(Revised version)

Can Mathematical Structure and Physical Reality be the Same Thing? – An attempt to find the fine structure constant and other fundamental constants in such a structure

Pinhas Ben-Avraham Thebock, 12/5 Rehov Rashi, Elad 48900, Israel1

March 18, 2009; Revised October 7, 2009

Keywords: Electric charge, fractional charge, fine structure constant, fundamental constants, dimensionality, uncertainty principle, mathematical universe, physical reality, Mach’s principle.

Abstract

We try to demonstrate a simple mathematical structure’s properties as an observable physical reality or toy universe. Commencing from properties of an n-dimensional Euclidean structure 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 geometrical environment. We discuss the implications of dimensionality and try to find a reasonable minimum amount of interpretation to let the mathematical structure resemble an observable physical reality without “plugging in” constants. Instead, we only “plug in” some elementary concepts of physics we try to keep to a minimum. We discuss, without any claim to completeness, in what way the mathematical structure could be conceived as a physical reality or whether it could be a physical reality. In this exercise we find the fine structure constant to be the most naturally emerging constant.

1. Introduction

In 2006/7, Frank Wilczek [1, 2] stated that fundamental constants in physics, like for example interaction constants are purely numerical quantities whose values cannot be derived from first principles, meaning, they are not derivable from equations describing certain physical theories, let alone real phenomena that also are not derivable from such equations without “plugging in” natural constants. He further stated that these natural constants make up the link between equations and reality, and their values cannot be determined conceptually.

Arthur Eddington [3] tried for the greater part of his later life to find a geometrical principle to describe physics on the basis of the fine structure constant’s 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 are needed to explain the masses of all the particles. He refers to MacGregor [5, 6, 7] who shows in three papers that the masses of the particles of the Standard Model depend solely on the electron mass and the fine structure constant’s numerical value in natural units. Nottale et al. [8, 9] propose a model of “scale relativity” that solves the problem of the divergence of charges or coupling constants and self-energy with the fine structure constant, α = 1/137, on the electron scale. They attempt to devise a geometrical framework in which motion laws are completed by scale laws. From these scale laws they obtain standard quantum mechanics as mechanics in a non-differentiable space-time2. In particular, in reference [8] Nottale demonstrates a derivation of the fine structure constant by “running down” the formal QED inverse coupling from the electron scale (Compton length) to the Planck scale by using its

1 Email: pinhas_ben_avraham@hotmail.com, Tel. +972-50-863.91072 They do not arrive at a discrete space-time, but rather postulate it.

renormalization group equation3. The numerical value achieved by this procedure is pretty close to reality. A shortcoming of this approach is it yields different values for the “bare charge” or “bare coupling”. Again, he needs to refer to experimental observation to choose the “correct” or “physical” of the three possible solutions. Furthermore, specific length scales like the Compton and the Planck length have to be “plugged in” to come up with realistic values for the coupling constants he determines. Similarly, Garrett Lisi [10] needs to choose the symmetry breaking and the action by hand to achieve an otherwise compelling proposal for a “Theory of Everything” matching the Standard Model. Other approaches to derive the numerical values of coupling constants, and in particular the fine structure constant, border on numerology or other “esoteric” approaches bearing little resemblance of physical reasoning that can be derived from observational experience underlying the construct of the mathematical structures proposed.

In our approach we try to avoid any input of numerical values for interaction or coupling constants, but resort only to some fundamental concepts of elementary physics where necessary. By allowing generalized dimensionality we include the possibility of a fractal picture of space-time that seems to be, at least tentatively, justified by phenomena such as Brownian motion and zitterbewegung, the latter of the two showing true fractional dimensionality, and by quantum theory itself that proposes the Planck length and Planck time as a smallest scale. It shall, however, become clear in the course of our treatment of the underlying mathematical structure we have assumed that such phenomena are the result of the underlying mathematical structure.

The introduction of additional dimensions in Kalutza-Klein theories or string theory as well as 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 coupling constants?

2. What role plays dimensionality in the sense of Hausdorff’s extended view on dimensionality and fractional dimensionality in physical interactions?

We attempt to shed light onto these questions considering some properties of spaces seen as mathematical structures containing, resembling or being such physical interactions without claiming the identity of our structures with physical reality as such. We try to keep the physical reality as simple as possible to see how much “physical law” in form of properties of the underlying structure such simplistic example can produce, and how much additional input in form of mathematical structure or its properties is needed to make our structure be a realistic “toy” universe.

Max Tegmark [11] proposed in 2007 a mathematical universe hypothesis stating “Our external physical reality is a mathematical structure”, based on the assumption that “There exists an external physical reality completely independent of us humans”. He argues for the equivalence 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 to encode numerically elements of language defining or describing mathematical entities or (partial) structures, at least one information theoretical problem remains: one need to agree

3 Such equation needs physical insight to be “derived”. A merely mathematical reasoning without reference to phenomena or physical concepts is impossible.

2

on the encoding. We do have no proof of a “natural” encoding mechanism that would be provably inevitable by emerging from the structure itself as a “by nature preferred encoding mechanism”. We hold against the quest for an absolutely mathematical nature of physical reality that human language and its content may well be translated into mathematical symbolism or “language”, but cannot be immune against a decidedly willed, random or even illogical treatment of that physical reality by humans. Furthermore, any distinctions within the structure are arguably man-made, except they would “automatically” emerge from the structure itself. Thereby the choices made what to look for inside the structure may be also arguably man-made. Besides this caution we find it enormously interesting to try to build a mathematical structure “from scratch” that describes or resembles a physical reality. We are not insisting on what is the “ultimate scratch”, but are interested whether we will be able to argue in favor of an identity of mathematical structure and physical reality.

We will try in the following to investigate a mathematical structure resembling a physical reality using a simple example for such a reality. A central question we shall try to answer is whether and how such structure can provide us with numerically acceptable unique values for, say, conditions of minimal physical (inter-) action. The choice of our example cannot be completely arbitrary and random. Hence, we try to determine from the properties of a simple structure 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 arbitrarily many 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 look the same in any arbitrarily chosen locality of that space. By the introduction of time we introduce a structure similar to Minkowski space, but we shall use complementary spaces such as momentum space as a basic structure to arrive there. In section 3 where we also try to define what is movement and how time-like coordinates arise from it. In sections 4 to 6 we construct such complementary spaces and demonstrate some properties of “position space” and “velocity space”6, taking into consideration “acceleration space”, all in particular dependent on dimensionality. We use the conditions we found in those sections to derive a possible physical interaction in section 5. In sections 6 and 7 we attempt a discussion about the physical meaning of dimensionality and a relativity of space-volume in n dimensions and try to give an interpretation of a possible dependency of observed physical interactions on dimensionality by discussing velocity or momentum densities in different dimensions for identical movements taking “acceleration space” and “jerk space” into consideration, to finally conclude in section 8 with a discussion of our findings and try to assess how much interpretation is necessary to find the physical reality in the mathematical structure. In a brief outlook we try to suggest a program for systematically exploring avenues towards the development of a TOE based on purely geometric considerations.

