Emergence of Time and Physical Interactions

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Emergence of Time and Physical Interactions in FractionalN-dimensional Euclidean Space Volumes

Pinhas Ben-Avraham, 63 Rehov Kibbutz Galuyot, Ofakim 80300, Israel1

March 25, 2010

Keywords: Electric charge, emergence of time, fractional charge, fine structure constant,fundamental constants, fractional dimensionality, uncertainty principle, quantization of time,mathematical universe, physical reality, Machs principle.

Abstract

We try to demonstrate a simple mathematical structures properties as an observable physical reality or toy universe.Commencing from properties of an n-dimensional Euclidean structure we analyze the properties of such space andsome of its Fourier transforms related to velocity, acceleration and jerk as motion spaces. We develop the motion of a

point within that structure into a means to determine one or more interaction constants for this point in its (fractional) n-dimensional geometrical environment. We discuss the implications of dimensionality of motion spaces and try to find a

reasonable minimum amount of interpretation to let the mathematical structure resemble an observable physical realitywithout plugging in constants. Instead, we only plug in some elementary concepts of physics we try to keep to aminimum. We discuss, without any claim to completeness, in what way the mathematical structure could be conceivedas a physical reality or whether it could be a physical reality. In this exercise we find the fine structure constant to bethe most naturally emerging constant, and the other interaction constants dependent on it. It is also shown that motionspaces derived as Fourier transforms of a spherically symmetric n-dimensional Euclidean position space are Lorentzinvariant. Quantum mechanical effects like zitterbewegung and quantization of topological charges including theelectric charge are discussed in the light of n-dimensional space-volumes. In light of the possible fractality and withthat non-differentiability of motions we discuss scale-dependent effects and sensibility of the quantization of space andtime and their relationship to mass and charge, respectively. We find reasons for a direct dimensional relationship ofthe emergences of charge and time prior to space and gravity.

1. Introduction

In 2006/7, Frank Wilczek [1, 2] stated that fundamental constants in physics, like forexample interaction constants are purely numerical quantities whose values cannot be derivedfrom first principles, meaning, they are not derivable from equations describing certainphysical theories, let alone real phenomena that also are not derivable from such equationswithout plugging in natural constants. He further stated that these natural constants makeup the link between equations and reality, and their values cannot be determinedconceptually.

Arthur Eddington [3] tried for the greater part of his later life to find a geometrical principleto describe physics on the basis of the fine structure constants peculiar numerical value,1/137, to no avail. Koschmieder [4] uses lattice theory to explain the masses of the particles

of the Standard Model, concluding that only photons, neutrinos and electric charge areneeded to explain the masses of all the particles. He refers to MacGregor [5, 6, 7] who showsin three papers that the masses of the particles of the Standard Model depend solely on theelectron mass and the fine structure constants numerical value in natural units. Nottale et al.[8, 9] propose a model of scale relativity that solves the problem of the divergence ofcharges or coupling constants and self-energy with the fine structure constant, = 1/137, onthe electron scale. They attempt to devise a geometrical framework in which motion laws are

1 Email:[email protected], Tel. +972-50-863.9107, +972-8-996.3527
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completed by scale laws. From these scale laws they obtain standard quantum mechanics asmechanics in a non-differentiable space-time2. In particular, in reference [8] Nottaledemonstrates a derivation of the fine structure constant by running down the formal QEDinverse coupling from the electron scale (Compton length) to the Planck scale by using itsrenormalization group equation3. The numerical value achieved by this procedure is prettyclose to reality. A shortcoming of this approach is it yields different values for the barecharge or bare coupling. Again, he needs to refer to experimental observation to choosethe correct or physical of the three possible solutions. Furthermore, specific length scaleslike the Compton and the Planck length have to be plugged in to come up with realisticvalues for the coupling constants he determines. Similarly, Garrett Lisi [10] needs to choosethe symmetry breaking and the action by hand to achieve an otherwise compelling proposalfor a Theory of Everything matching the Standard Model. Other approaches to derive thenumerical values of coupling constants, and in particular the fine structure constant, borderon numerology or other esoteric approaches bearing little resemblance of physicalreasoning that can be derived from observational experience underlying the construct of themathematical structures proposed. Relativistically, one needs to plug in the velocity of lightas a maximum velocity to obtain Lorentz invariance. We shall determine whether and how

this invariance may be derived from geometrical considerations only.

In our approach we try to avoid any input of numerical values for interaction or couplingconstants, but resort only to some fundamental concepts of elementary physics wherenecessary. By allowing generalized dimensionality we include the possibility of a fractalpicture of space-time that seems to be, at least tentatively, justified by phenomena such asBrownian motion and zitterbewegung, the latter of the two showing true fractionaldimensionality, and by quantum theory itself that proposes the Planck length and Planck timeas a smallest scale. We shall attempt to show fundamental differences in the properties ofEuclidean position space and the Lorentz invariant motion spaces related to momentum(velocity), force (acceleration), and impact (jerk), and with that fundamental differences ofspace and mass versus time and charge. It shall, however, become clear in the course of ourtreatment of the underlying mathematical structure we have assumed that physicalphenomena are the result of the underlying mathematical structure, if so interpreted by anobserver.

The introduction of additional dimensions in Kalutza-Klein theories or string theory as wellas the above mentioned approaches seems to warrant two fundamental questions:

1. Is there a fundamental connection of space geometry to at least one of the couplingconstants?

2. What role plays dimensionality in the sense of Hausdorffs extended view ondimensionality and fractional dimensionality in physical interactions and the fabric of

space-time?

We attempt to shed light onto these questions considering some properties of spaces seen asmathematical structures containing, resembling or being such physical interactions withoutclaiming the identity of our structures with physical reality as such. We try to keep the

2 They do not arrive at a discrete space-time, but rather postulate it.3 Such equation needs physical insight to be derived. A merely mathematical reasoning without referenceto phenomena or physical concepts is impossible.

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physical reality as simple as possible to see how much physical law in form of properties ofthe underlying structure such simplistic example can produce, and how much additional inputin form of mathematical structure or its properties is needed to make our structure be arealistic toy universe.

Max Tegmark [11] proposed in 2007 a mathematical universe hypothesis stating Ourexternal physical reality is a mathematical structure, based on the assumption that Thereexists an external physical reality completely independent of us humans. He argues for theequivalence of a mathematical structure and the physical reality it describes and we observe,not merely the mathematical structure describing the physical reality. Despite his effort toencode numerically elements of language defining or describing mathematical entities or(partial) structures, at least one information theoretical problem remains: one need to agreeon the encoding. We do have no proof of a natural encoding mechanism that would beprovably inevitable by emerging from the structure itself as a by nature preferred encodingmechanism. We hold against the quest for an absolutely mathematical nature of physicalreality that human language and its content may well be translated into mathematicalsymbolism or language, but cannot be immune against a decidedly willed, random or even

illogical treatment of that physical reality by humans. Furthermore, any distinctions withinthe structure are arguably man-made, except they would automatically emerge from thestructure itself. Thereby the choices made what to look for inside the structure may be alsoarguably man-made. Besides this caution we find it enormously interesting to try to build amathematical structure from scratch that describes or resembles a physical reality. We arenot insisting on what is the ultimate scratch, but are interested whether we will be able toargue in favor of an identity of mathematical structure and physical reality.

We will try in the following to investigate a mathematical structure resembling a physicalreality using a simple example for such a reality. A central question we shall try to answer iswhether and how such structure can provide us with numerically acceptable unique valuesfor, say, conditions of minimal physical (inter-) action. The choice of our example cannot becompletely arbitrary and random. Hence, we try to determine from the properties of a simplestructure and first principles4 whether we can find a physical (inter-) action we can observe.

In section 2 we choose as a starting structure for our example Euclidean space5 in arbitrarilymany dimensions. To include fractional dimensions into our discussion we construct an n-dimensional structure with n a real number. We further allege all physical reality should lookthe same in any arbitrarily chosen locality of that space. By the introduction of time weintroduce a structure similar to Minkowski space, but we shall use complementary spacessuch as momentum space as a basic structure to arrive there. In section 3 where we also try todefine what is movement and how time-like coordinates arise from it. In sections 4 to 6 weconstruct such complementary spaces and demonstrate some properties of position space

and velocity space6

, taking into consideration acceleration space, all in particulardependent on dimensionality. We use the conditions we found in those sections to derive apossible physical interaction in section 5. In sections 6 and 7 we attempt a discussion aboutthe physical meaning of dimensionality and a relativity of space-volume in n dimensions and4 We try to limit these to the definitions of position, time, velocity, acceleration and higher time derivativesas specified in section 3.5 NOT space-time!6 Velocity space shall be at this stage identical with momentum space as we try not to define anything like amass yet.

