1
On Discrete (Digital) Physics: as a Perfect Deterministic Structure
for Reality - And the Fundamental Field Equations of Physics
Ramin Zahedi *
Logic and Philosophy of Science Research Group, Hokkaido University, Japan
Jan 7, 2015
In this paper I provide an analysis and overview of some notable
definitions, works and thoughts concerning discrete physics (a.k.a.
digital philosophy, digital physics or digital cosmology) that mainly
suggest a finite, discrete and deterministic characteristic for the physical
world, as well as, of the cellular automaton, which could serve as the
basis of a (or the only) perfect mathematical deterministic model for the
physical reality.
Moreover, as a confirmation, in Appendix 1 (Ref. [37]) I‟ve shown that
by linearization and then canonical quantization of the energy-momentum
relation, the general forms of all the main fundamental field equations of
physics, including the laws of the fundamental forces of nature (i.e.
gravitational, electromagnetic and nuclear fields equations), the
relativistic-quantum wave equations (such as the Pauli and Dirac
equations), and their generalizations, (mathematically) could be derived
on the basis of a new algebraic approach – where it is assumed that
certain physical quantities are discrete.
Keywords: Foundations of Physics, Ontology, Discrete Physics, Discrete Mathematics, Domain of Integers, Computational
Simulation.
“…I consider it quite possible that physics cannot
be based on the field concept, i.e., on continuous
structures. In that case nothing remains of my
entire castle in the air gravitation theory included,
-and of- the rest of modern physics.” A. Einstein
The concept and etymology of digital is distinct, or “discrete”. Digit and its derivatives
come from the Latin digitus, meaning finger. In discrete physics (a.k.a. digital philosophy,
digital physics or digital cosmology) it is usually supposed that space, time, physical states
and quantities and all the microscopic and fundamental physical processes are, ultimately,
finite, discrete and deterministic (principally, on the Planck scale).
* Email: [email protected], [email protected]
Copyright: CC Attribution-NonCommercial-NoDerivatives 4.0 International License.
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
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The main reason for this article is the rising interest of many great contemporary
scientists in this field (that nature is ”discrete” on the Planck scale) and in particular the
recent papers of one of the leading international physicists and Nobel laureate, Prof. Gerard
't Hooft [1-10].
1. Digital philosophy, and Discrete, Finite Physical World
The physical world has always been described by ordinary calculus and partial
differential equations, based on continuous mathematical models. In digital philosophy a
different approach is taken, one that often uses the model of cellular automaton (see the
next section) [15].
Digital philosophy grew out of an earlier digital physics that proposed to support much of
fundamental theories of physics (including quantum theory) in a cellular automaton
structure. Specifically, it works through the consequences of assuming that the universe is
a gigantic cellular automaton. It is a digital structure that encompasses all of physical
reality (including mental activities) as digital processing. From the point of view of
determinism, this digital approach to philosophy and physics gets rid of the essentialism of
the Copenhagen interpretation of quantum mechanics.
In fact, there is an ongoing effort to understand the physical systems in terms of digital
models. According to these models, the universe can be conceived as the output of a
universal computer simulation, or as mathematically isomorphic to such a computer, which
is a huge cellular automaton [16, 17, 18]. Digital philosophy proposes to find some ways of
dealing with certain issues in the philosophy of physics and mind (in particular issues of
determinism) [15]. In some sense in this discrete approach to physics, continuity,
differentiability, infinitesimals and infinities, are “ambiguous” notions. Despite that, many
scientists proposed discrete structures (based on the current theories) that can approximate
continuous models to any desired degree of accuracy.
Richard Feynman in his famous paper [29], after discussing arguments regarding some of
the main physical phenomena concluded that: all these things suggest that it's really true,
somehow, that the physical world is representable in a discretized way. It is worth to note
here also Einstein's view on continuous models of physics: I consider it quite possible that
physics cannot be based on the field concept, i.e., on continuous structures. In that case
nothing remains of my entire castle in the air gravitation theory included, -and of- the rest
of modern physics [30].
3
2. The Cellular Automaton
Proposals of discrete physics reject the very notion of the continuum and claim that
current continuous theories are good approximations of a true discrete theory of a finite
world. Typically such models consist of a regular “lattice” of cells with finite state
information at each cell. These lattice cells do not exist in physical space. In fact physical
space arises from the relationships between states defined at these cells. In the most
commonly studied lattice of cells or cellular automaton models, the state is restricted to a
fixed number of possibilities.
Firstly, cellular automaton models were studied in the early 1950s. Von Neumann
introduced cellular automata more than a half-century ago [21]. By standard definition, a
cellular automaton is a collection of stated (or colored) cells on a grid of specified shape
that evolves through a number of discrete time steps according to a set of certain rules
based on the states of neighboring cells. These rules are then applied iteratively for as
many time steps as desired. In fact, von Neumann was one of the first people to consider
such a model. The most interesting cellular automaton is something that von Neumann
called the universal constructor. The neat thing about cellular automata is that they don‟t
look exactly like computers and there are no such constructs like program, memory or
input. They look more like discrete dynamical systems and instead have functionally
similar but semantically distinct constructs like evolution rules, space, time and initial
conditions.
One of the most fundamental properties of a cellular automaton is a type of grid on which
it is calculated or computed. The simplest grid is a one-dimensional line. In two
dimensions, square, triangular and hexagonal grids can be considered. Cellular automata
can also be built on the Cartesian grids in arbitrary number of dimensions [22, 23]. Cellular
automata theory has simple rules and structures that are capable of producing a wide
variety of unexpected behaviors. For example, there are universal cellular automata that are
able to simulate the behavior of any other cellular automaton [24].
An increasing number of works on cellular automata related to philosophical arguments
are being presented by professional scholars interested in the conceptual implications of
their work. Among the interesting issues that have already been addressed through the
approach of cellular automata in philosophy of science are free will, the nature of
computation and simulation, and the ontology of a digital world [25].
4
3. Is Discrete Physics a Perfect Deterministic Model for Physical
Reality?
In the opinion of the author, the answer is affirmative [37]. The notion of nature as a
discrete form/structure (and a cellular automaton, like a computer simulation model),
seems to be supported by an epistemological desideratum and in the last half century many
great scientists have logically and reasonably proposed that the physical world might have
fundamentally a discrete and in addition a computational (numerical simulation) structure
[16, 17, 18, 20, 27, 28].
Richard Feynman had speculated that such discrete structures will ultimately
provide the most complete and accurate descriptions of physical reality [20]: it always
bothers me that, according to the laws as we understand them today, it takes a computing
machine an infinite number of logical operations to figure out what goes on in no matter
how tiny a region of space, and no matter how tiny a region of time. How can all that be
going on in that tiny space? Why should it take an infinite amount of logic to figure out
what one tiny piece of space/time is going to do? So I have often made the hypothesis that
ultimately physics will not require a mathematical statement, that in the end the machinery
will be revealed, and the laws will turn out to be simple, like the chequer board with all its
apparent complexities.
As we already noted, Prof. Gerard 't Hooft, a contemporary leading physicist, has also
published many papers on this subject in recent years. Particularly, he has tried to consider
questions, like:
- Can Quantum Mechanics be Reconciled with Cellular Automata Model?
- Obstacles on the Way Towards the Quantization of Space, Time and Matter -- and
Possible Resolutions,
- Does God Play Dice? (One of the Famous Einstein‟s Ontological Questions),
- The Possibility of a Local Deterministic Theory of Physics,
Here is one of the Gerard 't Hooft's discussions on the possibility of a local deterministic
theory of physics [26] (also see [9]): quantum mechanics could well relate to micro-physics
the same way thermodynamics relates to molecular physics: it is formally correct, but it
may well be possible to devise deterministic laws at the micro scale. Why not? The
mathematical nature of quantum mechanics does not forbid this, provided that one
carefully eliminates the apparent no-go theorems associated to the Bell inequalities. There
are ways to re-define particles and fields such that no blatant contradiction arises. One
must assume that all macroscopic phenomena, such as particle positions, momenta, spins,
and energies, relate to microscopic variables in the same way thermodynamic concepts
5
such as entropy and temperature relate to local, mechanical variables. The outcome of
these considerations is that particles and their properties are not, or not entirely, real in the
ontological sense. The only realities in this theory are the things that happen at the Planck
scale. The things we call particles are chaotic oscillations of these Planckian quantities.
t‟Hooft in his most recent paper [9], (see also [10]), where discussing the mapping
between the Bosonic quantum fields and the cellular automaton in two space-time
dimensions, concluded that: "the states of the cellular automaton can be used as a basis for
the description of the quantum field theory. These models are equivalent. This is an
astounding result. For generations we have been told by our physics teachers, and we
explained to our students, that quantum theories are fundamentally different from classical
theories. No-one should dare to compare a simple computer model such as a cellular
automaton based on the integers, with a fully quantized field theory. Yet here we find a
quantum field system and an automaton that are based on states that neatly correspond to
each other, they evolve identically. If we describe some probabilistic distribution of
possible automaton states using Hilbert space as a mathematical device, we can use any
wave function, certainly also waves in which the particles are entangled, and yet these
states evolve exactly the same way. Physically, using 19th century logic, this should have
been easy to understand: when quantizing a classical field theory, we get energy packets
that are quantized and behave as particles, but exactly the same are generated in a cellular
automaton based on the integers; these behave as particles as well. Why shouldn't there be
a mapping"?
Of course one can, and should, be skeptic. Our field theory was not only constructed
without interactions and without masses, but also the wave function was devised in such a
way that it cannot spread, so it should not come as a surprise that no problems are
encountered with interference effects, so yes, all we have is a primitive model, not very
representative for the real world. Or is this just a beginning"?
He also mentions in his paper concerning three space-time dimensions (for which there is
a special interest and emphasis in the literature and relating to the physical reality of three
dimensional sub-universe [11, 12, 13, 14]: "the classical theory suggests that gravity in
three space-time dimensions can be quantized, but something very special happens; … now
that would force us to search for deterministic, classical models for 2+1 dimensional
gravity. In fact, the difficulty of formulating a meaningful „Schrodinger equation‟ for a 2+1
dimensional universe, and the insight that this equation would (probably) have to be
deterministic, was one of the first incentives for this author to re-investigate deterministic
quantum mechanics as was done in the work reported about here: if we would consider any
classical model for 2+1 dimensional gravity with matter (which certainly can be
formulated in a neat way), declaring its classical states to span a Hilbert space in the sense
described in our work, then that could become a meaningful, unambiguous quantum
system".
6
In addition, contemporary British physicist, John Barrow states: we now have an image
of the universe as a great computer program, whose software consists of the laws of nature
which run on hardware composed of the elementary particles of nature [19].
As a special but important case concerning Bell's inequalities, t‟ Hooft points out, Bell
has shown that hidden variable theories (that the quantum particles are, somehow,
accompanied by classical hidden variables that decide what the outcome of any of possible
measurements will be, even if the measurement is not made) are unrealistic. We must
conclude that the cellular automaton theory - the model of t‟ Hooft (see [8, 9]) - cannot be
of this particular type. Yet, we had a classical system and we claim that it reproduces
quantum mechanics with probabilities generated by the squared norm of wave functions.