2. N-dimensional Euclidean space4 We try to limit these to the definitions of position, time, velocity, acceleration and higher time derivatives as 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 a mass yet.

3

For a (geometrical) object or its motion to be described or to take place, a certain minimum volume of space is necessary even if we follow Mach’s and Leibniz’s argumentation in favor of 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 invisible space and time. Mach’s idea suggests that the Newtonian way of thinking about the working of a universe, which is still deep-rooted, is fundamentally wrong. The Newtonian philosophy describes 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 of that space and time which is seen as a holistic interplay of space and its contents. This means the actual structure of space and time is determined by the dynamics and spatial distribution of its contents. We will see in this treatise how such space can emerge from a very simplistic dynamics7. Depending on the nature of such dynamics, complementary spaces will play an important role in demonstrating “physicality”8.

In regard to scale, we do not assume any scale but define the length of elementar movement as one and the resulting time interval also as one. We want, for the moment, not too strictly adhere to Mach’s principle but allow a spherical space in n dimensions enclosing our object or its movement. To avoid more restrictive assumptions we allow highest possible symmetry of our space which is spherical symmetry. We also choose to allow arbitrarily many dimensions n (real number), and our space shall be Euclidean. We reserve the right to further generalize as we progress building our structure. It shall be understood that space with n = 0 can contain a point, n =1 a line, n = 2 a surface and n ≥ 3 a voluminous object. For the word “volume” we want to allow besides a conventional voluminous geometrical object an area of a surface and the length of a line as a volume; only a point without any motion shall have zero volume. We will see the reasons for our choices during our construction process. We further generalize dimensionality to n [12].

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

, with n = 2k

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

we get for our spherical volume element of radius r and dimensionality n

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

4

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

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 2 4 6 8 10

n

0

0.25

0.5

0.75

1

r

As we can see, the voluminosity of our n-dimensional sphere behaves counter-intuitively. The volume reaches a maximum for unit radius and decreases to zero for large n. Furthermore the dimension where the maximum volume occurs, increases with increasing radius.

In such a space we can describe the positions of points or objects relative to each other and arrive at a description of dynamical behavior of a system of objects by looking at their velocities and positions relative to each other. We agreed above that we want to enclose such an object or system of objects by a suitable sphere representing a geometric space spanned up

5

by the “physical” action9. We will see later that for our considerations it is sufficient to simply look at the volumes of such enclosing spheres. We remind the reader about such spheres being chosen for the convenience of having highest possible symmetry. Whether they can stand the test of being or resembling a physical reality, we shall see later.

3. Motion

We need to agree on the following facts as philosophically necessary to describe a space emerging from a point. Let us assume that there exists a point in no environment that we let move over a length one to create a straight line that we want to consider as the radius of our n-dimensional sphere we discussed above. Our initial point shall have no physical or other attributes attached to it other than that of a point resting. We further agree that we can move our point from one to another position as we decide. The familiar definitions of “position”, “velocity” and “acceleration” shall hold, but we do not want to introduce definitions like “force”, “momentum” or “energy” at this stage. Any other properties of the point like “mass” or “charge” shall also be un-defined “unknown labels”. We only allow mathematical entities to exist together with our three “physical” definitions as follows:

1. Position as a vector x = (x1, x2, … , xn);2. Velocity shall be a vector v = dx/dt;3. Acceleration shall be a vector a = dv/dt, and its further time derivatives.

The time shall be denoted by t and higher time derivatives of a shall be considered for non-uniform accelerations of our point. The concept of time has to be introduced as a comparison of the motion of our point relative to a clock-mechanism which imposes a formidable problem in so far as uncertainty is concerned. For convenience, we shall regard the time as a continuum to allow differentiability and integrability, but for a realistic picture of physical reality we would have to assume, strictly speaking, a clock with infinitely high frequency10.

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

The velocity of the moving point can only be determined, if one knows at least two different positions at two separate instants of time11. John Wheeler 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-mâché construction 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 for r and t is justifiable.

9 One could argue the action of a moving point to be “mathematical” as well.10 We do not want to indulge in fundamental discussions about the nature of time in this paper, but we point out that any definition of time should be dependent on motion, if we accept the 2nd law of thermodynamics 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 and space interval one, if it is considered to rest at the beginning of the movement.

6

If one regards a static position of a point as zero dimensional, it can be at any position relative to another point at rest in any dimensionality. If we construct a velocity space in n dimensions, both points will be resting at the origin of that space. This means according to Mach’s principle that velocity space does not exist for those static points or is represented by one point. When the one point changes position moving relative to the other at some not necessarily constant velocity, the moving point will be able to construct a “velocity-sphere” in n dimensions12. In position space such movement will be represented by a line of minimum one dimension which is a co-dimension in position n-space, because the line can be existent in many dimensions. In velocity n-space a point with uniform velocity existent in many dimensions will be represented by a resting point in that velocity space and must have a minimum co-dimension of one in position space. This implies that any movement represented by less than one co-dimension in position space is un-physical or at least physically questionable for now. We want to restrict this implication for the moment until we have discussed the meaning of fractional dimensions in the context of movement. To effect any interaction13, a minimum volume in spatial and velocity space is necessary, allowing for acceleration (change in position and velocity) at all times. From this we can conjecture that any change in velocity or any interaction needs to take place over at least one co-dimension within the respective n-spaces for position and velocity, if there are no effects present such as zitterbewegung. Hence, any motion connected to an interaction constructs a minimum volume of position and velocity space as well.

The current view of Mach’s principle in the context of general relativity that one creates a problem with handling a space-time metric, in particular concerning problems of masses relating to space-time curvatures, can be weakened by our above assertion of a minimum volume 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-time curvature and the resulting assertion that all matter can be expressed by the geometrical structure of space-time, one has to accept also that dynamics should be expressible in terms of changes of that very structure which in our case is a change in radius with time. Those changes, however, are constraint naturally by the relationship between the space “hosting” dynamics, momentum space14, and that “hosting” position, spatial space. Changes of this structure are a critical issue, whether one can assume a mathematical structure to be a physical reality. Only in Mach’s 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 be described as Δx = x2 - x1. In Euclidean space we can connect the two positions with a straight line, and in other types of space with a geodesic line. To define another distinction, because we consider one point moving from one position to another, we need to introduce another label or coordinate, time. In n dimensions, this can be regarded as the construction of a quotient space of position change versus velocity change, fixing the time scale by implication. If the point is considered moving continuously from one position to the other, 12 Again, it is and remains the choice of the observer, how many dimensions he or she chooses to construct a 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 that interaction.14 This we simplified to velocity space as we have given no mass to our point.

