# Fiber bundle description of number scaling in gauge theory and geometry

## Abstract

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, , is followed by a description of gauge theory. A fiber at point of contains a pair of scaled complex number and vector space structures, for each in A space time dependent scalar field, , determines, for each the scaling value of the vector space structure that contains the value, of a vector field at Vertical components of connections between neighboring fibers are taken to be the gradient field, of Abelian gauge theory for these fields gives the result that is massless, and no mass restrictions for Addition of an electromagnetic field does not change these results. In the Mexican hat Higgs mechanism 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 pairs, for each and nonnegative real is zero everywhere. The 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 means that either it couples very weakly to matter fields, or that it is close to zero for all in a local region of cosmological space and time. It says nothing about the values outside the local region.

## 1 Introduction

The use of fiber bundles [1, 2] in the description of gauge theories and other areas of physics and geometry has grown in recent years [3]-[7]. In gauge theories connections between fibers in a bundle are described using Lie algebra representations of gauge groups. For Abelian theories the gauge group is with as a local gauge transformation. For nonabelian theories [8, 9] the gauge group is with the exponential of a sum over generators of as the local gauge transformation. Additional details are given in many texts about field theories [10, 11, 12].

In this work the effect of number scaling on gauge theories and some geometric quantities is described. The base space, , of the bundle is a flat manifold, either as Euclidean space or as Minkowski space time. The bundle fiber is expanded to include both scalar fields and vector spaces. This takes account of the fact that scalar fields are included in the description of vector spaces. For gauge theories a fiber at includes a complex scalar field, with For tangent bundles each fiber includes a real scalar field, with a tangent space, This expansion is achieved by use of the fiber product [1, 2] of two bundles. Because is flat all the fiber bundles considered here are product bundles.

Number scaling is accounted for by expansion of the bundle fiber to include pairs of scaled scalar fields and vector spaces for all scaling factors. The scaling factors are complex for gauge theories and real for geometric properties.

This work expands earlier work on number scaling by the author [17, 18, 19] in that fiber bundles play an essential role. Scaling is accounted for by use of a space or space time dependent real or complex valued scaling field. Vertical components of connections between fibers at neighbor locations on are based on this field.

This work also differs from the earlier work in that the vector scaling fields, as gradients of the scaling field are integrable. In earlier work vector scaling fields were used with no connection to scalar fields. As a result the integrability of the fields was open.

Scaling is by no means new in physics and geometry. It was used almost 100 years ago by Weyl [20] to construct a complete differential geometry.^{1}

The number structure scaling used here in the connections between fibers is different from these types of scaling. It differs from conformal transformations in that both angles between vectors and vector lengths are scaled. The scaling of angles may seem strange and counterintuitive. However, as will be seen, this does not cause problems. For example, the properties of number scaling are such that trigonometric relations are preserved under scaled parallel transfer from one fiber to another.

The basic difference between the scaling used here and the other types of scaling used in physics is that both quantities and multiplication operations are scaled. The scaling must be such that the axiom validity of the number and vector space structures is preserved under scaling. The type of scaling used here and in earlier work is closest to that in a recent paper in which functional relations between number structures of different types include the basic operations as well as the quantities [23].

The plan of the paper is to first describe the salient aspects of scaling for both number and vector space structures. Because of its importance to the rest of this work, it is done in some detail in Section 2. Section 3 comes next with a description of the parts of the theory of fiber bundles that are needed here. The bundle fiber contains pairs of scalar structures and vector spaces at all levels of scaling.

Connections between fibers are described in Section 4. A space or space time dependent scaling field is introduced to describe the vertical component of connections. The effect of this connection field on both scalar and vector structure valued fields and on scalar and vector valued fields is described.

The effect of number scaling fields on Lagrangians, actions, and equations of motion for Klein Gordon and Dirac fields is described next in Section 5. Abelian gauge theory for pure number scaling Lagrangians shows that the imaginary part of the gradient, , of the scaling field must have no mass. The real part of the gradient can have any mass including Inclusion of the electromagnetic field does not affect these mass conditions.

