Abstract
In this writing, we rely on the results of research writings, curricula and articles by physicists, chemists and biologists, analyze, summarize and add thereto our own ideas in order to obtain a monograph on the quantum of spacetime. The principal knowledge is still based on Einstein’s relativity theory and the quantum theory, along the line of the development of the loop quantum gravity theory combined with the string theory through such tools as the theory of linearized general relativity, the use of the projection operator formalism for research of the link between the mathematical Hilbert space and the physical Hilbert space; non-commutative geometry applied to non-commutative spacetime during research of the quantum space beneath Planck scales, teleparallel effective fields and bimetricism as well as dimensional analysis to demonstrate the following ideas: Space is a physical entity which has quantum structure and complies with the principles of quantum mechanics. Quantum of space is either incorporated or separated depending on the variation of the energy impulse tensor present in the space by time and the measure of smallest units (quanta) of space. The quantum scale ( with kernel reproduction in structure of physical Hilbert space) of the quantum spacetime corresponds with the scale curvature of the spacetime ( mathematiccal Hilbert space) in the Einstein equation. This means that spacetime is merely a mathematical representation of the physical spacetime with a variation of quantum scale. Geometry of the physical spacetime with kernel reproduction of quantum structure is the relativity geometry; thus, systems of matter structure are only products in a certain spacetime, espcialli with respect to biological structures.
Introduction
The general relativity theory and the quantum theory presented the two greatest knowledge achievements of the twentieth century. They profoundly transformed the basis of awareness in the physical and engineering world and attained great success in the description and research of nature. Within its respective scope of research, each theory has been successful to a surprising extent of accuracy. Between these two theories, however, there exist several vacancies which are to be filled. With scientific experience, we may hope that these two theories are actually part of a general picture and specific cases of a universal theory. Such universal theory must harmonically include those tow above-mentioned theories or, more concretely, the relativity theory will be quantized with the core idea being the quantum space-time. Our studies are becoming more and profound and in depth, yet a satisfactory and complete unity of those two theories remains a dream of scientists. However, in research documents and quantum mechanics curricula, calculations have been determined on the basis of the stability of space and time, therefrom research will be carried out into the state of evolution of physical aspects or space as well as their connections, interactions and evolutions.
It is the quantum theory that changed our perception of space and time, but quantum mechanics needs to have its concepts and principles expanded while researching the quantum of gravity where spacetime geometry contain fluctuations as typical values of its nature. A large number of theoretical physicists [2], [4], [6],[7], [8], [9], [10], etc. have, with their increasingly profound and comprehensive research, noted that geometry of spacetime, and even space, is not only a benchmark for comparison, not only a mathematical entity in description and research of nature, not only a passive stage to physical processes but, more than that and as intended by the author of this monograph, a physical entity in its full meaning. Spacetime is not a passive stage, neither is it an active one; rather, it is a subject participating in physical phenomena and processes as well as natural ones.
In the general relativity theory, Albert Einstein mentioned the dynamic role of spacetime geometry in his gravity equality [1]:
Where the metric tensor of space Newton
Through the above equality, the general relativity theory introduced a new vision on physics: physical encoded in geometry of spacetime. This is entirely different from Newton
More practically, it is also possible to state that space is also a physical entity and complies with physical laws, but certainly with features different from what mankind has ever discovered. Based on works by theoretical physicists, a logical conclusion has been drawn that: space is the very framework in matter structure or, in other words, due to the accumulation of energy, fundamental particles – vibration modes of various types of strings according to the string theory – in a single spacetime area, that combination is called matter. Matter is space and physical events happening to matter also happen to space. Matter is only relatively closed structural systems in terms of energy and a relatively open one in terms of space. Matter does not have its own absolute space, and when motion is merely a translation of energy structure through space zones but does not carry its own space. Thus, the quantum of matter is certainly related to the corresponding feature of spacetime which is the quantum of spacetime.
So, is it possible to apply quantum mechanics in the research of spacetime? This question has been a headache to so many theoretical physicists and a fact is becoming more and more clear that textbook formulas of quantum mechanics are to some extent universal for the research of the quantum of space. However, theoretical laws of quantum mechanics have up to now regarded spacetime as given, as a slow and sluggish background and as a stage where physical processes evolve. Meanwhile, the relativity theory is not only the theory of gravity phenomenon but also one of the spacetime structure, the gravity dynamics encoded in the geometry of spacetime.
In quantum theory, three fundamental constants of nature are recognized: velocity of light transmission c, gravity constant G, and Planck constant h. By various ways in physics, three smallest fundamental scales have been determined:
Planck length:
Planck mass:
Planck time:
The various structures that we know in our universe is the dimensional scale of these fundamental scales. Physical knowledge shows us that the form of evolution of distinct degrees in physical theory is usually the consequence of potential transition, micro-level physics can be entirely different from macro-level physics. For a long time, physical trials and calculations have been correct at length scales much greater than Planck length. On such scale, the continuity of the universe picture is well described with readily available physical laws and theories. Yet, the problem is: is it possible to research the Planck length scale? Do time and space constitute at these scales? To what extent is spacetime quantized? And, if so, what is the natural quantum unit of space?
Currently, the string theory, or ST, and the loop quantum gravity theory, or LQG, are considered the two main approaches to the theories of quantum of gravity and quantum of space. These two theories are mutually complementary in many aspects. In ST, the central idea is the unity of interactive forces of nature while in LQG it is to find the unity of geometric quantum fundamental principles of space in which the principle of quantum mechanics is the most important guideline. In this writing, we rely on the results of research writings, curricula and articles by physicists, chemists and biologists, analyze, summarize and add thereto our own ideas in order to obtain a monograph on the quantum of spacetime. The principal knowledge is still based on Einstein’s relativity theory and the quantum theory, along the line of the development of the loop quantum gravity theory combined with the string theory through such tools as the theory of linearized general relativity, the use of the projection operator formalism for research of the link between the mathematical Hilbert space and the physical Hilbert space; non-commutative geometry applied to non-commutative spacetime during research of the quantum space beneath Planck scales, teleparallel effective fields and bimetricism as well as dimensional analysis to demonstrate the following ideas:
- Space is a physical entity which has quantum structure and complies with the principles of quantum mechanics.
- Quantum of space is either incorporated or separated depending on the variation of the energy impulse tensor present in the space by time and the measure of smallest units (quanta) of space.
- The quantum scale ( with kernel reproduction of quanta in structure of physical Hilbert space) of the quantum spacetime corresponds with the scale curvature of the spacetime ( mathematiccal Hilbert space) in the Einstein equation. This means that spacetime is merely a mathematical representation of the physical spacetime with a variation of quantum scale.
- Geometry of the physical spacetime with kernel reproduction of quantum structure is the relativity geometry; thus, systems of matter structure are only products in a certain spacetime, espcialli with respect to biological structures.
This monograph aims to clarify the basis of quantum physics in the unity of universe, applied as a theoretical guideline for explaining a number of issues of modern physics and biology.
I. Space and quantum of space
1. Space is a physical entity
Though unable to feel space, we may still in real life know of its existence around us and even within our bodies. For a long time, the perception that space is blank vacuum and even nothing dominated philosophy as well as science. As such, despite the fact that in physical formulas such quantities as distance, area and volume are decisive parameters, they merely make mathematical contributions for comparison.
Albert Einstein’s relativity theory made a great breakthrough by no longer regarding space as an absolute parameter in physical processes and by stating that space also deforms when containing an object which has weight in it. The curvature of space, in addition to its purely mathematical significance, took on the additional role of creating a dynamics of gravity among objects. The space-time complex unified in the four-dimensional spacetime is a mathematical representation of profound physical significance.
However, modern research works have come closer to considering space a physical entity. In theoretical physics, the loop quantum gravity theory would be possible without the recognition of quantum of geometry of space. In the string theory, quantum fluctuations of space are the basis for the assumption that different strings can have different modes of vibration around space directions creating fundamental particle families.
Within this writing, we will present some experimental evidence to demonstrate that space is not only a mathematical entity with physical significance but a real physical entity.
1.1 Energy of vacuum
In the curricula of the theory of quantum fields, vacuum is also considered as a field containing dynamics in which pairs of particles and anti-particles are continuously generated and are cancelled when properly matching. From the theoretical forecast by Paul A.M Dirac, later works by Richard Feymann, J.S Schwinger and Shinichiro Tomonaga demonstrated that a “blank” space was in fact more complicated than what we had ever perceived in the definition of vacuum. Pairs of particles generated in vacuum do not have the same nature as the particles and anti-particles of the fundamental particle families with static particle mass or with no static mass at all that we already know. Pairs of particles, as we call them for the time being – generated in vacuum do not exist long and are thus called “virtual”.They are actually results of a continuous transformation of the fundamental state of energy of the physical space entity. Later, in research works of space quantum, physicists unanimously regarded it as a fluctuation quantum of space.
Due to the excessively rapid mutual destruction of particle pairs, it is practically impossible to measure energy of vacuum by current instruments and techniques. It seems that space quantum not only complies with the principles of quantum mechanics including the bimetric particle wave principle, the Heisenberg principle of indeterminacy, etc but also has other principles due to the firm cohesion of structural quantum, which makes it impossible for space-space to be pierced or broken. In special cases when space is close to the event horizon of black hole, in S. Hawking’s viewpoint, there will appear the radiation on the edge of black holes due to the longer existence of virtual particles as a result of the fact the their fellow particles are drawn into the black holes. It is necessary to brief the readers on our viewpoint when considering black hole only as a sufficiently large space quantum as the merge of quantum takes place gradually when the quantum level varies; as such, black hole radiation (if any) will be different from S. Hawking’s viewpoint which acknowledges the nature of black hole as the Schwarzchild solution which is solved from the Einstein equation in the general relativity theory. In accordance with that, space only curves and quanta of that space do not have undergo any substantial changes in its fundamental energy state without a variation of the degree of freedom due to space quantum. Even when there is such variation, that is not enough to create quantum jumps to give rise to radiation on the edge of black holes as mentioned above. We will come back to this matter in the later part of this writing.
