# Elementary Process Theory a formal axiomatic system with a potential application as a foundational framework for physics supporting gravitational repulsion of matter and antimatter.

код для вставкиСкачатьAnn. Phys. (Berlin) 522, No. 10, 699 ? 738 (2010) / DOI 10.1002/andp.201000063 Elementary Process Theory: a formal axiomatic system with a potential application as a foundational framework for physics supporting gravitational repulsion of matter and antimatter Marcoen J. T. F. Cabbolet1,2? 1 2 Center for Logic and Philosophy of Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium Institute of Theoretical Physics, Kharkov Institute of Physics and Technology, Akademicheskaya str. 1, 61108 Kharkov, Ukraine Received 15 May 2010, revised 28 June 2010, accepted 18 July 2010 by F.W. Hehl Published online 16 August 2010 Key words Gravitational repulsion, dynamics, individual processes, foundations of physics. Theories of modern physics predict that antimatter having rest mass will be attracted by the earth?s gravitational field, but the actual coupling of antimatter with gravitation has not been established experimentally. The purpose of the present research was to identify laws of physics that would govern the universe if antimatter having rest mass would be repu?lsed by the earth?s gravitational field. As a result, a formalized axiomatic system was developed together with interpretation rules for the terms of the language: the intention is that every theorem of the system yields a true statement about physical reality. Seven non-logical axioms of this axiomatic system form the Elementary Process Theory (EPT): this is then a scheme of elementary principles describing the dynamics of individual processes taking place at supersmall scale. It is demonstrated how gravitational repulsion functions in the universe of the EPT, and some observed particles and processes have been formalized in the framework of the EPT. Incompatibility of Quantum Mechanics (QM) and General Relativity (GR) with the EPT is proven mathematically; to demonstrate applicability to real world problems to which neither QM nor GR applies, the EPT has been applied to a theory of the Planck era of the universe. The main conclusions are that a completely formalized framework for physics has been developed supporting the existence of gravitational repulsion and that the present results give rise to a potentially progressive research program. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction 1.1 Research question, its historical background and relevance to science The purpose of the present investigation was to search for an answer, as rigorous and complete as possible, to the following research question: Which elementary principles might underlie the hypothesis that antimatter such as positrons, antiprotons and antineutrons has positive rest mass, but will be repulsed by the gravitational field of the earth? Theoretical studies on the interplay of antimatter and gravitation have been performed since the 1950?s. Since then, various theoretical arguments against a mutual repulsion of matter and antimatter have been discussed in the scientific literature. Below the main arguments will be repeated summarily; for an extensive review the reader is referred to the literature, cf. [1?4]. ? E-mail: marcoen.cabbolet@vub.ac.be or m.cabbolet@liberalitas.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 700 M. J. T. F. Cabbolet: Elementary Process Theory The perhaps most compelling argument against a gravitational repulsion of matter and antimatter is that it would violate the law of conservation of energy, which Morrison inferred from a Gedanken experiment in [5]. Nieto and Goldman, however, showed that Morrison?s Gedanken experiment does not forbid a gravitational repulsion outside the framework of Einstein?s General Relativity (GR), cf. [1]; in addition, Chardin and Rax have shown that Morrison?s idea doesn?t even forbid gravitational repulsion inside the framework of GR, cf. [3]. Another argument that was initially thought to rule out anomalous gravitational behavior of antimatter is that of Schiff, who stated in [6], [7] that such is impossible on quantum fieldtheoretic grounds. However, Nieto and Goldman showed in [1] that Schiff?s renormalization procedure is invalid, and that the argument is thus inconclusive. A third argument is that Good in [8] inferred from the decay of neutral K0 meson that the gravitational interaction of antimatter can not possibly deviate from that of matter. However, Good?s argument is criticized for using absolute potentials, and Chardin and Rax showed that, when using relative potentials, ?CP violation in the kaon system may be explained by antigravity?, cf. [3]. In short, the arguments of Morrison, Schiff and Good have been refuted. The first main theoretical argument that has thus far not been refuted is that the principle of equivalence in GR equates gravitational mass mg and inertial mass mi : of the latter it has experimentally been established that it has a positive value for both matter and antimatter. In GR, inertial mass is not just identical to rest mass m0 , but depends on the momenta in x-, y-, and z-direction according to mi = m20 + p2x + p2y + p2z . Thus, in the framework of GR the following relation always holds for any particle having rest mass: mg ? m0 > 0 (1) The second main theoretical impossibility argument, still standing, is that TCP-invariance of the Standard Model, which is based on Quantum Mechanics (QM), predicts the same mass for a particle and its antimatter counterpart. Thus, in research programs aimed at extending the Standard Model with a quantum theory of gravitation there is no reason to believe that the gravitational mass mg,p? of an antiproton p? would have the opposite sign of the gravitational mass mg,p of a normal proton p. In other words, it is assumed that mg,p? = mg,p > 0 (2) Obviously, (1) and (2) exclude the possibility that, for example, antiprotons have negative gravitational mass ? a situation which would occur if antiprotons would be repulsed by the gravitational field of the earth. The expressions (1) and (2), however, are assumptions and no 100% guarantees: the coupling between antimatter and gravitation, namely, has thus far not been established experimentally ? direct measurements on, for example, antiprotons are extremely difficult if not downright impossible because of the much larger electromagnetic couplings; the production and manipulation of neutral antimatter, in particular anti-hydrogen, requires further perfectioning before a measurement of the gravitational coupling is possible. The crux is thus that none of the theoretical impossibility arguments is decisive, which is also mentioned as a conclusion in [4]: the current state of affairs is, thus, that in virtually all fundamental research programs in physics it is assumed that gravitation is attraction only, but this is not known. In addition, the cornerstones of modern physics, that is, QM and GR, correspond with mutually exclusive world views; for an extensive discussion on the differences of QM and GR see the literature, e.g. [9]. Up untill the present day the differences between QM and GR have never been resolved, that is, the post-World War II research programs in physics have not produced an advancement of knowledge about the fundamental workings of the universe that has brought a unification of GR and QM any closer: it seems that either GR is wrong, or QM is wrong, or both GR and QM are wrong. Given that there is no evidence at all that GR or QM are applicable to the supersmall scale, cf. [10?12], it is currently thus absolutely not excluded that the actual interactions causing gravitational and quantum effects occur at a much smaller scale than the areas of application of GR and QM, so that GR and QM are not fundamental but merely emergent in their area of application. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 701 Now if a negative coupling between antimatter and gravitation would be established ? and suggestions for experimental research in this direction have been published, cf. [13] ? then the following relation holds between rest mass m0 and gravitational mass mg in case of antimatter (e.g. antiprotons): ?mg ? m0 > 0 (3) From the contradiction of (3) with (1) and (2) it is thus clear that fundamental research programs in modern physics will experience great difficulties in incorporating the experimental results if it would be established that antimatter is repulsed by the gravitational field of the earth: contemporary theories of modern physics are then clearly falsified as the true foundations of physics, and a new framework for physics would then be required. The relevance of the present research question to science is thus that identifying fundamental principles underlying a gravitational repulsion of matter and antimatter could lead to an improvement of the fundamental description of any observed process in terms of elementary constituents and individual processes: one ought to realize that the world functions entirely different than currently thought if antimatter were repulsed by the gravitational field of the earth. The outline of this article is as follows. The remainder of this general introduction, which is intended for a broad audience, presents the main results (Sect. 1.2), outstanding open issues (Sect. 1.3) and main conclusions (Sect. 1.4) in non-technical language. Section 2 presents the key steps in the development of the main result, the Elementary Process Theory. The main body of the article is dedicated to detailed technical descriptions (Sect. 3) and discussions (Sect. 4) of the results. 1.2 Main results, application to real world problems and relevance to technology In the course of this investigation it became clear that new physical concepts (primitive notions) had to be introduced and that new physical principles had to be formulated mathematically to describe gravitational repulsion. The axiomatic method was applied, which in the present case meant that primitive notions were formalized in mathematical language but without reference to any concrete mathematical structure, that derived notions were defined in terms of primitive notions using a monoid structure, that axioms were formulated as well-formed closed expressions in mathematical language using the newly defined formalism, and that interpretation rules were defined which translate the formal axioms into elementary physical principles underlying the hypothesis. It turned out that generalized principles could be formulated; the resulting scheme of seven elementary principles is called the Elementary Process Theory (EPT). The whole construct together forms a formal axiomatic system, all theorems of which are statements about physical reality ? as a side note, it is also a ?closed system? as meant by Heisenberg, cf. [14]. Mathematically, the axioms of the EPT together determine a universe that evolves in a nonlinear way, and the physical interpretation then yields a fundamental understanding of the individual processes that take place at supersmall scale in the universe governed by the EPT: it is clear of what kinds of components this universe consists at supersmall scale, and it is clear how new components are formed from existing ones by discrete transitions in individual processes. To this is attached a notion of physical completeness: the EPT describes the creation of every component of the universe ? there are no other individual processes then those described by the EPT. The EPT thus finds itself in the realm of process philosophy, i.e. the view that physical reality is best understood as a process; historically, the Greek philosopher Heraclitus of