7

our time coordinate can be considered continuous as can its path. Since we have not agreed on a particular scale or system of units, we want to define this movement as having length one in position space and length one on the time coordinate. We remind ourselves again of Wheeler’s 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 remains unobserved. If we, however, consider the point resting in its first position and then covering unit length in unit time, the start velocity will be zero and the velocity in the second position will be one, if the point is uniformly accelerated over a time interval of one. The mean velocity over the distance will be ½. 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 the acceleration changes on the way but remains over the unit time interval at unit value, we do not know the exact relationship between position and velocity. The velocity known between the two positions is always between zero velocity and the end velocity in the accelerated case, since the point rests in its first position and reaches the second position in unit time. If we do not know whether and how the point is accelerated, the uncertainty of velocity lies between the mean value and one, in this case it will be ½, if the position and time differences are precisely known. For |Δx| = 1 we will induce an uncertainty of |Δv| = ½, so that their product becomes ½. We will show later, how this relatively sloppy estimation of uncertainty can be more rigorously derived from purely geometrical considerations and first principles.

Above we agreed that only mathematical structure in form of Euclidean space exists in form of an n-dimensional sphere constructed by the displacement of a point representing its radius. Whether we decide to move the point to a unit sphere surface with constant velocity or accelerated from rest leaves us no choice regarding the introduction of movement, meaning, if we have only a resting point that we want to move and define its displacement as our radius, we have to start at velocity zero and produce with that an acceleration. To measure the position of a point while moving, it is not necessary to bring it to 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 element with 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 the following scenario: let us assume, |j| = 1, then a(t) = ∫0

1j dt = 1t =1, and v(t) = t2/2 with x(t) = t3/6. Vice versa, we need a mean jerk <j> of 6 to reach length one in unit time. Now we can introduce infinitely many “introductions” of the motion in question and will end up with x( t) tn for reaching length one. Could this be a quantum jump? – We will suggest an answer later when we know more about uncertainties, but one thing is sure: for higher order jerks we get nonlinear acceleration and with that chaotic behavior of the equation of motion that applies, and even the uncertainty relations between position and acceleration or jerk behave 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 in n-dimensional spherically symmetric space we need to define a velocity space corresponding to our position space. We need to look at the velocity change over unit length and time once more. Let us look at a simple case:

8

If the acceleration is known as one, the integral of dvdt equals ½. If Δx = 1 and Δv = ½, then their product will be half, with x = v2/2 from Fx = max = mv2/2 for starting from zero velocity and static zero position. Hence, Δx Δv = ½.

A change in position of length one in a time interval of one means a velocity over that distance of one. This is only valid, if the velocity is considered constant over the time interval in question. For an accelerated motion of our point, the velocity reaches one at the end point of the interval, so that for a = 1 = const. the mean velocity <v> = ½. Since only two positions are known for position and velocity, there is no way in telling whether the motion is accelerated or not. Hence, the velocity can lie between the two extremes of ½ and 1, and the uncertainty of v becomes ½.

Furthermore, an uncertainty in mathematical structure of a similar type exists also in the context of complementary n-spaces. The complementary spaces can be expressed as Fourier transforms of the spaces representing lower time derivatives than themselves, so that a position space can be transformed into a velocity (momentum) space, transforming into the time domain. We have argued above that our point moves in an n-dimensional spherical volume. This volume is a function of radius and dimensionality. According to our construct of a velocity space being the Fourier transform of our spatial volume function, we argue that for n-dimensional displacement or movement from rest there exists an n-dimensional displacement or movement in velocity space. If this is the case, we need to determine minimum conditions of both volumes for enabling such movement in n dimensions. Above we have analyzed the uncertainty relation for a movement of unit length through 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 a mathematical structure like our Euclidean n-sphere volume and its Fourier transform. For a simple real space displacement and its transformation there is a minimum uncertainty:

For normalized, the Fourier transformation is also

normalized, according to Plancherel’s theorem. The dispersion about zero is

, and

, according to [15].

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

, [16].

This value is the general mathematical uncertainty for complementary variables. The numerical value for such uncertainty can be determined for any structure and its transformation. One can therefore state for complementary mathematical sub-structures that if one of them is precisely known, the other is only known in a very imprecise way or not at all. Hence, it is questionable whether the complementary structure has any reality at all [17]. Anyway, we can say if both structures are known and have reality, both structures are showing a dispersion of accuracy. For that reason we may allege a slightly blurred structure.

9

If the precisions of both position and velocity are equal, we have a noise or “blurring” of the structure of 7.957% for both of them.

A fundamental question arises, how to accommodate uncertainty in our mathematical structure and how to interpret it in physical reality. If, as alleged at the beginning, the mathematical structure not only represents physical reality but is it, the introduction of dynamics in the mathematical structure creates complementary variables (observables) and with that uncertainty arises, where the uncertainty of one sub-structure determines the uncertainty of its complementary sub-structure, and hence, is observer-dependent. If we then want to quantify such uncertainty, we can do this in two ways:

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

2. By examining the fractional dimensional behavior of the structure and deducing probability distribution functions from them taking behaviors such as random walk or zitterbewegung into consideration.

The very impossibility to assign to each position of our moving point a velocity lies in the fact that the distance the point covers to exhibit a velocity can be regarded as unit length no matter how short this distance becomes. Even by introducing differentials we end up with uncertainties 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 of time, the product of the dispersion integrals will always be the same, meaning, the uncertainty is self-similar regarding length and time scaling. It is well known that random walk, noise, zitterbewegung and 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.

A further consideration is the role of space as a mathematical structure. We have assigned a volume to both position and momentum or velocity space, employing the conditions of uncertainty derived from purely mathematical reasoning. We further analyze the resulting product function of p or v dependent on x or r and n15. As a minimum velocity or momentum we take ½ as the minimum velocity of our point determinable by observation. We arrive at the following results:

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

,

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

15 Since we have no mass defined, there shall be equivalence of p and v as well as x and r.

10

.