Although it is not yet possible to directly measure the magnitude of vacuum energy variation, we can still prove its existence through the Casimir control experiment or the Lamb experiment while studying line spectrum.
The Casimir effect and the control experiment:
This effect in quantum mechanics was forecast by Hendric Casimir in 1948 and was experimentally verified by Marcus Sparnay’s experiment in 1958.
Placing two sheets of specially flat metal in vacuum extremely close to each other (less than 0.1 micrometer) and using special instruments, its was found that pressure appeared to push them closer to each other.
The cause of this phenomenon is an effect represented as follow: In vacuum, there is a continuous occurrence and self-destruction of pairs of virtual particles which, despite their short-lived appearance, are of the nature of particle wave. Between the two metal sheets, particles of low energy move at wavelength greater than distance have no chance of being created. Meanwhile, in the external space where there is limit, various types of particles of all wavelengths have a chance to occur. Accordingly, as the number of external particles is greater than that of internal ones, pressure appears to push the two metal sheets closer to each other.
Later on, this effect was experimentally confirmed by other authors such as Stewe Lamsreaux (1997) and Mohideen and Anushree Roy (1998) using higher. The calculation results were as follows: Casimir force Fc with respect to the surface area of metal sheet A gave rise to Casimir pressure pc represented by the formula:
Where h is the Planck constant which was found out by Dirac; c is light speed and d is the distance between the two metal sheets. The experiment results obtained were that when d was equal 190 nm, then pc was equal to 1 pa; and when d was equal to 11 nm, then pc was equal to 100 pa.
It is helpful to recall that in quantum theory describing the movement of fundamental particles, the wave function Broglie is used for representation [28]. If we acknowledge that the pairs of particles described in the Casimir effect and later in S. Hawking’s hypothesis of black hole radiation are free micro-particles (though short-lived) carrying energy E = mc2 and with impulse P = mv as well as regarding particles as wave with energy
Where
The wave vector has its form as;
The Broglie wave function with respect to appearing particles has its form as:
In can be inferred that the condition for particles to be generated between the two metal sheets in the experiment is that the magnitude of the distance must be equal to a integer multiple times of the particle’s wavelength, d = n x wavelength with n being a positive integer. If the wavelength has some specific value, such value must be equal to a positive integer sum of some structural unit of space which is very likely to be the quantum of space. If calculation is made in the length Planck, we will then have:
Such opinions will give rise to such remarks as: first, space is also an entity of quantum structure; second, Planck length is a distance generated from m space quanta and m is very likely to be equal to 1 at all times; third, below the Planck level, space has properties that require more special mathematical tools for its survey and description
as the most fundamental concepts of geometry such as points, distances, area and volume, etc. are no longer compatible.
1.2 The expansion of space:
In a period from 1924 to 1929, through the data gathered from the analysis of light spectrum emitted from various galaxies, astronomer Edwin Hubble realized that the further the light is taken from the galaxy, the more its spectrum moves to the red [28]. This can be explained by the Dopler effect on the change of frequency of wave when the wave source moves closer to or further from the receiver. The further the relative movement of the wave source is, the lower the frequency of the obtained wave is.
Considering the movement of galaxies away from earth at a velocity of v, we obtain light emitted from it with declining frequency, which means the spectrum obtained during analysis has an increasing infrared band (movement toward the red). Therefrom, it is possible for us to calculate their relative velocity:
V = H.r
Where: H is the Hubbler constant
R is the distance between the earth and the galaxy
The earth, however, is certainly not the center of universe. As such, the proper conclusion to be drawn when explaining the Hubbler law is no that galaxies “flee” from us but that the space of universe is expanding and the expansion speed is increasing great.
According to [48], two astronomical research groups, namely High-Z-Team (z is the physical symbol for the movement toward the red), which conducted observation from the Siding Spring Observatory since 1995, and the Supernova Cosmology Project – founded in 1988 and has observed from the Lawrence Berkely National Laboratory in California, focused their attention on the explosions of the dual star system Supernova with strong radiation being emitted and lasting for a few weeks up to maximum before diminishing and becoming neutron stars.
An analysis of the results obtained from these two independent observation groups confirmed the movement toward the red in light spectrum emitting from supernovas lasting during the explosion process. That is evidence demonstrating the expansion of space as mentioend above.
However, when observing the explosions of type 1 a supernovas taking place 4 to 7 billion years ago, both of these research groups obtained surprising results: supernovas only had illumination equal to three-quarters of the calculated degree.
Thus, of the many hypothesis given to explain this phenomenon by such authors as Craig J.Hogan, Robert P. Kirshner and Nicholas B. Suntzeff (2000), we see that the hypothesis that vacuum energy (Die Kraft des Vakuums) has a variation and influences back on the gravity of universe and thus curbs the speed-up of the expansion of vacuum is the most probable one.
2. Quantum mechanics in respect of spacetime:
A lot of theoretical physics works have sought to apply quantum mechanics, which has been very successful in the research of the fundamental physical fields, atomic physics and matter structure, etc, to the research of the nature of spacetime.Though at various level and despite using different methods, the authors share the same starting point which is the acknowledgement of the quantizable structures of space. However, as the quantum of space has no large degree of freedom due to the fixed background among them, quantum mechanics needs to develop more special tools. Quantum is here examined by the various physical states in the geometric structural levels of space: point, abstain, spacelike surface and volume which have their own fluctuation of values, the quantum state of the quantum unit structure of space in a unified complex of spacetime and the relation between them. With such thinking, the quantum theory is absolutely entitled to be the right way for changing mankind’s awareness complex of spacetime and moving toward the unity of physics.
The currently available process of generalization of quantum mechanics into the multi-direction lattice elastic system conserves the coherent state among components but always has a fluctuation of the degree of freedom of internal energy like the space of universe, we have to refer to the characteristic factors of an equivalent physical system in the following sample problem:
- The research object is a closed system structured from the smallest units (quanta) which are homogeneous and isotropic;
- Bigger and non-homogenous particles are those constituted by smaller ones when coming under the effect of some sufficiently large energy operator
- The measure of perturbation distinguishes between big particles described by the incorporation in accordance with the principle of superposition.
Up to now, no physical evidence is available for us to conclude whether the universe is limited or unlimited, whether the expansion of space by time is applicable to the entire universe or to the region where we are and whether the standard universe theory with the Big Bang as the core is true or not. Yet, with what we have obtained in astronomy, such as the movement toward the red as mentioned earlier, background of residual radiation, theoretical inference and calculations that the diameter of universe ca. 5-10 billion years, on generality, we may see it as a closed system in order to apply the fundamental principles of quantum mechanics to the research of space.
Normally, the quantum theory supposes the determination of the natural state of spacetime also by means of a quantum state equation , then the Schrodinger equation of quantum mechanics used for description of the quantum relation between the necessary quantities is as follows:
In canonical conditions, the quantum state has mapp
Time t in the Schroedinger equation is the time by various concepts. In non-relative theories t is the absolute time of Newton
In the above-mentioned second law, the mapping of the quantum state is determined in the Hilbert space. In the field theory or particle mechanics, the Hilbert space is structured from the components of the field or the location of the particle in a physical space. With that perception, the Hilbert space
Despite the fact that in physical theory, calculations are made in n dimensions, we – in this writing remain loyal to the Minkowski n = 3 +1 which is applied in Einstein’s relativity theory: 3 space dimensions and 01 time dimensions. As such, the space quantum theory considered canonically for the 4 dimensional spacetime is proper for the spacetime in the relativity theory and naturally recognizes the reference systems and fundamental principles of quantum mechanics – relativity theory as its theoretical basis. Physical phenomena and processes are considered by the graph outlined in the 4 dimensional space in accordance with the normal Lorentzian symmetry. The path of particles and the impulse dynamics transmitting in the quantized spacetime or of the physical spacetime are still considered by their Feymann probability sum obtained in time and represented in diagram.
And is mathematically represented through the integral expression:
The integral on the left is one of all the transmission lines of the research object from A with time t = 0 to B with time t = T.
Amplitude is the sum including all transmission means from A to x in a period of time from t = 0 to t. Next, we have amplitude la
In quantum mechanics, the above functions determine the transformation state of particle wave in stable space, i.e. the fluctuation of energy or the structure of space are not yet taken into account. From the above approach for the space itself but through the amplitude of impulse transmission, we assume that the structure comprises small and fine particles of space. It is necessary, however, to demonstrate the stable and coherent Lorentzian character of spacetime geometry so as to determine the direction of space and time quantities, the foliating family of space in the (3 + 1) complex through which its evolutions of quantum state and the history of small particles are individual values calculated in each time slice of the space line in the study space.
In his writing [19], James B.Hartle held that when generalizing the conditions for application of the quantum theory for the quantum spacetime, at a low energy level, the theoretical effect of the gravity quantum is an explanation for the general relativity theory when the spacetime metric is intertwined with the field of energy matter. As a matter of fact, in applicable cases, it is necessary to make adjustments due to the nature of matter having structural frame is the space itself. Demonstration of the gravity quantum following some particular principle as well as how the quantum mechanism of the interaction of the dynamics interacts with the space potential energy of space will provide sample documentation so that the quantum theory will also be an effective tool when studying spacetime for which we always hope in nature of its quantum structure.
