A Very Simple Model Concerning the Unified Field Theory Basing on the Kronig-Penney-Model

In this paper, a very simple novel model is presented concerning the unified field theory (“theory of everything”). In the scope of this novel theory, it is assumed that matter, space and time are quantized. It is assumed that the space is subdivided into cubic elementary cells (space quanta), and in each of its eight corners a Delta potential is positioned. That means the Delta potentials are equidistantly arranged, so that the Delta potentials are forming a lattice similar to a crystal lattice in solid state physics. The novel theory is analogue to the Kronig-Penney model well-known in solid state physics: a crystal lattice comprises Delta potentials arranged equidistantly to one another, so the lattice space can be considered as being quantized by an array of equally spaced Delta potentials or the lattice space is divided into cubic elementary cells (space quanta). But instead of electrons, material quanta are inserted into the cubic elementary cells or space quanta. So the material quanta are not freely vibrating (unbound state), but are vibrating in a bound state with discrete energy levels separated by an energy gap. This is due to the presence of the array of Delta potentials. In the frame of this novel theory the Schrödinger Equation for the Kronig-Penney-Model is not solved by differentiation, but the Schrödinger Equation is integrated yielding the formula ( ) ( ) 2 0 – kin x na E E zV ψ = = , by whose discussion the existence of an energy gap is revealed. This energy gap is responsible if the material quantum occurs as light quantum (photon) or mass quantum.


Introduction
In the frame of the unified field theory research ("theory of everything"), there exist two theories competing one another: the first one is called the string theory and the second one is called the loop quantum gravity theory.
According to the string theory the universe consists of freely vibrating objects called strings with the dimension in the order of Planck length [1]- [10]. At the moment mainly two kinds of strings are discussed: the first one comprises open strings featured by a segment with two endpoints and the second one comprises closed strings featured by a closed segment similar to a loop e.g. like a circle [11].
A further development of this theory is called Superstring theory [12]. A generalization of string theory is the so-called M-theory [13] [14].
In 1974, K. Wilson has developed a gauge theory featured by regulisation of space and time [15].
According to the loop quantum gravity theory, the space and time is quantized, while the space is structured by a very fine network of woven finite loops [16] [17] [18] [19] [20].
D. Vaid has tried to combine both well-recognized theories the string theory and the loop quantum gravity theory into a common theory, so both theories are not competing one another but are completing one another [21].