Ephesus (▒ 550 - 480 B.C.) was the first to use this approach. A nontrivial formal aspect is that the EPT is formalized in an abstract mathematical-logical framework and not in a concrete set-theoretical domain. As a result, individual constants of the EPT referring to constituents of physical reality have as value ?a set?, without that set being specified. This enabled the application of the concept of a designator to physics, instead of the concept of a representation. To elaborate, consider that in a theorem of the EPT the constant ? occurs. Mathematically, the term ? is an abstract set, and by the explicitly given interpretation rules this set ? designates a constituent of the physical world. However, because the set ? is not further specified, it does not represent the state of the component in question. In other words: from the formalism it is clear to which component of the physical universe a www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 702 M. J. T. F. Cabbolet: Elementary Process Theory given designator refers, but the designator contains no further information such as position, momentum or mass of that component. Compare the term ?the queen of the Netherlands?: it is obvious (at least, in 2010) to whom this term refers, yet this term contains no further information such as position, bodyweight of the person designated. As a result, the EPT has a higher degree of abstractness than QM and GR, where representations are used. For example, the quantum-mechanical wave function of an electron is a representation of its quantum state: it enables the calculation of the expectation values of momentum and position. This degree of abstractness adds a feature of generality to the EPT: the principles of the EPT apply to the components involved regardless of their position, mass, momentum, etc. In addition, at this degree of abstractness the principles of the EPT are of great simplicity: the workings of the universe would thus be a lot simpler if gravitation would not be attraction only. The universe of the EPT then consists of a world and an antiworld, and components of this universe are simultaneously a constituent of the world and a constituent of the antiworld; the latter is what would be observed of this component by a (hypothetical) observer in opposite time-direction. For comparison, Feynman?s interpretation of a positron is, that a positron e+ is an electron e? traveling backwards in time [15]. The universe of the EPT is then an isolated, heterogeneous system, whose different phases consist of five kinds of ultimate constituents called ?phase quanta?. The latter are forms of energy: rest mass then relates to the amount of energy contained in a particlelike phase quantum, and gravitational mass to the amount of energy contained in a wavelike phase quantum. Motion of nonzero rest mass entities like electrons, positrons, (anti)protons, and (anti)neutrons is stepwise in this universe: nonzero rest mass entities, located motionless at a certain position in the state of a particlelike phase quantum, undergo a discrete state-transition to move in the state of a wavelike phase quantum, and become located at a new position in the state of a particlelike phase quantum after another series of discrete state-transitions. The main characteristic of the universe of the EPT is that for all interactions the individual processes involved are essentially the same ? the EPT thus brings about unification of individual processes. In one such individual process, a nonzero rest mass entity only once absorbs energy from its surroundings, and only once emits energy to its surroundings: the framework of the EPT is thus consistent with the idea that there really is only one kind of interaction ? there is no such thing as an electromagnetic interaction or a gravitational interaction, there are only electromagnetic and gravitational aspects of a single cosmic interaction. In addition, the universe, governed by the EPT, is neither deterministic nor probabilistic: the EPT contains an elementary principle of choice, which enables the universe of the EPT to be endowed with volition. On the basis of the EPT it can be explained how antimatter particles such as antiprotons can be repulsed by the gravitational field of the earth: the general picture is then that matter attracts matter, antimatter attracts antimatter, and matter and antimatter repulse each other. If this would be observed experimentally then the contemporary foundations of physics, GR and QM, would not be applicable to explain the observations. Furthermore, a variety of observed processes has been formalized in the framework of the EPT: it is demonstrated that the motion of an electron orbiting an atomic nucleus, the motion of a neutron gravitating towards earth, the decay of a neutron, the formation of deuterium and the annihilation of a proton/antiproton pair can be described mathematically with the same scheme of elementary principles ? the EPT. Currently, there is no other framework in physics in which all these processes can be formalized. In addition, the EPT is applied to a theory of the Planck era of the universe. This is a major open problem in physical cosmology to which neither GR nor QM applies, that is, for which no solution has been formulated on the basis of GR or QM; the theory based on the EPT offers a new perspective regarding the so-called Horizon Problem. The EPT is critically confronted with GR and QM: contradiction of QM and GR with the EPT is proved mathematically. In the context of two consecutive measurements, whereby an electron is observed first at position xa and next at position xb , the radical difference between the EPT, QM and GR comes to expression in the difference between the three views on how the electron has got from xa to xb . According to GR, continuous motion has taken place. The electron was observed as a particle at position xa , then moved as a particle on a continuous trajectory from xa to xb , where it was again observed as a particle. So c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 703 Fig. 1 Illustration of the classical concept of continuous motion of GR (a) and the accepted concept of wave motion in QM (b). In both figures the two points xa = (xa , ya ) and xb = (xb , yb ) where the electron was observed are depicted in an xy-plane. Figure 1a shows the trajectory x(t) along which the electron has moved as a particle according to GR. Figure 1b shows three regions U90% (ti ) for intermediate points in time ti with ta < t1 < t2 < t3 < tb . Each region U90% (ti ) represents an area U with a probability of p(U ) = 90% of finding the electron within that area at that time ti according to QM. according to GR, in between the two measurements the electron was at every point of time in the state of a particle with a definite position. See Fig. 1a for an illustration. According to QM, wave motion has taken place. It is true that the electron was observed as a particle at positions xa and xb , but in between the measurements the electron wasn?t really anywhere: instead, at every intermediate time t the electron had in every region U of the whole space X a possibly nonzero probability p(U ) of being found there; for a region U this real valued probability p(U ) can then be calculated using the wave function ?t of the electron at the time t and the formula p(U ) = ?t (x)?t? (x)dx (4) U where x is a position vector, ?t (x) the complex valued image of x under ?t , and ?t? (x) its complex conjugate. According to QM there is no point in asking what the electron looked like in between the measurements: one just has to trust in the mathematics with which this kind of probabilities can be calculated. See Fig. 1b for an illustration. According to the EPT stepwise motion has taken place. The electron was in the state of a particlelike phase quantum at the positions xa and xb . In between the measurements the electron was finitely many times in such a particlelike state at a definite position and moved in the state of a wavelike phase quantum from one particlelike state to the next one. See Fig. 2 for an illustration. Thus, in short, according to GR the electron had a definite position all the time in between the measurements; according to QM the electron had no definite position in between the measurements; and according to the EPT the electron had finitely many times a definite position in between the measurements. Positive results of the research program based on the EPT thus may render the research programs based on QM or GR degenerating, that is, may lead to the rejection of GR and QM. In addition, experimental verification of a gravitational repulsion of matter and antimatter would have possibly far-reaching consequences for technology: a repulsive aspect of gravitation would namely be applicable to the generation of a vertical displacement without fossile fuel. That offers perspectives for future technologies for the conversion of energy, but of course in this early stage no statement can be made about its efficiency. 1.3 Outstanding open issues The crux is that there is currently insufficient proof that the seven elementary principles of the EPT indeed correspond with physical reality. But that does not imply that further research can not yield sufficient proof: www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 704 M. J. T. F. Cabbolet: Elementary Process Theory Fig. 2 Illustration of the new concept of stepwise motion of the EPT. In the xy-plane the five positions xa , x1 , x2 , x3 , and xb , where the electron exists in a particlelike state, are shown, and three regions U90% (i), U90% (ii) and U90% (iii). These three regions represent the area?s at three intermediate points of time, for which t2 < ti < tii < tiii < t3 , where 90% of the energy is concentrated in internal wave states of the wavelike phase quantum, that effects the motion from the particlelike phase quantum at x2 to that at x3 . although the supersmall scale is not directly experimentally accessible in a laboratory, the scientific method still applies. A concrete mathematical model of physical reality, based on the EPT, can namely be used to predict how a physical system, governed by the principles of action of the EPT, will behave in the long run. In a research program, aimed at showing the correspondence between the EPT and physical reality in this way, three key problems ? all still open ? have to be solved; these will be discussed below. The first major open issue is that the EPT has currently no concrete mathematical model: the terms of the EPT are defined without reference to any concrete mathematical structure. The first objective of further research is therefore to develop a concrete mathematical model M of the EPT, that is, an interpretation of the terms of the EPT in a concretely defined set-theoretical structure (such as the space of all functions from a well-defined position space X to some field F ), such that the following first-order conditions are satisfied: (i) M |= Ai for any of the seven axioms A1 , . . . , A7 being the translation of the seven axioms of the EPT in terms of the model M ; (ii) M |= Pj for any of n formulas P1 , . . . , Pn being a formulation of n empirical premises in terms of the model M ; (iii) M |= H for a formula H, being a formulation of the hypothesis of Sect. 1.1 in terms of the model M. This is a non-standard mathematical-physical problem; the notion of a model M of a theory T is wellknown, cf. [16]. Note that such a model M is necessarily inconsistent with GR because of (iii). From logical consistency of the EPT, consistency of the EPT with experiment, and the completeness theorem of first-order logic it follows that such a mathematical model M of the