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try to give an interpretation of a possible dependency of observed physical interactions ondimensionality by discussing velocity or momentum densities in different dimensions foridentical movements taking acceleration space and jerk space into consideration, tofinally conclude in section 8 with a discussion of our findings and try to assess how muchinterpretation is necessary to find the physical reality in the mathematical structure. In a briefoutlook we try to suggest a program for systematically exploring avenues towards thedevelopment of a TOE based on purely geometric considerations.

2. Space and Time

2.1. N-dimensional Euclidean Space

For a (geometrical) object or its motion to be described or to take place, a certain minimumvolume of space is necessary even if we follow Machs and Leibnizs argumentation in favorof the non-existence of absolute space and time. Mach insisted that science must deal with

genuinely observable things which made him deeply suspicious of the concepts of invisiblespace and time. Machs idea suggests that the Newtonian way of thinking about the workingof a universe, which is still deep-rooted, is fundamentally wrong. The Newtonian philosophydescribes objects of the universe contained in a space-time that exists before anything else.The Machian idea takes the power from space and time and gives it to the actual contents ofthat space and time which is seen as a holistic interplay of space and its contents. This meansthe actual structure of space and time is determined by the dynamics and spatial distributionof its contents. We will see in this treatise how such space can emerge from a very simplisticdynamics7. Depending on the nature of such dynamics, complementary spaces will play animportant role in demonstrating physicality8.

In regard to scale, we do not assume any scale but define the length of elementar movementas one and the resulting time interval also as one. We want, for the moment, not too strictlyadhere to Machs principle but allow a spherical space in n dimensions enclosing our objector its movement. To avoid more restrictive assumptions we allow highest possible symmetryof our space which is spherical symmetry. We also choose to allow arbitrarily manydimensions n (real number), and our space shall be Euclidean. We reserve the right to furthergeneralize as we progress building our structure. It shall be understood that space with n = 0+ with being a very small positive number unequal to zero, can contain a point, n = 1 aline, n = 2 a surface and n 3 a voluminous object. For the word volume we want to allowbesides a conventional voluminous geometrical object an area of a surface and the length of aline as a volume; only a point without any motion shall have zero volume. We will see thereasons for our choices during our construction process. We further generalize dimensionality

to n

[12].

For example, in Nottales scale relativity particles are identified with a family of fractaltrajectories within scales smaller than the particle itself like the Planck or Compton scales,described as the geodesics of a non-differentiable space-time. These trajectories have fractalinternal structures, meaning they are explicitly dependent on at least one scale variable

7 We do not, however, adhere rigorously to Machs principle.8 Cf. sections 4 to 6.

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characterizing resolution. If we consider an electron in an electromagnetic potential, itswave functions phase contains the products of fundamental quantities like position, time andangle together with their conjugates or complementaries momentum, energy and angularmomentum as they are related through Noethers theorem and Heisenbergs uncertaintyrelations. The conjugate variables are the conservative quantities that originate from space-time symmetries. The gauge transformation for the electric charge comes out here as aconjugate of the charge itself. The question is, if scale relativity allows the derivation of theelectric charge from the properties of space-time alone, then it should be possible todetermine within space and time a fundamental fractal relationship between particleproperties, quantizations and fundamental limits like the speed of light, but what is thefundamental ingredient of such a relationship between fundamental properties of space andphysical interactions and particle properties? According to Nottale and Feynman the quantumbehavior of physics is a manifestation of fractal geometry of space-time, in the same way asgravitation is, in Einsteins theory, a manifestation of the curvature of space-time. Animportant part in such non-differentiable but continuous space-time is played by curves,surfaces, volumes, and more generally spaces of dimension n being a real number. To allowphysics or interactions, as well as dynamics, the properties of such n-dimensional space-time

need to be discussed as they are generally not well understood, in particular, if n is a realnumber.

To illustrate n-dimensional space including its fractal properties, let us consider a space ofposition that is isotropic and homogeneous: the spherically symmetric n-space. To satisfy theconditions of interaction, we need to provide volumes where these interactions can happen.

Before we embark into any reasoning about (inter-)actions, we discuss the behavior of thevolume of a sphere as a function of its radius and of dimensionality without suggesting orassuming a special metric or gauge invariance we normally would use to describe physics. Aspherical volume element of radius one (unit radius) is described by Hamming [13]:

!

)(),(

2

k

rrCnrV

kn

n

==

, with n = 2k

Since(k+ 1) = k!, n will be even for integerk. Generalizing n yields a function V(r, n)that is continuous and differentiable in respect to radius and dimensionality includingfractional dimensions. With

)1()1!2

+=+(= nkk

we get for our spherical volume element of radius rand dimensionality n

)

)(),(

2

2 2

n

n

rnrV

+( 1

=

as its volume. For unit radius this yields a dependency of the volume from dimensionality asshown in Fig. 1a, and Fig.1b shows a plot ofV(r, n).

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5 10 15 20n

1

2

3

4

5

V

Figure 1a

Figure 1b

0 2 4 6 8 10

n

0

0.25

0.5

0.75

1

r

0

2

4

6

8

V

0.

0.5

0.75

1

Figure 1c

As we can see, the voluminosityof our n-dimensional spherebehaves counter-intuitively. Thevolume reaches a maximum forunit radius and decreases to zerofor large n. Furthermore, thedimension where the maximum

volume occurs increases withincreasing radius. If we look atdimensions n < 1 and r< 1, weencounter more counter-intuitivebehavior of the volume function.As we can see from Figure 1cand from V(r, n), at r= 0 and n =0 the equation has no solution.

6

0

0.25

0.5

0.75

1

r

-0.5

0

0.5

1

n

0

0.5

1

1.5

2

Vs

0.25

0.5

0.75

1

r


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Hence, the existence of a point or a point-like object in a spherically symmetric environmentat zero dimensions does not exist. The point-like object can only exist there either with aninfinitesimally small radius or infinitesimally small dimensionality. In the immediateneighborhood ofr= n = 0 the volume function has a very steep slope and jumps from zerovolume to V= 1 for positive n. There are no solutions forr= 0 and n 0. Forn = 0 and r 0the volume is always equal to one. In one dimension, our intuition is rectified by a pointbeing a sphere of radius zero with zero volume. So we have established the existence of apoint with zero volume and zero radius occurs only at n > 0. The existence of differentiablelines occurs at n 1, that of differentiable surfaces at n 2, and that of differentiablevolumes and hyper-volumes at n 3, which is straight forwardly understood. What is lessunderstood is the occurrence of sets of finite separated lines at dimension, that of sets offinite separated surfaces and sets of separated three dimensional volumes between one andtwo dimensions. Here we see that deterministic fractals exist in the neighborhood of integerdimensions, and that some of them may require a higher dimensional environmentto be ableto exist. Examples are two and three dimensional Cantor dust. There the elements of the dustare two or three dimensional by themselves, but the set itself has a lower fractionaldimension. This means, the respective volumes for such sets can be determined by a smallest

sphere enclosing the set and having the same dimensionality as the set.

In such a space we can describe the positions of points or objects relative to each other andarrive at a description of dynamical behavior of a system of objects by looking at changes oftheir positions relative to each other. We agreed above that we want to enclose such an objector system of objects by a suitable sphere representing a geometric space spanned up by thephysical action9. We will see later that for our considerations it is sufficient to simply lookat the volumes of such enclosing spheres. We remind the reader about such spheres beingchosen for the convenience of having highest possible symmetry.

2.2. The Role of Time

Asking the question What is time? is as old as mankind. At a fundamental level of physicsspace and time are basic ingredients for the understanding of any process, any dynamics andany interaction, but the fundamentality of space-time may be not enough to understand the principle nature of dynamics or interaction, as such space-time is a combination offundamentally different basic ingredients: space allocating position, and time allocatingchange. A timeless quantum mechanics for instance is possible without destroying thedescriptive power of the theory, but relativity cannot be timeless without becomingmeaningless. Noethers theorem shows us that times translational symmetry is identical tothe conservation of energy while spaces translational symmetry is identical to theconservation of linear momentum. The rotational symmetry of space is identical with theconservation of angular momentum. Here we see that Noethers theorem is directly related to

conjugate or complementary observables related to each other in quantum mechanics byHeisenbergs uncertainty relationship, and mathematically by Fourier transformations. Tohenceforth see time as orthogonal to position space coordinates is acceptable, but to fullytreat time and space coordinates on an equal footing remains very questionable.