Quantum states, and in particular entangled quantum states, are perfectly legitimate to
describe statistical distributions. But to understand why Bell's inequalities can be violated
in spite of the fact that we do start off with a classical deterministic, discrete theory (e.g.
based on the cellular automaton) requires a more detailed explanation (see [8]). There is
also a complete explanation regarding the collapse of the wave function via the cellular
automaton structure [7, 8].
An immense and relatively newer research field of physics is loop quantum gravity,
which may lend support to discrete physics, also assumes space-time is quantized [32-36].
From the historical perspective it is worth to note that one of the first ideas that “the
universe is a computer simulation” was published by Konrad Zuse [16]. He was the first to
suggest (in 1967) that the entire universe is being computed on a huge computer, possibly a
cellular automaton. In his paper he writes: that at the moment we do not have full digital
models of physics … which would be the consequences of a total discretization of all
natural laws? For lack of a complete automata-theoretic description of the universe he
continues by studying several simplified models. He discusses neighboring cells that
update their values based on surrounding cells, implementing the spread and creation and
annihilation of elementary particles. He writes: in all these cases we are dealing with
automata types known by the name "cellular automata" in the literature, and cites von
Neumann's 1966 book: Theory of self-reproducing automata [16, 31].
7
4. Some Remarks
From the above discussions and arguments some logical/ontological questions naturally
arise. Are we part of a computer simulation? Are there some advanced civilizations, who
have created this huge simulation?, In other words, if we discover that we are existing in a
sort of computer simulation, naturally and logically, we can ask, who has created it and is
running this simulation, and also for what reason(s)?, Are we a part of a vast scientific and
social experiment? Does it made sense to reason that this simulation was created by others?
The ontological structure of a discrete-finite model of reality needs further research. One
prospect would be searching for phenomena which cannot be predicted, calculated and
described (theoretically/experimentally) according to current quantum theories and other
fundamental theories of physics, but could be demystified only by discrete structures.
Gerard t 'Hooft in one of his remarkable articles concerning discrete models (describing
by integers) of real world emphasizes that [38]: "In modern science, real numbers play
such a fundamental role that it is difficult to imagine a world without real numbers.
Nevertheless, one may suspect that real numbers are nothing but a human invention. By
chance, humanity discovered over 2000 years ago that our world can be understood very
accurately if we phraze its laws and its symmetries by manipulating real numbers, not only
using addition and multiplication, but also subtraction and division, and later of course also
the extremely rich mathematical machinery beyond that, manipulations that do not work so
well for integers alone, or even more limited quantities such as Boolean variables. Now
imagine that, in contrast to these appearances, the real world, at its most fundamental level,
were not based on real numbers at all. We here consider systems where only the integers
describe what happens at a deeper level. Can one understand why our world appears to
be based on real numbers? The point we wish to make, and investigate, is that
everything we customarily do with real numbers, can be done with integers also".
In particular and as a confirmation, in Ref. [37] (see Appendix 1) has been proven that
the general forms of all the main fundamental field equations of physics, including the
laws of the fundamental forces of nature (i.e. gravitational, electromagnetic and nuclear
field equations, and “only” these categories of fields), the quantum wave equations
(such as the Pauli and Dirac equations), and their generalizations, mathematically and
uniquely, could be derived based on a new algebraic approach – where it is assumed
that certain physical quantities are discrete (i.e. having the integer values).
8
References:
[1]- G. 't Hooft, `"Quantum Mechanics and determinism," in Proceedings of the Eighth Int.
Conf. on "Particles, Strings and Cosmology, Univ. of North Carolina, Chapel Hill, Apr. 10-
15, 2001, P. Frampton and J. Ng, Eds., Rinton Press, Princeton, pp. 275 - 285; ITP-
UU/01/18, SPIN-2001/11, arXiv:hep-th/0105105; id., Determinism beneath Quantum
Mechanics, presented at “Quo vadis Quantum Mechanics?”, Temple University,
Philadelphia, (September 25, 2002), ITP-UU-02/69, SPIN-2002/45, arXiv:quant-
ph/0212095, 2002.
[2]- G. 't Hooft, “The mathematical basis for deterministic quantum mechanics, in Beyond
the Quantum,” World Scientific, Th. M. Nieuwenhuizen et al, ed., pp.3-19, arXiv:quant-
ph/0604008, 2006
[3]- G. 't Hooft, "Quantum Gravity as a Dissipative Deterministic System," Class. Quant.
Grav. 16, 3263, 1999.
[4]- G. 't Hooft, "Determinism in Free Bosons," Int. J. Theor. Phys. 42, 355, 2003.
[5]- G. 't Hooft, "Entangled quantum states in a local deterministic theory," 2nd Vienna
Symposium on the Foundations of Modern Physics (June 2009), ITP-UU-09/77, SPIN-
09/30; arXiv:0908.3408v1 [quant-ph], 2009.
[6]- G. 't Hooft, "Classical cellular Automata and Quantum Field Theory," in Proceedings
of the Conference in Honor of Murray Gell-Mann's 80th Birthday "Quantum Mechanics,
Elementary Particles, Quantum Cosmology and Complexity", Singapore, February 2010,
H. Fritzsch and K. K. Phua, eds., World Scientific, pp 397 - 408, Repr. in: Int. J. Mod.
Phys. A25, no 23, pp. 4385-4396, 2010.
[7]- Gerard 't Hooft, "Quantum Mechanics from Classical Logic," Journal of Physics:
Conference Series 361, 012024, IOP Publishing, 2012.
[8]- G. 't Hooft, “How a wave function can collapse without violating Schrodinger's
equation, and how to understand Born's rule,” ITP-UU-11/43, SPIN-11/34,
arXiv:1112.1811 [quant-ph], 2011.
[9]- Gerard 't Hooft, "The Cellular Automaton Interpretation of Quantum Mechanics. A
View on the Quantum Nature of our Universe, Compulsory or Impossible?,"
arXiv:1405.1548v2, Jun 2014.
[10]- Gerard 't Hooft, "Duality between a deterministic cellular automaton and a bosonic
quantum field theory in 1+1 dimensions," arXiv:1205.4107 [quant-ph], 2012.
[11]- S. Deser, R. Jackiw and G. 't Hooft, “Three-dimensional Einstein gravity: dynamics
of at space,” Ann. Phys. 152 p. 220, 1984.
[12]- G. 't Hooft, “Classical N-particle cosmology in 2+1 dimensions,” Class. Quantum
Grav. 10, S79-S91, 1993.
[13]- G. 't Hooft, “Cosmology in 2+1 dimensions”, Nucl. Phys. B30 (Proc. Suppl.) pp.
200-203, 1993.
[14]- G. 't Hooft, “Quantization of point particles in (2+1)-dimensional gravity and space-
time discreteness,” Class. Quantum Grav. 13, pp. 1023-1039, arXiv:gr-qc/9601014, 1996.
9
[15]- E. Fredkin, “An Introduction to Digital Philosophy,” International Journal of
Theoretical Physics, Vol. 42, No. 2, February 2003.
[16]- Konrad Zuse, “Rechnender Raum,” Elektronische Datenverarbeitung, Vol 8., pp.
336–344, 1967.
[17]- K. Zuse, “Calculating Space,” Cambridge, MA: MIT Press, 1970. (for some
historical details about Zuse‟s works see also: K. Zuse, “The Computer - My Life,” Konrad
Zuse (et al.), Springer Verlag, Berlin, 1993).
[18]- K. Zuse, “The Computing Universe,” International Journal of Theoretical Physics,
21 (6–7), pp. 589–600, 1982.
[19]- John D. Barrow, "New Theories of Everything," Oxford University Press, 2008.
[20]- Richard Feynman, "The Character of Physical Law," Messenger Lectures, Cornell
University, p.57, 1964.
[21]- J. Von Neumann, “The General and Logical Theory of Automata,” in Cerebral
Mechanisms in Behavior: The Hixon Symposium, New York: John Wiley & Sons, 1951.
[22]- G. Rao Venkatesh, "Digital Philosophy: Are Cellular Automata Important?,"
Ribbonfarm Inc. (www.ribbonfarm.com), 2007.
[23]- S. Wolfram, “A New Kind of Science,” Champaign, IL: Wolfram Media, 2002.
[24]- P. Gacs, "Reliable Cellular Automata with Self-Organization." J. Stat. Phys. 103, pp.
45-267, 2001.
[25]- Francesco Berto, Jacopo Tagliabue, “Cellular Automata , # Cellular Automata as
Models of Reality” The Stanford Encyclopedia of Philosophy, 2012.
[26]. Gerard 't Hooft, “Does God Play Dice, Physics World,” December 2005.
[27]- E. Fredkin, “A New Cosmogony,” in Phys. Comp. '92: Proceedings of the Workshop
on Physics and Computation, IEEE Computer Society Press, pp. 116–121, 1993.
[28]- S. Wolfram, “Statistical Mechanics of Cellular Automata,” Reviews of Modern
Physics, 55: 601–644, 1983.
[29]- Richard Feynman, “Simulating Physics with Computers,” International Journal of
Theoretical Physics, Vol. 21, pp. 467-488, 1982.
[30]- Pierre Speziali, (ed.) “Albert Einstein–Michele Besso: Correspondance 1903-1955,”
Hermann, Paris, 1972.
[31]- John von Neumann, “The Theory of Self-reproducing Automata,” A. Burks (ed.),
Univ. of Illinois Press, Urbana, IL, 1966.
[32]- Zizzi, Paola, "A Minimal Model for Quantum Gravity," Mod. Phys. Lett. A20, pp.
645-653, 2005.
[33]- Zizzi, Paola, "Computability at the Planck Scale," arXiv:gr-qc/0412076, 2005.
[34]- A. Marzuoli, M. Rasetti, "Spin Network Quantum Simulator," Phys. Lett. A306, pp.
79-87, 2002.
[35]- A. Marzuoli, M. Rasetti, "Computing Spin Networks," Annals of Physics 318: pp.
345-407, 2005.
10
[36]- F. Girelli, E. R. Livine, "Reconstructing Quantum Geometry from Quantum
Information: Spin Networks as Harmonic Oscillators," Class. Quant. Grav. 22: pp. 3295-
3314, 2005.
[37]- Ramin Zahedi, "On the Logical Structure of the Fundamental Forces of Nature: A
New Deterministic Mathematical Approach," Hokkaido University Publs., Japan, January
2015 [A submitted and accepted research project, Ramin (A.) Zahedi, (On "Foundations of
Physics"), Japan, 2014 – 2015; And an expanded version of my earlier published articles in
the Bulletin of the Lebedev Physics Institute, Russian Academy of Sciences, New York,
Springer-Verlag, 1997]; https://www.scribd.com/doc/265058014. (See Appendix 1)
[38]- G. 't Hooft, "Relating the quantum mechanics of discrete systems to standard
canonical quantum mechanics," Foundations of Physics , Springer-Verlag, 44 (4), 2014;
http://arxiv.org/abs/1204.4926.