EPT must exist, but that does not necessarily imply that such a model is easy to find. More precisely, it is questionable whether there are enough experimental data (to be translated in empirical premises Pj ) to develop a model covering the weak and strong interactions. The intention is therefore to first develop a mathematical model of a single long-distance interaction having gravitational and electromagnetic aspects. The second major open issue is that there is currently no mathematical proof that QM and GR are approximations of the EPT in their respective areas of application. A direct such proof is mathematically impossible, because the EPT is defined in an abstract mathematical setting, while QM and GR on the other hand are each formulated in a concrete set-theoretical domain. The proof in question has to be done in two steps. The first step is to develop a concrete mathematical model M of the EPT, as outlined above: this yields an interpretation of the abstract expressions of the EPT in a concrete set-theoretical domain. The second step is then to prove that the mathematical model M of the EPT adheres to the principle of correspondence, that is, that the mathematical expressions of QM and GR are approximately true in the mathematical model M of the EPT in their area of applicability. This too is a non-standard mathematicalphysical problem. The third open issue is that experimental facts have to be produced which can not or not easily be incorporated in the research programs based on GR or QM. This boils down to experimentally verifying c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 705 predictions which have to be derived from a mathematical model of the EPT by means of formal deduction. For that matter, an interesting experiment is the measurement of the coupling of antimatter with gravitation: if it would be established that this coupling is negative, that is, if it would be established that antimatter is repulsed by the gravitational field of the earth, then the currently mainstream research programs will find themselves in trouble ? if, on the other hand, it would be established that antimatter is attracted to earth, then there is no compelling reason to continue research in the direction of the EPT. Another interesting option is to give a quantitative, physical explanation of the observed expansion of the universe; currently this can be described by manipulating the cosmological constant, but without giving an underlying cause of the change. An idea in this direction is presented in Remark 4.3.2.1. 1.4 Conclusions The main conclusion is that an answer to the research question, which fundamental laws of physics might govern the universe if gravitational repulsion exists, has been formulated mathematically. The EPT, a scheme of elementary principles governing the universe at supersmall scale, has been rigorously formalized in a newly defined mathematical-logical framework, and it has been demonstrated that a repulsion of antimatter by the gravitational field of the earth can be explained on the basis of the elementary principles of the EPT. In addition, it has been shown that the EPT adheres to the general principle of relativity, that is, that the elementary principles of the EPT are the same for all observers. A second conclusion is that the EPT determines a fundamentally new research field. The elementary principles of the EPT are not formulated on the basis of QM nor on the basis of GR: the EPT is fundamentally different from QM and GR. It is not only the case that the EPT is formalized in an entirely different mathematical setting than QM and GR, but also the world view of the EPT differs from the world views of GR and QM: the EPT doesn?t use the classical notions of particles and fields (applied in GR) nor the notions of wave functions and operators (applied in QM), but instead uses the primitive notion of a phase quantum; in addition, while GR corresponds to a concept of continuous motion and QM to a concept of wave motion, the EPT corresponds to a concept of stepwise motion. In short, the EPT can not be incorporated in the current research programs based on QM and/or GR. A third conclusion is that the EPT currently has the status of a protoscientific theory. That is, it is true that the EPT is speculative, but it nevertheless satifies traditional criteria of quality of conceptual unitarity, logical consistency, mathematical rigor, and physical completeness, and in addition is testable according to the scientific method of Lakatos (refined falsificationism). That means that further research in this direction may not lead to an improved understanding of physical reality, but it may also lead to the establishment of the EPT as a scientific theory. For that matter, the research program based on the EPT must be both theoretically and empirically progressive compared to the research programs based on GR or QM. Currently the EPT has been applied to solve the Planck era of the universe and to formalize a wide range of observed processes which can not be formalized in the framework of any other theory, but that is not enough to render the research programs based on GR and QM degenerative: for that matter, further results, both theoretical and experimental, have to be produced. The bottom line is that a completely formalized framework for physics, containing fundamental laws of nature governing the supersmall scale, has been put forward and that this gives rise to a potentially progressive research program. Further research in this direction may yield a proof that QM and GR are not applicable to the supersmall scale and thus are not fundamental ? such further research is then recommended: the intention is, after all, to find the truth, and nothing but the truth. 2 Method The EPT has been developed in a dialectical process; in the last phase the axiomatic method has been applied. In this section the main steps in the development of the EPT are given, but the subject is not treated exhaustively: the position is taken that the merit of the EPT ? like any other theory ? is to be found www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 706 M. J. T. F. Cabbolet: Elementary Process Theory in results obtained by testing its correspondence to reality, and not in an endless elaboration on all the thinking steps that led to the EPT ? all the more because the notion of proof does not apply since the EPT has not been deduced by means of formal deduction. In fairly recent literature a repulsion of antimatter moving in the gravitational field of matter has been predicted, cf. [17], but a classical approach was used: this was not accepted, because nonclassical behavior of electrons has already been established ? positrons, the antimatter counterparts of electrons, then also exhibit such behavior. As a solution towards laws of physics which might underlie (3), the case was considered that rest mass m0 and gravitational mass mg are characteristics of different physical states: then, namely, m0 and mg do not necessarily have to be the same in sign or in absolute value. Having taken into account that motion of electrons is proven to have wavelike aspects, this led to the development of a concept of stepwise motion, according to which nonzero rest mass entities (protons, electrons, neutrons, their antimatter counterparts, etc.) move from one particlelike state ? which is absolutely at rest ? to a next in a wavelike state. The particlelike state being absolutely devoid of motion necessitated a departure from the classical concept of a particle, which is an object whose dimensions are neglectable compared to its motion. It is mentioned that the idea for stepwise motion has been suggested already in 1937 by Van Dantzig, who wrote: ... matter could be considered as discontinuous in time as well as in space. Let us see to what consequences this would lead. Using the usual illustration in spacetime, a particle would not be represented by a curve (worldline) but by a sequence of world-points, which will be called ?flashes? [18]. This ?more or less vague suggestion? of Van Dantzig (as he called it himself) was, however, never developed further to a mathematical representation. In the present case, rest mass m0 then is a characteristic of the particlelike state, and gravitational mass mg a characteristic of the wavelike state. In the end, this turned out to give the following relation between rest mass m0 and gravitational mass mg : |mg | ? m0 > 0 (5) The special case (3) is then consistent with the general case (5). The research was then focussed on identifying principles according to which nonzero rest mass entities would transform from a particlelike state to a wavelike state, and back into a similar particlelike state. However, it became clear that such principles could not be formulated in the framework of QM. In orthodox QM, namely, the quantum state of a nonzero rest mass entity ? say, an electron ? is represented by a wave function, but it is not the case that the electron in question is a wave. In the present case, however, the electron is a wave, at least temporarily, in the process of stepwise motion. It has got into a wavelike form by a discrete transition, which is certain to happen regardless whether one is observing the electron or not. That is, the actual state of the wavelike form may be influenced by the observation, but the discrete transition, by which the electron transforms from a particlelike into a wavelike form, takes place independent of observation. Because such a transition is discrete and not continuous, it can not be described by the Schro?dinger equation; and because the transition does not necessarily require a measurement it would be inappropriate to describe it as a discontinuous collapse into a definite state upon a measurement. Thus, continuing towards principles governing such discrete transitions would necessitate a departure from orthodox QM. It therefore turned out that principles governing gravitational repulsion could not be described in terms of wave functions and operators: it was necessary to introduce some new concepts, in particular the notion of a ?phase quantum?. It is, however, the case that a mathematical model of the supersmall scale has to be nonlocal in order to do the same predictions as QM on subatomic scale, as was shown by Bell in [19]. As a solution towards laws of physics that are in accordance with this result, the case was considered that the wavelike states, in which nonzero rest mass entities move from one particlelike state to the next, are nonlocal, that is, have their spatial extension instantaneously. In addition, it was considered that in the ontological interpretation of QM by Bohm all particles are accompanied by a wave which guides the particles? motion, cf. [20] and [21]. It is, however, not the case c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 707 that the wave function of an electron is interpreted as a real wave such that the electron is the wave at any point: it is merely the case that the electron moves on a continuous trajectory governed by the quantum potential, an object that is ontologically a different object than the electron and that derives from the wave function of the electron. Thus, because in the present case the electron is a wavelike object during its motion, and because motion is stepwise and not continuous, any fundamental principles underlying this stepwise motion clearly had to be formulated outside the framework of Bohmian QM. Furthermore, the observation was taken into account that photons are deflected by the gravitational field of the sun, cf. [22]. Assuming the hypothesis mentioned in Sect. 1.1, photons then would have to be an entirely different kind of