To give meaning to time, we need to consider change. In a universe or space where there isno content that exhibits any motion or not any object in such space displays any relative

9 One could argue the action of a moving point to be mathematical as well.

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motion or change in at least one property relative to any other object that space contains, timeis meaningless and cannot be measured within that space. From now we want to identifychanges of properties of objects with motion in property spaces. For example a colorchange would happen to be represented by a motion in color space. Hence, we need to definetime differently from position space and its properties. We can measure the distance ofobjects, say, points in our toy universe by using a rigid rod as a unit length, but if we wantto measure time, things are not that simple. First, the measurement of position space is direct,we actually can measure, strictly speaking, nothing else but positions of objects contained ina space or the changes of their relative positions. Second, time cannot be measured but viathe observation of changes in general. This means, time can only be measured indirectly. Allour time measuring devices such as clocks, pendula, and any other frequency renderingdevices are using repeated uniform displacement in space as a measure of time units. Fromthis follows that at least two positions in space frequented by a moving object are necessaryto define a time interval. This time interval can be divided into infinitely many small timesub-intervals that can, purely geometrically become infinitesimally small, but never are ableto reach the properties of a point. Also, such time intervals cannot be separated from eachother, so to speak into time and timeless intervals, while space can be divided into object-

occupied and non-occupied volume elements. In other words, space can be discontinuouswhile time cannot; meaning, times minimum dimensionality is one while space can have lessthan one dimension. This is very important in regard to fractional dimensionality of physicalprocesses, and it implies that in any fractal geometry physics makes only sense if positionand time coordinates (space and time) are not freely interchangeable, and instants of timeare non-existent.

2.3. Space-Time

If we want to construct a space-time in fractal geometry, we need to consider time as alwayscontinuous while space can have any properties regarding differentiability and continuity.Therefore, a minimally dimensional space-time has one dimension, if we consider it emptybut comparable to some other space containing a clock. What is the minimum dimensionalityof a fractal space-time if it contains a minimal clock? This is a very delicate question andneeds careful attention.

In the immediate neighborhood of zero dimensions a length-less zero-volume object can onlybe realized as something point-like while continuous change needs besides the one time-likedimension at least a fractional dimension that allows the existence of a minimal one-dimensional Cantor dust consisting of two points moving infinitesimally slow relative to eachother in a neighborhood of positions that, if filled by points, would produce a line.Theoretically we can state that such a space-time would exist in n > 1 dimensions inclusive oftime but not in exactly one dimension. This is valid if we assume spherical symmetry to

allow the full validity of Noethers theorem in fractional dimensions. Hence, a minimumspherically symmetric space-time in n > 1 dimensions must be asymmetric in regard todimensionality. We will see in section 7 what this exactly means in mathematical andphysical terms. Furthermore, in a fractional dimensional or fractal space the meaning of timeneeds to be more precisely defined in regard to processes of change. We need to discuss notonly time as a result of motion in different complementary spaces, but also its role in theprocess of becoming rather than being. Since a one-point universe cannot yield time, and

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a point cannot exist as a sphere with zero radius in zero dimensions, there is a need to answerthe following questions:

If the universe emerged from a singularity, was this singularity a point or an emptyset?

How can a point singularity (point in infinity) evolve from zero dimensions? How does time emerge from a singularity in position space? Is frequency a time-like volume that emerges with the emergence of motion-like

volumes as a volume in negative dimensions of such motion-like spaces or theirquotient spaces or is there an interaction that has a closer link to the emergence oftime?

How do motion spaces emerge from changes in position or property space?

Time and its reciprocal, frequency, are in classical and quantum physics regarded as scalarswhile positions and their time derivatives are denoted by vectors. However, we have an arrowof time resulting from thermodynamics second law and from the information-theoreticalconditions of causality. This causality, challenged by quantum eraser and delayed choiceexperiments and their combinations, poses a formidable challenge with the questions:

Is negative time a motion in time affecting the arrow of time with time as a scalar ordo we have to introduce at least additional fractional dimensions to time andfrequency to allow time-reversal or do we have to regard time as a volume?

If time is a volume-like scalar, what is the nature of time-reversing or seeminglytime-reversing experiments?

This set of questions suggests we shall have to concern ourselves with information theoreticalissues as well as thermodynamic and statistical physics issues. At this stage, before we haveanalyzed the properties of motion spaces related to time-derivatives-of-position spaces these

questions will not be answerable. We shall now embark into a discussion of motion spaces tofacilitate a background to discuss the above open issues.

3. Motion

We need to agree on the following facts as philosophically necessary to describe a spaceemerging from a point which is, according to our deliberations above, non-existent as havingzero volume, zero dimensionality and zero extension (radius in the spherically symmetriccase) at once:

As we can see from Fig. 1 c, the point can exist as a sphere of zero volume and zero radius atany dimension greater than zero. It has co-dimension zero in all these n-dimensionalenvironments. The emergence of position and motion spaces from moving points and theirvolume constraints shall be the subject of our further investigation. Similarly, the emergenceof time and its properties as a result of the geometry of motion spaces both dimensionally andfrom their spatial and dynamic properties opens a new view of both relativistic and quantumproperties of motion. To make such multi-dimensional approach intelligible, we start with aone-dimensional motion of a point relative to another point at rest, and define the length ofsuch motion as the radius of an n-dimensional sphere. For the derivation of minimal

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interactions and maximum velocities and their time derivatives we shall primarily look atvolume properties of the different motion spaces and of position space, as well as atuncertainties arising mathematically and by measurement constraints.

Let us assume that there exists a point in no environment or space (single universe) that welet move over a length one to create a straight line that we want to consider as the radius ofour n-dimensional sphere we discussed above. Our initial point shall have no physical orother attributes attached to it other than that of a point resting. This resting we have, due tothe condition of a point having no volume, radius or dimension, to challenge in regard to ourdiscussion above. According to that, we deal with an object that has at least one of theseproperties unequal to zero, if we assume it spherical, unless we accept it as an empty set.There are three possibilities for the point to exist in no environment:

At n = 0 we require an infinitesimally small radius that lets the volume jump toone.

The radius can have any length at n = 0 and the volume will always be one. For zero volume and radius we require at least an infinitesimally small

dimensionality.

The first two conditions show that we can have unit volume emerging from a point in infinityor from a point at any distance from the empty singularity at the origin, if we consider zero-dimensional Cantor dust as residing in or slightly above zero dimensions. As we can see fromFig. 1 c, unit volume is reached between 0 < n 1 at about r= 1/2 very weakly dependent ondimensionality. Only the function how unit volume is reached from zero volume varies froma step function very nearn = 0 to a smoothened step function at n = to a linear function atn = 1. Furthermore, near zero dimensions the existence of one point at any distance from theempty origin creates a unit volume. This unit volume is thereafter, at higher dimensions, onlyachieved by the coming into existence of such point at a minimum distance of near zero to

2 in about 64 dimensions. Within radius one a significant volume is only available up to 23dimensions. Hence, we shall concentrate on the dimensional interval 0 n 23.

We further agree that we can move our point from one to another position as we decide or itis able to emerge at any other position while fading from its original position. The familiardefinitions of position, velocity and acceleration shall hold, but we do not want tointroduce definitions like force, momentum or energy at this stage. The termexistence shall be defined as a variable that enables emergence and fading at differentpositions in the case of dimensionalities smaller than one. This existence shall be dependenton time. Any other properties of the point like mass or charge shall also be un-definedunknown labels. We only allow mathematical entities to exist together with our threephysical definitions as follows:

1. Position as a vectorx = (x1,x2, ,xn);2. Velocity shall be a vectorv = dx/dt;3. Acceleration shall be a vectora = dv/dt, and its further time derivatives shall also

exist.

The time shall be denoted by tand higher time derivatives ofa shall be considered for non-uniform accelerations of our point. The concept of time has to be introduced as a comparison

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of the motion of our point relative to a clock-mechanism which imposes a formidableproblem in so far as uncertainty is concerned. For convenience, we shall regard the time as acontinuum to allow differentiability and integrability, but for a realistic picture of physicalreality we would have to assume, strictly speaking, a clock with infinitely high frequency 10 toachieve that.