11
On the Logical Structure of the Fundamental Forces of Nature:
A New Deterministic Mathematical Approach
Ramin Zahedi *
Logic and Philosophy of Science Research Group, Hokkaido University, Japan
28 Jan 2015
Abstract
The main idea of this article is based on my earlier published articles (Refs. [1], [2], [3], [4], 1997). In this article
on the basis of a new mathematical approach (concerning the algebraic structure of the domain of integers) and the
assumption of discreteness of the relativistic n-momentum, by linearization and then canonical quantization of
the energy-momentum relation, we derive the general forms of all the main fundamental field equations of physics,
including the laws of the fundamental forces of nature, i.e. gravitational, electromagnetic and nuclear field equations,
as well as the relativistic-quantum wave equations (such as the Pauli and Dirac equations), and their generalizations.
These field equations have unique and distinct structures and are in the form of the complex tensor equations, and
represent the above categories of fields for dimensions D ≥2. In each tensor equation a term of the mass m0 (as the
rest mass of a field carrier particle, which naturally could be zero or non-zero) has appeared, as well as a separate
term of the external current (as the external source of the field). In fact, the tensor equations obtained in this work are
the expanded forms of the ordinary fundamental field equations including the Maxwell, the Einstein and the Yang-
Mills field equations, as well as the Pauli and Dirac equations and so on. Moreover, according to this new
mathematical approach, we derive a relativistic-quantum wave equation containing 4×4 real gamma matrices (as a
form of the Dirac equation) where it is only formulated in (1+2) dimensions. For (1+3) dimensions we obtain a form
of relativistic-quantum wave equation (structurally analogous to the Dirac equation) that contains 8×8 contravariant
matrices corresponding to Clifford algebra. This approach, along with graviton (with zero rest mass) predicts a
gravitational field carrier particle with non-zero rest mass. Moreover, based on the unique mathematical structure of
the fields equations derived, we conclude that magnetic monopoles (opposite electric monopoles) could not exist.1
Keywords: Foundations of Physics, Ontology, Discrete Physics, Discrete Mathematics, Domain of Integers, The Fundamental Forces of Nature.
PACS Classifications: 04.20.Cv, 04.50.Kd, 04.90.+e, 04.62.+v, 02.10.Hh, 02.10.Yn, 02.20.Bb, 02.90.+p, 03.50.-z, 03.65.Pm, 12.60.-i, 12.10.Dm, 12.10.-g.
1. Introduction
Let start with one of the greatest ontological questions: “Why are the universe and the fundamental
forces that are acting in it in the way, and form and shape, which we realize them?”; the forces that are
causers of all the movements and interactions in the physical world. In this article, we are going to
consider this question by a mathematical approach.
* Email: [email protected], [email protected].
1. https://archive.org/details/R.A.Zahedi1Forces.of.naturesLawsApr.2015, https://arXiv.org/abs/1501.01373;
Copyright: CC Attribution-NonCommercial-NoDerivatives 4.0 International License.
License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/
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Appendix 1.
12
This paper is based on my earlier articles (Refs. [1], [2], [3]), as well as my thesis work (1997) [4] (but in
a new expanded framework). Here on the basis of a new axiomatic approach (concerning the algebraic
structure of the domain/ring of integers) and the assumption of discreteness of the relativistic n-
momentum, by linearization and then canonical quantization of the energy-momentum relation, we
derive the general forms of all the main fundamental field equations of physics, including the laws of the
fundamental forces of nature, i.e. gravitational, electromagnetic and nuclear field equations, as well as
the relativistic-quantum wave equations (such as the Pauli and Dirac equations), and their generalizations.
These field equations have unique and distinct structure and are in the form of the complex tensor
equations and they represent the above categories of fields for all dimensions D ≥2. The main results of
this paper include:
1-1. Deriving the general mathematical forms of all the fundamental field equations of physics,
including the laws of the fundamental forces of nature, and also the relativistic-quantum wave equations
(for dimensions D ≥ 2), uniquely and distinctly, in the following (tensor) forms:
,0][ RD (1-1)
)(* GJRD (1-2)
,0][ ZD (2-1)
)(* NJZD (2-2)
,0][ FD (3-1)
)(* EJFD (3-2)
where
k
imD
0 , k
imD
0* (4)
and
0:0
,1
:000
k
gk
(5)
13
and m0 is the rest mass of a field carrier particle that appears in the above field equations, is the
(gauge) covariant derivative, k is the covariant n-velocity of a static observer; and where F is a
general field-strength tensor that represent the electromagnetic and nuclear (Yang-Mills) fields
(depending on the definition of the gauge covariant derivative), Z is also a general 3rd
order tensor that
could represent some fields, and R is the Riemann tensor for the gravitational field. In fact, each
system of tensor equations (1-1), (1-2) – (3-1), (3-2) could be divided into two subcategories depending
on mass m0 is zero or non-zero.
In addition, as we will show in Section 3, general equations (1-1) – (1-2) (representing the gravitational
field) give rise to the following field equations, first for two dimensional space-time (D = 2)we get
TR 8 (6)
ggTR 2
12
1 )8( (7)
gTK
imT )8(8 2
10
(8)
and for the higher dimensions (D > 2) we obtain
gKim
TRgR
02
1 8 (9)
where kK .
Equations (6) – (9) are equivalent to the Einstein field equations (for m0 = 0). In the meantime,
equations (6) – (9) also predict a gravitational field carrier particle with non-zero rest mass .
1-2. According to the unique structure of tensor equations (3-1) and (3-2) that represent the general form
of the Yang-Mills equations (including the Maxwell’s equations), we conclude that generally, there
could not be magnetic monopoles in nature. As some special cases of the above equations, we‟ll also
derive a relativistic-quantum wave equation containing 4×4 real gamma (Dirac) matrices (actually as a
form of the Dirac equation) where it is only formulated in (1+2) dimensions. For (1+3) dimensions we
obtain a relativistic-quantum wave equation (structurally analogous to the Dirac equation) that contains
8×8 contravariant matrices (matrices (152)) corresponding to the Clifford algebra.
We emphasize that all the above results are unique outcomes of a single mathematical approach which we
represent it in Section 2.
In the next section, as a new mathematical approach we describe the principles of the algebraic theory of
linearization based on the theory of rings/domains, and in Section 3 we show its applications in physics,
where we‟ll focus on (mathematically and in detail) “deriving” the general forms of all the main
fundamental field equations of physics, i.e. the laws of the fundamental forces of nature, and the
relativistic-quantum wave equations of physics in all dimensions D ≥ 2.
14
2. Theory of Linearization in the Domain of Integers: As a New
Mathematical Axiomatic Approach
The algebraic axioms of the domain of integers with binary operations ),,( usually are defined as
follows [5]:
- ,,...,, 321 aaa
- Closure:
, lk aa lk aa (10)
- Associativity: ,)()( rlkrlk aaaaaa rlkrlk aaaaaa )()( (11)
- Commutativity: ,kllk aaaa kllk aaaa (12)
- Existence of an identity element:
,0 kk aa kk aa 1 (13)
- Existence of inverse element (for addition):
0)( kk aa (14)
- Distributivity: ),()()( rklkrlk aaaaaaa
)()()( rlrkrlk aaaaaaa (15)
- No zero divisors: ,0)00( lklk aaaa (16-1)
Equivalently, the axiom (16-1) could be defined as:
...)0,0()0,0[( 222111 mmamma
0...)]0,0( 321 rrrr aaaamma (16-2)
If we just suppose
),(),...(][),(][),(][ 11311321121111 Zaaaaaa then equivalently, the
axioms (10) – (15) could also be written by square matrices (with integer components) as follows:
- ][ijkk mM ,
ijkm ,
njin ,...,3,2,1,: , ,,...,, 321 nnMMM
- Closure:
,nnlk MM nnlk MM (17)
- Associativity: ,)()( rlkrlk MMMMMM rlkrlk MMMMMM )()( (18)
- Commutativity (for addition): kllk MMMM (19-1)
15
- Property of the transpose for matrix multiplication:
T
k
T
l
T
lk MMMM )( (19-2)
where T
kM is the transpose of matrix kM .
- Existence of an identity element:
,0 kk MM knnk MIM (20)
- Existence of the inverse element (for addition):
0)( kk MM (21)
- Distributivity: ),()()( rklkrlk MMMMMMM
);()()( rlrkrlk MMMMMMM (22)
From the axioms (10) – (15), we can obtain the axioms (17) – (22) and vice versa.
In this article, we introduce the following algebraic axiom as a new property of integers, and we add it to
the axioms (17) – (22) (this new axiom is someway the generalized form of the axiom (16-2) and in fact,
the axiom (16-2) will be replaced with the axiom (23)):
Axiom 2-1. “ If we assume the algebraic form
s
q
r
p
pqpq bbF1 1
)( and the nn square matrices
][ijkk aA , where pqkij
H are some coefficients and
,1 1
s
q
r
p
pqpqkk bHaijij
),(, 11 ZHb pqkpq ij ,,...,3,2,1, nji ,,...,3,2,1 rk
,,...,3,2,1 rp ,,...,3,2,1 sq
then we have the following axiom:
,nnkM
)]0,0(...)0,0()0,0[[( 222111 rrr MMAMMAMMA
0)()])(...( 321 pqnnpqr bFIbFAAAA ” (23)
Remark 2-1. In (23), according to the arbitrariness of all the components of the nn matrix kM ,
without loss of generality, we may replace the nn matrix kM with a 1n matrix ,kM in each of
equations 0kk MA (with the same condition ,0kM but only with the “n” number of arbitrary
components). Note that the integer elements ijka are the “linear” forms of the integer elements pqb .
16
We can obtain the axiom (16-1) (or its equivalent, the axiom (16-2)) from the Axiom 2-1, but not vice
versa. Only for special case 1n , the set of axioms (17) – (23) becomes equivalent to the set of axioms
(10) – (16-2). Definitely, the Axiom 2-1 is a new axiom and in this section and the next section we‟ll
demonstrate some of its outcomes and applications.
Generally, there are the standard and specific methods, approaches and procedures for considering
and solving linear equations in the set of integers [7]. Since (on the basis of the Axiom 2-1) the
necessary and sufficient condition for an equation of the rth order such as 0)( pqbF (in the
domain of integers) is the transforming (or converting or in fact, “linearizing”) it into a system of
linear equations of the type 0kk MA (where 1:,0 nMM kk matrix), naturally, the main
application of the Axiom 2-1 will be the transforming the higher order equations into the
corresponding systems of linear equations.