matter than protons, because photons are identical to antiphotons: it can not be the case that one and the same photon is both attracted and repulsed by the gravitational field of the sun. The aforementioned observation was therefore not interpreted as a proof that photons are attracted by the gravitational field of the sun, but merely as a proof that the geometry of the vacuum is non-Euclidean. Given that photons travel with the speed of light and are emitted from a source, as a solution towards laws of physics that are in accordance with this observation it was considered that in the process of stepwise motion of a nonzero rest mass entity first a discrete transition occurs from a particlelike state at a definite position (characterized by rest mass) to a nonlocal wavelike state (characterized by gravitational mass), that next a discrete transition (collapse) occurs from the nonlocal wavelike state to a point-particlelike state at a next definite position, and that from there then a photon is emitted as a local wavelike phenomenon: the photon is then a different form of matter and has neither rest mass nor gravitational mass. Moreover, the observation was taken into account that the universe is expanding, cf. [23]. Now the concept of stepwise motion allows the subsequent rest masses of a proton to form a (strictly) decreasing sequence: if energy is to be conserved, then in every step the energy corresponding with a loss of rest mass would thus have to be emitted alongside a photon after the nonlocal wavelike state has collapsed as mentioned in the foregoing paragraph. Towards laws of physics that could in principle explain why the universe is expanding, it was considered that the energy emitted from the point-particlelike states led to the formation of space; a gradual decrease of rest mass of protons might then be the root cause of the expansion of the universe. It was also considered that in the modified quantum theory of Ghirardi, Rimini, and Weber it is postulated that microscopic systems are subjected to spontaneous localization processes at random times [24]. In the present case, however, stepwise motion occurs as the result of a series of different transitions, all of which are certain to happen: the chain of transitions together yields a physical process which is quite different from the localization processes suggested by this GRW quantum theory. Thus, any fundamental principles underlying the current concept of stepwise motion would also lie completely outside the paradigm of GRW quantum theory. 3 Results 3.1 Elementary Process Theory The EPT is contained in a formal axiomatic system, and is formalized within the mathematical-logical framework of set matrix theory. This framework is quite similar to the framework of Zermelo-Fraenkel set theory: it uses the same logical connectives, quantifiers, and predicate symbols. The difference is that in the framework of set matrix theory terms of the language are not necessarily sets: in the framework of set matrix theory besides sets also matrices with set-valued entries occur; these set matrices are in general not sets but can be elements of sets. The main features of the framework of set matrix theory are given in the Appendix; for a rigorous introduction see [25]. The essential part of the formal language for the EPT are the individual constants: these are the terms, specific for the EPT, which can occur in well-formed atomic formulas like t1 = t2 and t1 ? t2 . The remainder of the vocabulary is taken from the mathematical-logical framework within which the EPT is formalized. Thus, the (predicate) symbols ?=? and ???, the negation sign ?г?, the connectives ???, ???, ???, and ??? as well as the quantifiers ??? and ??? are the ones given www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 708 M. J. T. F. Cabbolet: Elementary Process Theory by the mathematical-logical framework of set matrix theory. In addition, the syntax, the set of rules on how to construct well-formed formulas using all these symbols, is also given by the mathematical-logical framework. To introduce the EPT, it thus suffices to introduce the individual constants of the EPT, the axioms that constitute the EPT, and the interpretation rules, by which theorems of the formal axiomatic system can be translated into statements about physical reality. The individual constants of the EPT are introduced in Sect. 3.1.1, the interpretation rules for the individual constants are given in Sect. 3.1.2; the axioms for the EPT together with some additional interpretation rules are presented in Sect. 3.1.3. In the remainder of the text, the symbol ?2? marks the end of a definition, axiom, interpretation rule, etc. where useful. 3.1.1 The individual constants of the EPT Definition 3.1.1.1. The individual constants of the language for the EPT are the following: (i) the infinite set of positive integers Z + = {1, 2, 3, . . .} (ii) the finite abelian group [ZN +] under addition modulo N , with ZN = {0, 1, 2, ., N ? 1} and (N ? 1) + 1 = 0 (iii) for every x ? ZN a section S?(x) = {k ? Z + |k < ?(x) + 1} ? Z + (iv) the nonabelian group [P ?], being the group of permutations on Z + (v) the commutative monoid [M +], with the following finite set G of generators: EP x NW x ?k ?k ?kx , ? one 2 О 1 set matrix EP x , one 2 О 1 set matrix N W x , one 2 О 1 set matrix ??k ??k ??kx N P x+1 LW x+1 ? ? one 2 О 1 set matrix N P kx+1 , and one 2 О 1 set matrix LW kx+1 for every x ? ZN ??k ??k and for every k ? S?(x) EP x EP x ?i ?k ? p(x, k) 2 О 1 set matrices EP x for every 2 О 1 set matrix EP x ??i ??k N P x+1 N P x+1 ?j ?k ? q(x, k) 2 О 1 set matrices N P x+1 for every 2 О 1 set matrix N P x+1 ??k ??j S x+2 ? ? and one 2 О 1 set matrix S kx+2 for every x ? ZN and for every k ? S?(x) ??k (vi) a subset ME of M , given by ??(? ? ME ? ? ? M ? ?E (?)) where ?E is a mathematical formalization of the unary predicate letter E (Exists) on M . (vii) for every set N P ?xk a set ?xk for which N P ?xk ? ?xk (viii) a function fC so that fC (?xk ) = NP ?xk for all ?xk 2 t1 t4 t2 t5 ? ? t3 t6 In addition, an agreement on notation is made for expressions of the type : , t1 t2 t3 these denote an expression , , ? R. For example, in the elementary principle t4 t5 t6 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 709 of nonlocal mediation 3.1.3.8 the subformula NW x EP x N P x+1 ?k ?k ?k , EP x , N P x+1 ? R. NW x ??k ??k ??k NW ?xk NW x ??k : EP ?xk EP x ??k ? ? ?x+1 k N P x+1 ??k NP stands for 3.1.2 Interpretation rules for the individual constants of the EPT The formulas that can be deduced within the axiomatic system containing the EPT are in themselves just mathematical expressions with no physical meaning. Thus, to convert the theorems of the system into statements about the physical universe a set of interpretation rules is given. The interpretation rules that concern the individual constants of the language for the EPT are given below in Table 1. These interpretation rules make use of primitive notions, which are described in a number of remarks. Additional interpretation rules that concern the axioms of the EPT are given in Sect. 3.3. Table 1 Interpretation rules for individual constants of the EPT; a superscript i refers to a Remark 3.1.2.i below. Symbol n k EP ?xk EP x ??k EP ?xi EP x ??i NW ?xk NW x ??k ?kx ??kx component of the universe, consisting of the extended particlelike phase quantum EP ?xk occurring in the world at the xth degree of evolution in the k th individual process from the xth to the (x + 1)th degree of evolution, and the conjugated extended particlelike phase quantum EP ??xk occurring in the antiworld4,10 EP x ?k subcomponent of a component EP x , consisting of the extended particlelike matter ??k quantum EP ?xi at the xth degree of evolution in the world, to be attributed to the ith monad, and the conjugated extended particlelike matter quantum EP ??xi at the xth degree of evolution in the antiworld, to be attributed to the ith monad5,6 component of the universe, consisting of the nonlocal wavelike phase quantum N W ?xk occurring in the world at the xth degree of evolution in the k th individual process from the xth to the (x + 1)th degree of evolution, and the conjugated nonlocal wavelike phase quantum N W ??xk occurring in the antiworld4,7,10 ?x+1 k N P x+1 ??k NP Interpretation degree of evolution1 (for n ? ZN ) numerical labels of the individual processes from the xth to the (x + 1)th degree of evolution2,3 (for k ? S?(x) ) component of the universe, consisting of the binad ?kx occurring in the world at the xth degree of evolution in the k th individual process from the xth to the (x + 1)th degree of evolution, and the necessarily conjugated binad ??kx occurring in the antiworld, cf. Example 3.1.3.19 component of the universe, consisting of the nonextended particlelike phase quantum N P x+1 ?k occurring in the world at the (x + 1)th degree of evolution in the k th individual process from the xth to the (x + 1)th degree of evolution, and the necessarily conjugated occurring in the antiworld4,10 nonextended particlelike phase quantum N P ??x+1 k www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 710 M. J. T. F. Cabbolet: Elementary Process Theory Table 1 (Continued). Symbol N P x+1 ?j N P x+1 ??j ?x+1 k LW x+1 ??k LW S x+2 ?k S x+2 ??k 0 0 Interpretation ?x+1 k subcomponent of a component N P x+1 , consisting of the nonextended ??k NP x ?j preceding the j th monad at the (x + 1)th particlelike matter quantum degree of evolution in the world, and the conjugated nonextended particlelike matter quantum N P ??xj at the (x + 1)th degree of evolution in the antiworld5 ME ?x+1 k NP component of the universe, consisting of the local wavelike phase quantum LW x+1 ?k occurring in the world at the (x + 1)th degree of evolution in the th k individual process from the xth to the (x + 1)th degree of evolution, occurring in the and the conjugated local wavelike phase quantum LW ??x+1 k antiworld4,8,10 g1 g?1 + ... + gn g?n component of the observable universe, consisting of the spatial phase quantum S x+2 ?k occurring in the world at the (x + 2)th degree of evolution after the k th individual process from the xth to the (x + 1)th degree of evolution, and the occurring in the antiworld4,9,10 conjugated spatial phase quantum S ??x+2 k 0 physical emptiness ( ? M) 0 component of the universe that is thesuperposition of these n subcomponents gi g1 gn , for any ,..., ?G g?i g?1 g? n x x existence predicate; for any ? M an expression ? ME thus x? x? x means that the component exists in the universe x? the set of parallel possible nonextended particlelike phase quanta at the (x + 1)th degree of evolution in the k th individual process from the xth to the (x + 1)th degree of evolution cf. Example 3.1.3.25. Thus, to elaborate on the formalism for sets designating constituents of world or antiworld: a Greek letter ? indicates that the constituent designated is a phase quantum and a Greek letter ? indicates that the constituent designated is a matter quantum; the left superscript indicates the type of phase quantum or matter quantum designated; the right superscript refers to the degree of evolution at which the constituent is formed; the right subscript refers to the individual process in which the constituent is formed; a bar over the symbol ? or ? indicates that a constituent of the antiworld is designated ? absence of such a bar thus automatically implies that the constituent designated is a constituent of the world. Remark 3.1.2.1. (degrees of evolution): The degrees of evolution are essential characteristics of the observable process of evolution. A degree of evolution is thus different from a moment in time, because a moment in time is a point on a linear continuum. It should be noted that the invariance of counting implies that all observers will find a structure isomorphic to [ZN +]: a degree of evolution n ? ZN is therefore an absolute value, that is, the same for all observers. