This work is based on the field of reference frames based on quantum
representations of real and complex numbers described in other work.
Here frame domains are expanded to include space and time lattices.
Strings of qukits are described as hybrid systems as they are both mathematical and physical systems. As mathematical systems they represent numbers. As physical systems in each frame the strings have a discrete
Schrodinger dynamics on the lattices. The frame field has an iterative
structure such that the contents of a stage

The need for a coherent theory of physics and mathematics together arises from considerations of the basic relationship between physics and mathematics. Why is mathematics relevant to physics [

The problem of the relationship between the foundations of mathematics and physics is not new. Some recent work on the subject is described in [

Another approach to this problem is to work towards construction of a theory that treats physics and mathematics together as one coherent whole [

In this paper a possible approach to a coherent theory of physics and mathematics is described. The approach is based on the field of reference frames that follows from the properties of quantum mechanical representations of real and complex numbers [

The use, here, of reference frames is similar in many ways to that used by different workers in areas of physics [

The use of reference frames here differs from those noted above in that the frames are not based on a preexisting space and time as a background. Instead they are based on a mathematical parameterization of quantum theory representations of real and complex numbers. In particular, each frame

Each reference frame contains a considerable number of mathematical structures. Besides

The goal of this paper is to take a first step in remedying this defect by expansion of the domain of each frame to include discrete space and time lattices. The lattices,

Representations of physical systems of different types are also present in each frame. However, the emphasis here is on strings of qukits,

Considerable space in the paper is devoted to how observers in a stage

The view of the contents of a stage

All this and more is discussed in the rest of the paper. The next section is a brief summary of quantum theory representations and the resulting frame fields [

Section

The last section is a discussion of several points. The most important one is that frame field description given here leads to a field of different descriptions of the physical universe, one for each frame, whereas there is just one. This leads to the need to find some way to merge or collapse the frame field to correspond to the accepted view of the physical universe. This is discussed in the section as are some other points.

Whatever one thinks of the ideas and systems described in this work, it is good to keep the following points in mind. One point is that the existence of the reference frame field is based on properties of states of qukit string systems representing values of rational numbers. However the presence of a frame field is more general in that it is not limited to states of qukit strings. Reference frame fields arise for any quantum representation of rational numbers where the values of the rational numbers, as states of some system, are elements of a vector space over the field of complex numbers.

Another point is that the three-dimensional reference frame field described here exists only for quantum theory representations of the natural numbers, the integers, and the rational numbers. Neither the basis degree of freedom

Finally, although understandable, it is somewhat of a mystery why so much effort in physics has gone into the description of various aspects of quantum geometry and space time and so little into quantum representations of numbers. This is especially the case when one considers that natural numbers integers, rational numbers, and probably real and complex numbers are even more fundamental to physics than is geometry.

In earlier work [

This purely mathematical representation of qukit strings makes no use of physical representation of qukit strings as extended systems in space and/or time. Physical representations are described later on in Section

The qukit (

A compact notation is used where the location

Strings are characterized by the values of ^{L}

Since one is dealing with quantum states of qukit strings, states with leading and trailing

The set of states so defined form a basis set that spans a Fock space

As already noted, the states

One can also describe gauge and base transformation operators on these states. Gauge transformations correspond to a basis change (

Quantum theory representations of real numbers are defined here as equivalence classes of Cauchy sequences of states of finite

The sequence

Here

The definition can be extended to sequences

Two sequences

Quantum equivalence classes of Cauchy sequences of states are larger than classical equivalence classes because each quantum equivalence class contains many sequences of states that have no classical correspondent. However no new equivalence classes appear. This is a consequence of the fact that each quantum equivalence class contains a basis-valued sequence that corresponds to a classical sequence of finite base

One can also define a canonical representation of each equivalence class as a sequence

Extension of the above to include quantum representations of complex numbers is straightforward. One method represents complex rational numbers by pairs of states of finite

Cauchy sequences of these state pairs are defined by applying the Cauchy condition separately to the component sequences of real rational number states and imaginary rational number states. Two Cauchy sequences