For Vp (p, n) we have integrated over the radius and arrive at a function of momentum and dimensionality. If we imply an uncertainty principle, we can argue that before the point moved there were neither position nor velocity or momentum space volumes available. With movement we enable at least a position volume element Vx with its complementary volume Vp. Before that both were zero, so that we can speak of Vx and Vp as ΔVx and ΔVp. If we accept our above reasoning for our two cases of uncertainty for accelerated and un-accelerated motion, we arrive at a generalized uncertainty relation 2 ΔVx ΔVp = 1. This yield

,

and solving for p representing momentum or velocity results in

.

If we set, as outlined above, p (r, n) = ½, and we consider an interaction constant α proportional to r2, we can obtain plots for p (√α, n)16. Our solutions will be complex, so we can plot the modulus, the real part and the imaginary part of the momentum or velocity.

5. Possible interaction for a momentum or velocity larger than ½ in the first 6 dimensions containing the purely real dimensions 1 and 5

Plotting the momentum (velocity) versus (in our units r = if we consider the generalized charges as one) and n renders for the first six dimensions a rather surprising result. In Fig. 2 one can clearly see the minimum mathematical uncertainty’s square-root emerging as a minimum around the fifth dimension. This value is not far away from the numerical value of the square-root

of the fine structure constant in natural units, , 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 2/3

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

11

0.24 0.525 1/34 - 6 1.1061 4.96 0.07826 1

Figure 2

0.02

0.04

0.06

0.08

r

0

2

4

6n

0

0.1

0.2p

0

0.1

0.2p

A search for the value of the fine structure constant’s square-root value renders a remarkable result. For the area between four and six dimensions we have solutions for p ≥ ½ as well as in the area between zero and two dimensions. Around five dimensions the area with positive real momentum for p ≥ ½ and the interaction resembling an electric charge, spans a little more than one co-dimension. Between zero and two dimensions we obtain the same conditions of a little more than one co-dimension around one dimension for ⅔ of an elementar electric charge, while ⅓ of a charge appears around ½ dimension with a co-dimension of a little less than one quarter co-dimension, as can be seen in Fig. 3.

Figure 3

12

0.03

0.04

0.05

r

0.250.5

0.751

1.251.5

n

00.050.1p

00.050.1p

Puzzling is the emergence of a numerical value of an elementar electric charge from the conditions given above and its nearness to the value of 1/4π (square root of the minimum mathematical 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 movement of our point that we let span up our space. The question arises why no other interaction constant emerges from our geometrical structure other than the fine structure constant. A further investigation rendered the same behavior for all odd dimensions greater than five (see Fig. 4).

0.04

0.06

0.08

r

0

10

20n

00.050.1p

00.050.1p

Figure 4

13

0.0265

0.027

0.0275

0.028

r

0.4

0.6n

00.010.02p

00.010.02p

Can the other known interaction constants be derived from the fine structure constant and what conditions we have to look at in our mathematical structure? Maybe if we look at momentum density as a measure of interaction-spaces and their minimum conditions, we can reach at least an estimate where to look for other interactions. This means also gauging the time 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 from geometrical considerations only, if one makes the simple assumption of constant acceleration, but its value still emerges in a very unexpected way, at least superficially. The value does not appear as any local minimum of α (n), but at a co-dimensional range between about 4.5 to 5.5 dimensions. The exact value of Δn being slightly larger than one may suggest an overlaying minimal zitterbewegung for such (inter-) action which would be very interesting to investigate further. The fractional dimensionality further suggests that for example 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 of pmin. = ½ over a constant acceleration within unit distance and time is a legitimate one in the sense of Wilczek’s condition of “minimum phenomenon contribution” to our structure. It is merely a logical consequence of our observability we have constrained to two instants of time. We need to remark that the deviation for pmin. at 5 dimensions from ½ is +0.01020489005 for the exact value of the fine structure constant, and the deviation of xmin. from one is -0.0728. This yields an overall error of the uncertainty at 5 dimensions of 0.16975%. This error’s contribution to the deviation of the co-dimensionality is negligible.

Surprising, however, is the fine structure constant’s emergence dressed as the elementar electric charge from an n-dimensional spherical position-momentum volume element, while all other constants do not appear. This may suggest a dominance of the fine structure constant over 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 this context not to be affected by the application of an uncertainty relation to Euclidean space at all. We therefore suggest exploring whether the induction of acceleration in form of higher derivatives of spatial motion may be related to the emergence of different interaction constants in different dimensions or whether momentum or velocity densities in different dimensions could be related to a length of motion similar to an uncertainty principle. If we assume for the latter case pmin. = ½ over unit length motion, we should be able to find a minimum interaction dependent on momentum or velocity density in different dimensionalities of our spherical space element. Since the volume changes with dimensionality and the distance in form of the radius not, we should be able to find some relationship like that.

14

To test our hypothesis we shall construct a momentum (velocity) density space we will relate to a length of motion. We determine the function for the volume of a momentum density space based on a Euclidean spherical volume element in n dimensions. It is

.

Assuming the same conditions as above, we can set the momentum ½. We assume further the proportionality of interaction constants to powers of r such as the fine structure constant and the gravitational constant being proportional to the square of the radius. We further assume generalized charges to be one and let the point bearing that set of unit charges move from its position at rest to the surface of our n-dimensional spherical volume element. The momentum density 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,

, while r and p are complementary observables underlying the same conditions as we 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 in all dimensions. This allows determining the radius of the smallest sphere in n dimensions enabling an interaction resulting in a movement over unit length and time.

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

,

which averages over the dimensions in question to about ½ (we only try here to get a rough estimate).

With

we can see for p = ½ that α = r5/8. This shows the dependence of the fine structure constant on five dimensions and that we need to divide our momentum volume by the real volume multiplied with its square root to norm five dimensions to the fine structure constant. If the other 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) = ½ and dividing by the volume of a smallest sphere with radius ½ with the condition mentioned above

15

.

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

5 10 15 20 25n

-20

-15

-10

-5

log r

Figure 5

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 six dimensions around n = 5, followed by the numerical value for weak interaction between eight and ten dimensions around n = 9. The numerical value for the square root of the gravitational interaction related to the fine structure constant emerges around n = 21 which is the sixth dimension with purely real solutions for momentum. It appears from these results that in this structure only odd dimensions and their surroundings yield “ground state velocity” or momentum, because they have real solutions.

It seems that first of all the fine structure constant is the dominating constant that exists in all dimensions as a result of the uncertainty of the complementarity of momentum and position space. Only in regard to momentum densities (Poynting vector) on a constant momentum density 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 as “labels” like electric charge or mass on an elementar particle. The dominance of the fine structure constant suggests Lorentz invariance, so that vmax. = 1 = c. This implies for p = 2 the introduction of an additional term that could be mass. For now, we leave this and any relativistic implications to speculation to be investigated in a later publication.