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 711 Remark 3.1.2.2. (individual processes): In the universe governed by the EPT, the observable process of evolution is composed of many individual processes simultaneousy; to quote Von Neumann: ?man generally perceives the sum of many billions of elementary processes simultaneously, so that the leveling law of large numbers completely obscures the real nature of the individual processes [26]?. The term ?individual process? is used in its intuitive sense. At any degree of evolution x, there are ?(x) of such individual processes. Remark 3.1.2.3. (labels of individual processes):The numbering of individual processes is not necessarily the same for all observers: there are ?(x) individual processes from the xth to the (x + 1)th degree of evolution, but every observer is free to use any number k ? S?(x) to refer to any particular individual process from the xth to (x + 1)th degree of evolution. There is, however, always a permutation in the set P of permutations on Z + such that the numerical labels of one observer can be transformed into the numerical labels of another observer. In addition, all elementary principles of the EPT turn out to be of the form ?k ? S?(x) (?(k)), so the elementary principles are applicable to every individual process, regardless of the value of the variable k which an observer uses to refer to that particular individual process. The point is that the actual numerical value, assigned by an observer to the label k for the k th individual process from the xth to the (x + 1)th degree of evolution, is not important for the description of what happens during the individual process: it is merely the case that maintaining the same label throughout one such process is both necessary and sufficient. Remark 3.1.2.4. (phase quanta): A ?phase quantum? is the smallest possible amount of a phase, where the term ?phase? is used in its thermodynamic sense of a distinctive portion of matter in a system. To describe it further, a phase quantum is an elementary energetic constituent of the universe, in which an amount of energy (the product of a scalar measure and an energetic unit) is distributed over a certain spatial extension. The word ?quantum? is thus used in the sense of the smallest possible amount of something: it is not possible to produce less than a phase quantum. Loosely speaking, every phase quantum is a form of energy, with ?energy? used as a primitive notion. The idea is to model a phase quantum ? mathematically as an element of a normed vector space containing functions from the set of all positions X to the field F : ?= C и?и?и? (6) ? = |?| (7) Here ? ? F is a real number representing the amount of energy distributed in the phase quantum ?, ? ? F X is a characteristic function which has the value 1 on a position x ? X if and only if the position x is an element of the spatial extension of the phase quantum ? and the value 0 else, and ? ? F X represents the distribution of the energy over the spatial extension ??1 (1). C is a constant ensuring (7). Furthermore, it is always the case that the spatial extensions of the conjugated phase quanta in world and antiworld are one and the same. In fairly recent literature, the idea of a universe in which world and antiworld are spatially separated has been suggested by Sannikov in [27]. Remark 3.1.2.5. (particlelike matter quanta): Of every particlelike matter quantum, the spatial extension is bounded. In addition, it has to be taken matter quanta are static, that is, do not that particlelike EP x ? move at all. Furthermore, in a subcomponent EP kx it is always the case that the amount of energy ??k distributed in the matter quantum EP ?xk is positive, while the amount of energy distributed in the conjugated matter quantum EP ??xk is negative. The amount of energy distributed in the matter quantum EP ?xk is the rest mass of the mth monad at the xth degree of evolution. In case a particlelike matter quantum is nonextended, the bounded spatial extension is limited to a single point and has therefore an empty interior. The notion of a nonextended particlelike matter quantum thus bears some resemblance with the notion of a point-particle in classical mechanics. However, one should not confuse the notion of a particlelike matter quantum with the classical notion of a particle: the term ?particle? is used in classical mechanics for www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 712 M. J. T. F. Cabbolet: Elementary Process Theory material objects whose dimensions can be neglected in describing its motion, while the term ?particlelike matter quantum? is used in the EPT for material objects which are absolutely devoid of motion and whose dimensions may not be neglectable ? which might be the case at the supermall scale. Remark 3.1.2.6. (monads): The right subscript index m of conjugated matter quanta EP ?xm and EP ??xm does not refer to the mth individual process from the xth to the (x + 1)th degree of evolution, but to the mth monad. Such a monad can be an electron, a positron, a proton, etc. The term ?monad? is used here in the sense of an ultimate, indivisible, noumenal being (i.e. a being that is not a phenomenon); this meaning is not exactly the same as the meaning of the term ?monad? in Leibniz? monadology or in the work of others who applied it. The point here is that one and the same electron (or any other nonzero rest mass entity for that matter) has different states of being at different degrees of evolution: one has to distinguish between the state of being of the electron (a variable phenomenon) and the electron in itself (a constant noumenon). Thus, e.g. ?electron? is a proper name of an monad (noumenon), while EP ?xk is a definite description of a material object (phenomenon). Different observers can number the monads differently, but given any q numerical labels of monads there is always a permutation ? ? P such that these numerical labels are identical to ? (1), . . . , ? (q). Remark 3.1.2.7. (nonlocal wavelike phase quanta): A nonlocal wavelike phase quantum N W ?xk is a wavelike phenomenon of finite duration, in which an amount of energy is distributed changeably over a continuous spatial extension, while the internal time-direction in the conjugated phase quantum N W ??xk is opposite. Here time is a linear continuum, enabling the numbering of the internal wave states in the direction from earlier to later. A nonlocal wavelike phase quantum has its spatial extension instantaneously, so that the principle of locality, being that an object can not directly exert influence at a distance, does not hold at such an event: hence the adjective ?nonlocal? in the name of these phase quanta. Remark 3.1.2.8. (local wavelike phase quanta): A local wavelike phase quantum is a wave phenomenon of continuously changing spatial extension; the speed of change of the changing spatial extension is identical to the speed of light. One of the fundamental differences with nonlocal wavelike phase quanta is that the local wavelike phase quanta adhere to the principle of locality, that is, do not exert instantaneous influence at a distance. Remark 3.1.2.9. (spatial phase quanta): A spatial phase quantum is a constituent of the vacuum, contributing directly to its spaciousness. The spatial phase quanta are transcended by nonlocal wavelike phase quanta, and together these two kinds of phase quanta form a homogenous phase, which has a non-Euclidean geometry; this homogenous phase acts as a carrier for the local wavelike phase quanta. Remark 3.1.2.10. (invariance of the type of phase quanta): It is emphasized that different observers will assign the same kind of form to a phase quantum: this is an invariant feature. The total number, five, of kinds of phase quanta (extended particlelike, nonextended particlelike, local wavelike, nonlocal wavelike, and spatial) corresponds ? with respect to that number five ? to the philosophy of Aristotle, who contended that there are five elements of nature: earth, fire, air, water and aether (quinta essentia). 3.1.3 The axioms constituting the EPT Below the axioms that together constitute the EPT are introduced; some additional interpretation rules are given. The axioms are supplemented with a number of postulates: these postulates can not be formally deduced from the axioms, but are included in the physical meaning of these principles. To put it in other words: the axioms are synthetic propositions, while the postulates are analytic propositions. In addition to these axioms, the following holds for the finitely generated monoid [M +]: x x? , y y? ?M ? x x? c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim + y y? = x+y x? + y? (8) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 713 g g? Note that this thus implies that the set of all entries g and g? of all generators ? G also generates a monoid under addition; in fact, it is a group when g + g? = 0. Postulate 3.1.3.1. ?x ? ZN ?k ? S?(x) ?? ? P ?p ? Z + Postulate 3.1.3.2. ?x ? ZN ?k ? S?(x) ?? ? P ?q ? Z + EP ?xk EP x ??k ?x+1 k N P x+1 ??k NP EP EP = ?x?(1) + . . . + ?x?(p) EP ??x?(1) + . . . + EP ??x?(p) = NP NP ?x+1 ? (1) + . . . + ??x+1 ? (1) + . . . + NP NP ?x+1 ? (q) ??x+1 ? (q) Using Table 1, Postulate 3.1.3.1 thus means that an extended particlelike phase quantum EP ?xk in the world is always a superposition of finitely many particlelike matter quanta EP ?x?(i) . The distinguishability of extended particlelike phase quanta among the existing phases is thus a direct consequence of the locality and the boundedness of the spatial extension of the extended particlelike matter quanta. It is emphasized that if two or more extended particlelike matter quanta form an extended particlelike phase quantum, then these matter quanta are centered at different positions, see Fig. 3 for an illustration. Postulate 3.1.3.2 is in similar to Postulate 3.1.3.1 in that it means that every nonextended particlelike phase quantum N P ?x+1 k N P x+1 ?? (j) . the world is composed of finitely many nonextended particlelike matter quanta Fig. 3 Illustration of the idea of an extended particlelike phase quantum EP x ?k , composed of two extended particlelike matter quanta EP ?x?(1) and EP ?x?(2) ; these matter quanta can not be seen as separate phase quanta (consider the case of a deuterium nucleus, composed of a proton and a neutron, cf. Example 3.1.3.19). Postulate 3.1.3.3. for ? = EP, N P , or N W, LW, S, ? x ? x ?k ?k 0 ?x ? ZN ?k ? S?(x) ? ME ? ? x = ? x 0 ??k ??k x Postulate 3.1.3.4. for any component ?M x? x x ? ME + x? x? ?i Postulate 3.1.3.5. for any n different nonzero components of the type ??i ?n ?1 ?n ?1 ? ME ? . . . ? ? ME ? + ...+ ? ME ??1 ??n ??1 ??n ? x ?k ? x ??k , The Postulates 3.1.3.3, 3.1.3.4 and 3.1.3.5 concern the existence predicate. Postulate 3.1.3.3, which is a scheme of five formulas, just means that any existing component, consisting of a phase quantum in the world and a phase quantum in the antiworld, differs from physical emptiness. Postulate 3.1.3.4, which is a scheme of formulas, asserts that no component exists which is a superposition of a component and itself. www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 714 M. J. T. F. Cabbolet: Elementary Process Theory Postulate 3.1.3.5, which is a finite scheme, asserts that if each of n different components, consisting of phase quanta in world and antiworld, exist,then their superposition exists. t3 t1 t2 ? Before introducing the axioms, it is reiterated that expressions : are de? t?1 t?2 t?3 t1 t2 t3 fined as a relation between , and , see Sect. 3.1. That is, t?1 t?2 t?3 t2 ? t3 t1 t2 t3 t1 : ? , , ?R (9) t?1 t?2 t?3 t?1 t?2 t?3 ? for some set R ? M О M О M As a set, the ternary relation R on M is completely determined by the elementary principles 3.1.3.6, 3.1.3.8, 3.1.3.10 and 3.1.3.12 of the EPT: a separate definition of R is therefore omitted. Axiom 3.1.3.6. (Elementary of Nonlocal Equilibrium): Principle EP x NW x ?k ?k 0 ? : EP x ?x ? ZN ?k ? S?