A(x) is integrable, then, expressing r y,x as an integral along a straight line path from x to y gives, (Eq. 39) r y,x,i (57) Here x i = x·î and y i = y ·î are the components of x and y in the direction i. The last equality assumes that the components of A commute with one another. 6 The subscript x indicates that the integrals are defined onR x .
The presence of A(x) affects the expression of a wave packet state ψ as given by Eq. 43. In this case the wave packet expansion of ψ x is given by where r y,x is given by Eq. 57. This result is obtained by noting that for each y the local representation of H y ,C y onH x ,C x , with scaling factor included, isH that is the same vector as ψ(y)|y is inH y , is denoted by ψ(y) r x · r x |y r x . Here ψ(y) r x is the same number value inC ry,x x as ψ(y) is inC y and |y r x is the same vector inH ry,x x as |y is inH y . This follows from the observation that y r x and y are the space positions associated with |y r x and |y inH ry,x x andH y respectively. Scalar vector multiplication inH ry,x x is shown by · r x .
The corresponding state onH x is obtained by noting that ry,x r y,x |y x = r y,x · x |y x .
(59) This is the result shown in Eq. 58. The use of (· r x ) x = ·x ry,x , Eq. 32, is based on the requirement thatH ry,x x satisfies the same Hilbert space axioms [17] as does H x .
It must be emphasized that the usual predictions of the quantum mechanical properties of wave packets, with A(x) = 0 everywhere, do a good job of prediction of experimental results. So far, quantum mechanical predictions and experiments have not shown the need for the presence of A. This shows that the effect of the A field must be very small, either through the values of the field itself or by use of a very small coupling constant, g, of the field to numbers and vectors. This would be accounted for by replacing A in Eqs. 34 and 57 by g A.
In this sense the presence of A is no different than the presence of the gravitational field. In theory, a proper description of quantum mechanics of systems should include the effects of the gravitational field. However, it can be safely neglected because the field is so small, at least far away from black holes where quantum physics is done.
Another feature of Eq. 58 is the dependence on the reference point x. This can be removed by restricting the integration volume to a region of space, excluding x, where the region contains essentially all of ψ. This is what one does in any experiment since ψ is prepared in a region that does not include the observer.
The removal of x dependence then follows from expressing Eq. 58 as a sum of two terms, one as the integral over V and the other over all space outside V : The subscript, W, on the second integral means that it refers to integration over all space outside V. Here x is a point in W . Assume that V is chosen so that the integral over W can be neglected. Then This equation has a problem in that the correspondence transforms are extended from any point in V to a point outside V. However these transforms are restricted here to apply within space or space time integrals, and not outside the integration volume. This can be fixed by choice of a point z on the surface of V and replacing r y,x by U x,z r y,z . The factor r y,z accounts for the correspondence transform from a point y in V to a point z on the surface of V , and U x,z is a unitary operator that parallel transforms the result from z to x. Then one has The subscript z on ψ x,z indicates a possible dependence on the choice of z on the surface of V . Figure 2 illustrates the setup for two points y, u in V . Figure 2: Representation of scaling factors in the integrals from point z on the surface of V to points y and u. The direction implied in the order of the subscripts of r is opposite to the direction of the parallel transformations taking the integrand factors from y and u to z. U x,z parallel transforms ψ z to ψ x,z . This is the same vector inH x as ψ z is inH z .
This result shows that the wave packet representation is independent of x, such as an observers location, provided it is outside of V . However, to the extent that A cannot be neglected, the representation does depend on the location of z. To see what this dependence is, let w be another point on the surface of V . Then, following Eq. 62, ψ x,w is given by (63) For the comparison, at x, of ψ x,z with ψ x.w , it is sufficient to compare, inH z , the parallel transformation of ψ w to z with ψ z , inH z . The parallel transformation of ψ w to z is given by Use of the fact that parallel transforms of numbers and and vectors from y to w and then to z are the same as transforms from y to z gives Since A is integrable, one can write (r y,w ) z = (r y,z r z,w ) z = r y,z (r z,w ) z to obtain (ψ w ) z = (r z,w ) z z,V r y,z ψ(y) z |y z dy z .