In this section, based on (23), essentially, we‟ll obtain the systems of linear equations that correspond to
the second order equation of the form: ,0)( pqbF and also some of the higher order equations. In the
methodological point of view, firstly, for obtaining and specifying a system of linear equations that
corresponds to a given equation of the type 0)( pqbF (defined in (23)), we assume and consider the
minimum value for n (the size number of nn matrices kA ). Secondly, by replacing the components of
the matrices kA with the linear forms
s
q
r
p
pqpqkk bHaijij
1 1
, we calculate the product
r
k
kA1
, and then
we put it equal to the matrix nnpq IbF )( . Then using this (obtained) equation, we can calculate the
coefficients pqkijH (which are independent of elements pqb ). Through, easily, the coefficients pqkij
H are
calculated and obtained by routine and standard methods of solving the relevant equations in the set of
integers. Thirdly, the algebraic forms
s
q
r
p
pqpq bbF1 1
)( (24)
via some certain rules and linear transformations, could be transformed into the algebraic forms of the
type
s
iiii
r
p
iiiiis
r
prcBccccG
1,...,,, 1
...321
321
321),...,,,( (25)
In continuation, by some examples we will show how the forms
s
q
r
p
pqpq bbF1 1
)( could be
transformed into the forms ),...,,,( 321 sccccG (trough some linear transformations). Furthermore, as
we‟ll show that, exceptionally, the second order forms of (24) could be transformed into the following
quadratic forms (by similar linear transformation)
s
ii p
iii
s
ii p
iiiss ppdBcBddddccccG
1,
2
11,
2
1
321321
21
21
21
21),...,,,,,...,,,( (26)
17
Remark 2-2. Moreover, concerning the formula (24), we have
s
q
r
p
qrqrpq
s
q
r
p
qrpq
s
q
r
p
qrpq dcbdbcb1 1
)1()1(
1 1
)1(
1 1
)1( 0)()0,0( (24-1)
and
s
q
r
p
qrpq
s
q
r
p
qrpq tcbcb1 1
)1(
1 1
)1( 0)(0 (24-2)
where the parameter t
is an arbitrary integer, with condition 0t . Below we write the systems of linear
equations that correspond with some special cases of the following equation (that according to the Axiom
2-1, one system for each case is enough):
0)(
1 1
s
q
r
p
pqpq bbF
(24-3)
that has been indicated in (23). Some special cases of (24-3), which we will consider below, in particular
include the second order equations with the different number of the elements, and some of the higher
order equations.
For ,2,,...3,2,1 rs equation (24-3) becomes as follows, respectively,
,02111
1
1
2
1
bbbq p
pq (27)
,022122111
2
1
2
1
bbbbbq p
pq (28)
,0231322122111
3
1
2
1
bbbbbbbq p
pq (29)
,02414231322122111
4
1
2
1
bbbbbbbbbq p
pq (30)
;025152414231322122111
5
1
2
1
bbbbbbbbbbbq p
pq (31)
.
.
Now a matrix equation (here we mean a system of linear equations) corresponding to (27) (according to
axiom (23)) is
00
0
2
1
0
0
m
m
f
e (32)
18
where 210110 , bfbe , and where we have
200
0
0
0
0
21 )(0
0
0
0Ife
e
f
f
eAA
(32-1)
Similarly, for (28) we have the following matrix equation
0
00
00
00
00
4
3
2
1
01
10
01
10
m
m
m
m
ee
ff
fe
fe
(33)
where 221121210110 ,,, bfbebfbe ;
Using (33) we may get equivalently, the following matrix equation for (28)
02
1
01
10
m
m
fe
fe (34)
where 221121210110 ,,, bfbebfbe , and where we have
21100
01
10
01
10
21 )( Ifefeee
ff
fe
feAA
(34-1)
A system of linear equations corresponding to (29) is
0
00000
00000
00000
00000
00000
00000
00000
00000
8
7
6
5
4
3
2
1
021
012
210
120
021
012
210
120
m
m
m
m
m
m
m
m
eff
eee
eff
eff
fee
fff
fee
fee
(35)
where 232132221121210110 ,,,,, bfbebfbebfbe ;
from (35) we can obtain the following matrix equation for equation (29),
19
0
0
0
0
0
4
3
2
1
021
012
210
120
m
m
m
m
fee
fff
fee
fee
(36)
where 232132221121210110 ,,,,, bfbebfbebfbe , and where we have
4221100
021
012
210
120
021
012
210
120
21 )(
0
0
0
0
0
0
0
0
Ifefefe
eee
eff
fef
fef
fee
fff
fee
fee
AA
(36-1)
Similarly, the matrix equations corresponding to (30) and (31) are obtained as follows,
for (30) we get
0
0000
0000
0000
0000
0000
0000
0000
0000
8
7
6
5
4
3
2
1
0321
0312
0213
0123
3210
3120
2130
1230
m
m
m
m
m
m
m
m
feee
feff
feff
feff
fffe
feee
feee
feee
(37)
where ;,,,,,,, 243143232132221121210110 bfbebfbebfbebfbe and where we have
20
21 AA
,)(
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
0000
833221100
0321
0312
0213
0123
3210
3120
2130
1230
0321
0312
0213
0123
3210
3120
2130
1230
Ifefefefe
eeee
eeff
eeff
eeff
ffff
feef
feef
feef
feee
feff
feff
feff
fffe
feee
feee
feee
(37-1)
and for (31) we obtain
0
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
feeee
feeff
feeff
feeff
feeff
feeff
feeff
fffff
fffee
fffee
fffee
feeee
fffee
feeee
feeee
feeee
(38)
where 254154243143232132221121210110 ,,,,,,,,, bfbebfbebfbebfbebfbe ,
and where we have
21
21 AA
164433221100
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
)(
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
Ifefefefefe
eeeee
eeeff
eeeff
eeeff
eeeff
eeeff
eeeff
effff
fffef
fffef
fffef
feeef
fffef
feeef
feeef
feeef
feeee
feeff
feeff
feeff
feeff
feeff
feeff
fffff
fffee
fffee
fffee
feeee
fffee
feeee
feeee
feeee
(38-2)
22
Similarly, systems of linear equations with larger sizes could be obtained for equation (24-3), where
.2,,...3,2,1 rs
In general, size of the square matrices of these matrix equations (corresponding with the second order
equations of the type (24-3), i.e. for 2r ) is ss 22 . But exceptionally, this size is reducible to
11 22 ss (only for the case of the second order equations), as it was for equations (29) – (31).
In general, the size of the square matrices in the matrix equations that correspond with the general cases
of equation (24-3), is ss rr . For all values of rs , , these matrix equations (according to (23), and
corresponding to the general cases of equation (24-3)) are derivable and calculable.
Meanwhile, as we previously noted, any of square matrices (i.e. the matrices kA in (23)) in equations
(32) – (38) and so on, is just one of the possible matrices, which we may obtain. Definitely, there are
other matrices (different ones, but with the same size and similar structure) which we may obtain for
constructing other equivalent cases (based on (23)) of equations (31) – (38); we just selected those
individual matrices (in (32) – (38)) because of their particular structures, and their special applications
that will be showed in the next section.
As some examples of the third order cases of equation (24-3), the systems of linear equations
corresponding with two equations
,312111
1
1
3
1
bbbbq p
pq
(39)
;322212312111
2
1
3
1
bbbbbbbq p
pq
(40)
respectively, are
,0
00
00
00
3
2
1
0
0
0
m
m
m
g
f
e
(41)
0
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
0000000
9
8
7
6
5
4
3
2
1
01
01
01
10
10
10
01
01
01
m
m
m
m
m
m
m
m
m
gg
ff
ee
gg
ff
ee
gg
ff
ee
(42)
23
where 321221121310210110 ,,,,, bgbfbebgbfbe .
The size of the square matrix in the matrix equation that corresponds to the next 3rd order equation, i.e.
,03
1
3
1
q p
pqb is .2727 Concerning the fourth order cases of equation (24-3), such as
,41312111
1
1
4
1
bbbbbq p
pq
(43)
;4232221241312111
2
1
4
1
bbbbbbbbbq p
pq
(44)
respectively, the systems of linear equations corresponding to them are
,0
000
000
000
000
4
3
2
1
0
0
0
0
m
m
m
m
h
g
f
e
(45)
0
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
00000000000000
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
01
01
01
01
10
10
10
10
10
10
10
10
01
01
01
01
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
hh
gg
ff
ee
hh
gg
ff
ee
hh
gg
ff
ee
hh
gg
ff
ee
(46)
where .,,,,,,, 421321221121410310210110 bhbgbfbebhbgbfbe
24
Similarly, as we noted above, for the fifth and the higher order cases of (24-3), with the larger number of
unknown elements, we get the matrix equations containing the square matrices with size .ss rr
Meanwhile, we can use the following linear relations (as the general rules) for transforming the second,
the third, fourth and the higher order cases of the form (24) into the form (25); e.g. for the second order
,1
2
11,
2
121
21
s
q p
pq
s
ii p
iii bcBp
(47)
we have
s
i
isisss
s
i
iii
s
i
i cBbcbcBbcbcBbcb1
21
1
222212
1
121111
2
22
2
222
2
2,...,,,,, ; (48)
and for the third order:
s
q p
pq
s
iii p
iiii bcBp
1
3
11,,
3
1
,321
321 (49)
we have
,,,,,,1
1223222112
1
1131121111
3
333
3
3
s
i
iii
s
i
i cBbcbcbcBbcbcb
,,,,...,,,,...,1
1)1(31)1(2)1(11
13211
3
33222
3
3
3
s
i
iissssssssi
s
i
sissss cBbcbcbcBbcbcb
.,,,...,1
)(3)(2)(1
3
33222
s
i
ississssscBbcbcb
(50)
Similarly, for transforming the fourth order and the higher order cases of equation (24) into (25), we can
define some linear transformations such as (48) and (50). However, if necessary, it is possible for
transforming (24) into (25), we define other linear transformations as well.
25
Remark 2-3. According to the particular applications of the second order cases of equation (24-3) in
the next section (Section 3), here we consider, analyze and introduce some of the properties of
matrix equations (34), (36), (37), (38).
First, we consider the following equation (the left-hand side is a special case of (26)):
n
ji
jijiij ddccB0,
0)( (51)
Where .jiij BB Now using matrix equations (34) (for 1n ), (36) (for 2n ), (37) (for 3n ) and
(38) (for 4n ) and so on, as well as the linear transformations of the type
n
j
jjiji dcBe0
),(
,iii dcf or
,
.
.
.
.
.
.
22
11
00
3
1
0
nnn dc
dc
dc
dc
f
f
f
f
(52-1)
;
.
.
.
...
.
.
.
...
...
...
.
.
.