According to Hartle [19], the pivotal idea is that when studying space quantum, the object is no longer quantum if spacetime but that of spacetime. To concretize that, we incorporate the above quantum process into the closed space of the 4 dimensional geometrical universer, represented by metric is the closed 3-dimensional fixed manifold. For convenience, we limit
It is necessary to emphasize that we are loyal to the guideline that the results of quantization of spacetime structure in accordance with the principles of quantum mechanics will still have to ensure suitability with the solution of the Einstein equation in the general relativity theory. Especially, due to the geometric nature in the content of the theory, it is impossible to omit the quantized geometric structure in respect of Riemann geometry represented in the Lorentzian form which is in its nature in accordance with the relativity theory mathematically.
According to B. Hartle, the three ingredients of the quantization theory for spacetime geometry which are required include:
- The operation of small quantum structures by 4 dimensional metric and matter field conformation in M;
- Partitioned larger dimensional structures of the (spacetime metric + matter field) complex into the 4 dimensional diffeomorphism invariant
- Cohesion according to the structure of function
And the coherence function as:
Where is the new state of space.
According to Lars Anderson [20], the Einstein equation is the Euler-Lagrange equation in the Lagrangian form for gauge symmetry and thus it is also satisfied in the Lorentzian norm as well as in the Yang-Mill equation and can be considered as a mutually dependent development system of equations. Under ideal conditions, the Einstein equation obtained solution as a hyperbolic system, is in its nature as a perturbed complex of spacetime harmonic coordinates and is considered a complex of wave coordinate. As such, the Einstein equation in theory becomes a quasi linear system of the wave equation. On the other hand, the solutions of the systems of equation obtained from the Einstein equation are also viewed as a system of elliptic equations with appropriate selected variables. From that demonstration, we can assume that the core equation of the relativity contains both elliptic and hyperbolic characteristics, depending on constraint equations and the Cauchy problem.
Mathematically, techniques and ideas from geometric analysis always play a central role in mathematical problems which need to consider the relativity theory. As a matter of fact, when viewing space in its nature as a physical entity of quantum character structured from natural quanta which is tightly coherent but has freedom degree for the fluctuation of quantum, compliance with the principle of indeterminacy of quantum mechanics as mentioned above gives the correct solutions in the Einstein equation, it is helpful for us to think of the general problem of space quantum which techno-mathematically corresponds with the submanifold geometric survey and foliation with minimal surfaces, marginally trapped surface with the general meaning being that curving hypersurface has a curve of zero. This remark was demonstrated by L. Anderson [20] as follows:
Considering a hypersurface N in the Euclidean space
mapping to function u. The area of surface N is given by
N is stationary with the mapping to A if u satisfies the equation:
Hypersurface N is determined as the graph of u when solving the above equation and is the minimal area with the compactly deformed supplementary mapping; it is called the minimal surface. With n ≤ 7, the solutions of the above equation determined in the overall
The above equation and the more general form ensure the meaning of the curved surface quasi linear equation of the quadratic elliptic surface. When solving the Plateau problem in the Riemann space [20] obtains the following consequence: With M being a complete manifold, Riemann of the number of dimensions n ≤ 7 and being its margin, N is the smoothness covering the manifold inside it. Now, with the obtained consequence, we consider is covered with the same size as a spacelike hypersurface in spacetime with being the first-degree and quadratic basic form creating as the expected combination Cauchy with respect to the Einstein equation.
Several authors have considered problem for the general case with the Einstein universe constant equals 0, we will then have:
As the Einstein tensor of V.
Toward expanding the meaning of the Einstein equation for the quantum spacetime, we find Wayne R. Bomstad and John R.Klauder’s manner [16] of quantization of gravity quantum as follows very satisfactory. According to these authors, the linearization of the gravity field according to the traditional variables of geometrodynamics is generated by the perturbation of metric tensor around the flat background. When we can represent the relativity theory in the Hamiltonian form, this indicates the coveriance of Einstein – Hilbert Lagrangian which can be separated in the (3 + 1) form and encoded in representing the equation proposed by Arnowitt R., Deser S and Misner C. (ADM), which describes the evolution of the 3-dimensional hypersurface in the 4-dimensioanl space. The dynamic variation here indicates the 3-metric symmetry of the hypersurface and its canonical dynamics density creates the flat and canonical space phase. In calculating the original data of metric tensor
Where the N interval and the shift vector
It determines the relation between the internal nature and the external curve through the 3-dimensional scale Rici and it is in proportion to the canonical momentum. The trivial diffeormorphism in the beginning is exaggerated by diffeormorphism constraint and through which the quantity is trivially displaced in the spacelike of hypersurface. The metric tensor and the conjugate dynamics are spread around the flat background as follows:
Where is the Euclidian 3-dimensional metric. The extent of fluctuation of the lapse and shift is spread in accordance with the following expression:
The means for expressing the orthogonal decomposition with respect to the symmetry of tensors are the three forms of perturbed oscillation: horizontal oscillation, vertical oscillation and traceless oscillation, representing fluctuation-dependent relation. Each
Where
2 degrees of freedom at
The authors of writing [16] demonstrated that the form of the polarized symmetric tensor also has Hamiltonian linear dependence and thus, their variable function can be represented in the form of Hamiltonian harmonic oscillation for transverse-traceless variables and quadratic functions of dependent variables
There is actually a structure of harmonic oscillation with the perturbation of degrees of freedom. Variables of vertical oscillation are also proved to linearly fluctuate with the nature of harmonic oscillation. According to Klauder, any change in unconstrained theory for quantitative combination of physical structures needs to have a proper tool in quantization. The tool for us is the use of projection operator E for seeking the basic relation between the Hilbert space in mathematical representation and the quantum state in physical spacetime, represented in the equality:
Where the projection operator is the function of the sum of square of the dependent operator with the presence of parameter
And
Of course, the said projection operator also complies with all the properties of a standard projection operator, namely:
When the cohesion states of metric elements of this projection operators are found, it is possible for us to obtain a physical Hilbert space through the appropriate reproducing kernel; in this writing, that is the quantum reproduction of space.
To implement the physical Hilbert space structure with the reproduced components, Bomstad and Klauder followed the two following steps: first, the reproduction of particles is adjusted by the smallest parameters
With the selection of the appropriate projection operator for the cohesion state of the structural element, the physical Hilbert space will have its state of quantum structure reproduced; in case the momentum impulse tensor is present in it, the reproduction process of particles will be a cancelled reproducing of kernel:
Where
The physical Hilbert space here is constituted by a limited number of points of the entire Cauchy sequence in the standard form [16]:
This structure is called the Hilbert space with kernel reproduction. The problem arising is: if we can find an appropriate function for representing the space quantum, we can then characterize the physical Hilbert space in our theory. The first task is to find an appropriate projection operator as well as determining the cohesion state of quanta, through which it would be possible to the forma accuracy of the kernel reproduction process applied in the linearization of quantum gravity. For simplicity, we examine the case of canonical operator and constraint relations in the ideal space as well as surveying and measuring self-adjustment in a given box; thanks to that, each position of the space with operators shall be represented as:
Where L is the width of the box,
can be used as the space momentum representation in the canonical commutation relations.
The next step is to represent operators in their Fourier multiple expressions. The conditions are given as in the ADM equation: let be given a combination vector
And as such, each operator can be written in its Fourier factor expression as:
where
And therefrom the cohesion state has its form as:
As stated earlier, the assumed cohesion state in the physical Hilbert space if of great significance to our next analysis. In general, there exist in parallel the firm cohesion states of space elements in the following form:
For traceless fluctuating oscillations represented by the component:
They are spread to adjacent elements in the manner represented in expression in 2 independent polarization states:
which symbolizes 2 levels of freedom and independence in the quantization of the gravity phenomenon from the viewpoint of the Einstein equation. In the expression giving only the transverse traceless oscillation variables, the Hamiltonian form is represented by the operator
With this Hamiltonian form, the transverse traceless oscillation is represented as a combination of the harmonic oscillator. The standard vector with respect to the cohesion state represents the standard harmonic oscillation of the basic form. The cohesion state is written by Kaluder in the form of Weyl operator as follows:
In which is contained the overall Hlbert space base with the transverse traceless oscillation
in the form of Weyl operator with the standard vector. For the verticl oscillation with trace of the Hilbert space
Thus, there are three combination operators for 3 degrees of freedom due to the determination of each point in the elastic space. The common state representing the diffeomorphism constraint through the Hamiltonian operator is:
including operations in all three components of the quantum space. For constraint state equations, the relations of Hamiltonian operators that we have stated have the results as follows:
It confirms the independence and the presence of commutation between component operators in describing the oscillation of space quanta.
The past surveys by Bomstad and Klauder [16] intended to confirm the cohesion state of quantum space structure and represented in the products of the direct physical Hilbert space as follows: (48)
thus through the mapping method between the physical Hilbert space and its image in the representation of the mathematical Hilbert space demonstrated that: the structural state which has always had the firm cohesion of the physical space always has variable degrees of freedom and partition walls, this phenomenon can be represented by the constraint operator with the following characteristics:
- having subspaces with the lattice quantum structure and coherent states
- oscillation has the standard Hamiltonian harmony in all 3 dimensions
- component state operators are commutative
Through quantum operators [16], the authors directly demonstrated that: the reproduction of particle structure in the presence of energy operators impacting the basic structural energy is considered a natural characteristic of the Hilbert space. The projection operator, established through constraints in demonstrating the linearity of the gravity force, represents through spectrum the sum of the squares of the relations:
Where is a parameter of extremely small value but different from zero, and X with respect to gravity linearization is a function operator of each point on the momentum lattice and is represented as:
With respect to the constraint of the gravity linearization, projection operators are very likely to be commutative, between the projection operator
Where
is the function of the correction parameter
As the projection operator for the transverse variation, and
is the projection operator for the longitudinal variation.