This said we can now investigate how we can describe the movement of our point thatconstructs our sphere. Thereby we do not scale any lengths except that the observedmovement shall end at length of radius one and the two known positions shall be at r= 0 andr= 1 at t0 and t1 respectively.

The velocity of the moving point can only be determined, if one knows at least two differentpositions at two separate instants of time11 which have to be treated as the ends of a timeinterval that is divisible into smaller ones, but not into point-like instants of time. JohnWheeler remarked in his article Law without Law [14]: What we call reality consists of a few iron posts of observation between which we fill in by elaborate papier-mchconstruction of imagination and theory. Thus, we have to consider two separate points in

space as well as in time as the minimum information we can obtain to determine a velocity,and hence, our assumption made above forrand tis justifiable.

If one regards a static position of a point as zero dimensional, it can be at any positionrelative to another point at rest in any dimensionality except zero. If we construct a velocityspace in n dimensions, both points will be resting at the origin of that space. This meansaccording to Machs principle that velocity space does not exist for those static points or isrepresented by one point. This point lies in the origin of the function Vp (p, n) as opposed tothe resting point in Vs (r, n) which does not exist. When the one point changes positionmoving relative to the other at some not necessarily constant velocity, the moving point willbe able to construct a velocity-sphere in n dimensions12. In position space such movementwill be represented by a line of minimum one dimension which is a co-dimension in positionn-space, because the line can be existent in many dimensions. In velocity n-space a pointwith uniform velocity existent in many dimensions will be represented by a resting point inthat velocity space and must have a minimum co-dimension of one in position space. Thisimplies that any movement represented by less than one co-dimension in position space is un-physical or at least physically questionable for now, because it cannot move but only fade inone position and emerge in another when dimensions lower than one are concerned. We wantto restrict this implication for the moment until we have discussed the meaning of fractionaldimensions in the context of movement. To effect any interaction13, a minimum volume inspatial and velocity space is necessary, allowing for acceleration (change in position andvelocity) at all times. From this we can conjecture that any change in velocity or anyinteraction needs to take place over at least one co-dimension within the respective n-spaces

10 We do not want to indulge further in fundamental discussions about the nature of time in this paper, butwe point out that any definition of time should be dependent on motion, if we accept the 2nd law ofthermodynamics as the origin of the arrow of time we observe classically.11 We can, in the simplest case have a uniform velocity or a velocity reaching the value 1 after time andspace interval one, if it is considered to rest at the beginning of the movement.12 Again, it is and remains the choice of the observer, how many dimensions he or she chooses to constructa spherical volume element with a radius determined by the displacement of a point in position and time.13 For any interaction (or physics) to take place, change in motion must be allowed to observe thatinteraction.

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for position and velocity, if there are no effects present such as zitterbewegung. Hence, anymotion connected to an interaction constructs a minimum volume of position and velocityspace as well. If the acceleration changes we also have a voluminous acceleration spacewhich is represented by the Fourier transform of velocity space.

After all these considerations we are left with position space and time derivative spaces thatare Fourier transforms and describe motion of any kind. They do not have to contain time asa separate coordinate to describe the physics and its constraints happening inside positionspace. The volume functions of such position and motion spaces make, as we shall see later,the construction of a space-time obsolete. From products of complementary spaces we canderive uncertainty relations and construct interaction spaces and their volume functions in ndimensions.

The current view of Machs principle in the context of general relativity that one creates aproblem with handling a space-time metric, in particular concerning problems of massesrelating to space-time curvatures, can be weakened by our above assertion of a minimumvolume of both types of spaces being required for any interaction or being constructed by

those interactions. If one further accepts the equivalence of energy density and space-timecurvature and the resulting assertion that all matter can be expressed by the geometricalstructure of space-time, one has to accept also that dynamics should be expressible in termsof changes of that very structure which in our case is a change in radius with time. Thosechanges, however, are constraint naturally by the relationship between the space hostingdynamics, momentum space14, and that hosting position, spatial space. Changes of thisstructure are a critical issue, whether one can assume a mathematical structure to be aphysical reality. Only in Machs sense this would be correct.

4. Some properties of position and motion

Let us take our point and move it from position x1 to position x2. This movement can bedescribed as x = x2 - x1. In Euclidean space we can connect the two positions with a straightline, and in other types of space with a geodesic line. To define another distinction, becausewe consider one point moving from one position to another, we need to introduce anotherlabel or coordinate, time. In n dimensions, this can be regarded as the construction of aquotient space of position change versus velocity change, fixing the time scale byimplication. If the point is considered moving continuously from one position to the other,our time coordinate can be considered continuous as can its path. Since we have not agreedon a particular scale or system of units, we want to define this movement as having lengthone in position space and length one on the time coordinate. We remind ourselves again ofWheelers remark cited above, which implies that if the point moves through positions x1 and

x2 at a constant velocity, this velocity can have any value in between these points and remainsunobserved. If we, however, consider the point resting in its first position and then coveringunit length in unit time, the start velocity will be zero and the velocity in the second positionwill be one, if the point is uniformly accelerated over a time interval of one. The meanvelocity over the distance will be . According to our above assertion the spheres in our n-dimensional spaces will be built by giving a radius to position and velocity spheres. If theacceleration changes on the way but remains over the unit time interval at unit value, we do

14 This we simplified to velocity space as we have given no mass to our point.

12


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not know the exact relationship between position and velocity. The velocity known betweenthe two positions is always between zero velocity and the end velocity in the acceleratedcase, since the point rests in its first position and reaches the second position in unit time. Ifwe do not know whether and how the point is accelerated, the uncertainty of velocity liesbetween the mean value and one, in this case it will be , if the position and time differencesare precisely known. For |x| = 1 we will induce an uncertainty of |v| = , so that theirproduct becomes . We will show later, how this relatively sloppy estimation of uncertaintycan be more rigorously derived from purely geometrical considerations and first principles.

Above we agreed that in our case only mathematical structure in form of Euclidean spaceexists in form of an n-dimensional sphere constructed by the displacement of a pointrepresenting its radius. Whether we decide to move the point to a unit sphere surface withconstant velocity or accelerated from rest leaves us no choice regarding the introduction ofmovement, meaning, if we have only a resting point that we want to move and define itsdisplacement as our radius, we have to start at velocity zero and produce with that anacceleration. To measure the position of a point while moving, it is not necessary to bring itto a halt. Hence, we do not worry about what happens to our spherical space in its totality

after the introduction of movement but decide only to look at a spherical volume elementwith maximum radius one within the evolving space.

We can now further argue that besides acceleration introduces a velocity to a resting point,acceleration also needs to be introduced by a jerk j = da/dt. This would produce thefollowing scenario: let us assume, |j| = 1, then a(t) = 01j dt= 1t=1, and v(t) = t2/2 with x(t) =t3/6. Vice versa, we need a mean jerk of 6 for the point to reach length one in unit time.Now we can introduce infinitely many introductions of the motion in question and will endup with x(t)tn for reaching length one. Could this be a quantum jump? We will suggest ananswer later when we know more about uncertainties, but one thing is sure: for higher orderjerks we get nonlinear acceleration and with that chaotic behavior of the equation of motionthat applies, and even the uncertainty relations between position and acceleration or jerkbehave chaotic themselves. We will see this towards the end of the paper.

At this point of our construction of a mathematical structure describing accelerated motion inn-dimensional spherically symmetric space we need to define a velocity space correspondingto our position space. We need to look at the velocity change over unit length and time oncemore. Let us look at a simple case:

If the acceleration is known as one, the integral of dvdtequals . If x = 1 and v = , thentheir product will be half, with x = v2/2 from Fx = max = mv2/2 for starting from zero velocityand static zero position. Hence, x v = .

A change in position of length one in a time interval of one means a velocity over thatdistance of one. This is only valid, if the velocity is considered constant over the time intervalin question. For an accelerated motion of our point, the velocity reaches one at the end pointof the interval, so that fora = 1 = const. the mean velocity = . Since only two positionsare known for position and velocity, there is no way in telling whether the motion isaccelerated or not. Hence, the velocity can lie between the two extremes of and 1, and theuncertainty ofv becomes .