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

16

In our results above we can clearly see that the interaction constants are never found in a way that they occur at integer values for n with p = ½. We alleged a superposition of zitterbewegung to explain this behavior. We find, by the same token, the numerical value of e.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 our point have the velocity ½ with co-dimension one at the experimental numerical value of the constant? – Zitterbewegung might be the answer, but how can we show any supporting evidence for such a possibility in our mathematical structure that is purely geometric? The geometries of velocity and position spaces give enough volume for such an effect, but we could also allow a different type of motion added instead of the zitterbewegung, e.g. some regular vibration or the like. As we will see below, this bears the difficulty that a(r, n) is not a continuous 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 vibrating moving 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 therefore treat acceleration and jerk analogously as Fourier transforms of velocity and acceleration respectively. Thus we can conjecture position, velocity, and acceleration and jerk to be complementary to each other. Velocity is complementary to position, acceleration is complementary to velocity and position, and jerk is complementary to acceleration, velocity and position, so that uncertainty relations between all of their pair wise combinations can be established. To obtain expressions for the volumes of acceleration and jerk we Fourier transform Vp to Va and Va to Vj as follows:

17

Figure 6

0.08

0.082

0.084

r

0 0.5 1 1.5

n

0

51020

11021

a

0

51020

11021

a

For a qualitative discussion of the results we first present a plot of Vx Va = ½ (Figure 6), where 1/2 denotes the uncertainty. We obtain at n ≤ ⅔ (upper dimension of ⅓ of the electric charge) a large acceleration space of a ≈ 1021. For n ≥ ½ and a > 0 we obtain a relatively random distribution of real solution “patches” for the acceleration. We can clearly see that in the region occupied by ⅓ charge, below ½ dimensions there is no space for acceleration, while at n > ½ there is a strongly chaotic behavior of the function a(r, n), reaching acceleration values of over 1037 within unit distance.

18

0.08

0.082

0.084

r

4.5 4.75 5 5.25 5.5

n

0

5000

10000

15000

20000

a

0

5000

10000

15000

20000

a

Figure 7

If we then look further at the conditions our acceleration minima (a = ½) show in the same dimensionality, and if we notice the “patched” allowed paths of our point having an acceleration, we see that our point needs slight dimensional changes to cover its path through acceleration space. These changes look random like a “dimensional percolation” rather than a straight path, and thus we can expect zitterbewegung that will for larger r cover two dimensions and resemble Brownian motion. This type of motion is suggested by the properties of the available space constructed by our moving point.

The acceleration plots show a constraint to constant acceleration only between n = even + ½, while around odd dimensions (reminder: every second odd dimension is purely real) the acceleration space allows (or even suggests) strong chaotic accelerations and with that zitterbewegung. In Figure 7 the overlap regions of the constant acceleration regions in a(r, n) with the regions of p ≥ ½ in p(r, n) are very small and occur very closely around the experimental numerical values of the interaction constants (error ~1.8%). In the other regions where zitterbewegung dominates, an additional velocity or momentum component needs to be added to our half momentum. It is remarkable that the interaction constant is determined

19

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

0.01

0.02

0.03

p

by constant acceleration and not by the minimum r ~ of the momentum (velocity) p(r, n) ≥ ½, where the acceleration a(r, n) shows chaotic behavior. Vj shows as well chaotic behavior and is dimensionally discontinuous.

A short discussion of one possible scenario referring to the initiation of acceleration by a jerk function alone or by a jerk initiated by a snap may direct to some fundamental ideas about motion and interaction. What does the geometry of the spherical n-dimensional space element tell us about interactions, minimum time intervals and minimum lengths? – An instant of time, for example, cannot be determined at a ground state with zero energy. Time would spread to infinity. According to Machian ideas time as a result of motion of points without further properties is therefore not determinable without the knowledge of two positions. If we do not know the energy and angular momentum of a Newtonian system, we need at least three instants of time to reconstruct the space-time where Newton’s laws are fulfilled. In a Machian system, however, two instants of time suffice, and the two configurations can be “best matched” to recover the information [20]. This still does not give us an absolute minimum time or space interval, but we know that Δx and Δt cannot be zero for two distinguishable 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, any interaction 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 it does to our point. Therefore we determine the product volume of position and acceleration under those conditions and get

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. We subtracted 6 from the acceleration so that only values equal or bigger than 6 are shown in our plot. We can clearly see the region where a ≥ 6 which is necessary to transport our point over unit length in unit time will limit the smallest length for each dimension below which 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% zitterbewegung we found to be 20 to 24 which is the region where we find the square root of the gravitational interaction constant as shown above. In our system of units this minimum length is very close to the Planck length. The same order of magnitude acceleration that allows the transportation of the point to v = c = 1 we find in the appropriate dimensions of electroweak interaction at the Compton length scale. Below those lengths the acceleration would lead to superluminal speeds reversing the charge-parity-time product or violating Lorentz invariance. It seems to follow that for each interaction type there is a minimum length set by the limit of maximum velocity c.

20

1.2

1.4

1.6

1.8

2

r

0 20 40 60 80 100n

0

0.25

0.5

0.75

1

a

0

0.25

0.5

0.75

1

a

Figure 8

Klinkhamer [21] argues for a fundamental length scale not necessarily equal to the Planck scale that is related to a non-vanishing vacuum energy density or cosmological constant. If there is no direct presence of matter or non-gravitational fields this fundamental length can be different from the Planck length. He further alleges that a sub-Planckian space-time structure determines certain effective parameters for the physics over distances of the order of the Planck length or larger. Seiberg [22] states that gravitational interactions cause a black hole at r < lPlanck. From the calculations of section 5 we saw that from five dimensions onwards the momentum becomes larger than ½ at a length scale of the order of the electric charge’s numerical value. This lies within the Planck length as well as all the other fundamental constants’ numerical values found in section 6a. The exception is the strong interaction, but it lies 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 of reservoir for interactions. According to Seiberg’s statement we may regard the domains below the critical lengths found for different dimensions as a formation length for different “charges” characteristic for the fundamental interactions. If we take the black hole idea for gravitation seriously, we might as well generalize this for all other interactions and propose a scenario where length-like dimensions swap into time-like dimensions. There then remains for 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 negative dimensions as time-like. Probing this, we found that in negative dimensional space the acceleration reaches an average of 6 over a time interval of about one in -21 to -25 dimensions which corresponds to 20 to 24 dimensions in positive dimensional space. The point acquires zero acceleration at t = 1. This means after such time interval we have force free movement along one spatial dimension. After this time the acceleration within these dimensions reaches values below 6 so that it can be transposed into positive space. With that happening sequentially through all relevant dimensions, the point may acquire all its

21

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

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 preferred interaction governs which dimension. It seems that if the jerk j = 6 (in Fig. 9 j = 0 is equivalent to j = 6) over minimum a length of one continuously, the dimensional maximum for that condition lies just below 10 dimensions, suggesting dominant electroweak and strong interactions, leaving gravitation untouched. As we can see from Fig. 9, gravity shows only a tenth of the length of a jerk present in the first ten dimensions. Additionally the strength of the jerk becomes weaker with increasing dimension. This clearly means a delay for the point to reach over the Planck length in the gravitational dimension.