Through the above equations, it is obvious that: the value of the regularization parameter depends on the factor k. In order to reduce this dependence, according to Bomstad and Klauder [16], it is simple to make a choice a sufficiently larger value as the starting limit for or ks. For example, a sufficiently large value shall be chosen so that we can obtain a k as:
With the lattice-dependent parameter. It is necessary to repeat that at k = 0, the lattice character is generally accepted to be omitted, which is intended to facilitate calculation during the reproduction of particles.
With the separated operators which have just been obtained, the general equation for reproduction of particles in the physical Hilbert space as follows:
From following calculations, the two authors of the article demonstrated that: contributions by the transverse and longitudinal variation do not help reproduce the space structure as it does not help formulate the phase space with the degree of freedom through value k-space point with the result that the reproduction of particles only characterizes in the one-dimensional Hilbert space which, as we know, is not physically significant. Meanwhile, dynamic degrees of freedom are totally dependent on the complex of transverse traceless variation of the space metric and the dynamic energy field, creating a free field in the quantum lattice structure and Hamiltonian harmonic oscillation in each lattice point.
From the work by Bomstad and Klauder which has just been mentioned, the consequence obtained is the connection between the Einstein equation and the physical nature of the gravity phenomenon in quantum space, the mechanism of particle reproduction and quantum reproduction by each order of quantum ensuring that the quantum space lattice always maintains the integrity of the structure with the connection factor being stable within a certain value range 0 1 when there is impact by some energy tensor. However, the isotropy and homogeneity of the space has been breached, thus diverting the movement of the fundamental particles with mass, resulting in the fact that their dimensional energy matter structure system is also diverted as determined by the calculation of the general relativity theory with respect to gravity interaction. It would hasty for us to discuss this in depth right now; it is, however, necessary to emphasize that it is due to the phenomenon of space quantum reproduction that the concept of space “point” changed its nature if we consider a single point as a space quantum in the physical Hilbert space and, for the same reason, it does not breach teh standard Lorentzian symmetry in that space.
II. The quantum structure of space
1. The loop quantum gravity theory
We have covered a distance along with experimental physicists, astronomers as well as theoretical physicists to place our faith in the quantum of space. Yet, what is the structure of quantum of space or, to put it more accurately, of spacetime, like? This has drawn the attention of a large number of researchers including [2] [4] [5] [6] [7] [8] ][10] [11] [14] [21] [26], etc. Authors pursuing various research ways have come up with various theories such as: the loop quantum cosmology with the center being the loop quantum gravity theory, the spin network theory, the spin of quantum foam theory and the string theory.
We earlier mentioned one of the most fundamental concepts of physical space geometry - point. This concept in quantum geometry, however, is quite different from the traditional concept in mathematical geometry and everyday intuition of the minimal size of a point. In quantum spacetime, point coincides with a structural unit and the physical –quantum function of space and is a limit singularity of all experimental examination tools. Under such viewpoint, under point we can only survey more special and purely theoretical research tools which, according to us, can be the quantum of geometry, the loop quantum of gravity theory and non-commutative geometry with which the later parts will deal with. However, in quantum spacetime, other dimensional concepts of quantum geometry bear a relativity meaning, depending on the various local structural-energy states and in the Lorentzian transformation so as to canonically ensure the presence of some necessary parameters or, to put it in other words, the apparent geometry of quantum spacetime must be the relativity geometry.
From the works [2] [3] [4] [6] [12] by A. Ashtekar, M.Bojowald et al, we notice that the loop quantum of cosmology is the cosmological scale of the loop quantum gravity of which the foundation if geometric quantum with the canonical quantization of the relativity theory when using the variables given by Ashtekar. In the furmula form, the gravity effect obtained from the constraint relation between the standard theory and the fundamental classical fields through representation by SU(2) connections [4], as well as spin connections create an outward curve due to the combination with the energy density
Where and
Where e is the curvature in
With S being the surface in and
In each space structural unit, the variations of the above variables have SU(2)-Gauss independence relation and are considered diffeomorphism constraints, it requires stability when the space varies, and for the Hamiltonian constraint, it represents the performance of dynamics in this structural model. As the variations of direct diffeomorphism in the fields are in fact diffeomorphism invariants, they can thus be represented using the classical Poisson algebra and the standard symmetry is always ensured in the quantization of geometric space [2] [3] [4].
In order to understand the staring idea of the LQC theory, Martin Bojwald [4] interpreted on the basis of classical algebra when considering the connections in each structural unit of space as a combination of mathematical structures. This consequently facilitates the examination of the variation as well as the cohesion of connections creating the quantum space lattice or, more accurately, the connections qualified as the smallest structural units of space. Examination is done to the simplest case which is the unique state 1 (A) = 1 space supposing that there is only one connection – one component – independence and nothing else even time. With all the higher states, there will appear the universe with complicated relations considered though the holonomies as the mutiplication of operators. From activities which combine discrete mathematical images in time and it is very likely that states at some extent integrate into groups or higher structures, then they are no longer trivial and represent the characteristics of the examined space. The interesting thing here, however, is that the edges of triads play an important role on the holonomies structure chosen and only determined the cylindrical functions by way of such holonomies. Holonomies generated as a result of the cohesion state of the sides of the triad in the holonomies are arranged according to a certain law which generate gaphs in space
While any properties of connections are only shown through the characteristics of the last holonomies state, the full state of space is considered the projective limit of the whole combination of graphs. Thus, the examined space is not limited to the relations of individual connections but in the expression content of high-degree arbitrary dimensional structure of the groups of composite connections of dimensions with time. This structure results in the natural diffeomorphism invariance integrated into the space quantum and is called the intrinsic Ashtekar- Lewandowski multiplication, it determines the Hilbert space characteristics of LQG.
Fluxes showing the linkage forms of connections and their operation processes are expressed through derivative operators. From the states of the holonomies of high-degree combinations of the connections, flux operators can be obtained on the following principle: The flux operator combined with surface S as a considerable impact on the empty state of the lable in graph g only when S and g intersects. When such operation is caused by the sum of intersections, mutual contributions of the integrated values of the edges will be encoded in the spin values of label j. According to [2] [4], the above mutually related operators are differential operators of intersection value sequence o surfaces S with state graphs and are integrated in integral value degrees. Along that guideline, the differential operator obtained from the discrete spectrum transformed to a geometric operator other than the density structure of triads and space area or volume.
In the LQG theory, Ashteka and Vanderloot held that the technique of the quantum field theory is represented on the metric basis, for example, when arranging the divergence of the field in its Fourier sequence and the defined state of vacuum as well as particles, this no longer holds true in the case of gravity quantum, where metrics must also be viewed as operators. In this theory, the basic variables for the quantization theory are closely causally to its resulting expressions, the connections and the density of triads can be naturally blurred throughout the curving section and surfaces do not need to use the basic metric expressed in the Hilbert space. The stability in diffeomorphism invariant is maintained, which means the basic independence theoretically is not altered during space deformation even when there are different selections of the complex of units to be represented. That is the fundamental property of LQG and is recognized as a factor of vital importance for the quantization of space geometry.
The symmetry of space quantum can be started in LQG to the extent of the fundamental state operator. First of all, it is necessary to formulate the isotropy of space in the connections and variable triads of a, here the role of scale factors is replaced with the density of complex triads p with
When representing the state of quantum evolution in the LQG theory, the first thing to look at is the quantum evolution equations which are the quantization of Friedmann equation [4]. They are modernized in the Hamiltonian constraint through the state representation equation in the form:
which is the sum of the calculated values of representing the state of triads, similar to earlier . As for constraint operators, operators are used for the synapsis of p related to in the Friedmann equation. From the exponential function of c, lying in the basic operation state when transforming to the label state, no transforms to the critical variable state in the label of function and we have the following differential function:
With the monovalent volume and the Hamiltonian matter field operator
One of the important conclusions in the gravity quantum of LQG is the non-existence of singularity through the demonstration as follows: the above dependent equation does not contain the common evolution in combination with time but can be acknowledged as a evolutionary representation of internal time as in place of the continuous variable a we have label
The intuitional image in this part can be obtained from the observation of the evolution in decreases when
That the quantization is not broken down results in the classical description of spacetime being no longer entirely true. Based on the sign of
An issue of importance in the space quantum theory that requires further discussion is the representation of wave functions and the relation thereof to time. In the normal representation of quantum mechanics, the wave function determines probability with respect to participating quantities conducted by observations outside the quantum system on the spacetime background. On the contrary, gravity quantum and cosmology in accordance with the idea of the quantum treatment theory for the entire universe are in the first place not observations outside the quantum system but right inside spacetime. Thus, the research of wave functions in the quantum cosmology theory is one of complexity but necessity. To some extent, we can divide the issues of the classical part and the spacetime quantum part in quantum mechanics principles by doing research separately. The extent of the classicality is determined in the attitude toward the degree of freedom of the examined object is trivial with respect to a separate matter structure system but is of significance to the evolution of system interactions.
So, how can the space quantum system be represented using the classical quantum principles? The problem is that the quantum system can also be described using wave functions and in similar situations when the approximation of the evolution in space quantum is also examined as with matter quantum. The effect is the encoding along the relative guideline: the wave function of the system contains in full all the information on all the appropriate examinations contained in it relating to the determined probability of the degrees of freedom on the condition that they have some obvious value.