Quantum theory representations of real and complex numbers differ from classical representations in two important ways. One is the presence of the gauge freedom or basis choice freedom. This is indicated by the

The other difference is based on the observation that states of qukit strings are elements of a Hilbert space or a Fock space. From a mathematical point of view these spaces are vector spaces over the fields of real and complex numbers. It follows from what has been shown that

The third degree of freedom arises from the free choice of the base choice for the humber representation. This choice, denoted by the

These three degrees of freedom can be combined to describe a three-dimensional reference frame field. Each reference frame

Because the iteration degree of freedom is directed, it is useful to use genealogical terms to describe the iteration stages. Frames that are ancestors to a given frame

From a mathematical point of view there are several possibilities for the stages. The frame fields can have a finite number of stages with both a common ancestor frame and a set of terminal frames. The fields can also be one way infinite either with a common ancestor and no terminal stage or with a terminal stage and no common ancestor, or they can be two way infinite. They can also be cyclic. These last two cases have no common ancestor or terminal frames.

Another way to illustrate the frame field structure is to show, schematically, frames emanating from frames. Figure

Schematic illustration of frames coming from frame

Three different viewpoints of the real and complex numbers as frame bases are of use here.

An observer in a parent frame

The main consideration of this paper is the proposed use of the reference frame field as a possible approach to a coherent theory of physics and mathematics. This ensures that quantum theory representations of natural numbers, integers, rational numbers, and real and complex numbers will play a basic role in the theory.

So far the reference frames contain mathematical systems. These include quantum theory representations of numbers, qukit strings, and representations of other mathematical systems as structures based on the different types of numbers. Physical theories and systems are not present in the frames. The reason is that there are no representations of space and time in the frames. These are needed for theories to describe the kinematics and dynamics of systems moving in space and time.

This must be remedied if the frame field is to be an approach to a coherent theory. One way to fix this is to expand the domains of the frames to include physical systems and descriptions of their dynamics.

A first step in this direction is to expand the domain of each frame in the field to include discrete lattices of space and time. The reason for working with discrete instead of continuum space and time will be noted later.

To be more specific, each frame

It should be noted that, to an observer in a frame, the points in each space and time lattice in a frame

A restriction on the lattices in frames is that the values of

Even though these requirements might seem restrictive, they are sufficiently general to allow lattices of arbitrarily small spacing and arbitrarily many points. Also they can be used to describe sequences of lattices that become continuous in the limit. An example of such a sequence is given by setting

It follows from this description that the points of a lattice,

So far each frame contains in its domain, space and time lattices, and strings of qukits that are numbers. It is reasonable to expect that it also contains various types of physical systems. For the purposes of this paper the types of included physical systems do not play an important role as the main emphasis here is on qukit string systems. Also in this first paper descriptions of system dynamics will be limited to nonrelativistic dynamics.

It follows from this that each frame includes a description of the dynamics of physical systems based on the space and time lattices in the frame. The kinematics and dynamics of the systems are expressed by theories that are present in the frame as mathematical structures over the real and complex number base of the frame. This is the case irrespective of whether the physical systems are particles, fields, or strings or have any other form.

One reason that space and time are described as discrete lattices instead of as continua is that it is not clear what the appropriate limit of the discrete description is. As is well known, there are many different descriptions of space and time that are present in literature. The majority of these descriptions arise from the need to combine quantum mechanics with general relativity. They include use of various quantum geometries [

The fact that there are many different lattices in a frame each characterized by different values of

So far the domain of each frame contains space and time lattices, many types of physical systems (such as electrons, nuclei, and atoms) and physical theories as mathematical structures based on the real and complex numbers. These describe the kinematics and dynamics of these systems on the lattices. Also included are qukit strings. States of these strings were seen to be values of rational numbers. These were used to describe real and complex numbers as Cauchy sequences of these states.

Here it is proposed to consider these strings as systems that can either be numbers, that is, mathematical systems, or be physical systems. Because of this dual role, they are referred to as hybrid systems. As such they will be seen to play an important role.