22
11
00
210
2222120
1121110
0020100
3
1
0
nnnnnnn
n
n
n
n dc
dc
dc
dc
BBBB
BBBB
BBBB
BBBB
e
e
e
e
(52-2)
we obtain the following matrix equations that correspond to equation (51), respectively
,0)( 10000 mdcB (53)
26
,0
)(
)(
2
1
00
1
0
1
11
1
0
0
m
m
dcdcB
dcdcB
j
jjj
j
jjj
(54)
,0
0)()(
0
)()()(0
)(0)(
4
3
2
1
00
2
0
2
2
0
1
001122
22
2
0
1
2
0
0
11
2
0
2
2
0
0
m
m
m
m
dcdcBdcB
dcdcdc
dcdcBdcB
dcdcBdcB
j
jjj
j
jjj
j
jjj
j
jjj
j
jjj
j
jjj
(55)
,0
0000
0000
0000
0000
0000
0000
0000
0000
8
7
6
5
4
3
2
1
0321
0312
0213
0123
3210
3120
2130
1230
m
m
m
m
m
m
m
m
feee
feff
feff
feff
fffe
feee
feee
feee
(56)
where
.,)(
,,)(
,,)(
,,)(
333
3
0
33
222
3
0
22
111
3
0
11
000
3
0
00
dcfdcBe
dcfdcBe
dcfdcBe
dcfdcBe
j
jjj
j
jjj
j
jjj
j
jjj
(56-1)
27
0
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
feeee
feeff
feeff
feeff
feeff
feeff
feeff
fffff
fffee
fffee
fffee
feeee
fffee
feeee
feeee
feeee
(57)
where
.,)(
,,)(
,,)(
,,)(
,,)(
444
4
0
44
333
4
0
33
222
4
0
22
111
4
0
11
000
4
0
00
dcfdcBe
dcfdcBe
dcfdcBe
dcfdcBe
dcfdcBe
j
jjj
j
jjj
j
jjj
j
jjj
j
jjj
(57-1)
We must note that there are not the same linear transformations such as (51-1) - (51-2) (as exceptionally,
they exist for (51)) for the third and the higher order equations of the form
n
kji
kjikjiijk dddcccB0,,
0)( ,
n
lkji
lkjilkjiijkl ddddccccB0,,,
0)( , … . (58)
28
In addition, by the following choices
,
...
.
.
.
...
...
...
210
2222120
1121110
0020100
nnnnn
n
n
n
BBBB
BBBB
BBBB
BBBB
B
;
.
.
.,
.
.
.,
.
.
.,
.
.
.
3
1
0
3
1
0
2
1
0
2
1
0
nnnn f
f
f
f
F
e
e
e
e
E
d
d
d
d
D
c
c
c
c
C (59)
we can rewrite the transformations (52-1) and (52-2) as follows
,),( DCFDCBE (60)
here from (60) we also get
)(),( 12
112
1 FEBDFEBC (61)
where 1B is the inverse of B ; according to equation (51), B is a symmetric matrix; also we assume
that 0det B . The relations (60) and (61) indicate that there is an one to one correspondence between
the components of DC, and the components of FE, ; using this property, and the relations (60) and
(61), we will determine the solutions of the system of linear equations (54) – (57), on the basis of the
solutions of equations (34), (36), (37) and (38).
29
Now utilizing the standard and specific methods of solving the systems of homogeneous linear
equations in integers [7], we can analyze and also obtain the parametric solutions for systems of
homogeneous linear equations (34), (36), (37) and (38) (for unknowns ie and if ).
First, it is worth to note that a natural parametric solution of a general homogeneous linear equation of the
type
01
n
i
ii xa (62)
in integers is as follows [7, 8]:
1
1
),1,...,3,2,1(,n
j
jjnjnj kaxnjkax (63)
where the parameters ik are arbitrary integers, and we suppose that 0na , and furthermore, if ix and
ix ),...,3,2,1( ni be two solutions of equation (62), then ii xx
and ixt
(where t is a non-zero
integer) also are the solutions of (62) such that
0(1
n
i
ii xa , )01
n
i
ii xa 0)(1
i
n
i
ii xxa (63-1)
00)(11
n
i
ii
n
i
ii xaxta (63-2)
Now for equations (34) we get the following general solutions (where we supposed 02 m )
121211110220 ,,, mkfmkemkfmke (64)
where the parameters 2121 ,;, mmkk are arbitrary integers.
For system of equations (36) we get (where we supposed 04 m )
322324121331142121120430 ,,,,, mkmkfmkemkmkfmkemkmkfmke (65)
where the parameters 4321321 ,,,;,, mmmmkkk are arbitrary integers. Moreover, using (24-1), (24-2), and
(65) we may also obtain the following general solutions for (36) (where we supposed 0,0 44 km ):
.,,
,,,
322321441213311
244212112034430
mkmkfmkmkemkmkf
mkmkemkmkfmkmke
(66)
where the parameters 43214321 ,,,;,,, mmmmkkkk are arbitrary integers.
30
For system of equations (37), the following general parametric solutions are obtained (where we supposed
08 m , and also there appears an additional condition for the parameters im (see below)):
.,,,
,,,,
63523438137351242822
72611418313122130840
mkmkmkfmkemkmkmkfmke
mkmkmkfmkemkmkmkfmke
(67)
where the parameters 4321 ,,, kkkk are arbitrary integers. Moreover, the parameters im should satisfy the
following equation (as a necessary condition for the parameters im , that comes out from the system (37)
in the course of obtaining (67)):
073625184 mmmmmmmm (68)
Since the parameter 4m does not appear in (67), the condition (68) e.g. could be solved by the following
choices:
.,,
,,,
,,1
776655
332211
73625148
umumum
umumum
uuuuuumm
(69)
Now using (24-2) and the relations (67) and (69), the following solution for (37) is determined
.,,,
,,,,
635234313735124222
726114131312213040
ukukukftkeukukukftke
ukukukftkeukukukftke
(70)
where the parameters 7653214321 ,,,,,;,,,, uuuuuukkkkt are arbitrary integers, and )0( t .
Similarly, we obtain the following solutions for matrix equation (38) (where we supposed “ 016 m ”, and
there also emerge five additional conditions for the parameters im (see below)):
.,
,,,,
,,,,
1241131025541614
14413310135316231541321112521632
15314212115116415132231401650
mkmkmkmkfmke
mkmkmkmkfmkemkmkmkmkfmke
mkmkmkmkfmkemkmkmkmkfmke
(71)
31
where the parameters 54321 ,,,, kkkkk are arbitrary integers. In addition, the parameters im should satisfy
the following equations (as necessary conditions for the parameters im , that come out from the equations
of system (38)):
.
,
,
,
,
131214111510169
135113102168
145123101167
155122111166
153142131164
mmmmmmmm
mmmmmmmm
mmmmmmmm
mmmmmmmm
mmmmmmmm
(72)
In like manner, since the parameters 98764 ,,,, mmmmm don‟t appear in the solutions (71), the conditions
(72) could be solved by the following choices:
.
,,,
,,,
,,,
,
,
,
,
,
,1
1515
141413131212
1111101055
332211
1312141115109
1351131028
1451231017
1551221116
1531421314
16
um
umumum
umumum
umumum
uuuuuum
uuuuuum
uuuuuum
uuuuuum
uuuuuum
m
(73)
Using the relations (71) and (73) and (24-1), (24-2), the following solution for (38) is obtained
.,
,,,,
,,,,
12411310255414
1441331013532315413211125232
153142121151415132231050
ukukukukftke
ukukukukftkeukukukukftke
ukukukukftkeukukukkuftke
(71-1)
where the parameters 151413121110532154321 ,,,,,,,,,;,,,,, uuuuuuuuuukkkkkt are arbitrary integers, and
)0( t .
32
Similarly, the parametric solution of the matrix equation (with size 3232 ) corresponding to equation,
,05
0
i
ii fe (74)
similar to (64), (65), (70), (71-1), will be gotten. However, there appear sixteen additional conditions for
parameters im (where these conditions include sixteen homogenous second order equations, and each
equation contains only four terms, similar to (68) and (72)). These conditions could be solved with some
specific choices for parameters im , similar to (69) and (73). In general, the parametric solution of the
system of linear equations corresponding to the second order equation of the form
00
n
i
ii fe (75)
will lead to )12
)1(2(
nnn
number of conditions for parameters im (including the four terms
homogenous second order equations), and these conditions could be solved by some specific choices for
parameters im , similar to (69) and (73), and ultimately, solution of (75) could be obtained and specified.
Meanwhile, the solutions (64), (65), (70), (71-1) could also be represented as follows, respectively
;0
0
,
1
2
1
1
1
0
1
2
1
0
k
k
u
u
f
f
tk
tk
e
e
(76)
where we just supposed ( tm 2 ) and ( 11 um ).
;
0
0
0
,
1
2
3
32
31
21
2
1
0
12
2
3
2
1
0
k
k
k
uu
uu
uu
f
f
f
tk
tk
tk
e
e
e
(77)
where we just supposed ( tm 4 ) and ( 332211 ,, umumum ).
33
;
0
0
0
,
1
2
3
4
563
572
671
321
3
2
1
0
1
2
3
4
3
2
1
0
k
k
k
k
uuu
uuu
uuu
uuuo
f
f
f
f
tk
tk
tk
tk
e
e
e
e
(78)
.
0
0
0
0
0
,
1
2
3
4
5
1011125
1013143
1113152
1214151
5321
4
3
2
1
0
1
2
3
4
5
4
3
2
1
0
k
k
k
k
k
uuuu
uuuu
uuuu
uuuu
uuuu
f
f
f
f
f
tk
tk
tk
tk
tk
e
e
e
e
e
(79)
If we define the matrix K as
1
2
3
.
.
.
k
k
k
k
K
n
(80)
where we suppose 0K , then using (60) and (61) we get the following sets of relations
34
;0det,0
,,
,)(
,)(
12
1
12
1
BK
UKFtKE
KUtBD
KUtBC
(81)
;0det,0
,)(2
,,)(2
,)(2
,))((
11
11
11
111
BC
CUtBK
CUtBMF
CUtBtE
CUtBUtBD
(82)
.0det,0
,)(2
,)(2
,)(2
,))((
11
11
11
111
BD
DUtBK
DUtBMF
DUtBtE
DUtBUtBC
(83)
In point of fact, the formulas (81) are the (integer) parametric solution of equation (51), where the
matrices UKDCB ,,,, have been defined by the relations (59) and (76) – (80), and where the parameter
t is an arbitrary integer ( 0t ). On the other hand, the formulas (82) and (83) show that, if we suppose
35
that ic (or id ), (where ni ,...,3,2,1 ) are given values, then we can calculate the values id (or ic ) in
terms of them and the matrices UB, .
We should note here that the general conditions (68) and (71) (for parameters im ) could be solved
by other main approach as well:
Since the parameter 4m does not appear in the solutions (67), the condition (68), easily, will be solved by
the following choices
,04 m (84-1)
),0(: 88 mparameterIntegerarbitraryanm (84-2)
0736251 mmmmmm (84-3)
where equation (84-3), according to the solutions (65) and (66), generally is solved as follows
)0(:,0
,,,
,,,
884
322371331621125
413422431
mparameterIntegerarbitraryanmm
vuvumvuvumvuvum
vumvumvum
(85)
and also a more symmetric solution of the type
).0(:
,0,,,
,,,
88
4322371331621125
144132442234431
mparameterIntegerarbitraryanm
mvuvumvuvumvuvum
vuvumvuvumvuvum
(86)
where the parameters 43214321 ,,,;,,, vvvvuuuu are arbitrary integers. By replacing the values of im ,
(from the relations (85) or (86)) in formulas (67), and taking into account formulas (84-1) and (84-2), we
get a new general parametric solution for matrix equation (37).