0.25

0.5

0.75

1

r

5 10 15 20

n

00.250.5

0.75

1

j

00.250.5

0.75

1

j

Figure 9

We can interpret this further in the sense that the strong and electroweak forces thermalize long before gravity comes into the play outside the Planck length. This means the gravitational energy would remain within the Planck scale until the electromagnetic part of our point reaches the Compton scale and acquires mass as its gravitational part leaves the Planck scale. Speculating further, the not yet thermalized gravitational degrees of freedom remain inside a very small volume for a longer time than the degrees of freedom of the standard model forces. They cannot leave this volume element inside the Planck scale, but have to overcome a volume inflation of a factor of 4×1023 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 other forces reach the Compton length and gravity comes out of the region where the energy (acceleration) space is larger than necessary to accelerate our point’s gravitational degrees of freedom to c, but the jerk to do that is not strong enough to achieve this, our spatial volume around the gravitational degrees of freedom stays small (1.5) against the spatial volume around the standard model degrees of freedom (2×10240) at the Compton scale. The geometry became with that: three real dimensions with their surroundings spanning up a six dimensional spheroid wrapped into a six dimensional spheroid with hardly any volume but a

22

high negative energy density. This could be interpreted as a possible cause for inflation. To test this interpretation our “one point moving” scenario to make up an n-dimensional sphere needs to be modified to an energy density model similar to existing inflationary models, but this 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 not fundamentally change the above scenario. We think, however, it may be worthwhile to examine the issue of inflation further in a different paper.

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

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

Interaction strong Electromag. weak spin spin gravitationNumerical value

√αr2 or αr9.98 1/√137.036 8.3×10-4 1.3×10-10 5×10-16 4.18×10-23

Purely real dimensions

0<n<2 4 < n < 6 8<n<10 12<n<14 16<n<18 20<n<22

Table 2

Besides the four fundamental forces we found around the dimensions 13 and 17 orders of magnitude for interactions that could resemble the Lorentz invariance violating spin dependent interaction constants predicted by Arkani-Hamed et al. [23]. Insofar as spin is concerned, we have not yet made attempts to find conditions for the induction of spin in this structure besides the numerical values as we became aware of Arkani-Hamed’s work during the compilation of this paper. We included the numerical values of his predictions, because we find it highly interesting that they appear seemingly “at the right spots” but regard these dimensions with caution.

7. Some suggestions on the question of dimensionality

It seems disturbing that the electric charge and its fractions appear within the realm of six dimensions at places that are anything else but straight forward integer dimensions. The fact that the smallest fraction of ⅓ charge appears explicitly around half a dimension and has a co-dimension of little less than about a quarter suggest that the existence of a “point charge” is questionable in any dynamic system. Since any interaction constant needs motion or dynamics to be able to be determined, by experiment or by mathematical technique, the terms “complementarity” and “uncertainty” become a key feature of both mathematical and physical reality. For the dimensionality of a moving point to be determined, we need to consider that we need one dimension to enable movement of a point which has zero dimensions as such, if at rest. The minimum value of the uncertainty between position and momentum is ½. Hence, we can conjecture that a “point particle” in the conventional picture needs to be replaced by a point-like geometrical object of a dimension between 0.4 and 0.66 dimensions. Remarkable here is the asymmetry around ½ dimensions. The exact meaning of ½ dimensions is not very clear, but may be derived from the meaning of 1.5 dimensions, as we will see. If this point would move without uncertainty in position, it would create at least a second point which creates a line between the point’s positions before and after movement.

23

If now some noise or zitterbewegung is added to its movement due to uncertainty, the total co-dimension becomes greater than one. If the point stays in its position with uncertainty and thus exhibits dimensionality of smaller than one and greater than zero, a quarter co-dimension may be explained by its uncertainties in velocity (momentum), position (radius) and dimensionality. One could say, the point tries to move and become a line. The nearer the dimensionality gets to co-dimension one the greater the probability that it actually will move. A quarter co-dimension suggests that ⅓ of a charge cannot be isolated on a particle which is experimentally verified, while ⅔ of a charge has a little more than one co-dimension and can move alone which is also in agreement with experiment (quark isolation problem).

We have used a product of momentum difference and difference in position to determine the dimensions where the interaction radius corresponds to the experimentally determined charge. Hence, the uncertainty in position needs to correspond to an uncertainty in dimension as well. In this picture, a point’s dimensionality between zero and one dimension denotes its readiness to move at the minimum average speed of ½.17

A further interesting issue is that because of the fractional dimensions involved in our determination of minimal interaction constants, we need to consider defects in the space structure which automatically will lead to defects in the time structure. Rowlands [24] points out some very interesting aspects on continuity and divisibility of space and time. In the conventional perception time seems to be infinitely divisible. At least two arguments may be considered against such an allegation. First, there arises the very well known paradox of Zeno of Elea, and second, if we would try to infinitely divide time we would need to construct a periodic motion of infinite frequency which means infinite energy would be needed to drive such motion. Space, on the other hand, gives no rise to the allegation that it is not infinitely divisible as such, but this becomes a fundamental issue if such space contains dynamic systems or, in the Machian sense, is dynamically evolving due to the dynamics of its contents. In the definitions of velocity, acceleration, jerk or snap, time is the independent variable, and space is the dependent variable. As we have seen above, interactions are taking place where irregularities like zitterbewegung are involved and acceleration in regards to the radius of our toy universe becomes noisier the smaller our structure is. According to Seiberg’s allegation mentioned above, we swapped space-like coordinates into time-like ones and tacitly accepted this also happening to fractional dimensions. If we accept this, we have to accept defects in time and allow fractal time. In such case it is essential to discuss the effect of defects in time on the physics that is happening around such defect. If we allow fractional dimensions for space, our situation is somehow clear and the consequences for experimental predictions are known, but if we allow a deviation from the one-dimensionality of time, those consequences are not entirely clear. If we, however, define the arrow of time or the flux of time as a consequence of the thermodynamic behavior of the distribution of matter in space, a fractal time dimension becomes thinkable. This may result in a discontinuous and heterogeneous flux of time.