According to Martin Bojowald [4], if the examination of the degree of freedom plays a specially important role for the evolution in different quantum space, then it is the expression of internal time: it is not the absolute time lying outside the quantum system as in classical quantum mechanics; neither is it a complex of time as in the relativity theory which is altered when the transformation takes place. As such, it is a characteristic representing the physical degree of freedom of evolution determined by dynamic laws and it indicates how different degrees of freedom change in their interactions with each other. From that viewpoint, it is obvious that no further external examination is needed to read time with its role as a clock or another measuring device. Thus, internal time depends on the very object chosen for examination and different problems will arise upon the selection of the starting point for structural systems at various levels. In the case of singularities of classical physics, where spacetime indefinitely shrinks, it is the very evolution of spatial volume that contains the quantum significance of internal time in the direct or inverse direction. With that significance, the wave function determines how it relates to the matter field which alters the gravity interaction that we regard as the representation of the variation of the degree of freedom of the quantum space, resulting in the shrinkage or expansion of universe. In this case, the progress of the local spacetime is encoded in the wave function and . The above authors demonstrated that the differential function in = 0 and thus, the relative probability is determined with respect to all the internal time without break-down. According to Bojowald [4], we already have a basis to understand how the wave function is significance in quantum space, but the calculation means approximately true for use are not yet known. Below is a possibly rational guideline in our opinion:
We usually use the Hilbert space to determine the fundamental operator and the quantized Hamiltonian constraint to determine the wave functions
This is called the internal dynamic integration which is used for readjusting the quantum theory into the spacetime quantum. Yet, unlike in quantum mechanics where internal dynamic integration is an approximate representation of wave function; in gravity quantum, the internal dynamic integration of the very space must be forecast in an entirely different way. The reason for this is that the equation of quantum evolution in internal time is a constraint equation other than the equation of evolution in the external absolute time parameter. The problem is which compatible relation has the classical critical value of the theory which contains the variations of small scales of the wave function. This means that in the differential equation,
A very small quantum scales in the research of quantum spacetime, the conventional geometric concepts no longer seem to be satisfactory, especially symmetry. According to [2], [3] and [4], universe quantization must take into account the symmetric system, especially in case of gravity quantum. Before the discussion on quantum scales, we will examine the isotropy of universe at connection level and triad surfaces of a which take turn to vary. The role of the scale factor is now replaced with the density of complex triads p with
In an entirely theoretical form, this state is usually written in the representative connection as the function of holonomy. Hereafter is a simplified diagram of the isotropic symmetry group leading to the orthogonal state being complex functions of isotropic connections c, calculated in the works by A.Ashtekar, M.Bojowald and J. Lewandowski
in this state, the elementary variables p and c are represented by:
with the following properties:
1.
2. Operator
3. Only the exponential function of c is represented, not c directly.
This statement results in a number of issues which need to be explained: first, the classical Poisson relations between the elementary variables are in fact represented by adjustment, taking the Poisson bracket into the divergent commutator at
In terms of mathematical representation, the basic steps in the research of quantum geometry as a basis for gravity quantum, Bojowald [4] demonstrated as follows:
Let be given a classical algebra of examination
is considered the unit operator in the Hilbert space.
In the Schroedinger representation, the Hilbert space of the function square was used with limited . Then, the elementary operator was:
which are in fact a representation of the fundamental dependent commutative relation. Moreover, the function operator family of s and t is continuous and can be derived when s equals 0 and t equals 0 orderly:
This is a conventional representation of quantum mechanics which, according to the Stonevon Neumann theory, is unique in the comdition where
In quantum cosmology, we meet situations other than in quantum mechanics due to energy density. We cannot directly examine at a sub-quantum structural level but can only do that indirectly through perceptive information, such as the singularity issue where there is a high concentration of energy is the most prominent property, for research. As we have earlier forecast, the information obtained at the Planck scale of the basic concepts of space geometry which are flatness, continuity and commutativeness or is represented in a form satisfying the commutative condition of Weyl algebra. The formulas, equations in quantum mechanics are effectively calculated with such mathematical tools, but it is also for these that quantum mechanics meet with the obstacles of the classical singularity when surveying sub-Planck scale levels or, in other words, in order to describe the quantum of mathematical spacetime, special developments are required. As we already know, in the universe, any jump from this fundamental state of nature to another is a causative outcome of the jump of potential energy. In the matter field world, the geometric tools that we have in hand for description are acceptable, yet the changes in energy-structure (quantum geometry) – time at the level of space quantum are closely interwound and thus require newer mathematical tools or, more accurately, mathematics must now be physical mathematics. This unity is the root of the uniform theory in physics as well as for the entire universe. A good demonstration of this idea is the general relativity theory of which the heart is the Einstein equation where Rieamann geometry and Minkowski geometry are the tools providing a uniform image of mathematics and physics: gravity interaction potential energy – one of the four fundamental interactions of nature – is encoded in a directly proportional way in the Riedmann curvature tensor of the local space caused by the energy impulse tensor of the matter field present in that space. Recently, the advent of non-commutative geometry has been hailed by physicists and is expected to be an effective mathematical tool for research of space quantum at the most fundamental structural levels. We also try to get to know the physical significance of this special mathematical tool within our intended purpose.
2. Non-commutative geometry in the research of space quantum:
In [9] and [25], according to author Pierre Martinetti, when applying geometry in the examination of gravity quantum, non-commutative geometry is understood as a geometric form of quantum space, which means in space (spacetime or some space phase), the most fundamental arrangement may no longer be commutative but be in another standard guideline:
Where is a constant or by Lie algebra representation:
The issue to be discussed is: is the significance of this geometric form in physics the nature of spacetime quantum or is it just geometric mathematics of quantum space? Right from the first observation into the non-commutative fields, what the basic concepts such as point, distance as well as geometric representation tools such as the differential structure, homology or spin actually are? It is from these basic concepts which are expanded and made more flexible that allow us to obtain a dialectic description of the space quantum structure.
According to A. Commes [9], the starting point of non-commutative geometry was the extension of Riedmann spin manifold geometry which was then completed as a specific case of commutativeness and named non-commutative mathematical theory. Part of it was generalized into geometry in an expression containing spectrum data. The first proposal by Connes indicated that the geometric information of the Riedmann spin manifold can be recovered from algebra data through spectral triples including Algebra A, the Hilbert space H and operator D satisfying strict conditions. Algebra A here is algebra of a homogeneous function in manifold and is commutative. This means the complex of spectrum triples and A are commutative with the spin manifold:
Riedmann spin geometry Linear algebra
The allowed tool in order for the right side of the arrow which is algebra to be directed toward the left side is not necessarily commutative, it remains appropriate even when algebra is not commutative. This means non-commutative geometry is the object of mathematics generated from the spectrum triples (A, H, D) where algebra A is not necessarily non-commutative:
Non-commutative geometry Non-commutative algebra
This indicates that the entire quantum spacer given in physical literature can be represented in the spectral expression. However, the future research of physics in space must be further examined and, to some extent, must be closely associated with this new mathematical tool; the physics of quantum space will sooner or later be faced with the mathematical tool as proposed by the Connes theory. As the simplest example, the conventional concept of distance is no longer suitable in describing quantum space as the determination of a geometric point has been generalized very differently.
The zero level of geometry of the capacity to specify the level of distinction between two points considered adjacent to each other and that is the very object of topology geometry. Therein, it has been proved that the topological information of the compact space X is wholly contained in C (X) algebra of the overall function continuous in X. When C (X) algebra is commutative:
With the involution*:
And standardization:
Thus constituting algebra C(X), knowing C*-algebra of A through the construction of compact space; A is thus understood as the algebra of the continuous function in X. Therefrom, we have the diagram:
Commutative algebra C* with unit A
On the physical standpoint, the transformation from space to algebra is of great importance. Point x of X can be considered the purpose for the function f in the following diagram to have value or equivalently and to acquire the property as the object for determination of the order of members considered:
Where the left side refers to classical physics (space at first) and the right side approaches quantum mechanics. This relation, however, is yet to acquire the meaning of quantum mechanics considering that we have only carried out commutative examinations [9]. Now that we start with non-commutative algebra A to constitute an object called the non-commutative space Y, which means that A acts as a function in Y. However, that property is not sufficient as an appropriate means to obtain geometric information from non-commutative algebra. As a way of addition, we will research the state algebra by applying linearly and variably from A to C with a positive and canonical value ( (I) = 1 when I is the unit of A):
Set S (A) of the states of unit C-algebra has a convex form, which means
Where
According to P.Martinetti and A.Connes [9], non-commutative geometry is structured by the following spectral triple: of A, and D is the boundless operator in H. These factors must satisfy the following conditions:
- Satisfy the premises of Riedmann spin geometry in non-commutative algebra expressions
- Can be extended into the non-commutative properties, namely: 1. The number of directions of space. 2. The smoothness of the coordinates. 3. The generation of bundle nature of manifold spins. 4. The non-commutative condition of D is the Dirac first-degree differential operator, 5. The Pincare algebra formula is duality between r and (n-r) homology groups of the n-dimension manifold; 6. Combination is chirality and commutatively appropriate. For determination of direction, the seventh condition is the real structure so as to permit the enhancement of the natural group structure to the spin bundle group. We have the following diagram:
Where L (M,S) is the space of the Riedmann spin manifold compact, M and D are the conventional Dirac operators. This diagram satisfies the axiom of the spectral triple, its Riemann spin manifold is non-commutative geometry. Vice versa, the spectral triple with commutative A fully determines the Riedmann spin manifold and the therein geodesic distances is defined as:
In the general case, the points are corrected as the states
And it should be noted that this formula is also true for the pure and incomplete states of A. The two definitions of distance as given are not different when determining the geodesic distance of the Riedmann structure in M. Thus, the classical definition of distance as the measurement of the shortest path is no longer appropriate. Non-commutative geometry makes it possible to represent manifold which contains independent complexes broken down into sections with relative boundaries with each other. In the simplest case which is the geometric form:
Where m is the entire elements represented by
which is the orthogonal matrix:
Then, the two adjacent states of A are
The two points of space can be distinguished by the representation:
B.Iochum, T. Krajewski, P. Martinetti and J.Geom demonstrated that in non-commutative geometry, the distance between two points determined as above - as the only difference is in the perturbation amplitude of value compared with z. It is indefinite in terms of value while having only one constant valued in Euclidean geometry [9]. To put it more accurately, distance in Euclidean geometry is equivalent to distance value indefiniteness in non-commutative geometry and coincides with each other when the value of the distinction amplitude compared with z is unchanged.