The subscript, z, denotes parallel transformation to z, of mathematical elements that are not in z . No subscript appears on r y,z as it is already a number value inR z . This shows that (ψ w ) z differs from ψ z by a factor, (r z,w ) z = (r w,z ) −1 , Eq. 41. The difference is preserved on parallel transformation to x in that (ψ w ) x = ψ x,w differs from (ψ z ) x = ψ x,z by a factor (r z,w ) x .
If the effect of A is small, then it is useful to express r y,x as an expansion to first order in the exponential. For example the expression for ψ x in Eq. 58 becomes, Hereν is a unit vector along the direction from x to y. The first term of the expansion corresponds to the usual case with A equal to 0 everywhere. The x dependence arises from the second term, which gives the correction due to the presence of A.
The restriction of the integration to a finite volume V, as in Eq. 62, removes the dependence on x in that ψ x is the same vector inH x as ψ y is inH y provided y is not in V . Expansion of (r y,z ) x to first order in small terms shows that the z dependence arises from the A containing term as in Eq. 68.
The dependence on z can be appreciable because z can be any point on the surface of V . What is interesting is that this dependence can be greatly reduced by using the properties of actual measurements to minimize the effect of A on the predicted expectation value.
Consider a position measurement on a system in state ψ x , Eq. 58. The expectation value for this measurement, calculated at x, is given by This expectation value 7 is an idealization or what one does. It does not take account of what one actually does. Typically, position measurements are done by dividing a volume of space up into cubes and measuring the relative frequency of occurrence of the quantum system in the different cubes. A measurement consists of a large number of repetitions of this measurement on repeated preparations of the system in state ψ. Assume the cubes in space have volume ∆ 3 where ∆ is the length of a side. Then outcomes of the repeated experiment are "yeses" from the cube detectors whose locations are denoted by triples, j, k, l of integers. Each "yes" means that the location is somewhere in the volume of the responding detector cube located at position, z j,k,l = j∆, k∆, l∆. The local availability of mathematics means that z j,k,l is a triple of numbers inR z j,k,l .
The presence of parallel and correspondence transformations enables physical theory to express exactly what is done experimentally. Eq. 58 for ψ x is replaced by an expression that limits integrals with scaling factors to the cube volumes and parallel transports these integrals to a common point x where they can be added together. The result, ψ ′ x , is given by Here the sum is over all cubes. Each cube is labeled by a point z = z j,k,l on the cube surface. Each integral over the volume, V j,k,l = ∆ 3 , of cube, j, k, l, is a vector inH z . Within each integral, the r factor scales the values of each integrand at point w to values at z. U x,z parallel transforms the integrals at different z to a common point x.
The theoretical expectation value for the experimental setup described here is given by The effect of the r factor is smaller here than it is in the expectation value using ψ x . The reason is that it is limited to integrations over small volumes. This representation of the prediction is supported by the discussion on mathematical and physical commerce. The "no information at a distance" principle requires that the information contained in the outcomes of each position measurement, as physical systems in "yes" or "no" states for each point z j,k,l , be transmitted by physical means to x where the results of the repeated measurements are tabulated. The tabulation is all done at x. No factor involving A appears in the transmission or tabulation.
As noted, the effect of A appears only in the integrals over the volumes ∆ 3 . In these integrals, to first order, Since the integral is limited to points within the volume ∆ 3 , it is clear that as ∆ → 0, the integrals for each cube also approach 0. This shows that the effect of the A field diminishes as the accuracy of the measurement increases. In the limit ∆ = 0, the A field disappears and one gets the usual theoretical prediction without A present. However, the Heisenberg uncertainty principle prevents the limit, ∆ = 0, from actually being achieved.