The Lagrangian for the Mexican hat potential with the scaling field present is used in subsection 5.5 to find the effect of the Higgs mechanism on the scaling field. The field combines with a Goldstone boson to give a massive vector boson, a photon, and a Higgs field. The field remains with any mass possible. The very speculative possibility that might be the gradient of the Higgs field is noted.

The effect of scaling on some geometric quantities occupies Section 6. Both curve lengths and the geodesic equation are derived. It is seen that scaling affects the proper time of clocks carried along a path.

Experimental restrictions on the possible physical properties of and are discussed in Section 7. The lack of direct physical evidence for the presence of the and fields means that the values of these fields, in a region of cosmological space and time occupiable by us as observers and by other intelligent beings with whom we can effectively communicate, must be too small to be observed. There are no restrictions on the values of these fields outside the region. The region should be small with respect to the size of the observable universe.

Section 8 concludes the paper. It is noted that number scaling has no effect on comparisons of computations with one another or with results of measurements carried out at different space time points.

It should be noted that the idea of local mathematical structures is not new. It has been described in the context of category theory in which locality refers to structure interpretations in different categories [24]. Here locality refers to locations in space time. Locality in gauge field theories has also been discussed [25], but not with respect to mathematical structures.

## 2 Number scaling

A brief but explicit description of number scaling is in order as it is basic to all that follows. One begins with the observation that mathematics is based on a collection of structures of different types of mathematical systems with relations or maps between them [13, 14]. A structure consists of a base set, and a few basic operations, relations, and constants for which a relevant set of axioms are true. These are referred to as models in mathematical logic [15, 16].

Examples of structures include the different types of numbers, (natural integers, rational, real, complex), vector spaces, algebras, etc. The usual structures for the different types of numbers and the axioms sets they satisfy are listed below.

Here and in the following, an overline, such as in denotes a structure. No overline, as for , denotes a base set. The complex conjugation operation has been added as a basic operation to as it makes the development much easier.

As usually used, these structures conflate two distinct concepts, that of numbers as elements of the base sets, and that of number values acquired by the base set elements as part of a structure containing the base set. Natural numbers are the easiest set in which to see the distinction between number and number value.

To understand this consider the even natural numbers. These are just as good a structure for the natural numbers as are all the natural numbers. The relevant structure is given by

(1) |

Here is a subset of consisting of every other element of . An explicit example of is obtained from Von Neumann’s characterization of the natural numbers in set theory as the sets, for Here consists of the sets, Here denotes the empty set.

Comparison of the structures, and shows that the element of that has value in has value in In general, the element of that has value in has value in The subscript on values denotes the structure to which the values belong.

This is the simplest example of the distinction between numbers as base set elements and number values of the base set elements in structures. It extends to structures that are described by where

(2) |

This structure is based on the fact that the multiples of are also a model of the natural number axioms. Here one sees that an element of has a value in that is times the value it has in If is a factor of then elements of are also elements of , and an element of has a value in that is times the value has in Conversely the value of an element (in and ) in is times the value of in

This concept of value change or scaling can be extended to the other components of to define a structure that expresses the values that and have in . The resulting structure is defined by where

(3) |

The scaling of multiplication is not arbitrary. It is done to satisfy the requirement that satisfy the axioms of arithmetic if and only if and do. Addition is unchanged. Note that addition and multiplication are operations on number values, not on base set numbers.

The structure describes a relative scaling between the values of the elements of in and This relative scaling can be extended to any pair of numbers where is a factor of One substitutes for everywhere in the above description.

The description of number scaling for the natural numbers applies with minor changes to number scaling for the other types of numbers. Scaled structures for the rational, real, and complex numbers are given by,^{2}

(4) |

(5) |

(6) |

Here is a rational, real, and complex scaling factor for each of the corresponding number types.

These structures differ from those for the natural numbers in that the base set is the same for all scaling factors. This is a consequence of the fact that these number types are all fields; they are closed under division.

It is often useful to define a generic number structure,

(7) |

Here and are the sets of basic operations, relations, and constants for the structure of type For the number types considered here, with correspondingly natural, rational, real or complex.