36
As another similar case, since the parameters 98764 ,,,, mmmmm do not appear in the parametric
solutions (71), the conditions (72) (for the parameters im , as general outcomes of equations of the system
(38) in the course of determining (71)) is solved by the following choices as well
,098764 mmmmm (87-1)
),0(: 1616 mparameterIntegerarbitraryanm (87-2)
,0145123101 mmmmmm (88-1)
,0155122111 mmmmmm (88-2)
,0153142131 mmmmmm (88-3)
,0113135102 mmmmmm (88-4)
.0141113121510 mmmmmm (88-5)
As equation (88-5) could be derived from (88-1), (88-2), (88-3) and (88-4), we will not consider it in the
relevant next calculations.
Referring to the solutions (65) and (66) (for equation (36) that corresponds to quadratic equation (29)),
respectively, equations (88-1), (88-2), (88-3) and (88-4) are solved as follows,
first, using the solutions (65) we get
(88-1)
;,
,,
,,
433415523
244214532
322313541
vuvumvum
vuvumvum
vuvumvum
(89-1)
37
(88-2)
;,
,,
,,
433415515
144112532
311311541
vuvumvum
vuvumvum
vuvumvum
(89-2)
(88-3)
;)(,
,,
,)(,
422414515
144112523
211210541
vuvumvum
vuvumvum
vuvumvum
(89-3)
(88-4)
.,
,)(,
,)(,
322313515
133111523
211210532
vuvumvum
vuvumvum
vuvumvum
(89-4)
By the definitions of the type 2211 , vvvv , the solutions (89-1) – (89-4) could also be simplified.
Now using the solutions (66), we get the general solutions (with more symmetric forms) for equations
(88-1) – (884) as well, respectively,
(88-1)
;,
,,
,,
43341525523
24421435532
32231345541
vuvumvuvum
vuvumvuvum
vuvumvuvum
(90)
38
(88-2)
;,
,,
,,
43341515515
14411235532
31131145541
vuvumvuvum
vuvumvuvum
vuvumvuvum
(91)
(88-3)
;)()(,
,),(
),()(,
42241415515
14411225523
21121045541
vuvumvuvum
vuvumvuvum
vuvumvuvum
(92)
(88-4)
.),(
),()(,
,)()(,
32231315515
13311125523
21121035532
vuvumvuvum
vuvumvuvum
vuvumvuvum
(93)
By simplifying the relations (90) – (93), ultimately, we get the following set of the general solutions for
equations (88-1) – (88-4):
39
).0(:
,,
,,
,,
,0,0,0
,0,
,0,
,,
1616
433415244214
322313144112
311311122110
987
651155
452253
3553245541
mparameterIntegerfreeam
vuvumvuvum
vuvumvuvum
vuvumvuvum
mmm
mvuvum
mvuvum
vuvumvuvum
(94)
where the parameters 5432154321 ,,,,;,,,, vvvvvuuuuu are arbitrary integers. By replacing the values of
im (from the relations (94), in formulas (71)), and taking into account formulas (87-1) and (87-2), we
get the general parametric solution for matrix equation (38) ■.
Similarly, for the systems of linear equations corresponding to (75), with more variable elements (i.e.
larger values of n in (75)), the similar conditions and relations to (84-1) – (84-3) and (87-1) – (87-2) and
(88-1) – (88-5) and so on, could be chosen, ultimately. Then, based on them, we can generally determine
the parametric solution for the system of linear equations corresponding to each specific case of quadratic
equation (75).
40
3. Deriving the General (and Unique) Forms of the Fundamental Field
Equations of Physics, Including the Laws of (all) the Fundamental Forces
of Nature and the Relativistic-Quantum Wave Equations
If we assume that in the special relativity condition, the components of the n -momentum are discrete1,
i.e. have integer values in the invariant and the energy-momentum relations
,
ppgppg (95)
2
00
0002
0 )()(g
cmgcmppg
(96)
where g are constant coefficients, and pp , are the components of the momentum vector in two
reference frames, then the relations (95) and (96) are special cases of the algebraic relation (51); and
consequently they, necessarily, should be linearized (on the basis and framework of axiom (23)) and
transformed into systems of linear equations. Hence, using the matrix relations (53) – (57), we get the
following unique systems that correspond to the relations (95) and (96); first, for (95) we have,
respectively ( is are the integer parameters similar to the parameters im in the matrix relations (53) – (57)),
0)( 100
00 sppg (97)
0)(
)(
2
1
00
1
11
0
s
s
ppppg
ppppg
(98)
where 1,0 ;
0
0)()(
0
)()()(0
)(0)(
4
3
2
1
00
21
001122
22
10
11
20
s
s
s
s
ppppgppg
pppppp
ppppgppg
ppppgppg
(99)
where 2,1,0 ;
------------------------------------------------------------------------------
1. There are many modern standard and consistent quantum (relativistic) theories in physics in which there are assumed that some physical
essential quantities are discrete, like lattice field and gauge theories, quantum gravity theories, lattice QCD, and many other well-known theories.
41
0
0000
0000
0000
0000
0000
0000
0000
0000
8
7
6
5
4
3
2
1
0321
0312
0213
0123
3210
3120
2130
1230
s
s
s
s
s
s
s
s
feee
feff
feff
feff
fffe
feee
feee
feee
(100)
where 3,2,1,0 and ,073625184 ssssssss
(100-1)
.),(
,),(
,),(
,),(
333
3
3
222
2
2
111
1
1
000
0
0
ppfppge
ppfppge
ppfppge
ppfppge
(100-2)
0
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
feeee
feeff
feeff
feeff
feeff
feeff
feeff
fffff
fffee
fffee
fffee
feeee
fffee
feeee
feeee
feeee
(101)
42
where we have
,153142131164 ssssssss (101-1)
,155122111166 ssssssss (101-2)
,145123101167 ssssssss (101-3)
,135113102168 ssssssss (101-4)
;131214111510169 ssssssss (101-5)
.),(
,),(
,),(
,),(
,),(
444
4
4
333
3
3
222
2
2
111
1
1
000
0
0
ppfppge
ppfppge
ppfppge
ppfppge
ppfppge
(101-6)
and 4,3,2,1,0 .
…
For (96) we get (where we suppose 00
00,0:0
g
cmpp ), respectively
0)( 100
00
00
s
g
cmpg (102)
0
)(
)(
2
1
00
00
1
100
0000
s
s
g
cmppg
pg
cmgpg
(103)
43
where 1,0 ;
0
)(0
0)(
)(0
0)(
4
3
2
1
00
00
21
00
0012
2
1
00
0000
1
2
00
0000
s
s
s
s
g
cmppgpg
g
cmppp
ppgg
cmgpg
ppgg
cmgpg
(104)
where 2,1,0 ;
0
0000
0000
0000
0000
0000
0000
0000
0000
8
7
6
5
4
3
2
1
0321
0312
0213
0123
3210
3120
2130
1230
s
s
s
s
s
s
s
s
feee
feff
feff
feff
fffe
feee
feee
feee
(105)
where 3,2,1,0 and
,073625184 ssssssss (105-1)
.,
,,
,,
),(),(
33
3
3
22
2
2
11
1
1
00
000
00
0000
0
pfpge
pfpge
pfpge
g
cmpf
g
cmgpge
(105-2)
44
0
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
00000000000
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
04321
04312
04213
04123
03214
03124
02134
01234
43210
43120
42130
41230
32140
31240
21340
12340
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
feeee
feeff
feeff
feeff
feeff
feeff
feeff
fffff
fffee
fffee
fffee
feeee
fffee
feeee
feeee
feeee
(106)
where we have
,153142131164 ssssssss (106-1)
,155122111166 ssssssss (106-2)
,145123101167 ssssssss (106-3)
,135113102168 ssssssss (106-4)
;131214111510169 ssssssss (106-5)
45
.,
,,
,,
,,
),(),(
44
4
4
33
3
3
22
2
2
11
1
1
00
000
00
0000
0
pfpge
pfpge
pfpge
pfpge
g
cmpf
g
cmgpge
(106-6)
and 4,3,2,1,0 .
…
In this section we will use the geometrized units [9], the Einstein notation, and the following (sign)
conventions:
- The Metric sign convention: )...(
- The Riemann and Ricci tensors:
RR
R
- The Einstein tensor (sign): ...8)( 21 TRgR . (107)
It is worth to note that concerning the relations (97) – (101) for the components p and p , we can use
the relations (81) – (83) and the solutions (76) – (79) for deriving the linear transformations between two
reference frames. In other word, the general forms of the linear transformations (equaling to Lorentz
transformation) between two reference frames, directly, are determined from the relations (97) – (101)
for all dimensions.
Now basically, we use the relations (102) – (106) for deriving - uniquely and completely – the
fundamental field equations of physics, including the laws of the fundamental forces of nature and
the relativistic-quantum wave equations in all dimensions. For this goal in principle, we canonically
quantize the relations (102) – (106). That’s all. Thus, as a principal substitution rule we substitute the
following canonical covariant operators (containing quantum mechanical operators) by their equivalent
quantities in the relations (102) – (106) including p ,
g (constants), is (parameters):
- Covariant n-momentum operator: ip (108)
- General components of the metric tensor: g g (109)
- Components of supposed field tensors: ...;;;ˆ ZFsi (110)
46
In fact, we show (and accept, in principle) that the general forms of the fundamental fields equations
of physics, including the laws of fundamental forces of nature and the relativistic-quantum wave
equations in all dimensions, are derived by canonical quantization of the unique linearized forms (the
relations (102 )–(106) - obtained on the basis of axiom (23 )) of the energy-momentum relation (96).
According to the structures, the compositions and “the number” of equations of the systems (102) – (106),
we conclude that there exist only three kinds of anti-symmetric tensors whose components could be
substituted by the parameters is (except in equation (102), which is a special and trivial case, see below)
and they transform the systems (102) – (106) to the general tensor equations. These tensors are a 2nd
order, a 3rd
order and a 4th order tensors. The 4
th order tensor of these tensors just matches the Riemann
tensor R (as a basic mathematical tensor, which is necessary for calculating and specifying the
components of the metric tensor g ); other two tensors that could be written as Z
and F
(necessarily) should be two anti-symmetric tensors with respect to the indices , such as:
FFZZ , .
Thus firstly, corresponding to relation (102) – (104) (which don‟t contain any condition for parameters
is ), we get the following tensor equations, respectively:
(the tensor equation corresponding to relation (102) is a special and trivial case, where the Riemann
tensor vanishes in that case; so below we just write it down, assuming a tensor such as F
that is
substituted by 1s and where we assume 100 g )
(102) 0*
FD
(111-1)
where 0 , and 011
00 ˆ,1 Fssg
.