The presence of the A field affects other quantum mechanical properties of systems. For example, the description of the momentum operator with A = 0, replaces Eq. 48 by D j,y ψ is given by altering Eq. 47 to read D j,y ψ = r y+dy j ,y ψ(y + dy j ) y − ψ(y) dy j .
Here r y+dy j ,z ψ(y+dy j ) y is the number value inC y that corresponds to ψ(y+dy j ) inC y+dy j .
Using the fact that r y+d j y,y = e Aj (y)d j y and expansion to first order in the exponential gives, The momentum components become p A,j,y = i y y D j,y = i y y (∂ ′ j,y + A j (y)).
This expression is similar to that for the canonical momentum in the presence of an electromagnetic field. Note that A j (y) is pure real. The expressions for the Hamiltonian for a single particle remain as shown in Eq. 49 except that ∂ ′ is replaced by D. For example Eq. 49 becomes with D y,j given by Eq. 76. Inclusion of scaling factors into the two particle state entangled by momentum conservation is straightforward. This is achieved by including scale factors in the two particle space integral, x (dz 1 ) x (dz 2 ) x , in Eq. 55. The result is Here r z1,x and r z2,x are given by Eq. 57.

Gauge theories
One approach [22] to gauge theories already makes partial use of the local availability of mathematics with the assignment of an n dimensional vector space to each x. Here the vector space is assumed to be a Hilbert space,H x , at each x. ThisH x is quite different from that discussed in the previous section in that the vectors inH x refer to the internal states of matter fields. Matter fields ψ are functionals where for each space time point x, ψ(x) is a vector inH x .
The freedom of choice of a basis [21,22] in eachH x is reflected in the factorization, of a parallel transform operator, U y,x , [24] fromH x toH y where y = x +νdx is a neighbor point of x. 8 The unitary operator V y,x expresses the freedom of basis choice. As such it is an element of the gauge group U (n) with a Lie algebra representation [27,28] Sum over repeated indices is implied. The τ j are the generators of the Lie algebra su(n) and the Ω j µ (x) are the components of the n different gauge fields, Ω j (x). Ξ(x) is the gauge field for the U (1) factor of U (n).
The covariant derivative of the field, ψ, is expressed by Here V µ,x is the µ component of V y,x . Expansion of the exponential to first order in small quantities gives Coupling constants, g 1 and g 2 , have been added. The definition of ∂ ′ µ,x ψ is essentially the same as that given in Eq. 47. It is given by Here The covariant derivative, Eq. 83, accounts for the local availability of mathematics and the freedom of basis choice. It does not include the effects of scaling factors for numbers. This is taken care of by replacing V µ,x ψ(x + dx µ ) x in Eq. 82 by r x+dz µ ,x V µ,x ψ(x + dx µ ) x . This is a vector in the local representation,H r x , Eq. 32, ofH y onH x .
Expansion of the exponentials to first order adds another term to D µ,x in Eq. 83. One obtains [18,20] A coupling constant, g r , for A(x) has been added. The coupling constants, and i are all number values inC x . The physical properties of the gauge fields in D µ,x are obtained by restricting the Lagrangians to only those terms that are invariant under local and global gauge transformations [28]. For Abelian gauge theories, such as QED, Ω(x) is absent. Invariance under local gauge transformations, Λ(x), requires that the covariant derivative satisfy [28] D ′ µ,x is obtained from D µ,x by replacing A µ (x) and Ξ µ (x) with their primed values, A ′ µ (x), Ξ ′ µ (x). The presence of the primes allows for the possible dependence of the fields on the local U (1) gauge transformation, Λ(x) where Use of Eq. 84 and separate treatment of real and imaginary terms gives the following results: [28] This shows that the real field A is unaffected by a U (1) gauge transformation. It also shows that Ξ µ (x) transforms in the expected way as the electromagnetic field.
As is well known the properties of the Ξ field show that it is massless. The reason is that a mass term for this field is not locally gauge invariant [22,28].
Unlike the case for the Ξ field, a mass term can be present for the real A field. This suggests that it represents a gauge boson for which mass is optional. That is, depending on what physical system A represents, if any, the presence of a mass term in Lagrangians is not forbidden.