Support for this proposal is based on the observation that the description of qukit strings as both numbers and physical systems is not much different than the usual view in physics regarding qubits and strings of qubits. As a unit of quantum information, the states of a qubit can be

Also it is reasonable to expect that the domain of a coherent theory of physics and mathematics together would contain systems that are both mathematical systems and physical systems. The hybrid systems are an example of this in that they are number systems, which are mathematical systems, and they are physical systems.

Let

The different

The presence of the sign qubit is needed if the states of the hybrid systems are to be values of rational numbers. Since the qubit also corresponds to the

Note that there is a change of emphasis from the usual description of numbers. In the usual description, strings of base

As was noted, the states of the

The description of the hybrid systems as strings of qukits is one of several possible structures. For example, as physical systems that move and interact on a space lattice

Whatever structure the hybrid systems have, it would be expected that, as bound systems, they have a spectrum of energy eigenstates described by some Hamiltonian

The gauge variable has been removed because the requirement that (

The existence of a Hamiltonian for the

One way to resolve this problem is to let the energy of a hybrid system state with no leading or trailing

One consequence of this association of energy to rational number values is that to each Cauchy sequence of rational number states of hybrid systems there corresponds a sequence of energies. The energy of the

It is not known at this point if the sequence of energies associated with a Cauchy sequence of hybrid system states converges or not. Even if energy sequences converge for Cauchy sequences in an equivalence class, the question remains whether or not the energy sequences converge to the same limit for all sequences in the equivalence class.

The above description is valid for one hybrid system. In order to describe more than one of these systems, another parameter,

Pairs of hybrid systems are of special interest because states of these pairs correspond to values of complex rational numbers. The state of one of the pairs is the real component and the other is the imaginary component. Since these components have different mathematical properties, the corresponding states in the pairs of states of hybrid systems must be distinguished in some way.

One method is to distinguish the hybrid systems in the pairs by an index

As might be expected, the kinematics and dynamics of hybrid systems

The Hamiltonian can be expressed as the sum of a Hamiltonian for the separate systems and an interaction part as in

The question arises regarding how one should view

One way to shed light on this question is to examine physical representations of number tuples in computers. Their

Here it is assumed that

This picture is supported by the actual states of computers and their computations. The background potential well matrix that contains the

This picture of each frame containing physical systems and a plethora of different hybrid systems and their tuples may seem objectionable. However, one should recall that here one is working in a possible domain of a coherent theory of physics and mathematics together. In this case the domain might be expected to include many types of hybrid systems that have both physical and mathematical properties. This is in addition to the presence of physical systems and mathematical systems.

So far it has been seen that each frame,

Since this is true for every frame, it is true for a frame

One consequence of the relations between frames at different iteration stages is that entities in a frame, that are seen by an observer in the frame as featureless and with no structure, correspond to entities in a parent frame that have structure. For example, elements of

It is useful to represent these two in-frame views by superscripts

The distinction between elements of a frame and their images in a parent frame exists for other frame entities as well. The state

The use of stage superscripts and subscripts applies to other frame entities, such as hybrid systems, physical systems, and space and time lattices. A hybrid system

The state

The eigenvalue equivalence class is a base

If

A similar representation holds for parent frame views of point locations of lattices. Let

The view or image of

The space point locations

A useful way to select a unique constant sequence is to require that all states in the sequence be a unique state of the hybrid system

It follows from this that the locations of space points

A similar representation of the time points in the lattices is possible for the real rational number time values. In this case the time values of a lattice are seen in a parent frame as rational number states of a hybrid system.

The above description of a stage

This strongly suggests that each space point image

This shows that set of all parent frame images of the

Figure

Stage

It is of interest to compare the view here with that in [

The differing views of hybrid systems as either number systems or physical systems may seem strange when viewed from a perspective outside the frame field and in the usual physical universe. However it is appropriate for a coherent theory of physics and mathematics together as such a theory might have systems that represent different entities, depending on how they are viewed.