Theoretical Contemplation
In the scope of the novel theory it is assumed that matter is quantized. In the following this matter quantum is also called material quantum. Besides it is assumed that space is also quantized: the space is subdivided into cubic elementary cells confined by eight corners, and in each corner a Delta potential is located.
That means the Delta-potentials are equidistantly arranged or the Delta potentials are forming a lattice similar to a crystal lattice in solid state physics. In the following these cubic space elementary cells are called space quanta. Also it is assumed that the time is also quantized (time quantum).
The novel theory is analogue to the Kronig-Penney model well-known in the solid state physics [22] [23] [24]: The equidistantly spaced positively charged atom cores are represented as Delta potentials, thus a crystal lattice is contemplated which contains Delta potentials arranged equidistantly to one another, so the lattice space can be considered as being quantized by an array of equally spaced Delta potentials or the lattice space is divided into cubic elementary cells (corresponding to space quanta of the novel theory). In solid state physics this yields the Schrödinger Equations as a system of coupled differential equations.
The Schrödinger Equations for the Kronig-Penney-Model is solved by differentiation, that means as an approach a complex exponential function is applied yielding a dispersion function leading to the existence of an energy gap.
But by the novel theory instead of electrons, material quanta (e.g. in form of zero-dimensional punctual mass objects) are inserted into the cubic elementary cells or space quanta. So the material quanta are not freely vibrating (unbound C. Wochnowski Journal of High Energy Physics, Gravitation and Cosmology state), but are vibrating in a bound state with discrete energy levels separated by an energy gap. This is due to the presence of the array of Delta potentials. It is important to mention that the material quanta are not considered as strings at this point.
In this case of the novel theory, instead of solving the Schrödinger Equation by differentiation the Schrödinger Equation is solved by integration, that means the Schrödinger Eqauation is integrated and yields the formula , by whose discussion the existence of an energy gap is revealed. This energy gap is responsible if the material quantum occurs as light quantum (well-known als photon) or mass quantum (until now not experimentally confirmed).
It is assumed that space is subdivided into cubic space quanta which are confined by Delta potentials in its corner points. This is equivalent with a space model traversed by an array of equally spaced (equidistant) Delta-potentials which is comparable to a common simple cubic crystal lattice with a lattice constant "a" described by the Kronig-Penney model in solid state physics. Between the Delta potentials material quanta occur which can be resting or moving with or without acceleration. The material quanta are not free or unbound but bound by the array or lattice of Delta potentials. The material quanta are described by the probability distribution : the material quanta are located either between two adjacent Delta potentials or at the site of the Delta potentials. For the purpose of simplicity it is supposed that exactly one material quantum is located between two Delta potentials, so in every cubic space quantum exactly one material quantum exists.
Thus the corresponding Schrödinger equation is as followed: But instead of solving the Schrödinger equation by an complex exponential function leading to the dispersion relation, now the Schrödinger equation is integrated from −∞ to +∞ after having been multiplied with the complex conju- is overall a steady and differentiable function, a short calculation yields the formula: Now it is assumed that between every two adjacent Delta potentials exactly one material quantum is located. So due to ( ) Thus we discuss the graph as follows: the zero point of the graph (point of intersection between the graph and x-coordinate (axis of abscissae)) is denoted as E 0 , then the physically reasonable solutions are located in the area between 0 0 kin E E E ≤ ≤ = between the minimal kinetic energy E kin = 0 (the kinetic energy E kin is zero) and the maximal kinetic energy E kin = E = E 0 (kinetic energy E kin is equal to the total energy E). Beyond E kin = E 0 , that means in the area of E kin > E 0 , the solutions are not reasonable, because firstly the kinetic energy E kin must not be higher as the total energy E and secondly In the first case we assume E = E kin , that means the total energy E is equal to the kinetic energy E kin . As mentioned above, the probability distribution ( ) 2 x na ψ = is zero at the Delta potential x = na, that means no material quantum is located at the Delta potential. Consequently, the material quantum is located not at the Delta potentials, but it is located between them due to the nor-   x a ψ = is zero that means the material quantum is located between two adjacent Delta potentials (Figure 2(a))) then the material quantum is representing a "mass quantum", while in the last case (if E kin = 0 and thus ( ) 2 x a ψ = is maximal that means the material quantum is located at the site of a Delta potential (Figure 2(f))) then the material quantum is representing a "light quantum" or photon.
One can compare the situation with the harmonic oscillator treated classically and by quantum mechanics (Figure 3)   Eventually one can interpreted this as something similar to an inverse (harmonic) oscillator.
Optionally one can cited that Schrödinger Equation yields that the material quantum starts vibrating only after the insertion into a cubic space quantum confined by Delta potentials as described above. Before insertion into the cubic space quantum, no vibration occurs. So one could eventually assert that the material quantum becomes a string only by insertion into a cubic space quantum due to the presence of the equally spaced Delta potentials which makes the material quantum vibrating. Without any Delta potentials in its surrounding, the material quantum is nothing else as a non-significant zero-dimensional punctual material quantum without any physically relevant features or characteristics and totally lost in space, time and universe.
This could be vaguely and faintly similar to an oscillator featured by a Higgs potential as well as to the Higgs postulation citing that a particle mass arises only by interaction between a particle and the Higgs field.

Conclusions
In the attempt to find an unified field theory, it is an usual way to contrive a novel field whose covariant derivation yields a field strength tensor and thus a Lagrange density. But until now this approach has not succeeded. This can be explained by an array of Delta potentials as the novel field as follows: A field or an array of Delta potentials cannot be covariently derived due to irregularities at the site of Delta potentials. But the Delta potentials are the relevant feature or characteristics of such a novel field, because elsewhere the poten-