(x) NW x ??k ??k 0 ? Interpretation Rule 3.1.3.7. The elementary principle of nonlocal equilibrium means that at every th degree of evolution x and in every of evolution to the next, an equilibrium process fromthat x degree EP x NW x ? ? occurs between the components EP kx and N W kx , which is not mediated by any existing phys??k ??k ical object, but occurs spontaneously. That is, when the extended particlelike phase quantum EP ?xk exists in the world, spontaneously a discrete transition EP ?xk ? N W ?xk occurs in the world; this is accompanied by a discrete transition N W ??xk ? EP ??xk in the antiworld. In every nonlocal equilibrium the genuinely new substance N W ?xk is created in the world: every instance of the principle 3.1.3.6 thus corresponds with an event causation in the world ? the transition EP ?xk ? N W ?xk (an event) causes the existence of N W ?xk . Axiom 3.1.3.8. (Elementary Principle Mediation): of Nonlocal NW x EP x N P x+1 ?k ?k ?k ? ?x ? ZN ?k ? S?(x) : EP x N P x+1 NW x ? ??k ??k ??k Interpretation Rule 3.1.3.9. The elementary principle of nonlocal mediation means that at every degree of evolution x and in every process from that xth degree of evolution to the next, the component NW x EP x N P x+1 ?k ?k ?k mediates an equilibrium between the components EP x and N P x+1 . That is, NW x ??k ??k ??k NW x EP x N P x+1 the phase quantum ?k causes a discrete transition ?k ? ?k in the world, while the phase EP x ?? ? in the antiworld. Physically, in the world quantum N W ??xk causes a discrete transition N P ??x+1 k k the nonlocal wavelike phase quantum N W ?xk , which evolved from the extended particlelike phase quantum EP x ?k , collapses into a superposition of a finite number of nonextended particlelike matter quanta N P ?x+1 , j N P x+1 which together form the nonextended particlelike phase quantum ?k occurring in the world at the (x + 1)th degree of evolution in the k th individual process from the xth to (x + 1)th degree of evolution. The function of the phase quantum N W ?xk is thus that it causes the transition from the phase quantum EP x ?k to the phase quantum N P ?x+1 : every instance is thus an agent causation. 2 k NW x For any x ? ZN and any k ? S?(x) , in a nonlocal equilibrium the genuinely new substance ?k is formed from the extended particlelike phase quantum EP ?xk . Now the amount of energy distributed in the newly created phase quantum N W ?xk is always larger than or equal to the amount of energy distributed in the extended particlelike phase quantum EP ?xk from which it originated: E(N W ?xk ) ? E(EP ?xk ) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (10) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 715 Here E(.) is a real-valued function. In case E(N W ?xk ) > E(EP ?xk ), the excess of energy is absorbed from the surroundings (vacuum). Laws of conservation of energy are discussed below, and in Sect. 4.2.2. The relation of E(N W ?xk ) with gravitational mass is given in Sect. 4.3.2. By the collapse of the nonlocal wavelike phase quantum N W ?xk the genuinely new substance N P ?x+1 k is created in the world. Energy is conserved in this collapse: ) = E(N W ?xk ) E(N P ?x+1 k (11) The newly created nonextended particlelike phase quantum N P ?x+1 is composed of q nonextended partik N P x+1 N P x+1 ?? (1) , . . . , ?? (q) : if q > 1 then the nonlocal wavelike phase quantum N P ?x+1 clelike matter quanta k has collapsed into multiple nonextended particle-like matter quanta N P ?x+1 at different positions. In addi? (j) N P x+1 , . . . , ? tion, the q nonextended particlelike matter quanta N P ?x+1 ? (1) ? (q) have a different spatiotemporal EP x location than the p extended particlelike matter quanta ??(1) , . . . , EP ?x?(p) constituting the extended EP x particlelike phase quantum ?k , from which the nonlocal wavelike phase quantum N W ?xk originated. See Fig. 4 for an illustration of the elementary principle of nonlocal mediation, using the case that both ) are simple, that is, are composed of a single matter particlelike phase quanta involved (EP ?xk and N P ?x+1 k quantum (so p = q = 1 and ?(1) = ? (1) in this case). Fig. 4 Illustration of a nonlocal mediation, by which in the world the nonlocal wavelike phase quantum NW x ?k effects a discrete transition from the extended particlelike phase quantum EP ?xk , composed of the one matter quantum EP ?x?(1) , to the nonextended particlelike phase quantum NP ?x+1 , composed of the one k EP x NP x+1 matter quantum NP ?x+1 . The matter quanta ? and ? have different spatiotemporal positions. ? (1) ? (1) ? (1) NW ?xk NW x ??k Given Postulates 3.1.3.1 and 3.1.3.2, in a nonlocal mediation a component mediates an EP x EP x ? ? equilibrium between a component EP kx , constituted of p subcomponents EP ?(i) and a com??k ??x?(i) N P x+1 N P x+1 ?? (j) ?k ponent N P x+1 , constituted of q subcomponents N P x+1 . The numbers p and q need not be ??? (j) ??k identical: in case p = q a nuclear reaction takes place in the world ? all possible nuclear reactions are covered by the principle of nonlocal mediation. Examples where p = q will be given later on. If no nuclear reaction takes place, then p = q and {?(1), . . . , ?(p)} = {? (1), . . . , ? (q)}. Axiom 3.1.3.10. (Elementary Principle ofLocal Equilibrium): N P x+1 LW x+1 0 ? ? ?k : N P kx+1 ?x ? ZN ?k ? S?(x) LW x+1 0 ? ??k ??k Interpretation Rule 3.1.3.11. The elementary principle of nonlocal equilibrium means that at every degree of evolution x and in every process from that xth degree of evolution to the next, an equilibrium www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 716 M. J. T. F. Cabbolet: Elementary Process Theory LW x+1 ?x+1 ? occurs between the components N P kx+1 and LW kx+1 , which is not mediated by any exist??k ??k ing physical object, but occurs spontaneously. That is, when the nonextended particlelike phase quantum N P x+1 ?k exists in the world, spontaneously a discrete transition N P ?x+1 ? LW ?x+1 occurs in the world, k k LW x+1 ?k is emitted; this is accompanied by a which has to be taken that the local wavelike phase quantum ? N P ??x+1 in the antiworld. 2 discrete transition LW ??x+1 k k NP With the sole exception of the case x+ 1 = 0 in a local mediation the genuinely new substance LW ?x+1 k is created in the world: in all these cases the principle 3.1.3.10 thus corresponds with an event causation in the world, in which the discrete transition N P ?x+1 ? LW ?x+1 (an event) causes the existence of k k LW x+1 ?k . The spatial extension of every such new local wavelike phase quantum then spreads out gradually, that is, with the speed of light. Thus, the principle of locality does hold for such an event: hence the adjective ?local? in the name of these phase quanta. Axiom 3.1.3.12. (Elementary Principle of Local Mediation): ?x ? ZN ?k ? S?(x) ?? ? P ?q ? Z + LW x+1 N P x+1 ?k ?k ? : N P x+1 LW x+1 ? ??k ??k EP EP ?x+1 ? (1) + . . . + ??x+1 ? (1) + . . . + EP EP ?x+1 ? (q) ??x+1 ? (q) Interpretation Rule 3.1.3.13. The elementary principle of local mediation means that at every degree of evolution x and in every process from that xth degree of evolution to the next, the compo LW x+1 N P x+1 ?k ?k nent LW x+1 mediates an equilibrium between the component N P x+1 and the component ??k ??k EP x+1 EP x+1 ?? (1) + . . . + ?? (q) causes a discrete tran. That is, the local wavelike phase quantum LW ?x+1 k EP x+1 ??? (1) + . . . + EP ??x+1 ? (q) EP x+1 ? EP ?x+1 ?? (q) in the world, while the phase quantum LW ??x+1 causes a dissition N P ?x+1 k k ? (1) + . . . + EP x+1 EP x+1 N P x+1 ??? (q) ? ??k in the antiworld. The function of the phase quantum crete transition ??? (1) + . . .+ LW x+1 N P x+1 ?k is thus that it causes the transition from the phase quantum N P ?x+1 = N P ?x+1 ?? (q) k ? (1) +. . .+ EP x+1 EP x+1 to a superposition ?? (1) + . . . + ?? (q) of q extended particlelike matter quanta: every instance is thus an agent causation. 2 Definition 3.1.3.14. ?x ? ZN ?k ? S?(x+1) ?x+1 k N P x+1 ??k NP ? MA ? ?x+1 k LW x+1 ??k LW : ?x+1 k N P x+1 ??k NP ? ? 0 0 ?x+1 k of M that are elements of a subset MA of M ; the N P x+1 ??k designated components occur in annihilation reactions, cf. Example 3.2.1.8. The mediation in Definition 3.1.3.14 is to be called an annihilating mediation. NP This defines a property for generators ?x+1 k ? MA , by a local mediation the genuinely new substance ??x+1 k EP x+1 EP x+1 ?? (1) + . . . + EP ?x+1 ?? (i) ? (q) is created in the world. The q extended particlelike matter quanta N P x+1 then arise at the location of the q nonextended particlelike matter quanta ?? (i) that make up the phase . That is, contrary to the case of a nonlocal mediation, in a local mediation no disquantum N P ?x+1 k placement occurs. It should be noted that in a local mediation the right subscript indices, occurring in the N P x+1 superposition N P ?x+1 ?? (q) = N P ?x+1 , are conserved in the superposition of the matter k ? (1) + . . . + EP x+1 EP x+1 ?? (1) + . . . + ?? (q) : the same monads remain involved. See Fig. 5 for an illustration. quanta NP Except for the case that NP c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 717 Fig. 5 Illustration of what happens in the world by a local mediation; to the left the nonextended particlelike phase quantum NP ?x+1 , composed of one matter quantum NP ?x+1 k ? (1) , is shown: this precedes the local and the local wavelike phase mediation. To the right the extended particlelike matter quantum EP ?x+1 ? (1) LW x+1 quantum ?k , gradually spreading out in space, are shown: these exist a?fter the local mediation. It has to be taken that the two matter quanta (before and after) occupy the same position. Furthermore, the energy of the phase quantum N P ?x+1 is conserved in the emitted local wavelike phase k EP x+1 quantum LW ?x+1 and the remaining superposition of matter quanta EP ?x+1 ?? (q) : k ? (1) + . . . + ) = E(LW ?x+1 ) + E(EP ?x+1 E(N P ?x+1 k k ? (1) + . . . + EP ?x+1 ? (q) ) (12) Thus, in any individual process energy is first absorbed from the surroundings in the discrete transition ?xk ?N W ?xk , then energy is conserved upon the collapse N W ?xk ? N P ?x+1 , and energy is emitted k upon the discrete transition to the surroundings in the form of a local wavelike phase quantum LW ?x+1 k N P x+1 LW x+1 EP x+1 EP x+1 ?k ? ?k + ?? (1) + . . . + ?? (q) . This is the mechanism of any interaction. It is thus the case that Eq. (10) is always valid for the energy E(EP ?xk ) distributed in an extended particlelike phase quantum EP ?xk , and Eq. (11) is always valid for the energy E(N P ?x+1 ) distributed in the nonextended k N P x+1 ?k : these are therefore different kinds of particlelike phase quanta. particlelike phase quantum EP Postulate 3.1.3.15. ?x ? ZN ?j ? Z + ?l ? S?(x+1) ?? ? P ?p ? Z + ? EP ?xj EP x ??j = EP EP ?x?(1) ??x?(1) EP ?xj EP x ??j ? EP 0 0 = ?xl EP x ??l = EP EP ?x?(1) + . . . + ??x?(1) + . . . + EP EP ?x?(p) ??x?