13


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Furthermore, an uncertainty in mathematical structure of a similar type exists also in thecontext of complementary n-spaces. The complementary spaces can be expressed as Fouriertransforms of the spaces representing lower time derivatives than themselves, so that aposition space can be transformed into a velocity (momentum) space, transforming into thetime domain. We have argued above that our point moves in an n-dimensional sphericalvolume. This volume is a function of radius and dimensionality. According to our constructof a velocity space being the Fourier transform of our spatial volume function, we argue thatfor n-dimensional displacement or movement from rest (accelerated motion) there exists ann-dimensional displacement or movement in velocity space. If this is the case, we need todetermine minimum conditions of both volumes for enabling such movement in ndimensions. Above we have analyzed the uncertainty relation for a movement of unit lengththrough unit time without scaling such units. We can see, similarly to our two cases above,that there is also an uncertainty of purely mathematical nature in the relation between amathematical structure like our Euclidean n-sphere volume and its Fourier transform. For asimple real space displacement and its transformation there is a minimum uncertainty:

For 1|)(| 2 =

dxxf normalized, the Fourier transformation )()( vfpf

= is also

normalized, according to Plancherels theorem. The dispersion about zero is

= dxxfxfD 220 |)(|)( , and

20016

1)()(

fDfD , according to [15].

So we can write for space and velocity a minimum mathematicaluncertainty of:

2

2222

16

1)|)(|)(|)(|(

dvvfvdxxfx , [16].

This value is the general mathematical uncertainty for complementary variables. Thenumerical value for such uncertainty can be determined for any structure and itstransformation. One can therefore state for complementary mathematical sub-structures thatif one of them is precisely known, the other is only known in a very imprecise way or not atall. Hence, in such a case it is questionable whether the complementary structure has anyreality at all [17]. Anyway, we can say if both structures are known and have reality, bothstructures are showing a dispersion of accuracy. For that reason we may allege a slightlyblurred structure, not to be confused with space-time foam. If the precisions of bothposition and velocity are equal, we have a noise or blurring of the structure of 7.957% forboth of them, but that depends strongly on the conditions of the respective experiment chosen

by an observer.

The fundamental question arises, how to accommodate uncertainty in our mathematicalstructure and how to interpret it in physical reality. If, as alleged at the beginning, themathematical structure not only represents physical reality but is it, the introduction ofdynamics in the mathematical structure creates complementary variables (observables) andwith that uncertainty arises, where the uncertainty of one sub-structure determines the

14


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uncertainty of its complementary sub-structure, and hence, is observer-dependent. If we thenwant to quantify such uncertainty, we can do this in two ways:

1. By introducing dispersion or probability distributions and their respective functionsand their relationship to each other;

2. By examining the fractional dimensional behavior of the structure and deducingprobability distribution functions from them taking behaviors such as Brownianmotion, random walk orzitterbewegunginto consideration.

The very impossibility to assign to each position of our moving point a velocity lies in thefact that the distance the point covers to exhibit a velocity can be regarded as unit length nomatter how short this distance becomes. Even by introducing differentials we end up withuncertainties being dependent on the dispersion of the function describing position. Hence,no matter how tiny we choose our distance covered by the point in an equally tiny amount oftime, the product of the dispersion integrals will always be the same, meaning, theuncertainty is self-similar regarding length and time scaling. It is well known that randomwalk, noise, zitterbewegungand the like are exhibiting fractional dimensions. In our further

investigation of the behavior of a moving point in n-dimensional space we shall analyze an n-dimensional generalized uncertainty relation and scale-invariance of our propositions.

A further consideration is the role of space as a mathematical structure. We have assigned avolume to both position and momentum or velocity space, employing the conditions ofuncertainty derived from purely mathematical reasoning. We further analyze the resultingproduct function ofp orv dependent onx orrand n15. As a minimum velocity or momentumwe take as the minimum velocity of our point determinable by observation. We arrive atthe following results:

The spherical position space volume element dependent on radius and dimensionality isdetermined by

)

)(),(

2

2 2

ns

n

rnrV

+( 1

=

,

as we have seen above. Its Fourier transform represents the velocity or momentum spacevolume and is determined by

)1(

)

2

sin()1(||2

),(2

1221

n

n

p

nnp

npV

n

+

+

=

+

.

ForVp (p, n) we have integrated over the radius and arrive at a function of momentum anddimensionality. If we imply an uncertainty principle, we can argue that before the pointmoved there were neither position nor velocity or momentum space volumes available. With

15 Since we have no mass defined, there shall be equivalence ofp and v as well ass and r.

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movement we enable at least a position volume element Vswith its complementary volumeVp. Before that both were zero, so that we can speak ofVs and Vp as Vs and Vp. If we acceptour above reasoning for our two cases of uncertainty for accelerated and un-acceleratedmotion, we arrive at a generalized uncertainty relation 2 Vs Vp = 1. This yield

01)1(

)s i n ()1(||)(22

2

212 22

1

=+

+

n

nnnnpr

n

,

and solving for p representing momentum or velocity results in

nnnn

n

rnrp

n

n

++

=

+

1

1

2

22

2

)1(

)1()csc()(2),(

22

1

)1(2

3 .

If we set, as outlined above, p (r, n) = , and we consider an interaction constant proportional to r2, we can obtain plots forp (, n)16. Our solutions will be complex, so wecan plot the modulus, the real part and the imaginary part of the momentum or velocity.

5a. Interaction for a momentum or velocity larger than in the first 6 dimensionscontaining the purely real dimensions 1 and 5

Plotting the momentum (velocity) versus (in our units r = if we consider the generalizedcharges as one) and n renders for the first six dimensions a rather surprising result. In Fig. 2 onecan clearly see the minimum mathematical uncertaintys square-root emerging as a minimum

around the fifth dimension. This value is not far away from the numerical value of thesquare-root of the fine structure constant in natural units, 0 3 5 9 9 9.1 3 7

1 , which is the elementar

electric charge in the same units. Results are summarized in Table 1.

Table 1

Dimensions Co-dimension

n for pmax Min. Fraction of

0 - 2 1.4217 0.72 0.02685 11.0875 0.64 2/30.24 0.525 1/3

4 - 6 1.1061 4.96 0.07826 1

16 In the following all plots have to be understood that p ~ v and r ~ .

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Figure 2

0.02

0.04

0.06

0.08

r

0

2

4

6n

0

0.1

0.2p

0

1

A search for the value of the fine structure constants square-root value renders a remarkableresult. For the area between four and six dimensions we have solutions forp as well as inthe area between zero and two dimensions. Around five dimensions the area with positivereal momentum forp and the interaction resembling an electric charge, spans a littlemore than one co-dimension. Between zero and two dimensions we obtain the sameconditions of a little more than one co-dimension around one dimension for of anelementar electric charge, while of a charge appears around dimension with a co-dimension of a little less than one quarter co-dimension, as can be seen in Fig. 3.

Figure 3

0.03

0.04

0.05

r

0.250.5

0.751

1.251.5

n

0

0.05

0.1p

0

17

0.0265

0.027

0.0275

0.028

r

0.4

0.6n

0

0.01

0.02p

0


18/42

Puzzling is the emergence of a numerical value of an elementar electric charge from theconditions given above and its nearness to the value of 1/4 (square root of the minimummathematical uncertainty of complementary space integrals) around the fifth dimension,while around one dimension the numerical values of fractional charges are emerging. The co-dimensionality slightly bigger than one hints to a slightly chaotic behavior of the movementof our point that we let span up our space. The question arises why no other interactionconstant emerges from our geometrical structure other than the fine structure constant. Afurther investigation rendered the same behavior for all odd dimensions greater than five (seeFig. 4).

0.04

0.06

0.08

r

0

10

20n

0

0.05

0.1p

0

Figure 4

5b. Characteristics of the limitation of velocity by momentum volume

Let us assume that the Fourier transformed space volume function belongs to a velocityspace, and a point moving over length one in unit time, spanning up a position space withradius one. The volume created by this motion is 2. Since in velocity space this motionproduces an end-velocity of one, one would expect the velocity one to be achieved within avolume Vs = . In reciprocal space this becomes a volume Vp = 2. As we can see in Fig. 5b,the intersection between the velocity space function (p, n) at volume 2 cuts off at v = 0.894 ofthe expected velocity. Only at a dimensionality of n = 1.4217 unit velocity is reached as a

maximum velocity. This dimensionality coincides with the co-dimensionality of the minimalelectromagnetic interaction around one dimension. We could therefore argue thatelectromagnetic interaction around one dimension reaching the maximum velocity c = 1 isonly possible, if there is a minimum dimensionality available that allowszitterbewegung.