The description of parent frame images of lattice space points and their locations as

Each component

If the component hybrid systems in a

At this point the specific dependence of the energy on the parent frame image of the lattice point locations is not known as it depends on the properties of the hybrid system Hamiltonian. Nevertheless the existence of energies associated with locations of points of parent frame images of a space lattice is intriguing. One should note, though, that this association of energy to space points holds only for parent frame images. It does not extend to lattices

This aspect is one reason why one needs to do more work, particularly on the merging of frames in the frame field. It is quite possible that, in the case of a cyclic frame field, some aspects of the association of energy with space points in the same frame will be preserved. Note that for cyclic frame fields the restriction that an observer cannot see ancestor frames or their contents must be relaxed. The reason is that ancestor frames are also descendant frames.

The description of the motion of hybrid systems and other physical systems in a stage

It follows that the stage

As a sum of kinetic and potential terms, the Hamiltonian,

In the above, the forward and backward discrete derivatives

This description of the time development and Hamiltonian for a system

The Hamiltonian

The forward and backward discrete derivatives are expressed by equations similar to (

This use of states of

The reason for these restrictions on the properties of space and time hybrid systems is that one expects the space and time used to describe motion of systems to be quite stable and to change at most very slowly. Changes, if any, would be expected to be similar to those predicted by the Einstein equations of general relativity.

It is to be emphasized that the work presented so far is only a beginning to the development of a complete framework for a coherent theory of physics and mathematics together. Not only that but one must also find a way to reconcile the multiplicity of universes, one for each frame, to the view that there is only one physical universe.

One way to achieve reconciliation is to drop the single universe view and to relate the multiplicity of frame representations of physics and mathematics to the different many physical universes views of physics. These include physical universes in existing in different bubbles of space time [

If one sticks to the single physical universe view, then the frames with their different universes and space and time representations need to be merged or collapsed to arrive in some limit at the existing physical universe with one space time. This applies in particular to the iteration stage and gauge degrees of freedom as their presence is limited to the quantum representation of numbers.

One expected consequence of the merging is that it will result in the emergence of a single background space time as an asymptotic limit of the merging of the space time lattices in the different frames. Whether the ultimate space time background is a continuum, a foam [

In addition the merging may affect other entities in the frames. Physical systems, denoted collectively as

One potentially useful approach to frame merging is the use of gauge theory techniques [

This look into a possible future approach emphasizes how much there is to accomplish. Nevertheless one may hope that the work presented here is a beginning to the development of a coherent theory. The expansion of the frames in the frame field to include, not only mathematical systems but also space and time lattices, and hybrid systems that are both mathematical systems and physical systems, seems reasonable from the viewpoint of a coherent theory of physics and mathematics together. One might expect such a theory to contain systems that can be either physical systems or mathematical systems.

The use of massive hybrid systems to be stage

Whether these descriptions of parent frame images as hybrid systems will remain or will have to be modified remains to be seen. However, it should be recalled that these images are based on the dual role played by values of rational numbers both as mathematical systems and as locations of components of points in the lattices. Recall that the notion of a point in a lattice is separated from the location of the point just as the notion of a number, as a mathematical system, is separated from the value of a number. This use of number and number value is different from the usual use in mathematics in that expressions, as

In conclusion, it is worth reiterating the last paragraphs at the end of the introduction. Whatever one thinks of the ideas presented in the paper, the following points should be kept in mind. Two of the three dimensions of the field of reference frames are present only for quantum theory representations of the real and complex numbers. These are the gauge or basis degree of freedom and the iteration stage degree of freedom. They are not present in classical descriptions. The number base degree of freedom is present for both quantum and classical representations based on rational number representations by digit strings.

The presence of the gauge and iteration degrees of freedom in the quantum representation described here is independent of the description of rational number values as states of qukit string systems. Any quantum representation of the rational numbers, such as states

Finally, the importance of numbers to physics and mathematics should be emphasized. It is hoped that more work on combining quantum physics and the quantum theory of numbers will be done. The need for this is based on the observation that natural numbers, integers, rational numbers, and probably real and complex numbers are even more fundamental to physics than is geometry.

This work was supported by the U.S. Department of Energy, Office of Nuclear Physics, under Contract no. DE-AC02-06CH11357.