(p) Postulate 3.1.3.15 means that every physically nonzero matter quantum E(EP ?x+1 ), created by a local j mediation, occurs as a (possibly only) subcomponent of an extended particlelike phase quantum EP ?x+1 . l From there, the principle of nonlocal equilibrium applies and a new individual process starts (the lth individual process from the (x + 1)th to the (x + 2)th degree of evolution). It should be noted that the www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 718 M. J. T. F. Cabbolet: Elementary Process Theory superposition EP ?x+1 ? (1) + . . . + EP ?x+1 ? (q) , occurring in the elementary principle of local mediation, EP x+1 ?l does not necessarily form a single phase quantum EP x+1 for some l ? S?(x+1) . ??l It follows from the elementary principle of local mediation that a nonextended particlelike phase quanEP x+1 precedes the superposition EP ?x+1 ?? (q) in the process of evolution in the tum N P ?x+1 k ? (1) + . . . + world. In this context, it should be mentioned that in the period after Newton also other notions of entities preceding usual observable matter have been postulated, such as for example the notion of ?hylogeneous momenta? by Von Helmholtz, cf. [28] and the notion of ?prematter? by Sannikov, cf. [29]. Definition 3.1.3.16. Let for any sets X and Y the set X\Y be defined by EP ??x+1 ? (1) + . . . + EP ??x+1 ? (q) ??(? ? X\Y ? ? ? X ? ? ? Y ) (13) That is, X\Y is the set of all elements of X that do not occur in Y . Axiom 3.1.3.17. (Elementary Principle of Formation ofSpace): LW x+1 S x+2 ?k ?k ? ME ? S x+2 ? ME ?x ? ZN \{N ? 2}?k ? S?(x) LW x+1 ??k ??k With the sole exception of the case x + 2 = 0, by the formation of space, cf. Axiom 3.1.3.17, the is formed from the phase quantum LW ?x+1 , which has to be taken as the genuinely new substance S ?x+2 k k formation of space itself (as in ?the Firmament?). The elementary principle of formation of space 3.1.3.17 thus determines a law of succession for the world, instances of which can not be reduced to an event causation (it?s a continuing process). Axiom 3.1.3.18. (Elementary ofIdentity Principle of Binads): x EP x NW x ?k ?k ?k ?x ? ZN ?k ? S?(x) = EP x + N W x ??kx ??k ??k Using Table 1, the interpretation of this elementary principle 3.1.3.18 is straightforward. Having introduced the indivisible building blocks of the universe governed by the EPT, in the next example this elementary principle of identity of binads can be applied to relate the building blocks of the universe of the EPT to observed objects such as protons, electrons, etc. Example 3.1.3.19. All electrons, all positrons, all free neutrons, all free antineutrons, all free protons, and all free antiprotons occur as binads ?kx , of which the extended particlelike phase quantum EP ?xk is simple, that is, is composed of a single extended particlelike matter quantum EP ?xi . The adjective ?free? is used in the sense of not bound by the strong force, and not participating in a nuclear reaction. EP x ?k is A 10 5 B boron nucleus occurs as a binad, of which the extended particlelike phase quantum EP x 2 composite, in this case composed of 10 extended particlelike matter quanta ?i . A 1 D deuterium nucleus occurs as a binad, of which the extended particlelike phase quantum EP ?xk is composite and composed of two extended particlelike matter quanta EP ?xi . Theorem 3.1.3.20. (Principle duality of the EPT): of particle/wave EP x NW x ?k ?k ?kx ?x ? ZN ?k ? S?(x) ? ME ? EP x ? ME ? N W x ? ME ??kx ??k ??k The proof is omitted. 2 EP x ?k and the Theorem 3.1.3.20 asserts that without any exception both the particlelike component wavelike component N W ?xk of an existing binad ?kx exist: the term ?binad? therefore has to be seen in contrast to the term ?monad? in the sense that a binad is not absolutely simple, that is, not without parts. In addition, a binad is a phenomenon while a monad is a noumenon. Thus speaking, an electron orbiting x(i) a nucleus occurs in the world at consecutive degrees of evolution x(1), x(2), . . . , x(n) as a binad ?k(i) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) x(i) for which ?k(i) = EP x(i) EP x(i) ?k(i) + NW 719 x(i) ?k(i) = EP x(i) ?j + NW x(i) ?k(i) : the right subscript index j of the term ?j is conserved: this right subscript index j refers to the j th monad, which in this case is the electron orbiting the nucleus. Because particlelike matter quanta have a definite position, Theorem 3.1.3.20 thus asserts that an electron orbiting a nucleus has countably many times a definite position. This also holds for all free protons, antiprotons, neutrons, and antineutrons: all have countably many times a definite position. Remark 3.1.3.21. The general deduction rule LW 0 z ? y y z x x ?1 ? MA , = LW 0 , : ? M, z? ? y? y? z? x? x? ??1 y z ? ME ? ? ME y? z? y z y x ? . The case , and , applies to all degrees of evolution and to all components y? z? y? x? LW 0 z ?1 MA is discussed in Example 3.2.1.8. The case = LW 0 is treated in Sect. 3.2.2. z? ??1 Axiom 3.1.3.22. (Elementary Principle of Choice): N P x+1 ?k ) fC (?x+1 k ?x ? ZN ?k ? S?(x) = N P x+1 N P x+1 ??k ??k Interpretation Rule 3.1.3.23. The interpretation of Axiom 3.1.3.22 is that every nonextended particlelike phase quantum N P ?x+1 , occurring in the world at the (x + 1)th degree of evolution in the k th process k th th from the x to the (x + 1) degree of evolution, is a choice of parallel possible phase quanta. Lemma 3.1.3.24. (Elementarychoice lemma): NW x EP x ?k ?k ) ? fC (?x+1 k ?x ? ZN ?k ? S?(x) : EP x NW x N P x+1 ??k ??k ??k ? P r o o f. Lemma 3.1.3.24 follows from the elementary principle of nonlocal mediation 3.1.3.8 and the elementary principle of choice 3.1.3.22 by substitution. The elementary choice lemma 3.1.3.24 explicates that the choice in question is made in a nonlocal mediation by the collapse of the nonlocal wavelike phase quantum N W ?xk , that has an internal time-direction in the direction of evolution. Assuming the elementary principle of choice means that the ideas of a deterministic universe and of a probabilistic universe have to be rejected. Example 3.1.3.25. To illustrate the notion of parallel possible nonextended particlelike phase quanta, let the following be given for the k th process from the xth to the (x + 1)th degree of evolution: EP x EP x EP x ?? (1) ?k ?k (14) = EP x ? EP x ? ME EP x ??? (1) ??k ??k At the instant when the discrete transition EP ?x?(1) ? N W ?xk takes place ? and this transition will certainly take place because of the elementary principle of nonlocal equilibrium 3.1.3.6 ? the nonlocal wavelike phase quantum in the world N W ?xk has not yet collapsed. Thus, at said instant the collapse can still happen at every point of the spatial extension of the nonlocal wavelike phase quantum N W ?xk : every a priori possible point of collapse now corresponds with a parallel possible nonextended particlelike phase quantum N P x+1 NW x ?k . At the instant when the nonlocal wavelike phase quantum ?k collapses, the nonextended , and thus the component particlelike phase quantum N P ?x+1 k www.ann-phys.org NP NP ?x+1 k ??x+1 k , is chosen. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 720 M. J. T. F. Cabbolet: Elementary Process Theory Remark 3.1.3.26. (causal laws): The five principles of action 3.1.3.6, 3.1.3.8, 3.1.3.10, 3.1.3.12, and 3.1.3.17 are causal laws: all other causal laws are hereby rejected as invalid at the supersmall scale. The elementary principle of identity of binads 3.1.3.18 and the elementary principle of choice 3.1.3.22 are not causal laws, but are still synthetic propositions. In addition, the elementary principles of the EPT describe every individual process in the universe governed by the EPT, and by that the creation of every fundamental building block of this universe: the EPT is therefore physically complete. 3.2 Applications of the Elementary Process Theory 3.2.1 Formalization of some observed processes in the framework of the EPT In this section, first the notions of simple, simplest, and complex individual processes will be formalized in the framework of the EPT, and the concept of stepwise motion will be discussed. Next, a variety of observed process will be formalized in the framework of the EPT: currently, there is no other framework in physics in which all these processes can be formalized. Definition 3.2.1.1. The k th individual process from the xth to the (x + 1)th degree of evolution is simple if and only if in this k th individual process the following expressions are true: EP x EP x ?? (1) + . . . + EP ?x?(p) ?k (15) = EP x for some ? ? P and p ? Z + EP x ??? (1) + . . . + EP ??x?(p) ??k Furthermore, NP NP EP ?x?(1) + . . . + ?x+1 ? (1) + . . . + ??x+1 ? (1) + . . . + NP NP EP ?x+1 ? (p) ??x+1 ? (p) ?x?(p) = ?xk : EP x NW x ??? (1) + . . . + EP ??x?(p) ??k N P x+1 EP x NW x ?? (1) + . . . + N P ?x+1 ?? (1) + . . . + EP ?x?(p) ?k ? ? (p) : EP x NW x N P x+1 ??k ? ??? (1) + . . . + EP ??x?(p) ??? (1) + . . . + N P ??x+1 ? (p) N P x+1 N P x+1 LW x+1 ? (1) + . . . + ?? (p) ?k ? 0 : N P ?x+1 LW x+1 N P x+1 ??? (1) + . . . + ??? (p) ??k ? 0 N P x+1 EP x+1 LW x+1 ?? (1) + . . . + N P ?x+1 ?? (1) + . . . + EP ?x+1 ? ?k ? (p) ? (p) : N P x+1 LW x+1 EP x+1 ? ??? (1) + . . . + N P ??x+1 ??? (1) + . . . + EP ??x+1 ??k ? (p) ? (p) 0 0 ? ? ?x+1 k N P x+1 ??k NP NW (16) (17) (18) (19) (20) in expression (17). 2 In a simple individual process no nuclear reactions take place. This property is captured in expression (17) by conserving the right subscript index of the matter quanta; if a nuclear reaction occurs, these right subscript indices are not conserved (see below for examples). All motion under the influence of longdistance interactions is due to simple processes, regardless whether gravitational or electromagnetic aspects are predominant. A simple individual process can, of course, be succeeded by a new simple individual EP x+1 EP x+1 ? (1) + . . . + ?? (p) of (19) satisfies for some integer process. In that case, the superposition EP ?x+1 EP x+1 ??? (1) + . . . + ??? (p) EP x+1 EP x+1 ? (1) + . . . + EP ?x+1 ?l ? (p) l ? S?(x+1) the identity EP ?x+1 = while the expressions (15)/(19) x+1 EP x+1 ??? (1) + . . . + EP ??? (p) ??l then hold for the lth individual process from the (x + 1)th to the (x + 2)th degree of evolution. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 721 Definition 3.2.1.2. The k th individual process from the xth to the (x + 1)th degree of evolution is a simplest individual process if and only if this k th individual process is a simple individual process for which the following expression is true: EP x EP x ?? (1) ?k = EP x for some ? ? P (21) EP x ??? (1) ??k This is a special case of (15), meaning that in a simplest individual process the extended particlelike phase quantum in the world is simple, that is, is composed of only one extended particlelike matter quantum. 2 To sketch what happens in the world during a simplest process, let?s assume that the k th individual process from the xth tothe (x +1)th degree of evolution is a simplest individual process; the starting EP x EP x ?? (1) ? = , cf. formula (22). The extended particlelike matter point is the component EP kx EP x ??? (1) ??k quantum has an observable spatiotemporal position, say, X. In this individual process, in the world first spontaneously a discrete transition EP ?x?(1) ? N W ?xk takes place, cf. formula (16) with p = 1, whereby the nonlocal wavelike phase quantum N W ?xk comes into existence. In the antiworld, then the discrete transition N W ??xk ? EP ??x?