(103)
,0][ FD (112-1)
)(* EJFD (112-2)
where 1,0,, ,and .,ˆ,ˆ )()()(
221011
E
v
EE DJssFss
(103)
,0][ ZD (112-3)
)(* NJZD (112-4)
where 1,0,,, , and .,ˆ,ˆ )()()(
221011
N
v
NN DJssZss
47
(103)
,0][ RD (112-5)
)(* GJRD (112-6)
where 1,0,,,, , and .,ˆ,ˆ )()()(
221011
G
v
GG DJssRss
(104)
,0][ FD (113-1)
)(* EJFD (113-2)
where 2,1,0,, , and
.,ˆ,ˆ,ˆ,ˆ )()()(
44213302221011
E
v
EE DJssFssFssFss
(104)
,0][ ZD (113-3)
)(* NJZD (113-4)
where 2,1,0,,, , and
.,ˆ,ˆ,ˆ,ˆ )()()(
44213302221011
N
v
NN DJssZssZssZss
(104)
,0][ RD (113-5)
)(* GJRD (113-6)
where 2,1,0,,,, , and
.,ˆ,ˆ,ˆ,ˆ )()()(
44213302221011
G
v
GG DJssRssRssRss
where in equations (111-1) – (113-6) we have
48
k
imD
0 , (114-1)
.0*
kim
D
(114-2)
,0:0
,1
:000
k
gk
(115-1)
,,0 0)()()(
Fk
imJII EEE
(116-1)
,,0 0)()()(
Zk
imJII NNN
(116-2)
.0,,0 )(*0)()()(
GGGG JDRkim
JII
(116-3)
In the systems of linear equations (102) – (104), the parameters is were just arbitrary integer parameters,
for which we substituted the components of tensors RZF ,, by them (based on the principal
canonical operators definitions (108) – (110)).
Before writing the tensor equations corresponding to the relations (105) and (106), which are similar to
tensor equations (112-1) – (113-6) (as we will show), we should at first mention the following two
mathematical remarks:
Remark 3-1. For the next systems of linear equations, i.e. (105) and (106) (and so on, i.e. for the higher
dimensions D >3), the situation for the parameters is is a bit different. There are some general conditions
for the parameters is (including quadratic equations (105-1) and (106-1) – (106-5)), that should be
considered and solved. In previous section, we dealt with this situation in two different ways. We had two
types of general solutions for these conditions; one includes the solutions of forms (69) and (73) (simply
they respectively would be the solutions of the conditions (105-1) and (106-1) – (106-5), where ii sm
). The other one includes the solutions of the forms (85) and (86) (simply they become the solutions of the
condition (105-1), where ii sm ), and the solutions of the forms (89-1) – (89-4) and (94) (that simply,
they also become the solutions of the conditions (106-1) – (106-5), where ii sm ). In all these solutions
for the conditions (105-1) and (106-1) – (106-5), the parameters is , necessarily, are written in terms of
the new parameters iu and iv , which we may represent them as the following general form
49
),...,,,;,...,,,( 321321 nmii vvvvuuuuHs (117)
where the parameters iu and iv are arbitrary integers. Now for dealing with this situation, we may reason
that because the components of tensor fields (corresponding to the operators is , according to the
definitions (108) – (110)) are substituted by the parameters is , and at the same time the parameters is are
not arbitrary parameters but are defined by (117) (where iu and iv are arbitrary parameters), we can
conclude that the operators is , naturally, should have the following form as well
)ˆ,...,ˆ,ˆ,ˆ;ˆ,...,ˆ,ˆ,ˆ(ˆ
321321 nmii vvvvuuuuHs (118)
where iu and iv are some operators that are substituted by the parameters iu and iv , and they should be
specified for the tensor components corresponding to the operators is . So, from the above conditions and
arguments, in principle, we may conclude that
)]ˆ,...,ˆ,ˆ,ˆ;ˆ,...,ˆ,ˆ,ˆ(ˆ[)],...,,,;,...,,,([ 321321321321 nmiinmii vvvvuuuuHsvvvvuuuuHs (119)
Meanwhile, according to the parametric solutions (69), (73), (85), (86), (89-1) – (89-4) and (94), the form
iH is not a unique form. By using (119), the form iH and iu and iv could be specified (in fact,
beforehand we will clarify which form(s) of iH are acceptable), such that they will be consistent with
the operators is (that correspond to the components of tensors RZF ,, ).
Remark 3-2. In connection with the above note and the relations (118) and (119), for specifying the
form iH , and the operators iu and iv in the structures of tensors RZF ,, , principally, we will
use the Riemann tensor R as the base, that on one hand it is necessary for specifying the metric
tensor g , and on the other hand it is a basal mathematical tensor with a certain structure.
50
Thus, on this basis and concerning the form iH in (117), (118) and (119), only the parametric solutions
of the type (86) (formally with respect to ii sm ), i.e.
).0(:
,0,,
,,
,,
88
43223713316
2112514413
2442234431
sparameterIntegerarbitraryans
svuvusvuvus
vuvusvuvus
vuvusvuvus
(117-1)
for equation (105-1), and also only the parametric solutions of the type (94) (formally with respect to
ii sm ), i.e.
).0(:
,,
,,
,,
,0,0,0
,0,
,0,
,,
1616
433415244214
322313144112
311311122110
987
651155
452253
3553245541
sparameterIntegerarbitraryans
vuvusvuvus
vuvusvuvus
vuvusvuvus
sss
svuvus
svuvus
vuvusvuvus
(117-2)
for equation (106-1) – (106-5) are acceptable; furthermore, by starting from the Riemann tensor and its
specific structure in (107) and using the relation
g , we get
)()(
R (120)
51
Now if we define the operator C , operating on a second order tensor Y as follows
)( YYYaYC (121)
where a is a non-zero constant, then we can rewrite (120) such as
])()[(1
gCCgCC
aR (122)
In addition, if we define the operator D as follows,
)(
CCD (123)
then the relation (122) for the Riemann tensor, ultimately, could be written as
)(1
gDgDa
R (124)
Meanwhile, the natural generalization of operator C , operating on an arbitrary thn order tensor
nA ...321
, is
)( ...]...[... 3214321321 nnn
AAaAC (125)
According to (122) and (124), the only and the most general form representing the structure of Riemann
tensor is
)ˆˆˆˆ(1
BABAa
RR (126)
where .ˆ,ˆ gBDA (127)
Now as we stated in Remark 3-1 and Remark 3-2, we use and apply the relation (126) (as a basal
criterion), and the relations (117) – (119) for determining the general formulation of the operators is (that
are substituted by the parameters is in systems (103) – (106)), which correspond to the components of
the Riemann tensor. That is, respectively
52
))};0(:
,(
),ˆ,ˆˆˆˆˆ{(:)103(
22
01101
)(
20110101
sparameterIntegerfreeas
BABAas
sBABAaRsa G
(128)
))};0(:
,,
,(),ˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,ˆˆˆˆˆ{(:)104(
44
1221320022
01101
)(
41221213
20020220110101
sparameterIntegerfreeas
BABAasBABAas
BABAassBABAaRsa
BABAaRsaBABAaRsa
G
(129)
))};0(:
,,,
,0,,,(
),ˆ,ˆˆˆˆˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,0ˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,ˆˆˆˆˆ{(:)105(
88
211271331632235
4033030220201101
)(
821121271331316
322323540330303
02202020110101
sparameterIntegerfreeas
BABAasBABAasBABAas
sBABAasBABAasBABAas
sBABAaRsaBABAaRsa
BABAaRsasBABAaRsa
BABAaRsaBABAaRsa
G
(130)
53
))}.0(:
,,,
,,,
,0,0,0,0,,0
,,,(
),ˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,0ˆ0ˆ,0ˆ,0ˆ
,ˆˆˆˆˆ,0ˆ,ˆˆˆˆˆ
,ˆˆˆˆˆ,ˆˆˆˆˆ{(:)106(
1616
122115311314322313
144112244211433410
9876400454
033032002201101
)(
1612212115
3113131432232313
1441411224424211
433434109876
400404540330303
20020220110101
sparameterIntegerarbitraryans
BABAasBABAasBABAas
BABAasBABAasBABAas
ssssBABAass
BABAasBABAasBABAas
sBABAaRsa
BABAaRsaBABAaRsa
BABAaRsaBABAaRsa
BABAaRsassss
BABAaRsasBABAaRsa
BABAaRsaBABAaRsa
G
(131)
where iA and iB are arbitrary parameters. Moreover, in the above formulas for parameters is , the
element a is just an arbitrary parameter ( 0a ). But in the above formulas representing is , the element
a will be specified later (that is ia ).
According to Remark 3-1 and Remark 3-2, and formulas (128) – (131), it is clear that for the form iH in
(117), (118) and (119), only the relations (117-1) and (117-2) are acceptable, which are consistent with
the formulas (126) and (128) – (131). Firstly, it is easy to show that the algebraic formulas representing
is in (128) and (129), are also consistent with previous algebraic conditions of the parameters is in the
systems (103) and (104) (where they just were arbitrary integers). For example, regarding the algebraic
formulas representing is in (129), by the following choices
.,,,
,)(),(),(
322322311311
312211221012210
skBskAskBskA
skkkkaskskBskskA
(132)
54
where the parameters iii skk ,, are arbitrary integers, directly, we can show that is are also arbitrary
parameters.
Secondly, concerning the formulas (130) and (131), we could show that by the following choices,
respectively,
);0(:,0
,,,,
,,,,
884
13132222
31314040
sparameterIntegerarbitraryanss
uBavAuBavA
uBavAuBavA
(133)
)0(:
,0,0,0,0,0
,,,,,
,,,,,
1616
98764
1414232332
3241415050
sparameterIntegerarbitraryans
sssss
uBavAuBavAuB
avAuBavAuBavA
(134)
the formulas (130) and (131), completely, accord with formulas (117-1) and (117-2) (and only with them,
as it should be).
Now according to the above notes and approaches, particularly, Remark 3-1 and Remark 3-2, and using
the relations (130) and (133), uniquely, (in the continuation of deriving the field equations of the
fundamental forces for the higher dimensions D >3) we derive the following general field equations,
respectively
(105)
,0][ FD (135-1)
)(* EJFD (135-2)
where 3,2,1,0,, and
55
.,ˆ,ˆ
,ˆ,ˆ,0ˆ
,ˆ,ˆ,ˆ
)()()(
881277
3166235544
303320221011
E
v
EE DJssFss
FssFssss
FssFssFss
(105)
,0][ ZD (135-3)
)(* NJZD (135-4)
where 3,2,1,0,,, and
.,ˆ,ˆ
,ˆ,ˆ,0ˆ
,ˆ,ˆ,ˆ
)()()(
881217
3166235544
303320221011
N
v
NN DJssZss
ZssZssss
ZssZssZss
(105)
,0][ RD (135-5)
)(* GJRD (135-6)
Where 3,2,1,0,,,, and
.,ˆ,ˆ
,ˆ,ˆ,0ˆ
,ˆ,ˆ,ˆ
)()()(
881277
3166235544
303320221011
G
v
GG DJssRss
RssRssss
RssRssRss
56
where the definitions and relations (114-1) – (116-3) are applied to equations (135-1) – (135-6) as well1.