In Kobelev’s papers [25] the fractality of time produces preferred coordinate systems, but we think this can be remedied by leaving the overall dimension of time an integer, meaning the fractality is localized. One possible solution to this dilemma is to consider an initial jerk or

17 Our calculations above rely often on mean values, because of the imposed constraints in observability. We remind the reader that we look at our length and time interval defined as one as the smallest discernable distance which is not scaled.

24

snap for an input of energy, but then we consider our toy universe as open to inputs from objects of negative dimensionality which could be regarded as reservoir spaces or sinks making the observable universe a dissipative system. We have such objects in the form of black and white holes or naked singularities available in our universe. The contents of such objects may reside inside the singularities, and could be interpreted as residing in negative dimensions, if one accepts the notion that negative dimensional space is contained in a singularity. Both positive and negative dimensional spaces together may then be regarded as a conservative system with an average integer time. An interesting feature of fractal space-time is its time asymmetry that is restored to symmetry if the fractal and non-differentiable features are taken out of this picture.

This speculative discussion reveals a very essential question: Is the presence of zitterbewegung a necessary requirement for time asymmetry? – If the answer is yes, this has far reaching consequences for how we need to look at the physics of our universe. Fractality and non-differentiability of time-related spaces that we represented as Fourier transforms can become a very simple explanation for time-asymmetry, uncertainty and similar features of the structure describing physics of the universe, but building such structure still requires observation and interpretation. Otherwise we have no right to assume that we see the emergence of for example the fine structure constant from the geometry of an emerging spherical n-dimensional space, and we have to assume Einstein’s relativity principles as valid.

Another point that we want to put our attention to in future work is the influence of the acceleration function a (r, n) and the jerk and snap function on the behavior of strings. In particular the acceleration’s irregular surface may cause some interesting chaotic behavior when applied to strings.

8. Conclusions and outlook

We have demonstrated the dependence of a purely mathematical uncertainty on dimensionality. From geometrical considerations we have arrived at numerical values for minimum space for movement and movement densities in n dimensions. We have not scaled our results to any physical size; nevertheless the results are somehow intriguing, if we scale to natural units. We have done this implicitly by scaling to one. We want to conclude with a discussion what the dimensional treatment of a moving point in n dimensions reveals and what it does not.

The moving point’s velocity or momentum can only be determined by two positions in space-time. If the movement is accelerated, one does not know what exactly happens between these two positions, mathematically and experimentally. We determined an emergence of the numerical value of the fine structure constant at co-dimension ≈1.1 around 5 dimensions in our representation. This reveals that the point needs at least one dimension to move plus some dispersion of that movement. To cover around one co-dimension between zero and two space dimensions we need at least ⅔ of that value of the fine structure constant. Between the same dimensions momentum density considerations revealed numerical values in the range of the strong interaction at mZ [2], where fractional electrical charges occur as charges of quarks. This should be further investigated but lies beyond the scope of this paper.

25

Furthermore we have to ask the question whether time is determinable with absolute exactness. The answer is no, because time is measured by clocks, and clocks have the intrinsic property of periodic movement to which we compare a position in space-time. Hence, space-time has to be “rough” and only differentiable in a “blurred” picture. If this is taken into account, our spherical position space element needs to have a rough surface as well which amplifies the effects observed on our allegedly smooth model.

Because of the minimum momentum and uncertainty considerations, it can be alleged, a resting point fulfilling both cannot exist. Hence, we cannot speak of a zero-dimensional object, if any interaction and with that any physics is concerned, also because any interaction requires motion or at least motion-like behavior of said object as below one dimension. Quantum mechanically, we would need a ground state of zero momentum (energy) for the realization of a resting point. For p = ½ this cannot be realized, unless we expand our sphere to infinite radius in zero dimensions. What this means is not entirely clear and shall be treated elsewhere. For now, we only want to suggest some speculative ideas which might be interesting for considerations such as the growth of a mathematical structure from nothingness that could represent or even be a physical reality.

Before we do that, we want to give an interpretation of dimensionality in position and velocity space. In our representation the movement at p = ½ alongside a particular dimension gets a real value only at odd dimensions while even dimensions are asymptotically approached. Between zero and two dimensions the case seems to be clear cut how to interpret what is allowed to happen to a movement of a point on a two-dimensional surface, while in higher dimensions we see a certain analogy, but is it a necessary or sufficient (or both) condition to generate an electric charge by a point in moving same within e.g. 4.5 and 5.5 dimensions at a little more than p = ½? -- What our analysis does not show is how exactly this object or its movement looks like when it makes up that charge. We only know that a slight zitterbewegung is involved besides a straight and smooth movement. It also does not show us why such charge necessarily should be quantized. The only plausibility is to look for standing waves in a resonator (space?), or just take multiples of unit intervals in position space and relate them to a set of dimensions as we have done with momentum density volumes. For that set of dimensions we take the largest radius equivalent to an interaction constant minimum as a measure for our volume element and extrapolate to the other dimensions.

We conclude that we found an interesting way to construct a mathematical structure around a very simple “phenomenon”, a moving point producing a unit n-sphere by its motion. By minimal input of phenomenology we succeeded to reliably find the numerical value for the fine structure constant in natural units, which also seems to be the most fundamental constant as it can be found without resorting to calculate momentum densities. Using the simple concept of position and some of its time derivatives we arrived at a (less reliable) way to determine numerical values of other interaction constants. A remarkable and somewhat surprising property of our “evolving toy universe” is the behavior of its shape when Fourier transformed into complementary spaces. By simple conceptual assumptions of limitations of observability it rendered at least the fine structure constant reliably and showed conditions for chaotic movement like zitterbewegung being included in that fine structure constant. We think we have shown a simplistic but viable example for a relatively naïve mathematical structure and minimal conceptual input, what richness lies in the structure’s (spherical

26

space’s) transformations, if interpreted. Without such interpretation there is no way of recognizing such structure as a (simplified) physical reality, and such interpretation has to be made by an observer. So, we come back to Wheeler’s signposts and the space between them: only if all the space between them can be filled with certainty, we can say we have a mathematical universe that is determinable without an observer and his or her participation. The very scalability of r = 1 in our model and the independence of the fine structure constant from this scaling shows at least in this model no reason for a Planck or Compton scale as they appear to us in meters, seconds and other arbitrary man-made units. The nature of time and our conclusion of the inevitable “roughness” of space-time, however fine that may be, it will be a finite value dependent on the means of the observer, forces us to assume mean values for position or for time derivatives of position. Uncertainty is mathematically ubiquitous even without quantum mechanics; it exists for complementary spaces and definitely for classical mechanics as well.

