On Discrete (Digital) Physics: as a Perfect Deterministic Structure for Reality - And the...

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


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


BABAasBABAas

BABAassBABAaRsa

BABAaRsaBABAaRsa

G

(129)

))};0(:

,,,

,0,,,(

),ˆ,ˆˆˆˆˆ,ˆˆˆˆˆ

,ˆˆˆˆˆ,0ˆ,ˆˆˆˆˆ


88

211271331632235

4033030220201101

)(

821121271331316

322323540330303

02202020110101


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


1616

122115311314322313

144112244211433410

9876400454

033032002201101

)(

1612212115

3113131432232313

1441411224424211

433434109876

400404540330303

20020220110101


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


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.

References

1. Zahedi, R. A. “Linearization Method in the Ring Theory”, Russian Academy of Sciences (RAS),

Bulletin of the Lebedev Physics Institute, New York, Springer-Verlag, No. 5-6, 1997.

2. Zahedi, R. A. “On Applied Aspects of the Ring Theory”, Russian Academy of Sciences (RAS),

Bulletin of the Lebedev Physics Institute, New York, Springer-Verlag, No. 3-4, 1997.

3. Zahedi, R. A. “On the Connection Between Methods of the Ring Theory and the Group Approach”,

Russian Academy of Sciences (RAS), Bulletin of the Lebedev Physics Institute, New York, Springer-

Verlag, No. 7-8, 1997.

4. Zahedi, Ramin A. “The Linearization Method Based on the Theory of Rings and its Applications in

Physics”, Russian Academy of Science (RAS) Publs., Russia, 1997.

( http://search.rsl.ru/en/catalog/record/49410 , http://dlib.rsl.ru/viewer/01000049410#?page=1 )

5. Durbin, John R. “Modern Algebra: An Introduction”, (3rd ed.), John Wiley and Sons, USA, 1993;

Blyth , T.S.; Robertson, E.F. "Groups, rings and fields: Algebra through practice," Book 3. Cambridge

University Press, UK, 1985.

6. Dickson, Leonard E. “History of the Theory of Numbers”, Volume II: Diophantine Analysis,

Mineola, Dover Publications, NY, USA, 2005.

7. Mordell, L. J. “Diophantine Equations”, Pure and Applied Mathematics, 30, Academic Press, 1969.

8. Ireland, K.; Rosen, M. “A Classical Introduction to Modern Number Theory”, 2nd ed., Springer-

Verlag, New York, USA, 1998.

9. Guidry, M. W. “Gauge Field Theories: An Introduction with Applications”, John Wiley and Sons,

USA, 1991.

10. Byrnes, J. (ed.) “Computational Noncommutative Algebra and Applications”, Springer-Verlag,

pp. 363-387, 2004.

11. Bourbaki, Nicolas “Algebra”, Berlin, New York, Springer-Verlag, 1988.

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