Value maps of base set elements to values they have in structures are useful. The value of an element, in the base set in is related to the value of in by

(8) |

Here is the same number value in as is in The equation shows that if is such that then

It is important to keep the nomenclature clear. Here and in the following, denotes a number value in . Also is the same number value in as is in However is not the value the number, as an element of the base set has in This value is given by

Let and be pairs of rational, real, and complex scaling factors. For rational, real, and complex numbers, the structures corresponding to are

(9) |

The structure representations for rational and real number structures hold only if and are both positive or both negative. If not, then is replaced by in the structures for and

The rational, real, and complex number structures in Eq. 9 can be expressed generically by where

(10) |

for and . For , is replaced by

The relation between structures, with a superscript only and those such as with both superscript and subscript show that The subscript index labels the basic operations and relations in the structure. The relations between and can be made explicit by a map, , of the components of onto the components of . One has

(11) |

where

(12) |

The properties of show that it is the identity on the base set of numbers. but number values are scaled by , The scaling of the operations is based on the following implications:

The inverse does not scale as This can be seen by replacing the inverse by the division operation where One has

The scaling of the basic operations^{3}

The fact that axiom validity is invariant under scaling of number structures can be extended to properties in general. If is a property of number values, then for each pair, of scaling factors, and number value, is true for if and only if is true for . is obtained from by replacing all basic operations used to describe in by the corresponding ones in

These aspects of number scaling emphasize the fact that the elements of the base set, must be separated from the values they have in different structures. However, there is just one number where the distinction between elements of and their values in a structure is not needed. This is the number This number has the same value, in all scaled structures. It is like a number vacuum in that the value, , is invariant under all scalings.

The properties of number scaling are such that the term in a power series in corresponds to in . Here is the same number value in as is in It follows from this that if is an analytic function in (for ) with values , then is the corresponding analytic function in with values in

Another nice property of number scaling is that equations are preserved. This follows from the equivalences of equations for general mathematical terms.^{4}

The lefthand equation is in , the middle one is in and the righthand one is in Also the number values, are the same values in as are in as are in

Vector spaces are also affected by scaling of the fields on which they are based. This is a consequence of the fact that vector spaces are always spaces over a scalar field. As such they cannot be considered in isolation. One begins with the usual representation of a normed vector space [31] over the scalar field,

(13) |

Here denotes scalar vector multiplication, and denotes the real valued norm of vectors. denotes an arbitrary vector.

Vector spaces can also be associated with scaled number structures. Let be a scaled scalar field with the scaling factor. Here is usually or with a real or complex scaling factor. is also possible with a rational factor. The components of the vector space structure associated with are given by

(14) |

The relation between the vectors and are defined by the requirement that has the same properties relative to as has relative to . For example must be the same number value in as is in If the vectors, and in are related by , then the corresponding vectors in are related by and conversely. Here is the same number value in as is in If are three vectors in then if and only if in .

These conditions show that and are the same vectors in as are in Note that and are vector space and scalar structures where the scaling factor is

One can also define the vector space associated with the scalar field It is defined by

(15) |

Here is the same vector in as is in as is in Also is the same number in as is in as is in

The scaling maps of the components of and onto those given by Eqs. 10 and 15 will be referred to here and in later sections as correspondences. For example, corresponds to the number value and corresponds to both in given in Eq. 10. The term, ’correspondence’, will also be used to describe the relations between the components of and to those of Eq. 10, and Eq. 15.

One sees from the above that operations do not commute with scaling. In the above the norm operation was done before the correspondence as

Reversing the order of the operations gives

These results are not the same unless is a nonnegative real number. For each case the decision on which comes first, scaling or operation, is based on the requirement of axiom validity preservation.

There is another definition of scaled vector space structures that satisfies the axiom validity preservation requirement. It is given by

(16) |

The difference between this structure and that of in Eq. 15 is that the vectors are not scaled.

This structure is not used here because it fails the equivalence between dimensional vector spaces and their representations based on complex numbers. Thus one has and but not

## 3 Fiber bundles

Fiber bundles have been much used to describe gauge theories and other areas of physics and geometry [3]-[7]. Here they will be used to describe the effects of number scaling on gauge theories and on some geometric objects. A fiber bundle [1, 2] can be described by Here and are the total and base spaces, and is a projection of onto . For each point, in , is a fiber, at , in .