We already have the basic demonstrations for the non-commutative geometric space structured from interconnected components, the first and foremost of which are the products of geometric spins with the structural points as earlier represented. This means that the distance between the 2 spectral triples:
The results is that
In case of the spin manifold M integrated by two space points.
The space of pure states is the two-sheet model space, two copies of M. Then, the distance is determined as the geodesics distance, equivalent to the metric:
with being the space metric in M. It is worth noticing that when the geodesic distance coincides with the distance in Euclidean geometry, then no difference would continue to exist between the two sheets.
Thus, in quantum space, the determination of the basic units of the conventional measurement is more general and depends on the quantum state of each different space region or different times and the geometry describing the phenomena in quantum space is also one of relativity, called relative geometry.
3. Dimensional analysis method and Planck units
In classical physics, Planck quantities became standards for calculations and they have been used as the most fundamental constant variables of nature. Their dimensional quantities create the world of which we are part.
The earlier presented evidence of this writing consolidates our faith in the quantum of physical space and even the most fundamental concepts of space geometry have become variable combinations when calculated in physical equations. With the quantum structure of space and time, when relativity has become general for all as we imply in this writing, then physical calculation quantities are no longer the same, including what we have considered the premise constants. However, in order to avoid difficulties in calculation, we still have to rely on the available fundamental units with some kind of correction suitable with the quantum degree values of each spacetime region. Dimensional analysis is one of such ways. However, attention is to be paid to the simplification of complex calculations which may be superficially logical but fail to indicate the nature of phenomena.
Planck scales can be calculated from the appropriate dimensional coordination of three natural constants which are the Newton
By taking logarithm, we obtain:
Solving the linear equation system in the logarithmic form the above variables by applying the Cramer rule [26]. The solution with respect to M:
Where
Is the factor determinant, we then have
With respect to L and T is calculated in the same way. Note that the equation system has only one solution when the value of is different from 0.
The equation for determining the value of Planck length is:
where K is the non-dimensional constant. We can in turn write its equivalent form:
In the expressions of M, L and T, we have:
We then find the values by the complex of the twos sides of the above equation to obtain . Therefrom, the Planck quantities are found as follows:
Comparing with the Planck scales in page 4 of this writing, we notice a difference by factors K, which means the Planck scale has different calculation values depending on the characteristics of various local parameters. This conclusion is very important in that it indicates the quantum nature of space. It is the space degrees with the effect of “reproduction” of space quantum that make the value of the smallest conventional units become relatively static.
The problem is whether the conventional quantities in physics mentioned above become really significant in nature, especially in spacetime. Several research works in quantum physics and quantum gravity have affirmed that nature is not constituted from random infinite parts but from the most fundamental units which are structural quanta. But, it must be at some structural combination level that the properties, which are the smallest quantities corresponding with the Planck scales, are indicated. To put it more accurately, it was from the concepts of constituting units and functions that the quantum physical theory was come up with right in Planck’s time. They continued to be generalized in the research of gravity and spacetime. Spacetime, however, is not quantized by the manner in matter quantization mechanics, but the quantum nature of space-time requires us to describe and study it only by the principles and rules of the quantum theory.
Dimensional analysis is a way of describe by the form of quantum rule: qualitative structures and functions and natural quantities, which have had the framework of space with various rules of distribution of energy density creating diversity at various levels, have always been the combination of basic units of relative independence which are considered the dimensions of each other. In its turn, quantum space also has such properties and the dimensions describing it in geometry cannot but have relativity. As such, relative space geometry is the very physical nature of the mathematical tool when researching nature in most the most essential manner. It is necessary to emphasize that the relative geometry that we refer to in this writing is not purely mathematical but is the geometry of spacetime which has the nature is the quantum physical entity containing internal energy and which has interactions with the matter ambient present in it.
III. Interaction between matter and quantum space
1. Impulse tensor in space
To demonstrate that matter is a structural combination (energy-space) and are, in essence, the forms of distribution, transmission and interactions of typical energy in spacetime. Each matter structure system does not possess its “own” space, space cannot displace independently but only the dynamic field “slides” on the space quantum structure, interacting with space in some stringent physicochemical relations. In the Einstein equation, that interaction is represented by the relation of spacetime tensors and energy-impulse tensors; the consequence of that interaction of the image of spacetime being bent in proportion to the magnitude of the object mass determined in the formula E = mc2.
The energy-impulse tensor is the representation of energy structure of object through the distribution of the density of electromagnetic energy-electromagnetic field. According to H. Ruder and M. Ruder [39], the density of electromagnetic in vacuum is represented by:
Using the transformation formula for the electric field E and magnetic field B, we can easily determine:
This result means that the electromagnetic energy density u is not Lorentzian scale. But it is known that u is 0 -0 component of the second-degree 4-dimensional tensor and can be transformed through the expression of force density by Minkowski which is:
With the assistance of the Maxwell expression, we have:
The last term through the exchange of parameters
As the Maxwell field equation of the internal field in 4-dimensional space has its form as:
We will then have:
The Minkowski force density is represented in the tensor form:
which is called the energy-impulse tensor. When replacing , we obtain:
We notice that i1{
We can see that the electromagnetic energy density is the 0-0 component of the energy-impulse tensor, its conservation in Lorentzian transformation is determined.
With the space-time component (j = 1, 2, 3), we find:
Where S is the Poyting vector and
With space-space component (I, j = 1, 2, 3), we obtain:
We have a remark about the physical significance of each component from metric {
1. Component 0-0 I the electromagnetic energy density u.
2. Component 0-j, (j = 1, 2, 3) is
3. Component j-0, (j = 1, 2, 3) is =
4. The space component – space is the very negative Maxwell potential energy tensor. It has a consequence which is the pressure of the electromagnetic field of flat space metric, which s the very basic significance of the Einstein equation (1).This indicates the interaction between space and the energy-impulse field. Yet, in order to understand the mechanism of that phenomenon, we will study a number of hypotheses as follows:
2. Teleparallel effective geometry and bimetricism
The close relation between the matter-energy field and space on the standpoint of quantum mechanics is actually the interaction between two quantum fields. The physical space in the “quantum lattice” structure where there is a distribution of internal energy density encoded in connections, the faces of triads, through the oscillation frequency and through transmission spins of perturbation of degrees of freedom, has the property of a quantum field. Despite its role as the skeleton for localizing and distributing the matter field at some point of time, with that nature of quantum field, space is certain to affect the internal variations of the matter field. It is possible to state that there is an alligatory uniform interaction between the quantum space and the energy matter field.
The Riedmann geometric examination method through the representation of teleparallel with the energy-impulse metric of matter is a guideline for research of theoretical physics which is likely to attain a unified theory, which has always been dream of many physicists.
Our following surveys are the extension of the ideas of L.C.Garcia de Andrade [20] who initiated the use of teleparallelism in the research of gravity field and the electromagnetic field. Yet, unlike them, we wish to refer to a more essential relation brought by the geometric teleparallel effect when applied into the teleparallel energy-impulse field with the space quantum field. Thus, it is necessary to separate the two issues although they are closely elated: First, the electromagnetic teleparallel effect and the temporary space area are the flat space inside the object; second, the teleparallel effect is determined outside the object which is expressed as the space geometric deformation as pointed out by the general relativity theory.
Apparently, even if the distribution density of the fundamental particles constituting matter reaches an infinite level, there will still exist an internal space between and in particles with or without static mass. It is through these spaces that the specific interactions and movements occur as electromagnetic interactions, strong interactions and weak nuclear interactions occur. In the research writing [17] (2001), Y.S. Kim demonstrated that the internal space of particles with and without static mass always had symmetry. Yet, in relative particles, symmetry complies with the Wigner little groups, symmetry on particles with mass is like that in the three-dimensional rotation groups and symmetry in particles without mass is the locally isomorphic form with respect to the 2-dimensional Euclidean group. It is noteworthy that the rotation of the degrees of freedom with respect to particles without mass result in its helicity and it is the two translational degrees of freedom which are symmetric that become the canonical degrees of freedom [17]. According to Y.S. Kim, this points out that the E(2) form symmetry of particles without static mass can exist as infinite momentum and particles with zero mass is the O(3) form symmetry of particles with static mass. With the demonstration that inside the particles, the symmetry of spacetime is always ensured, Y.S. Kim and Noz [17] remarked: the fundamental state of the harmonic oscillation wave function which has the Lorentz amplification nature and they found the internal harmonic oscillation formula which can be extended as from little groups O(3) is the oscillation form of great significance as it plays a pivotal role in the spacetime and matter quantum mechanic combinations.