Now based on Remark 3-1 and Remark 3-2, and the relations (117) – (119) and (123), (125), (127) and
(128) – (131), and (126) (as a basal and principal criterions for specifying the structures of the quantities
is , that correspond to the components of the field tensors), the following relations defining the structures
of tensors F and Z in all equations (112-1) – (113-6) and (135-1) – (135-6), necessarily, are
obtained and specified, respectively
)(
1QDQD
aF (136)
)(
1 HDHD
aSZ (137)
where Q and H are a scalar field and a vector field, and where we get ia . Furthermore, the
following gauge operators are obtained for the fields F and Z (or S ) as well
,
,:)(
QQCA
Aig
FE
(138-1)
],[
)(
AAig
AAFE
(138-2)
where )(Eg is the coupling constant;
and
],,[
,:
)(
)(
HHig
HHL
Hig
Z
N
N
(139-1)
)(
)(
LHLHig
LLSZN
(139-2)
where )(Ng is the coupling constant.
---------------------------------------------------------------------------- 1. Similarly, using the relations (131), (134) and (108) – (110), we may obtain the tensor equations that correspond to the system
(106). However, there might be appeared some limitations for the higher dimensions if we suppose the discreteness of the other essential
physical quantities such as space-time coordinates and so on.
57
Now in principle, we conclude that in the field equations (112-1) – (113-6) and (135-1) - (135-6) and
relations (138-1) – (139-2), where F is a general field-strength tensor that represent the
electromagnetic and nuclear (Yang-Mills) fields (depending on the definition of the gauge covariant
derivative), Z is also a general 3rd
order tensor that could represent some fields, and R is the
Riemann tensor for the gravitational field. In fact, each system of tensor equations (112-1) – (113-6) and
(135-1) – (135-6) could be divided into two subcategories depending on mass m0 is zero or non-zero.
Based on the unique structure of equations (112-1) – (112-2), (113-1) – (113-2) and (135-1) – (135-2),
that correspond to the general form of the Yang-Mills equations (including the Maxwell’s equations),
magnetic monopoles (opposite electric monopoles in connection with formula )()( E
v
E DJ ) could
not exist, in general.
The general form of the Einstein field equations could be also obtained from equations (112-6),
(113-6), (135-6). Using the (second) Bianchi identity and the conventions (107) we get
RRR (140)
Now from (140) and equations (112-6), (113-6), (135-6) and the assumption
)(8)(8 TgTgBTTJ (141)
where T is the stress-energy tensor ( TT ) and g is the metric tensor and B is a constant, we get
the following general gravitational field equation
qgKim
BTgTR
0)(8 (142)
where q is a constant value (that emerges naturally, when we obtain equation (142)), and kK
(where k has been defined in (115-1)), and KK , 0
K (due to its complex coefficient).
First, the equation (142) for (112-6) (concerning two dimensional space-time) takes the following form
,)8(8 210
gTKim
T
(143-1)
;)8( 21
21
ggTR (143-2)
TR 8 (144)
where 0,2 Bq and is the cosmological constant.
58
For equation (113-6) (concerning three dimensional space-time), the field equation (142) takes the
following form
gKim
TRgR
02
1 8 (145)
where 1,21 Bq and is the cosmological constant.
And concerning equation (135-6) (concerning the four dimensional space-time), field equation (142)
takes the following specific form as well
gKim
TRgR
02
1 8 (146)
where 2
1, Bq and is the cosmological constant. Equations (144), (145) and (146) are
equivalent to the Einstein field equations for m0 = 0.
In equations (142) – (146) we may have the (total) conservation law as follows1
0)( 0
Zim
T
(147)
Here we also emphasize that equations (112-5), (112-6), (113-5), (113-6), (135-5), (135-6), (143-2),
(145), (146) of the gravitational field, include two subcategories: for m0 = 0 (of the field carrier particle
i.e. graviton), and also for m0 ≠ 0 (of a presupposed massive gravitational field carrier particle, as the
above equations predict it).
---------------------------------------------------------------------------
1. Meanwhile, according to the second Bianchi identity, from equations (113-5) and (135-5) we may have 0:0, 0 abRim
ba
;
that in the case 00 m (as the rest mass of a presupposed gravitational force carrier particle), there may appear some additional conditions for the
components of the Riemann tensor.
59
In the special relativity conditions, tensor equations (112-1) – (112-4), (113-1) – (113-4) and (135-1) –
(135-4) also could be written in the forms
,0])[( 0 FImi
(148)
0])[( 0 ZImi
(149)
where is the gauge covariant derivative, I is the identity matrix, and where for two dimensional
space-time (concerning equations (112-1) – (112-4)) we have
.][,][
,01
10,
10
01
)(
10
)(
10
10
NE
ZZ
FF
(150)
It is worth here to note that (150) (together with (148) and (149)) as a generalized form, corresponds to
the Pauli equation, if we assume that ( 00 m ) and ( 0,0)()(
NE
);
and for three dimensional space-time (concerning equations (113-1) – (113-4)) we get
.][,][
,
0010
0001
1000
0100
,
0001
0010
0100
1000
,
1000
0100
0010
0001
)(
21
02
10
)(
21
02
10
210
NE
Z
Z
Z
ZF
F
F
F
(151)
Here it‟s also worth to note that (151) (together with (148) and (149)) as a generalized form, corresponds
to the Dirac equation, if ( 00 m ) and ( 0,0)()(
NE
);
60
and for the four dimensional space-time (concerning equations (135-1) – (135-4)) we obtain
,
00000100
00001000
00000001
00000010
01000000
10000000
00010000
00100000
,
00000010
00000001
00001000
00000100
00100000
00010000
10000000
01000000
,
00000001
00000010
00000100
00001000
00010000
00100000
01000000
10000000
,
10000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
32
10
.0
][,0
][
)(
12
31
23
30
20
10
)(
12
31
23
30
20
10
NE
Z
Z
Z
Z
Z
Z
Z
F
F
F
F
F
F
F
(152)
Here (152) (together with (148) and (149)) is also turned into the general form of the relativistic-quantum
wave equation for the four dimensional space-time, if ( 00 m ), ( 0,0)()(
NE
).
All the above “α“ - alpha matrices are real and contravariant (including previous matrices) and they
correspond to Clifford algebras [10, 11].
61
In fact, in some special cases such as: ( 00 m ) and ( 0,0)()(
NE
), equations (148) and (149) (as
well as equations (112-1) – (113-6) and (135-1) – (135-6), in general) are turned into the relativistic-
quantum wave equations that correspond to the Pauli and Dirac equations and so on. We emphasize that
matrices (151) together with formulas (148) and (149) are formulated only in (1+2) dimensions (here we
may also conclude that particles like electron and quarks are two dimensional (spatial) objects). For (1+3)
dimensions, we have to apply a relativistic-quantum wave equations (structurally analogous to the Dirac
equation) that contains the 8×8 contravariant matrices (152) corresponding to Clifford algebra.
4. Conclusion
In Section 2, since the set of algebraic axioms (17) – (23) for integers have been formulated in terms of
the square nn matrices (for an arbitrary n), we can conclude that for a complete representation of
algebraic properties of integers, necessarily and sufficiently, the square matrices :nn
nnnnijnnijnnij cba ,...][,][,][ should be applied; and ordinary (old) algebraic axioms (10) – (16-2)
that had been formulated in terms of the single elements: ,...,, 321 aaa – in fact, they are single
components of 11 matrices such as: )(),...(][),(][),(][ 11311321121111 Zaaaaaa – are not
sufficient for a complete description of the algebraic properties of integers.
In Section 3, by the assumption of discreteness of the relativistic n-momentum, and canonical
quantization of the unique linearized forms (i.e. the relations (102) – (106), obtained on the basis of
axiom (23) – as a new algebraic axiom – in Section 2) of the energy-momentum relation (96), we derived
the general forms of the fundamental field equations of physics, including the laws of the fundamental
forces of nature and the relativistic-quantum wave equations, that were the general tensor equations (112-
1) – (113-6), (135-1) – (135-6) and the relations (148) – (149). Concerning the fundamental forces, these
general field equations describe three main categories of the fields including gravitational,
electromagnetic and nuclear forces (for dimensions D ≥2). Furthermore, when we compare the derived
field equations with the general forms of the ordinary field equations of physics such as the Einstein field
equations and the Yang-Mills equations (including the Maxwell‟s equations) and so on (that they have
been formulated, generally, based on the empirical evidences), there emerge some generalizations for
these ordinary field equations – particularly, appearing a term of the mass m0 (as the rest mass of the
field carrier particle) in all these field equations. Moreover, in some special conditions these general
tensor equations were turned into the relativistic-quantum wave equations that corresponded to the Pauli
and Dirac equations. In particular, we derived the relativistic-quantum wave equation (151) that contained
4×4 real gamma (Dirac) matrices (as a form of the Dirac equation) and it was only formulated in (1+2)
dimensions. For (1+3) dimensions, we showed that we have to apply a relativistic-quantum wave
equation (structurally analogous to the Dirac equation) that contains the 8×8 contravariant matrices (152),
corresponding to Clifford algebra.
This approach along with graviton (with zero rest mass) predicts a gravitational field carrier particle
with non-zero rest mass as well. According to the unique structures of the field equations obtained (i.e.
Yang-Mills equations), we also concluded that generally, (opposite electric monopoles) magnetic
monopoles could not exist.
We emphasize again that the procedure of deriving the fundamental field equations of physics
(including the laws of the fundamental forces of nature and the relativistic-quantum wave equations (in all
dimensions D ≥ 2 ) was based on a new single mathematical approach (that was presented in Section 2,
concerning the algebraic structure of the domain/ring of integers), and the assumption of discreteness of
the relativistic n-momentum, and ultimately by canonical quantization of the unique linearized forms
(obtained on the basis of Axiom 2-1) of the energy-momentum relation.
62
As we mentioned, the derived field equations are unique, and one of the main goals of this article was
to show that the general field equations of (all) the fundamental forces of nature (uniquely and
distinctly) are derivable from certain mathematical arguments. The results obtained in Section 3,
demonstrate the efficiency of the algebraic theory of linearization (presented in Section 2, as a new
mathematical structure) and a wide range of its possible applications.
Acknowledgment
Special thanks are extended to Prof. and Academician Vitaly L. Ginzburg (Russia), Prof. and
Academician Dmitry V. Shirkov (Russia), Prof. Leonid A . Shelepin (Russia), Prof. Vladimir Ya.
Fainberg (Russia), Prof. Wolfgang Rindler (USA), Prof. Roman W. Jackiw (USA), Prof. Roger Penrose
(UK), Prof. Steven Weinberg (USA), Prof. Ezra T. Newman (USA), Prof. Graham Jameson (UK), Prof.
Sergey A. Reshetnjak (Russia), and many others for their support and valuable guidance during my
studies and research.
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