For a massive particle with unit mass this above scenario could be interpreted as follows: Ifwe assume that the Lorentz transformed mass of our unit mass particle is 2.2318 at a velocityof 0.894, the total momentum is 1.995 which is in the one dimensional case the volume of themomentum space. Taking rounding errors into consideration, we can claim this volume to be

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-1

-0.5

0

0.5

p

-0.5

0

0.5

1

n

-2

-1

0

1

2

1Vs

=Vp

-0.5

0

0.5

p

Figure 5b

about 2. This would show that our Fourier transform of the n-dimensional sphere volume isfor massive particles Lorentz invariant in at least one dimension. Furthermore, this showsclearly that the maximum velocity is c = 1.

Our interpretations show that Vs in this case is only half filled with action, and hence VpVs =, which is the generalized Heisenberg uncertainty minimum.

Why can this be argued? We have a point moving over a distance of one within a sphere ofradius one. The Fourier transform ofVsshows Vp = 2 forc = 1, meaning the diameter ofVs is2. Within this diameter we can start and end the motion over distance one wherever we wantinside the sphere with diameter 2. Only half of the sphere is used for n = 1 without affectingthe momentum volume. Hence, ifp is known, the uncertainty of position is .

Following this line of argumentation we can see that the minimum uncertainty and theobvious Lorentz invariance shows that for particles with low or no mass the velocity of lightcan only be reached in dimensions larger than one, depending on the mass of the particlewhich is related to dimensionality, with c reached finally at 1.4217 dimensions. This meansthat any particle with a smaller mass than unit mass moves with an overlay ofzitterbewegung

or random walk. In case of a massless particle the dimensionality equal to the co-dimensionality of electromagnetic interaction suggests a direct relationship with the nature ofthe elementar charge, zitterbewegung, and the volume functions of spherical position spaceand its Fourier transforms. In the course of our treatise we shall see acceleration spacevolume as a Fourier transform of velocity space volume produces besides these relationshipsa direct relationship to spin. However, let us first explore interactions other thanelectromagnetic interactions governed by the fine structure constant.

19


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Can the other known interaction constants be derived from the fine structure constant andwhat conditions we have to look at in our mathematical structure? Maybe if we look atmomentum density as a measure of interaction-spaces and their minimum conditions, we canreach at least an estimate where to look for other interactions. This means also gauging thetime to the same scale in all dimensions including the fractional ones.

6a. Momentum or velocity densities within a spherical n-dimensional space element

We found that the numerical value of the fine structure constant can be determined fromgeometrical considerations only, if one makes the simple assumption of constantacceleration, but its value still emerges in a very unexpected way, at least superficially. Thevalue does not appear as any local minimum of (n), but at a co-dimensional range betweenabout 4.5 to 5.5 dimensions. The exact value of n being slightly larger than one may suggestan overlaying minimal zitterbewegung for such (inter-) action which would be veryinteresting to investigate further. The fractional dimensionality further suggests that forexample an electric discharge almost never takes place on a straight line, but on an erratic

path. Additionally we want to argue that the boundary condition ofpmin. = over a constantacceleration within unit distance and time is a legitimate one in the sense of Wilczekscondition of minimum phenomenon contribution to our structure. It is merely a logicalconsequence of our observability we have constrained to two instants of time. We need toremark that the deviation forpmin. at 5 dimensions from is +0.01020489005 for the exactvalue of the fine structure constant, and the deviation ofxmin. from one is -0.0728. This yieldsan overall error of the uncertainty at 5 dimensions of 0.16975%. This errors contribution tothe deviation of the co-dimensionality is negligible.

Surprising, however, is the fine structure constants emergence dressed as the elementarelectric charge from an n-dimensional spherical position-momentum volume element, whileall other constants do not appear. This may suggest a dominance of the fine structure constantover all other known interaction constants so that

1. either all other interaction constants are dependent on it or2. the other interaction constants are independent from the geometry of space.

In particular, the other 1/r2 dependent constant, the gravitational constant, seems in thiscontext not to be affected by the application of an uncertainty relation to Euclidean space atall. We therefore suggest exploring whether the induction of acceleration in form of higherderivatives of spatial motion may be related to the emergence of different interactionconstants in different dimensions or whether momentum or velocity densities in differentdimensions could be related to a length of motion similar to an uncertainty principle. For the

latter case, if we assume pmin. = over unit length motion, we should be able to find aminimum interaction dependent on momentum or velocity density in differentdimensionalities of our spherical space element. Since the volume changes withdimensionality and the distance in form of the radius does not, we should be able to findsome relationship like that.

To test our hypothesis we shall construct a momentum (velocity) density space we will relateto a length of motion. We determine the function for the volume of a momentum density

20


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space based on a Euclidean spherical volume element in n dimensions. The momentum as afunction of radius and dimensionality at minimum uncertainty is

nnnn

n

rnrp

n

n

++

=

+

1

1

2

22

2

)1(

)1()csc()(2),(

22

1

)1(2

3 .

Assuming the same conditions as above, we can set the momentum . We assume further theproportionality of interaction constants to powers ofrsuch as the fine structure constant andthe gravitational constant being proportional to the square of the radius. We further assumegeneralized charges to be one and let the point bearing that set of unit charges move from itsposition at rest to the surface of our n-dimensional spherical volume element. The momentumdensity will therefore vary between zero at the center and one at the surface of the sphere.Here it is assumed that the velocity of the point changes linearly from zero to one. Hence,

)(trp , while randp are complementary observables underlying the same conditions aswe have established above for the finding of the fine structure constant.

To determine whether the other interaction constants somehow depend on the fine structure

constant we try to find the smallest volume required for an interaction that we norm to one inall dimensions. This allows determining the radius of the smallest sphere in n dimensionsenabling an interaction resulting in a movement over unit length and time.

A smallest sphere is in this case (n-1)-dimensional. According to [18] the radius R of thesmallest sphere in n dimensions enclosing an object with diameter one is given by

)1(2 +=

n

nR ,

which averages over the dimensions in question to about (we only try here to get a rough

estimate).

With

2

2

2

2 /2

r

rq

r

p =

we can see forp = that = r5/8. This shows the dependence of the fine structure constanton five dimensions and that we need to divide our momentum volume by the real volumemultiplied with its square root to norm five dimensions to the fine structure constant. If theother interaction constants really depend on the fine structure constant, at least dimensionally,we should find them by applying our generalized uncertainty relation.

Let us first look at unit momentum density. We obtain from p(r, n) = and dividing by thevolume of a smallest sphere with radius with the condition mentioned above

21


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n

n

n

n

n

n

nn

n

n

/1

2

5.0

1

)1(2

31

25

2

)1(

2

sin)1(2

22

)1(

)4/(

)(

2

+

+

++

+

+

+=

.

A semi-logarithmic plot over the inverse radius dependent on dimensionality obtained fromthe above conditions is shown in Fig. 5a. Here log r= log .

5 10 15 20 25n

-20

-15

-10

-5

log r

Figure 5a

A numerical value of about 10 for the strong interaction is obtained between zero and two

dimensions, around n = 1. The electromagnetic interaction follows between four and sixdimensions around n = 5, followed by the numerical value for weak interaction between eightand ten dimensions around n = 9. The numerical value for the square root of the gravitationalinteraction related to the fine structure constant emerges around n = 21-23 which contain thesixth dimension with purely real solutions for momentum. It appears from these results thatin this structure only odd dimensions and their surroundings yield ground state velocity ormomentum, because they have real solutions.

It seems that first of all the fine structure constant is the dominating constant that exists in alldimensions as a result of the uncertainty of the complementarity of momentum and positionspace. Only in regard to momentum densities (Poynting vector) on a constant momentumdensity surface in n dimensions it seems to appear dressed in different strengths of

interaction dependent on n. Hence, it can be that we can observe dimensions higher than 4 aslabels like electric charge or mass on an elementar particle. The dominance of the finestructure constant suggests Lorentz invariance, so that vmax. = 1 = c. This implies forp = 2 theintroduction of an additional term that could be mass, as we have already seen above.