A fiber bundle is a product bundle if the total space is a product as in Here is a fiber, and for all in . Here the interest is in product bundles because is flat as Euclidean or Minkowski spaces.

Here fibers are taken to be products of vector spaces and scalar structures on which the vector spaces are based. The corresponding product bundle is given by

(17) |

This bundle is a fiber product [1] of the bundles and

The fiber at each point, of in the bundle is given by

(18) |

Both and are local mathematical structures in that they are associated with a point of . The bundle is a good description of local scalar and vector space structures associated with each point of .

The fiber bundle based representation of local mathematical structures will be used throughout this work. Gauge theories make use of this representation [5, 6]. For these theories fibers at each point of contain gauge groups as structure groups acting on vector spaces. Principal frame bundles of vector space bases are associated with the vector space fiber bundles. However the scalar field associated with the vector spaces is taken to be a single complex number field not associated with any point of .

This is changed here by associating a scalar structure with the vector space at each point of . This is shown in the description of where the fiber at contains both local scalar and vector space structures, and The components of include as a map from to and as a map from to

A main point of this work is to use fiber bundles to describe the effect of number scaling on gauge theories and on some geometric objects. This can be done by expanding the fibers to include all scaled pairs of scalar fields and vector spaces. For gauge theories with complex scalars the pairs are for all complex numbers. and are given by Eqs. 6 and 14. For geometric objects the pairs are is a scaled real number structure and is a scaled tangent space.

Inclusion of number scaling in a fiber bundle description of gauge theories can be achieved by first defining fiber bundles,

(19) |

for all complex scaling factors, . Here is a fiber product as defined by Eq. 17.

One then defines a sum bundle by

(20) |

Here is the collection of all pairs, projects the total space onto . The inverse map is defined by

(21) |

Here and are defined from and as was done in Eq. 18. The projection is related to the individual projections, by

(22) |

All complex number structures, in the fiber at have the same base set, All possible number values for each number in are included in structures in . For each number in and number value , there is a scaled complex number structure in in which has as a value. The vector space structures for all at also have the same base set, .^{5}

A structure group, with elements, that are structure isomorphisms for any complex number can be defined on the fiber. The action of on is given by^{6}

(23) |

The group is a continuous commutative group with the inverse of The scaling factors, are all elements of the group, also is the structure group for the fibers at each point, of .

The bundle, is a principal fiber bundle. The reason is the group acts freely and transitively on the fibers in [5, 6]. No scaled structure is preferred over another. All are completely equivalent.

The definition of fiber bundles is quite general in that the fibers can contain much material. For example a fiber can include a scalar field structure, and the mathematical analysis based on A fiber can include integrals and derivatives of functions, charts of and other systems. In all cases the fiber bundle is represented here by The bundle is trivial on because is flat as Minkowski or Euclidean space. The range set of consists of local structures at all points of . This generalization will be made use of in the following sections.

## 4 Connections

Connections or parallel transports [25] play important roles in gauge theories and geometry. They account for the fact that, for vector spaces at different points of , a choice of a basis at point does not determine the choice at point, of . This is the ”Naheinformationsprinzip”, ’no information at a distance’ principle [25]. Applied to number scaling it says that the choice of a number scaling factor at point does not determine the choice at point

Connections enable one to compare values of quantities in fibers at different locations. These are needed to describe physical quantities such as derivatives and integrals on . They connect a quantity at an arbitrary scaling level in a fiber at to a quantity at a neighboring level in a fiber at a neighboring point of . The connection can be decomposed into two components, a horizontal one that connects a quantity at point to a quantity at the same level but at point and a vertical component that connects a quantity at point to a quantity at a different level at point .

Connections can be used to map scalar and vector structures in a bundle fiber at to a structure in a fiber at These are the simplest to understand. and are discussed first. The results are used to map number values and vectors inside structures in a fiber at to number values and vectors in structures in a fiber at neighboring points of .