In the teleparallel effect, the electromagnetic field is regarded as being inserted into the spacetime metric by way of the orthogonal structural frame to create a unified field which we view as the unique electromagnetic field. Thus, according to [10], in that guideline, only the non-cancelled combinations of the Cartan torsion are determined as the effect of the electromagnetic field and gravitational potential or that the factor state of space metric is represented in the expression of the effect similarly to the electric field with the potential of magnetic field. Thus, the Maxwell equation in vacuum can be obtained by the derivative of the electric field in the normal way. In the case of Riemann, we consider the electrostatic spacetime in the Einstein equation in vacuum with the linear approximation of the fields. Garcia de Andrade [20] states that the research of geometrical electrodynamics is where we obtain metric constituted from the electrodynamic equations and obtain the Maxwell equation in vacuum from the teleparallel theory. Following the same guideline, Einstein and Cartan dreamt of a unified field theory where the metric factor obtained from electrodynamic potential is unified into the combination of gravity potential complex and electromagnetic potential. Metrics here are actually viewed as representing the perturbation of the flat Minkowski spacetime. The Riemann electrostatic is constituted from the flat Riemann space or as the Minkowski vacuum electrodynamics in case of linear electrodynamics. When Riemann tensor is not canceled, we obtain the non-vacuum area where the Riemann curving tensor is represented by the density of area. And, the author pointed out that the Einstein equation in vacuum converts to conventional Minkowski electrodynamics from the static metric representing the electrostatic spacetime. Another interesting characteristic of this spacetime geometry is that it has singularities in the metric g00 = 0 in accordance with theory on the condition that the black hole occurs in the metric effect of the Riemann spacetime.
On the path of studying the unity of fields as the extended solution of the Einstein equation, M. Nieuwenhuizen [10], starting from field theory approaches in the Minkowski space, stated that the gravity energy of the momentum tensor is in essence obtained from the solutions of the Einstein equation. Therefrom, it can be represented as:
Resonant tensor = k x total energy-momentum tensor
where k is a constant. For the flat space of universe, the gravity energy is negative and cancels matter energy. When demonstrating the relativity of the gravity field, the author stated that bimetrics as a result of the connection between Riemann metric and Minkowski metric underwent a change in conjugate invariance. As we already know, when proposing the general relativity theory, Einstein did overcome the deadlock experienced by Lorentz and Poincare by the decisive step from flat space to the bent space relating to the energy-momentum tensor.
As to which energy-momentum tensor is appropriate in the gravity field which has been studied by several author, including Babak and Grishchuk [10] based on the viewpoint of the filed theory with respect to gravity, represented in the field tensor expression
And the Riemann tensor is determined as follows:
That is the appropriate guideline to encode the gravity field, allowing the representation on the one face by:
Thus, the linear way to encode one face into the flat space is not abnormal.
In [10], M. Nieuwenhuizen remarked that according to the Maxwell theory, gravity is a field in the flat space; afterwards, Nathan Rosen proposed the bimetricism theory where bimetric is a complex of the Minkowski and Riemann metrics with the specification as particle physics and as the specification of space. Rose acknowledged the covariant derivative
It is the very Christoffel symbol type that becomes the tensor in the Minkowski space.
Based on earlier studies by Landau-Lifshitz and Babk-Grishchuk, author Nieuwenhuizen determined the resonant tensor as follows:
And then he found the equation representing the bimetricism theory for the Einstein equation in the form:
when inserting the Einstein into the right side of the above equation, the author can write in the form of the Newton
Acceleration = Mass -1 x Force
To the following general formula:
Where
With the bimetricism theory as a complex of the Minkowski metric and the Riemann metric, Martin Nieuwenhuizen [10] continued by demonstrating that when we use the Friedmann – Lemaitre – Robertson - Walker (FLRW) metric to represent the universe:
Viewing space as flat with k = 0, U = 1 and V(t) =
which is an independent space. Combining all the above demonstrations, the author concluded that: the total density of cosmological energy is zero as the density of gravity energy is:
which is negative and opposite the total energy momentum of matter in accordance with the Friedmann equality. In other words, it seems that the universe that we are studying does not contain all the energy and, as discussed by the earlier part of this monograph, with the nature of the physical space, the energy missing in the earlier calculation is contained right in the quantum structure of space. It is dark energy which several cosmological studies have taken into account.
We proceed with Nieuwenhuizen’s demonstration when formulating the theoretical field with respect to the Einstein equation. If we see the local energy-momentum density as approaching the negligibly small level, then the Minkowski space is the pre-given and stable space, which means there exists some fundamental state without matter, like the region in front of the earth’s orbit which is almost empty and has the character of Minkowski as mentioned above and when the earth approaches will bring that space new properties of which the most clear are gravity field and the matter field.
With the remark that the Euclidean space is a special case of the Riemann geometry in each object for the entire universe, the demonstration by Nieuwenhuizen is understood as follows: the curvature of spacetime is the very geometric description of the gravity field in the flat space. The Einstein equation is described in the Minkowski space as follows:
The left side of the equation is geometrically regarded as creating the curving space with internal properties which make the gravity field appear as stated by the relativity theory. However, the demonstrations just mentioned show that with its quantum nature, space has properties in terms of structure and internal energy which comply with the principles of quantum mechanics as quantum field, the teleparallel effect shall have non-homogenous value and the result is that the gravity field also has relativity. The quantum state of the physical space changing in interactions can be easily described by geometry as in the Einstein equation. Space, however, remains flat due to the reproduction of quanta according to scale quantum with values being equal to the very scale curvature in the Einstein equation.
When the relativity theory of gravity is true, as when biometric is applied to represent the Minkowski metric, we acknowledge gravity as a field in the flat space as earlier stated by Maxwell.
We have covered a relatively long way to cement our belief that the universe is harmoniously unified in some level of organization where space, time and energy are blended, where mathematical, physical and chemical concepts are no longer separate ones and where nature itself adjusts its quantum structure – the self-nature structure – so that the value of the fluctuation of the degrees of freedom do not exceed the limit and so that the connection factor between quanta becomes infinite (in the M theory of the string theory, this effect is of great significance).
IV. CONCLUSION:
1. The Einstein equation under the condition of quantum spacetime
From equations (1),(17),(82),(90),(107),(115) let us conduct a number of examinations of the pertubative standard of gravity quantum. The technique required to be used here is the concept of non-dynamic background to separate the degrees of freedom of the gravity field in the metric term of background geometry
However, in our opinion, the representation of the tensor sum in the form of algebra geometry is:
In the preceding part, we already referred to scale quantum which we gave the symbol Y with kernel reproduction in structure of quantum spacetime ( physical Hilbert space) and which has a value equivalent to scale curvature R(mathematical Hilbert space) in the Einstein equation (1). Tensor Rici is the representation of the standard Riemann curving space state for the static homogenous and isotropic universe. As earlier demonstrated, the Riemann curving space in the Einstein equation is the mathematical representation of the Minkowski space in the event the internal energy of the quantum space is considered equal to zero; however, the quantum spacetime – due to its quantum physical nature – cannot be accurately determined as equal to zero, but equal to an equivalent value:
Where
We have together demonstrated the quantum nature of the cosmological space, the consequence of such awareness is a dynamic cosmological spacetime model in accordance with the principles of quantum mechanics of which the overall characteristic is the dynamic inhomogenous and unisotrpic universe. We recall that one of the conclusions of bimetric physicians is the relativity of gravity force due to the energy-impulse teleparallel relation of matter and the spacetime quantum field in which it is contained. Thus, the right side of Einstein equation will be:
Where is the energy-impulse tensor of matter according to theory, and
With the terms found, we replace into the Einstein equation in the above general relativity theory:
2. The nature of gravity interaction
In this Einstein equation extended for quantum spacetime, we emphasize the role of the scale quantum Y which is also called the space quantum size quantity, it has a value as the scale curvature of space under the impact of object mass. In reality, Y is a quantity which reflects the reproduction of quantum particles as we have presented in the previous part and we can describe it in the section of spacetime as follows:
We recall the dimensional analysis method and the Planck scale in the previous part, the reproduction of space quanta alters the internal structure in quanta due to changes in the linkage mode of space lattice connections and internal momentum spins but does not break the general cohesive network of the universe. As the characteristic of the relative geometry through the determination of point and distance is the dimension dependent on factor K on the Planck length formula:
Thus, the transmission manner of the fundamental particles in space is unchanged, but the nature of the transmission changes and diverts the movement orbit of all the structural particles and component interactive particles constituting objects and, of course, complies with the equivalent principle in Albert Einstein’s general relativity theory with gravity interaction. The magnitude of the gravity field is encoded in the quantum scale changes the internal structure of space, indicating the size and mass of the object in space and, vice versa, the size and mass of the object dictates the magnitude of the quantum scale and thereby determines the magnitude of the gravity field in the space containing that object.
Compared with the Einstein equation (1), equation (120) has the following differences:
- The description of interactions between the matter-energy field with the quantum space only needs to be done in the flat Minkowski space-time system
- The quantization nature of gravity interactions and its relativity
- Gravity interaction is different from the other 3 basic interactions of nature which are electromagnetic interaction, strong nuclear interaction and weak nuclear interaction in that there are no interactive transmission particles.
In canonical conditions, all the solutions found of the Einstein equation are also the solutions of this extended equation. There are, however, no singular and infinite solutions.
3. The matter structure system and quantum spacetime:
We are living in an era of unprecented scientific and technical development. The awareness of the world that we have accumulated is getting richer, more profound and more comprehensive. On the top of the physical evolution process, mankind with his developed brain has always desired to master the earth as well as himself, wishing to harness and control meteorological, geographic and biological that threat our existence. Yet, everyday, news of natural disasters, epidemics and accidents with increasing number and greater toll has given rise to the demand for more concerted efforts by scientists so as to work out more effective measures to harness the nature.