For nonabelian gauge theories, such as U (2) theories, Eq. 88 still holds. However there is an additional equation giving the transformation properties of the three vector gauge fields under local SU (2) gauge transformations. These properties result in the physical representation of these fields in Lagrangians as charged vector bosons [28]. The A and Ξ bosons are still present.

Physical properties of the A field from the gauge theory viewpoint
At this point it is not known what physical system, if any, is represented by the A field. Candidates include the inflaton field [29,30], the Higgs boson, the graviton, dark matter, and dark energy. One aspect that one can be pretty sure of is that the ratio of the A field -matter field coupling constant, g r , to the fine structure constant, α, must be very small. This is a consequence of the great accuracy of the QED Lagrangian and the fact that the A field appears in covariant derivatives for all gauge theory (and other) Lagrangians. As was noted, Ξ is the photon field. Inclusion of this field and a Yang Mills term for the dynamics of this field into the Dirac Lagrangian gives the QED Lagrangian [28].

Conclusion
This work is based on two premises: the local availability of mathematics and the existence of scaling factors for number systems. Local availability is based on the idea that the only mathematics that is directly available to an observer is that which is, or can be, in his or her head. Mathematical information that is separate from an observer, O x , at space time point x, such as a textbook or a lecturer at point y, must be physically transmitted, e.g. by acoustic or light waves, to O x where it becomes directly available.
This leads to a setup in which mathematical universes, x , are associated with each point x. If an observer moves through space time on a world line, P (τ ), parameterized by the proper time τ, the mathematics directly available to O P (τ ) at time τ is that in P (τ ) .
Each x contains many types of mathematical systems. IfS x is in x , then y contains the same system type,S y , and conversely. Each x contains the different types of number systems and many other systems that are based on numbers. Included are the real and complex numbersR x andC x .
Here the mathematical logical definition [12,13] of each type of system as a structure is used. A structure consists of a base set, basic operations, relations, and constants that satisfies axioms appropriate for the type of structure considered. Examples areR andC, Eq. 6 for the real and complex numbers.
For each type of number structure it is possible to define many structures of the same type that differ by scaling factors [19]. For each real number r, one can define structuresR r , Eq. 11, andC r , Eq. 13, in which a scale factor r relates the number values inR r andC r to those inR andC. The scaling of number values must be compensated for by scaling of the basic operations and relations in a manner such thatR r andC r satisfy the relevant axioms for real and complex numbers if and only ifR andC do.
The local availability of mathematics requires that one be able to construct local representations ofC y onC x . Two methods were described. One uses parallel transformations. These define or represent the notion of sameness between mathematical systems at different points. If F x,y is a parallel transform map fromS y ontoS x , then for each element, w y , inS y , w x = F x,y (w y ) is the same element inS x as w y is inS y . In this case the local representation, W x,ySy , of S y onS x isS x itself.
The other method uses what are called correspondence maps. These combine parallel transformations with scaling. The local representation ofC y onC x is C r x , which is a scaling ofC x by a factor r = r y,x . (From now onR is not explicitly mentioned as it is implicitly assumed to be part ofC.) The local representation of an element, a y , ofC y corresponds to the element r y,x a x inC x .
Here a x = F y,x a y is the same element inC x as a y is inC y .
It was seen that the scaling of numbers plays no role in the general use of numbers in mathematics and physics. This includes such things as comparing outcomes of theory predictions with experimental results or in comparing outcomes of different experiments. More generally it plays no role in the use of numbers in the commerce of mathematics and physics. The reason is that theory computations and experiment outcomes obtained at different locations are never directly compared. Instead the information contained in the outcomes as physical states must be transmitted by physical systems to a common point. There the states of the physical transmittal systems are interpreted locally as numbers, and then compared.