Besides the unfinished items “beyond the scope of this paper” mentioned above we want to suggest a few things worth looking at in the context of this little model that rendered α = 1/137 so surprisingly:

A paper by David Hestenes [19] tries an interpretation of quantum mechanics by zitterbewegung. Extending our little toy universe towards such an idea would be interesting. Another idea is the generalization of charges to Noether charges representing symmetries could help to understand symmetry breakings in a dimensional context. An action minimalization and such symmetry breaking could be helpful as a “conceptual” plug-in for Lisi’s TOE attempt.

Last not least we ask whether the constraint of minimum two space-time points in the primary structure of position space to determine any complementary spaces with all its uncertainties is a must for observer participation or not. We think it is, because of Wheeler’s signpost model and Mach’s principle where we define time by movement.

Finally, we can answer the question in the title as follows: mathematical structure and physical reality can well be the same thing, but will that structure ever be complete? – We doubt it, without observer participation in form of at least interpretation, not to speak of measurements, and encoding it into something we call “insight”, it may well be the same thing and even complete, but we will not recognize it for lack of completeness in our human way of using mathematics as a language. Some fill-ins between Wheeler’s signposts will always remain papier-mâché as long as mathematics is incomplete, at least for a TOE (this expression is also subject to a definition agreed upon by individuals by consensus – a compromise). Last not least we need to remark that all distinctions like Tegmark’s “reality independent of us humans” are man-made separations dependent solely on the man-made decision where to draw the line. In a real GUT or TOE those lines must be moveable at random, because all needs to be one and indistinguishable, otherwise it cannot be a GUT or TOE.

Acknowledgements:

27

The author thanks Ehud Duchovni of the Department of High Energy and Particle Physics of the Weizmann Institute of Science for the suggestion to look at the “strange geometry of n-dimensional objects” and Lorne Levinson, Moshe Kugler, Gad Maimon, Israel Oshry and Shlomo Makmel for many fruitful discussions. Special thanks go to Shlomo and his wife Ruth for their tremendous hospitality during the final writing of this paper.

References:

[1] F. Wilczek, Fundamental Constants, ArXiv: 0708.4361v1 [hep-ph] 31 Aug 2007[2] M. Tegmark, A. Aguirre, M. Rees, F. Wilczek, Dimensionless Constants and other

Dark Matter, ArXiv: Astro-ph/0511774v3 11 Jan 2006[3] A.S. Eddington, Fundamental Theory, Cambridge University Press, Cambridge 1946[4] E.L. Koschmieder, Theory of Elementar Particles, ArXiv: 0804.4848 [physics, gen-

ph] 15 May 2008[5] M. MacGregor, A “Muon Mass Tree” with α-quantized Lepton, Quark and Hadron

Masses, ArXiv: 0607233 [gen-ph] 20 July 2006[6] M. MacGregor, Electron Generation of Leptons and Hadrons with Reciprocal α-

quantized Lifetimes and Masses, ArXiv: 0506033 [gen-ph] 25 May 2005[7] M. MacGregor, The experimental lifetime α-quantization of the 36 metastable

elementary particles, ArXiv: 0806.1216 [gen-ph] 1 June 2008[8] L. Nottale, Scale Relativity, Fractal Space Time, and Quantum Mechanics, in Chaos,

Solitons and Fractals 4, 361-388, Pergamon Press 1994[9] L. Nottale, The Theory of Scale Relativity: Non-differentiable Geometry and Fractal

Space-Time, Computing Anticipatory Systems, CASYS 03 – Sixth Int. Conf. (Liège, Belgium, 11 – 16 Aug. 2003), Daniel M. Dubois (ed.), American Institute of Physics Conference Proceedings, 718, 68 – 95 (2004)

[10] A. Garrett Lisi, An Exceptionally Simple Theory of Everything, ArXiv: 0711.0770 [hep-th] 6 Nov 2007

[11] Max Tegmark, The Mathematical Universe, subm. to Found. Phys., ArXiv: 0704.0646v2 [gr-qc] 8 Oct 2007

[12] F. Hausdorff, Dimension und äuβeres Maβ, Mathematische Annalen, 157 – 179, 1918

[13] R.W. Hamming, Learning to Learn, Session 9, n-dimensional Space, Naval Postgraduate School, U.S. Navy, Feb. 2008.

[14] J.A. Wheeler, Law without Law, in J.A. Wheeler and W.H. Zurek, Quantum Theory and Measurement, Princeton University Press, Princeton, N.J. 1983, p. 182 -216

[15] M. Pinsky, Introduction to Fourier Analysis and Wavelets, Brook/Crole 2002[16] E. Stein, R. Shakarchi, Fourier Analysis – an Introduction, Princeton University

Press, Princeton, N.J., 2003[17] A. Einstein, P. Podolski, N. Rosen, Phys. Rev. 47, 777 – 780, 1935[18] H. Jung, Ueber die kleinste Kugel, die eine räumliche Figur einschliesst, Journal für

die reine und angewandte Mathematik 123, 241 – 257, 1901[19] D. Hestenes, The Zitterbewegung Interpretation of Quantum Mechanics, Found.

Phys. 20, No. 10, 1990, 1213 – 1232[20] J. Barbour, The End of Time, Weidenfeld and Nicholson, London 1999.[21] F.R. Klinkhamer, Fundamental length scale of quantum space time foam, JETP Lett.

86, 73 (2007), ArXiv [gr-qc] 0703009v5.

28

[22] N. Seiberg, 23rd Solvay Conference in Physics, Dec. 2005, ArXiv [hep-th] 0601234v1, 31 Jan 2006.

[23] N. Arkani-Hamed et al., Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse square law of force, ArXiv 0407034v3, 2004.

[24] P. Rowlands, A foundational approach to physics, ArXiv [physics] 0106054, 19 Jun 2001.

[25] L.Ya. Kobelev, Physical consequences of moving faster than light in empty space, ArXiv [gr-qc] 0001043v1, 15 Jan 2000.L.Ya. Kobelev, Can a particle’s velocity exceed the speed of light in empty space?, ArXiv [gr-qc] 0001042v1, 15 Jan 2000.

29