As a rule, at a high level of development, the boundary between sciences which used to be considered as distinct has been blurred. Such sciences have even become mutually assisting and served as a foundation for the advent of several new sciences. At present, biological studies have superficially been theorized to a great extent in the application of mathematical, physical and chemical research methods. Yet, this has merely been limited to the use of tools as the significance of the profound understanding at the level of microstructure (cosmology, astronomy, the relativity theory, the supermacro level (physics of fundamental particles, quantum physics) remains limited in the research of this fresh science. We, those who are trained to study biology, have been dreaming of the most profound biological theory that may serve as a lighthouse and that experiment is merely evidence on that path of deductive thinking. Obviously, as we enthusiastically remarked from time to time in the earlier part of this writing, the principles and processes at the level of quantum spacetime is a unity and interweavement from which all the current sciences and, in the future, biology and sociology, originate.
a. The locality of matter structure systems
An important consequence that we wish to emphasize in this writing:
- Matter is a combination of energy distributed according to certain structural rules and physiochemical interactions in some space and time region. That local quantum space structure determines the framework of matter. Thus, it is obvious that the interaction between matter energy field and the internal energy field in and adjacent to matter has a great impact on the stability of that structure system. In an universe with constantly varying spacetime (for the time being, we only have evidence – at least in the region in which we are living – that space is expanding), then matter structure systems only has rational stability in some expanse of space and time. That rule with respect to biological structure systems is evidential, indicated in the aging process and natural elimination.
- In parallel with the loss of adaptation of a number of creatures in biological evolution in the earth is an occurrence of new mutations which are more adaptive, the product of the new spacetime region and are selected to become new species. The irrationality in terms ofconsequential structure is the weakening of competition capacity. That is the way the law of natural selection of the Darwin
- Our earth is a system of matter structure with a large size compared with the Planck mass but is merely a small heavenly body in relation to other bodies in the solar system. As a matter of course, to some extent, it is affected by the spacetime regions of the universe which are always different due to its absolute movement. Disorders of the stratigraphic structure, the movement of the atmosphere, the variation of magnetism, etc. of the earth in the variation of quantum space in and around it are very likely to be important parameters in forecasting natural disasters.
- We have called several fatal diseases for which no cure has been found mal du siècle (disease of the century). That name is obvious evidence of the role of quantum spacetime in the generation of new viruses which are the cause of those diseases. From such awareness, medicine will work out more essential orientations for treatment and cure of similar diseases forecast to occur in the future.
b. Internal time and body clock in biological system:
You may still remember the demonstrations from earlier parts by theoretical physicists proposing the loop quantum gravity theory, of whom the outstanding representatives are A.Ashtekar and Martin Bojowald [2][3][4][6], etc. intended to demonstrate the theoretical consequence of the space quantum model with the nature as a starting point of internal time in variations arranging space lattice connections and energy-momentum spins inside the space quanta. In order to research this issue, a more detailed research work is needed. Within this writing, we forecast that the starting and regulation of biological would be impossible without the role of quantum variations of the internal space in the body; through the variation of the internal time, which is relative and plays the role of a body clock.
REFERENCES
[1]. J. Bicak. Einstein equations : exact solutions. Charles University Prague
gr-qc/0604102 v1 24 Apr 2006
[2]. A.Ashtekar. Quantum Mechanics of Geometry.CGPG-98/12-5
[3]. A. Ashtekar. Gravity, Geometry and the Quantum.
gr-qc/0605011 v1 1 May 2006
[4]. M. Bojowald. Loop Quantum Cosmology and Boundary Proposal.AEI-2003-
gr-qc/0312103 v1 23 Dec 2003
[5]. Y.Jack Ng , H. Van Dam . A geometrical derivation of the Dirac equation.
Uni. of North Carolina 4 Feb 2003
[6]. M. Bojowald, H.Hernandez, M. Kagan, P.Singh,A. Skirzewski. Hamiltonian
cosmological perturbation theory with loop quantum gravity corrections.
IGPG-06/9-7 gr-qc/0609057 v2 15 Sep 2006
[7]. A. Perez. Introduction to loop quantum gravity and spin foams. Proceeding of
the II International Conference on Fundamental Interaction Pedra
[8]. G.Amelino - Camelia. Planck-skale structure of spacetime and some
implications for astrophysics and cosmology. Uni.La Sapienza and
Sez. Roma ,Italy 8 Dec 2003
[9]. P.Martinetti. What kind of noncommutative geometry for quantum gravity ?
Instituto Superior Tecnico, Lisboa ,Portugal
Gr-qc/0501022 v2 29 Jun 2005
[10]. Theo M. Nieuwenhuizen. Einstein vs Maxwell: is gravitation a curvature of
space, a field in flat space, or both ? Uni. Of Amsterdam.The Netherlands
. Gr-qc/07040228 v1 2 Apr 2007
[11]. A.Mikovic. Spin network wavefunction and the graviton propagator.
[12]. M. Bojowald. Elements of Loop quantum cosmology. AEI-2005-025.
Gr-qc/0505057 v1 12 May 2005
[13].M .Bojowald. Loop Quantum Cosmology. Living Review in Relativity,8 Dec
2005. Max Planck Institute for Gravitation al Physics, Albert Einstein
Institute.Germany.
[14]. Y. Jack Ng . Hubble Meets Planck: Acosmic Peek at Quantum Foam.
Gr-qc/0701124 v1 22 Jan 2007
[15]. J.Kowalski - Glikman. Observer-independent quantum of mass. Uni. Of
[16]. Wayne R. Bomstad and John R. Klauder . Linearized quantum gravity using
the projection operator formalism. Uni. Of Florida
Gr-qc/0601087v1 20 Jan 2006
[17]. Y.S. Kim. Internal Space-time symmetries of Massive and Massless Particles and their Unification. Uni.of Maryland.USA.
Hep-th/0104051 v1 5 Apr 2001
[18]. M.Bojowald. Quantum Riemannian Geometry and Black Holes. IGPG-06/2-
2. AEI-2006-009. Gr-qc/0602100 v1 24 Feb 2006
[19]. James B.Hartle. Generalizing Quantum Mechanics for Quantum Spacetime.
Uni.of California.USA. Gr-qc/0602013 v1 2 Feb 2006
[20]. Lars Andersson. Geometric analysis and general relativity. Uni.of Miami
.USA. Gr-qc/0512145 v1 23 Dec 2005
[21]. Y . Jack Ng. Quantum Foam and Quantum Gravity Phenomenology. Uni.of
[25]. D. Bigatti. Non-commutative spaces in physics and mathematics. KUL-TF-
200-17.Belgium. Hep-th/00006012 v1 1 Jun 2000
[26]. D. Meschini Planck-scale physics: facts and beliefs. Uni.of Jyväskylä
[27]. G.Natta, M. Farina. Struktur und Verhalten von Molekülen im Raum.Verlag
Chemie.Weinheim.Germany,1976.
[28]. Eyvind H. Wichmann. Quanten Physik.Berkeley Physik Kurs.Vieweg Verlag.B raunschweig,1975.Germany.
[29]. W. Köhnen. Metrische Räume. Akademie verlag,Germany,1988.
[30]. Creig J. Hogan. Das k leine Buch vom Big Bang - Eine Kosmos Fibel. DTV
Verlag. April 2000.
[31]. G. Börner. Kosmologie. Fische Verlag. 10.2002.
[32]. Fred Adams, G. Laughlin. Die fünf Zeitalter des Universums.Ein Physik der
Ewigkeit. DTV Verlag,12.2002.
[33]. Barrow John. Der Ursprung des Universums vie Raum , Zeit und Materie
entstanden. W.G.Verlag, January 2000.
[34]. Greene Brian. Das elegante Universums.Superstrings, verborgene
Dimensionen und die Suche nach der Weltformel. B v T.,2.2002.
[35]. S.Weinberg. Der ersten drei M inuten. Der Ursprung des Universums Piper
Verlag ,12.2002.
[36]. H. Ruder, M. Ruder. Die Spezielle Relativitätstheorie.Vieweg Verlag,1993.
[37]. J.Stachel. Einsteins Annus mirabillis fünf Schriften, d ie Welt der Physik
revolutionierten. Rowohlt Verlag,4.2005.
[38]. S. Hawking. Eine kurze Geschichte der Zeit. Rowohlt Verlag, 1.1999.
[39]. S.Hawking, R.Penrose. Raum und Zeit. Rowohlt Verlag.,9.1998.
[40]. F.E.Taylor, J.A. Physik der Raumzeit. Eine Einführung in die spezielle
Relativitätstheorie. Spektrum Verlag.1994.
[41]. K.C. Cole. Eine kurze Geschichte des Universums A.T.Verlag.2002.
[42]. H.Fritzsch. Quarks, Urstoff unserer Welt. Pieper Verlag,1992.
[43]. Lee Smolin. Warum gibt es die W elt. Die Evolution des Kosmos D.T.V
[44]. B.Greene . Der Stoff, auf dem der Kosmos ist. Raum, Zeit und die
Beschaffenheit der Wirklichkeit.Siedler Verlag,2004.
[45]. S.Hawking. Einsteins Traum.Expeditonen an die Grenzen der Raumzeit.
Rowohlt Verlag,1997.
[46].J. Kleczek, P. Jakes. Das Weltall und unsere Erde. Dausier Verlag,1991.
[47]. Krauss,M.Lawrence. Neuer Antrieb für ein beschleunigtes Universum.
Spektrum der Wissenschaft,3.1999.
[48]. J. Craig Hogan, Rebert P.Kirshner,B.Suntzeff Nicholas. Die Vermessung der
Raumzeit mit Supernovae. Spektrum der Wissenschaft,3.1999.
[49]. J. Duff. Neu Welt theorien.Von String zu Membranen.
Spektrum der Wissenschaft,3.1998.
[50]. Bao Chau, Ngo. Hinh hoc dai so. Paris
[51]. Qui Tu, Pham; Dinh Thanh, Do.Co hoc luong tu. Hanoi
[52]. Huu Minh, Nguyen. Co hoc va Li thuyet tuong doi. Hanoi
[53]. Kim Quang, Ho; N.Kumar; Chi sing Lam. Moi ban lam quen voi vat li hien dai. Hanoi. 2005.