The number scaling connection is provided by a smooth valued field, , on . For each is a complex scaling factor. The values of are not elements of any complex scalar structure in a fiber. In this sense they are global in that they apply equally to all fibers. The field also acts at all levels, of structures in the fibers in

### 4.1 Scalar and vector structure valued fields

To see how the connection works it is useful to first consider a structure valued field Here is an arbitrary level in the fiber bundle. For a complex number structure valued field,

(24) |

for each in . For a vector structure valued field,

(25) |

These definitions show that for each and is unique in that for each just one scalar or vector structure value is possible for This simplifies the description of field quantities such as derivatives.

The derivative of makes use of the connections. The components of the derivative of have the form

(26) |

The limit is implied. Replacing and by their structure values gives

(27) |

The implied subtraction in the numerator does not make sense because structure subtraction can be defined only between structures at the same level within a fiber. It is not defined between structures at different levels in a fiber or between structures at the same level in different fibers.

Connections are used here to map to a structure in the fiber at and at the same level as is The horizontal component of the connection maps to the same structure, in the fiber at and the vertical component changes the level of to . The subtraction can then be carried out as both structures are at the same level in the fiber at

The horizontal component of the connection is taken here to be the identity.^{7}

(28) |

The vertical connection component makes use of the number scaling map in Eq. 10 where One also makes use of the fact that for any fiber level, , The result is given by

(29) |

Here

(30) |

has been used. To save on notation and have been replaced respectively by and .

The components of are obtained from Eq. 10. They are given by

(31) |

The subscript, is left off of the operations and number values in

Eq. 30 enables one to combine the terms in the numerator of the derivative. The result is

(32) |

Replacing and by their equivalents, and , and using a Taylor expansion of gives

(33) |

This shows that the effect of number scaling on structure field derivatives is independent of the scaling level, in the fiber bundle. The level cancels out because the ratio, or its inverse, appear as scaling factors for the structure components. and do not appear separately.

### 4.2 Scalar and vector valued fields

The results obtained for structure valued fields can be taken over to fields as sections on the fiber bundle, Let be a vector valued field where for each in , is a vector in in the fiber at One proceeds in a similar fashion to that used for the structure valued fields. The partial derivative is given by

(34) |

The first and second number terms are vectors in and

One first uses the horizontal component of the connection to map into the same vector, in as is in The vertical component is then used to map the result into the same vector in The map sequence gives the vectors,

(35) |

Replacement of in Eq. 34 by the right hand term of Eq. 35 and use of a Taylor expansion of gives

(36) |

as the result. This differs from the derivative for the structure valued field in that a partial derivative is also present and nonzero in general. Also, as was the case for , the result is independent of

This expression can be put in a more recognizable form by expressing the scaling field as the exponential of a pair of scalar fields as in

(37) |

This gives the final result,

(38) |

Here and are the gradients of and This replacement of by also applies to the derivatives of the structure valued field,

## 5 Gauge theories

The presence of number scaling extends the reach of gauge theories by adding another gauge group, , to those already used in the standard model and other areas of physics. The simplest case to consider is that in which is the only gauge group. This is an Abelian theory for pure number scaling.

The covariant derivative is given by Eq. 38. However it is instructive to obtain the derivative in another way well known [12] in gauge theory. The covariant derivative is given by

(39) |

Here is an element of the group . It accounts for the freedom of choice of number scaling factors in the mapping of in the fiber at to the fiber at Setting

(40) |

and expanding the exponential to first order in small quantities gives Eq. 38.

This approach is quite different from that used above to obtain Eq. 38. Eq. 40 introduces the vector fields, directly. There is no connection, implied or otherwise, of these fields to the gradient of a scalar field such as As a result, in this approach, it an open question whether none, one, or both of these fields are integrable or not. If the field is nonintegrable, then the only way to distinguish it from the electromagnetic field is to give it a different coupling constant in Lagrangians.

For these and other reasons the field, , is assumed here to be the gradient of a complex scalar field, This greatly simplifies expressions and derivations.

Here and from now on the subscripts on the field, are suppressed. The reason is that the quantities such as covariant derivatives are independent of the value of The descriptions can be shifted, with no change, from one level to another in All levels are equivalent.