6b. Conditions for acceleration inducing velocity and acceleration induced by a jerk

22


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In our results above we can clearly see that the interaction constants are never found in a waythat they occur at integer values for n with p = . We alleged a superposition ofzitterbewegungto explain this behavior. We find, by the same token, the numerical value ofe.g. the fine structure constant in a region where the co-dimension is slightly greater than one.One could argue, why should a strictly one dimensional interaction not be possible and ourpoint have the velocity with co-dimension one at the experimental numerical value of theconstant? Zitterbewegung might be the answer, but how can we show any supportingevidence for such a possibility in our mathematical structure that is purely geometric? Thegeometries of velocity and position spaces give enough volume for such an effect, but wecould also allow a different type of motion added instead of the zitterbewegung, e.g. someregular vibration or the like. As we will see below, this bears the difficulty that a(r, n) is not acontinuous function and with that a continuous vibration is not provided with enough space.It will be a chaotic vibration. A more extensive analysis of the chaoticity of such a vibratingmoving point (or string) is beyond the scope of this paper, but will be treated elsewhere.

We stated above that position and velocity are complementary observables, and we thereforetreat acceleration and jerk analogously as Fourier transforms of velocity and acceleration

respectively. Thus we can conjecture position, velocity, and acceleration and jerk to becomplementary to each other. Velocity is complementary to position, acceleration iscomplementary to velocity and position, and jerk is complementary to acceleration, velocityand position, so that uncertainty relations between all of their pair-wise combinations can beestablished. To obtain expressions for the volumes of acceleration and jerk we Fouriertransform Vp to Va and Vato Vjas follows:

( ) ( )

++

+

+

+

=

+

2sin)()1(1

2cos)1(1

2sin)1()(||

)1(

1

2221

2

nasigni

nnnna

V

nnn

n

na

( ) ( )

( ) ( )

]

[

2

sin)1(1)(

2

cos)1(1

2sin)1(1

2cos)1(1

2sin)1()(||

)1(2

1

22

22212

3

2

++

+

++

+

+

+

=

+

njsign

ni

ni

nnnnj

V

nn

nnn

n

nj

23


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Figure 6

0.08

0.082

0.084

r

0 0.5 11.5

n

0

5 1020

1 1021

a

0

5

For a qualitative discussion of the results we first present a plot ofVs Va = (Figure 6), where1/2 denotes the uncertainty. We obtain at n (upper dimension of of the electric charge)a large acceleration space of a 1021. Forn and a > 0 we obtain a relatively randomdistribution of real solution patches for the acceleration. We can clearly see that in theregion occupied by charge, below dimensions there is no space for acceleration, while atn > there is a strongly chaotic behavior of the function a(r, n), reaching acceleration valuesof over 1037 within unit distance (determined by calculation, but not depicted in thisqualitative plot).

24


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0.08

0.082

0.084

r

4.5 4.75 5 5.25 5.5

n

0

5000

10000

15000

20000

a

0

50

10

1

Figure 7

If we then look further at the conditions our acceleration minima (a = ) show in the samedimensionality, and if we notice the patched allowed paths of our point having anacceleration, we see that our point needs slight dimensional changes to cover its path throughacceleration space. These changes look random like a dimensional percolation rather than astraight path, and thus we can expect zitterbewegungor random walk that will for larger rcover two dimensions and resemble Brownian motion. This type of motion is suggested bythe properties of the available space constructed by our moving point.

The acceleration plots show a constraint to constant acceleration only between n = even + ,while around odd dimensions (reminder: every second odd dimension is purely real) theacceleration space allows (or even suggests) strong chaotic accelerations and with thatzitterbewegung. In Figure 7 the overlap regions of the constant acceleration regions in a(r, n)with the regions ofp in p(r, n) are very small and occur very closely around theexperimental numerical values of the interaction constants (error ~1.8%). In the other regionswhere zitterbewegungdominates, an additional velocity or momentum component needs tobe added to our half momentum. It is remarkable that the interaction constant is determined

25

0.08

0.082

0.084

r

4.5 4.75 5 5.25 5.5

n

0

0.01

0.02

0.03

p

0

0.

0


26/42

by constant acceleration and not by the minimum r ~ of the momentum (velocity)p(r, n) , where the acceleration a(r, n) shows chaotic behavior. Vj shows as well chaotic behaviorand is dimensionally discontinuous.

-1

0

1

a

1

2

3

n

0

2

4

6

Va

-1

0

1

a

Figure 7b

Fig. 7b gives a plot of the real part of the acceleration volume dependent on acceleration anddimensionality. Surprisingly, the function is not symmetric regarding positive and negativeaccelerations. Let us discuss some possible consequences of this property of the availableacceleration space. If we assume that negative acceleration denotes acceleration towards anobserver, and positive acceleration away from him, we can interpret the above graph as

follows: Around one and three dimensions the volume allows only negative accelerationswhile around two dimensions there is more positive than negative acceleration possible.Depending on the dimensionality of a particle path we can then determine the accelerationvolume or maximum possible acceleration change for any given acceleration or, in otherwords, we can determine the constraints of available space for accelerations dependent ondimensionality, and with that for different types of interactions. The two-dimensional caseshows an asymmetry that could be interpreted as being able to result in angular momentum.If we consider a small disk being accelerated in two dimensions within the full allowedvolume of this function, it will spin. Since lower dimensional objects need less volume toaccelerate than higher dimensional objects (or paths), as we can easily see from Fig. 7b, onlya full disk on a one-dimensional path will be accelerated uniformly. Any uncertainty,

zitterbewegung or random walk that has a dimensionality greater than 1.5 will lead toasymmetric acceleration for all parts of the path with n > 1.5. Hence, the acceleration patternover a certain path length or radius will determine in a, say, two-dimensional object, how andwhether its different outer edges will move relative to each other. To get an idea how suchaccelerations might be oriented, we shall look at the phase factor of our acceleration volume,dependent on acceleration applied and dimensionality (Fig. 7c).

26


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-1

0

1

a

1

2

3

n

-5

-2.5

0

2.5

5

Fa

-1

0

1

a

Figure 7c

We see the phase factor is zero in all integer dimensions while in fractional dimensions thereare strong asymmetries for fractional dimensions. On for example a 1.8 dimensionalaccelerated path, we will observe a lower phase factor for negative than for positiveacceleration. If we now compare this with the acceleration volumes real part in 1.8dimensions, we see a larger volume for positive than for negative accelerations. Dependinghow we define the directions of our fractional dimensional path, we see that left-rightaccelerations in addition to the one dimensional acceleration of the object that determines itsflight direction, will show an asymmetric pattern that leads to a screw-like dithering path.

Such a scenario was proposed by David Hestenes [19] in a totally different approach to showthat local circulatory motion ofzitterbewegungdetermines the spin of a particle. In addition,we can state that there is besides spin, also a general chirality involved that has a preferreddirection, namely left-handed. If we just qualitatively look at the phase factor asymmetry wejust talked about, we see larger phase factors in one preferred direction, in particular betweenone and two dimensions. A detailed analysis of this problem is beyond the scope of thispaper, but very well worth to investigate by simulations of different fractal trajectories inposition, velocity and acceleration space.

A short discussion of one possible scenario referring to the initiation of acceleration by a jerkfunction alone or by a jerk initiated by a snap may direct to some fundamental ideas aboutmotion and interaction. What does the geometry of the spherical n-dimensional space

element tell us about interactions, minimum time intervals and minimum lengths? Aninstant of time, for example, cannot be determined at a ground state with zero energy. Timewould spread to infinity, and according to our discussions above, it never can reach theproperties of a point-like instant of time. According to Machian ideas time as a result ofmotion of points without further properties is therefore not determinable without theknowledge of two positions and will always have the properties of a time interval. If we donot know the energy and angular momentum of a Newtonian system, we need at least threeinstants of time to reconstruct the space-time where Newtons laws are fulfilled. In a

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Machian system, however, two instants of time suffice, and the two configurations can bebest matched to recover the information [20]. This still does not give us an absoluteminimum time or space interval, but we know that x and t cannot be zero for twodistinguishable configurations, and hence, space-time itself underlies uncertainty principles.The quantization itself is determined by the products of the respective complementary space-volume functions and their dispersion relations, as we have seen above. Furthermore, anyinteraction is also dependent on space volume functions.