In this work, number scaling was limited to theory calculations that involve space time derivatives or integrals. Examples of this were described in quantum theory and in gauge theories. An example discussed in some detail was the expansion of a wave packet ψ = ψ(y)|y dy. Since ψ(y)|y is a vector inH y , the integrand has to be moved to a common point, x for the integral to make sense. This can be done by parallel transform maps which give or by correspondence maps which give Here the scaling factor, r y,x , is the integral from x to y of the exponential of the gauge field, A(y), as in Eq. 57. It was also seen that one can use both transform and correspondence maps to express the wave packet in a form that reflects exactly what one does in an experiment that measures either the spatial distribution or the position expectation value of a quantum particle. If the experiment setup consists of a collection of cube detectors of volume ∆ 3 that fill 3 dimensional Euclidean space, the outcome of each of many repeated experiments is a triple of numbers, j, k, l that label the position, j∆, k∆.l∆ of the detector that fired.
As was seen in the discussion of mathematical and physical commerce, the outcomes of repeated experiments must be physically transported to a common point, x where they are interpreted as numbers inR x for mathematical combination. It follows that the scaling factors are limited to integration over the volume of each detector. This results in the replacement of ψ x by ψ ′ x where, Eq. 70, ψ ′ x = j,k,l U x,z V j,k,l r w,z ψ(w) z |w z dw z .
The sum is over all cubes. z = z j,k,l is a point on each cube surface. Each integral is over the volume, V j,k,l , of each cube. The r factor, Eq. 72, scales the values of each integrand at point w to values at z. U x,z parallel transforms the integrals at different z to a common point x.
The state, ψ ′ x , differs from ψ x , or the usual quantum mechanical wave packet expression for ψ, in that it ties theory closer to experiment. It also reduces the effect of scaling to the sum of the effects for the volumes of each of the detectors. The effect is reduced because, for any point w in the sensitive volume of the experiment, the effect of A on the transform from w to x is limited to the part of the path in the detector volume containing w. As the detector volumes go to 0, so does the effect of A. The increase in the number of detectors as the volume of each gets smaller does not remove this effect.
In this sense the usual quantum theory wave packet integral for ψ is a limit in that it is independent of experimental details. Unlike the case for the usual representation of ψ, use of ψ ′ x to make predictions will give values that depend on experimental details. The fact that there is no indication, so far, of such dependence, at least to the accuracy of experiment, means that the effect of A(x) must be very small. Whether the small effect is due to small values of A itself or a small value of a coupling constant of A to states and matter fields, is not known at present.
The relation between ψ ′ x and the usual integral for ψ is further clarified by the observation that ψ x → ψ as the detector volume goes to 0. However, the Heisenberg uncertainty principle prevents experimental attainment of this limit.
It must be emphasized that the tying the wave packet integral to experiment details, as with ψ ′ x has nothing to do with collapse of the wave packet during carrying out of the experiment. ψ ′ x is just as coherent a state as is ψ. The gauge field A also appears in the expression for the canonical momentum. The usual expression for momentum p = j i ∂ j,x is replaced by, Eq. 73 where ∂ ′ j,x , Eq. 47, accounts for the local availability of mathematics. As a covariant derivative, D µ,x appears in gauge theories with additional terms. It was seen that the limitation of Lagrangians to terms that are invariant under local gauge transformations, [22,28], results in A appearing as a gauge boson for which mass is optional. This is the case for both Abelian and nonabelian gauge theories.
The physical nature of A, if any, is unknown. What is known is that the great accuracy of QED requires that the coupling constant of A to matter fields must be very small.
It must be emphasized that this work is only a first step in combining "mathematics is local" with the freedom of choice of scaling factors for number structures. An example of work for the future is to determine the effect of number scaling factors on geometry. It is suspected that the scaling factors may induce conformal transformations into geometry. More work also needs to be done on the effects of number scaling on quantum mechanics. An interesting question here is whether scaling factors are needed at all in classical mechanics.
Finally, one may hope that this work provides a real entry into the description of a coherent theory of physics and mathematics together. Such a theory would be expected to describe mathematics and physics together as part of a coherent whole instead of as two separate but closely related disciplines.