We will now discuss the scenario of a uniform jerk of strength 6 over unit time and what itdoes to our point. Therefore we determine the product volume of position and accelerationunder those conditions and get

n

n

nn

n

n

n

nn

n

n

n

n

nn

nrin

nnrin

nnnrnnnnr

n

nra

12

222

2

22

22222

2

1

)]1(2

sin)()()1(22

sin)1()()(22

sin

)1()(2

cos)()1(22

sin)1()(2

cos)(2[

21

),(

+

++

+

+

+

=

For the real part of a (r, n) we can plot 100 dimensions where the dimensions 1, 5, 9, possess real solutions only. This is shown in Fig. 8. According to the conditions evaluatedabove we subtract 6 from the acceleration so that only values equal or bigger than 6 areshown in our plot. We can clearly see the region where a 6 which is necessary to transportour point over unit length in unit time will limit the smallest length for each dimension belowwhich the acceleration will be higher than 6. Analyzing rmin.(n) we find a minimum at 40

dimensions (which is around the 10 th purely real dimension) of the order of magnitude of one.The lowest dimensions resembling unit length with 10 to 20% zitterbewegungwe found to be20 to 24 which is the region where we find the square root of the gravitational interactionconstant as shown above. In our system of units this minimum length is very close to thePlanck length. The same order of magnitude acceleration that allows the transportation of thepoint to v = c = 1 we find in the appropriate dimensions of electroweak interaction at theCompton length scale. Below those lengths the acceleration would lead to superluminalspeeds reversing the charge-parity-time product or violating Lorentz invariance. It seems tofollow that for each interaction type there is a minimum length set by the limit of maximumvelocity c.

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1.2

1.4

1.6

1.8

2

r

020

4060

80100

n

0

0.25

0.5

0.75

1

a

Figure 8

Klinkhamer [21] argues for a fundamental length scale not necessarily equal to the Planckscale that is related to a non-vanishing vacuum energy density or cosmological constant. Ifthere is no direct presence of matter or non-gravitational fields this fundamental length can bedifferent from the Planck length. He further alleges that a sub-Planckian space-time structuredetermines certain effective parameters for the physics over distances of the order of thePlanck length or larger. Seiberg [22] states that gravitational interactions cause a black holeat r< lPlanck. From the calculations of section 5 we saw that from five dimensions onwards themomentum becomes larger than at a length scale of the order of the electric chargesnumerical value. This lies within the Planck length as well as all the other fundamentalconstants numerical values found in section 6a. The exception is the strong interaction, but itlies well within the Compton scale and well within the region where a 6.

We may speculate that we can regard the physics within the Planck length as a sort ofreservoir for interactions. According to Seibergs statement we may regard the domainsbelow the critical lengths found for different dimensions as a formation length for differentcharges characteristic for the fundamental interactions. If we take the black hole idea forgravitation seriously, we might as well generalize this for all other interactions and propose ascenario where length-like dimensions swap into time-like dimensions. There then remainsfor all interactions only one spatial dimension the point can move on. This region can be

described as a mirror image of negative dimensions, where we can regard the negativedimensions as time-like. Probing this, we found that in negative dimensional space theacceleration reaches an average of 6 over a time interval of about one in -21 to -25dimensions which corresponds to 20 to 24 dimensions in positive dimensional space. Thepoint acquires zero acceleration at t= 1. This means after such time interval we have forcefree movement along one spatial dimension. After this time the acceleration within thesedimensions reaches values below 6 so that it can be transposed into positive space. With thathappening sequentially through all relevant dimensions, the point may acquire all its

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properties as a particle on its way to the Compton scale. As we will see, this includes alsospin.

Since the induction of acceleration is jerk, we need to determine what orders of magnitude jerk are available to transport the point into n-dimensional space and which preferredinteraction governs which dimension. It seems that if the jerkj = 6 (in Fig. 9 j = 0 isequivalent to j = 6) over minimum a length of one continuously, the dimensional maximumfor that condition lies just below 10 dimensions, suggesting dominant electroweak and stronginteractions, leaving gravitation untouched. As we can see from Fig. 9, gravity shows only atenth of the length of a jerk present in the first ten dimensions. Additionally the strength ofthe jerk becomes weaker with increasing dimension. This clearly means a delay for the pointto reach over the Planck length in the gravitational dimension.

0.25

0.5

0.75

1

r

5 10 15 20

n

0

0.25

0.5

0.75

1

j

0

0.

0

Figure 9

We can interpret this further in the sense that the strong and electroweak forces thermalizelong before gravity comes into the play outside the Planck length. This means thegravitational energy would remain within the Planck scale until the electromagnetic part ofour point reaches the Compton scale and acquires mass as its gravitational part leaves thePlanck scale. Speculating further, the not yet thermalized gravitational degrees of freedomremain inside a very small volume for a longer time than the degrees of freedom of thestandard model forces. They cannot leave this volume element inside the Planck scale, buthave to overcome a volume inflation of a factor of 41023 from r= 0.1 to r= 1. Since we talk

about negative energy here, and this process takes about 1020 Planck time units until the otherforces reach the Compton length and gravity comes out of the region where the energy(acceleration) space is larger than necessary to accelerate our points gravitational degrees offreedom to c, but the jerk to do that is not strong enough to achieve this, our spatial volumearound the gravitational degrees of freedom stays small (1.5) against the spatial volumearound the standard model degrees of freedom (210240) at the Compton scale. The geometry became with that: three real dimensions with their surroundings spanning up a sixdimensional spheroid wrapped into a six dimensional spheroid with hardly any volume but a

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high negative energy density. This could be interpreted as a possible cause for inflation. Totest this interpretation our one point moving scenario to make up an n-dimensional sphereneeds to be modified to an energy density model similar to existing inflationary models, butthis is beyond the scope of this paper.

To go further to a snap as the cause for our point to move and span up a space does notfundamentally change the above scenario. We think, however, it may be worthwhile toexamine the issue of inflation further in a different paper.

On the curve of Fig. 5 we can find the numerical values of the fundamental interactionconstants in Table 2.

Dimensional range 0 - 4 4 - 8 8 - 12 12 - 16 16 - 20 20 - 24

Interaction strong Electromag. weak spin spin gravitationNumerical valuer2 or r

9.98 1/137.036 8.310-4 1.310-10 510-16 4.1810-23

Purely realdimensions

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0.25

0.5

0.75

1

r

00.511.52

n

-1

-0.75

-0.5

-0.25

0

1t

Figure 10a

As we can easily see, the frequency is strongly dependent on length and dimension. If weanalyze what we are seeing here with the assumption that unit length is the Planck length,and unit frequency is the Planck frequency, this very frequency only emerges at thedimensionality where the full electric charge emerges at lowest dimensionality: n = 0.23.This is very remarkable, as the full frequency at any tangible radius or with that, size of aspace, emerges exactly there. Around zero dimensions, there is no frequency, as can be seenin Fig. 10b.

0.25

0.5

0.75

1

r

00.20.4

n

-1

-0.75

-0.5

-0.25

0

1t

Figure 10b

The full frequency exists at zero radius from about 0.05 to 0.23 dimensions. Closer to zerodimensions, the frequency falls rapidly to zero. As we have seen, at r= 0 and n = 0 thereexists only an empty set. This, of course, cannot oscillate. Leaving the radius of our positionsphere at zero, but increasing the dimension allows first a point to exist, and gives this pointthe possibility to oscillate, but not spatially. The only oscillation we can suggest for a singlepoint at smallest dimensionality is one that oscillates between existence and non-existence, orif we want to see this information-theoretically, a bit erasing and emerging periodically. The

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very profound question would be which of erasing and emerging takes place first. As weknow from Shannons information theoretical thermodynamics [26, 27], erasing a bitproduces heat. If we further conjecture that the empty set at the origin of our position spherevolume function contains one bit of information, namely that the set is empty, an erasure ofthat bit would produce heat and with that motion. The idea behind this is that time does notdescribe being but rather becoming. Since irreversible bit changes produce heat, we haveto discuss how dimensional fluctuations are caused by random bit changes that are erased inhistory, and how the emergence of an electromagnetic interaction could make space and timeemerging as well, but we want to do this only qualitatively, leaving a quantitative analysis foranother publication. Such emerging or erasing might, due to heat generation, lead to line-likeobjects or higher dimensional objects building according to the Machian view space.

The common dimensionality of frequency (time) and charge suggests a common feature:charge is quantized and in thirds not divisible. Time should, by its very nature of beingdimensionally line-like (as a time interval rather than an instant), be quantized in the sameway as charge. The full electric charge emerges at 0.23 dimensions and spans 1.4217dimensions. This dimensionality coincides with that where we reach c = 1 in our momentum

space volume function, hinti

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Emergence of Time and Physical Interactions