(1) has happened. Next, in this individual process in the world the nonlocal wavelike phase quantum N W ?xk collapses after a finite amount of time into a nonextended particlelike matter EP x quantum N P ?x+1 ?? (1) to N P ?x+1 ? (1) , thus in effect bringing about a transition from ? (1) ; the right subscript index ? (1) is thus preserved in this latter transition. In the antiworld then the opposite has happened in accordance with the elementary principle 3.1.3.8. Finally, in this individual process in the world the local wavelike phase quantum LW ?x+1 is emitted from the nonextended particlelike matter quantum N P ?x+1 k ? (1) , N P x+1 EP x+1 ?? (1) ? ?? (1) . The extended particlelike which immediately brings about the discrete transition EP x+1 ?? (1) , formed in this last phase of the individual process, has then an observable spamatter quantum tiotemporal position Y . In the antiworld then the opposite has happened in accordance with the principles 3.1.3.10 and 3.1.3.12. This individual process has then brought about the stepwise motion from the extended particlelike matter quantum EP ?x?(1) at position X to the extended particlelike matter quantum EP ?x+1 ? (1) at position Y . For an observer, who himself is subjected to the same individual processes, this is observed as the motion of, say, an electron from X to Y . Two factors have then contributed to the change in observable position ?X = Y ? X: (i) the displacement effected by the nonlocal wavelike phase quantum N W ?xk ; (ii) an intermediate change of the state of the vacuum by the formation of spatial phase quanta. Concerning (i), the larger the displacement ?X effected, the larger E(N W ?xk ) in (10). That is, the larger the displacement ?X effected, the more energy is absorbed from the surroundings of the matter quantum EP ?x?(1) . These surroundings are a two-phased heterogeneous vacuum system, consisting of a curved space (the first phase composed of nonlocal wavelike phase quanta and spatial phase quanta) and a second phase, carried by the curved phase, composed of local wavelike phase quanta. In such a simplest process a long-distance interaction having gravitational and electromagnetic aspects take place: the principle of the gravitational aspect is that the nonlocal wavelike phase quantum N W ?xk interacts with the curved space in the sense that the displacement ?X depends on the metric of the curved space (the nonlocal wavelike phase quantum N W ?xk ?sees? the metric) and simultaneously the metric of the curved space depends on the nonlocal wavelike phase quantum N W ?xk ; the principle of the electromagnetic aspect is that the displacement ?X depends on the state of the second phase (the nonlocal wavelike phase quantum N W ?xk ?sees? the field of nonlocal wavelike phase quanta) and on the intermediate change of state of the vacuum. Definition 3.2.1.3. The k th individual process from the xth to the (x + 1)th degree of evolution is complex if and only if it is not simple. 2 www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 722 M. J. T. F. Cabbolet: Elementary Process Theory th Suppose process from the xth tothe (x + 1)th degree of evolution starts with a com the k individual EP x EP x ? + . . . + EP ?x?(p) ? ponent EP kx = EP ?x(1) , and suppose that in this k th process the nonlocal ??? (1) + . . . + EP ??x?(p) ??k N P x+1 N P x+1 ??(1) + . . . + N P ?x+1 ?k ?(q) mediation yields a component N P x+1 = N P x+1 for some ? ? P . If the ???(1) + . . . + N P ??x+1 ??k ?(q) EP x ?? (i) individual process is complex, then the number (p) of subcomponents EP x of the component ??? (i) N P x+1 EP x ??(j) ?k is not necessarily the same as the number (q) of subcomponents N P x+1 of the compoEP x ?? ???(j) k N P x+1 ? nent N P kx+1 . In any complex individual process, the following holds for the right subscript indices ??k of these components: {? (1), . . . , ? (p)} ? {?(1), . . . , ?(q)} = ? (22) That is, the extended particlelike matter quanta EP ?x?(i) , entering a complex individual process, and the extended particlelike matter quanta N P ?x+1 ?(j) , formed in a complex individual process, can not be attributed to the same monads. Example 3.2.1.4. (electron in an electron shell): A sequence of the simplest processes applies to any electron orbiting any atomic nucleus in the world. At n consecutive degrees of evolution the electron is x+n?1 x , . . . , ?k(n) , for which the identity 3.1.3.18 is valid. then designated by the n corresponding binads ?k(1) Here the right subscript index k(j) has not necessarily the same value for every value of j, because the successor of the k th individual process from the xth to the (x + 1)th degree of evolution is not necessarily the k th individual process from the (x + 1)th to the (x + 2)th degree of evolution. However, because of Definition 3.2.1.2, for these n binads the following identities do hold for some i ? Z + : x ? ?k(1) = EP ?xi + NW x+1 ? ?k(2) = EP ?x+1 + i ?xk(1) NW ?x+1 k(2) and so forth. Thus, to indicate that the n binads are to be attributed to the same electron in itself, the right subscript i of EP ?xi is conserved in a sequence of simple processes. x = EP ?xi + N W ?xk(1) then reads: the binad occurring in the world in the k(1)th The equation ?k(1) individual process from the xth to the (x + 1)th degree of evolution consists of the extended particlelike matter quantum of the ith monad at the xth degree of evolution and the nonlocal wavelike phase quantum occurring in the world at the xth degree of evolution in that individual process. Thus speaking, in this sequence of simplest processes the electron orbiting an atomic nucleus has consecutively n definite spatiotemporal positions Xx , Xx+1 , . . . , Xx+n?1 : these are the positions where the n extended particlelike matter quanta EP ?ix+j for j = 0 to n ? 1 happen to find themselves. Example 3.2.1.5. (free neutron gravitating towards earth): A sequence of the simplest processes also y y+p?1 applies to any free neutron gravitating towards earth. A sequence of p consecutive binads ?l(1) , . . . , ?l(p) then designates the neutron at p consecutive degrees of evolution y, y + 1, . . . , y + p ? 1. To these p binads the following equations apply for some j ? Z + : y ? ?l(1) = y+1 ? ?l(2) = EP ?yj + EP NW ?y+1 + j ?yl(1) NW ?y+1 l(2) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 723 and so forth. In the process of gravitating towards earth, the neutron then has p consecutive, definite spatiotemporal positions Yy , Yy+1 , . . . , Yy+p?1 corresponding with the p extended particlelike matter quanta EP y ?j , . . . ,EP ?y+p?1 . 2 j x+n?1 x , . . . , ?k(n) will be very different from the p Comparing the last two examples, the n binads ?k(1) y y+p?1 binads ?l(1) , . . . , ?l(p) , and the n ? 1 leaps between the n consecutive positions Xx , . . . , Xx+n?1 will be very different from the p ? 1 leaps between the p consecutive positions Yy , . . . , Yy+p?1 . The point is, however, that the elementary principles of action are exactly the same for both processes. That is, at the degree of abstractness of the EPT, there is absolutely no difference between an electron orbiting an atom and a neutron gravitating towards earth: all simplest processes are the same. Remark 3.2.1.6. (photons): In any sequence of simplest individual processes, the nonzero rest mass entity has a spatial momentum in the world when leaping from the j th to the (j + 1)th position. Being motionless, the nonextended particlelike matter quantum at the (j + 1)th position has no spatial momentum. The aforementioned spatial momentum is then conserved by a photon: photons are local wavelike matter quanta, that occur in the world in local wavelike phase quanta. Assuming that the k th individual process fromthe xth to the (x + 1)th degree of evolution is a simplest individual process, the component LW x+1 ?k is then a superposition LW x+1 ??k ?x+1 k LW x+1 ??k LW ?kx+1 ??kx+1 = + ?kx+1 ??kx+1 (23) where ?kx+1 is the photon emitted in the k th individual process from the xth to the (x + 1)th degree of evolution, and ?kx+1 is another local wavelike matter quantum emitted in this process (cf. Remark 4.3.2.1 for some elaboration). Example 3.2.1.7. (decay of a neutron): For some ? ? P , let the ? (1)th monad be a neutron, the ? (2)th a proton, and the ? (3)th an electron. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP ?xk EP x ??k NW ?xk NW x ??k = EP EP ?x+1 k LW x+1 ??k LW : EP EP : ?x?(1) (24) ??x?(1) ?x?(1) ??x?(1) NP NP ? ? ?x+1 ? (2) + ??x+1 ? (2) + NP NP NP NP ?x+1 ? (3) ??x+1 ? (3) ?x+1 ? (2) + ??x+1 ? (2) + ? ? NP NP ?x+1 ? (3) (25) ??x+1 ? (3) EP EP ?x+1 ? (2) + ??x+1 ? (2) + EP EP ?x+1 ? (3) ??x+1 ? (3) (26) In this process, in the world the extended particlelike matter quantum EP ?x?(1) , that can be attributed to a EP x+1 neutron, has decayed in a superposition EP ?x+1 ?? (3) of two extended particlelike matter quanta, ? (2) + that can be attributed to a proton and an electron. This process can then by two simplest be succeeded EP x+1 EP x+1 ?? (2) ?? (3) individual processes, the starting points of which are the components EP x+1 and EP x+1 in ??? (2) ??? (3) accordance with (22). In the individual process described above in the world a decay of a neutron occurs according to the decay reaction n ? p+ + e? + ? as originally proposed by Pauli in [32]. The neutrino ? is a local wavelike www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 724 M. J. T. F. Cabbolet: Elementary Process Theory matter quantum, that exists in the local wavelike phase quantum LW ?x+1 emitted in this process: k LW x+1 ?k ?kx+1 ?kx+1 ?kx+1 = + + LW x+1 ??kx+1 ??kx+1 ??k ?»kx+1 (27) Here ?kx+1 designates the neutrino in the world. Example 3.2.1.8. (annihilation of a proton/antiproton pair): For some ? ? P , let the ?(1)th monad be a proton, and the ?(2)th an antiproton. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP x EP x ??(1) + EP ?x?(2) ?k = EP x (28) EP x ??k ???(1) + EP ??x?(2) N P x+1 EP x NW x ??(3) ??(1) + EP ?x?(2) ? ?k : EP x (29) N P x+1 NW x ? ???(1) + EP ??x?(2) ???(3) ??k N P x+1 LW x+1 ??(3) ?k ? 0 : N P x+1 (30) LW x+1 ???(3) ??k ? 0 In this individual process, in the world a proton and an antiproton are annihilated: such an annihilation process applies to any other pair of nonzero rest mass entities (such as electron/positron, neutron/antineutron, etc.). The right subscript index ?(3) in the designator N P ?x+1 ?(3) refers to a monad, which has the property that it immediately decays completely into a local wavelike phase quantum, cf. Definition 3.1.3.14 with N P x+1 N P x+1 ? ?k = N P ?(3) . The annihilating mediation (30) is thus a special case of a local mediation N P x+1 ??x+1 ??k ?(3) EP x+1 EP x+1 EP x+1 ?? (1) ?? (1) ??(3) 0 3.1.3.12 with q = 1 and EP x+1 = (and EP x+1 = EP x+1 ). 0 ??? (1) ??? (1) ???(3) Example 3.2.1.9. (formation of Deuterium):For some ? ? P , let the ? (1)th monad be a proton, the ? (2)th a neutron, the ? (3)th a proton of a Deuterium nucleus, and the ? (4)th a neutron of a Deuterium nucleus. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP x EP x ?? (1) + EP ?x?(2) ?k = EP x (31) EP x ??? (1) + EP ??x?(2) ??k N P x+1 EP x NW x ?? (3) + N P ?x+1 ?? (1) + EP ?x?(2) ?k ? ? (4) : EP x (32) N P x+1 NW x ? ??? (1) + EP ??x?(2) ??? (3) + N P ??x+1 ??k ? (4) N P x+1 EP x+1 EP x+1 LW x+1 ?? (3) + N P ?x+1 ? + ? ? ?k ? (4) ? (3) ? (4) : N P x+1 (33) LW x+1 EP x+1 ? ??? (3) + N P ??x+1 ??? (3) + EP ??x+1 ??k ? (4) ? (4) In this individual process, in the world a proton and a neutron have formed a deuterium nucleus. This complex individual process is then succeeded by a simple individualprocess, say the lth individual process from EP x+1 EP x+1 EP x+1 ? + ? ?l ? (3) ? (4) th th the (x + 1) to the (x + 2) degree of evolution, for which EP x+1 = EP x+1 . ??? (3) + EP ??x+1 ??l ? (4) The free neutrons and protons are different from (that is: have to be attributed to other monads than) the neutrons and protons bound in deuterium nuclei, because their rest masses add up differently: the rest mass of a deuterium nucleus is different from the sum of the rest masses of a free proton and a free neutron. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 725 Other examples of complex processes will be given in the next section. The variety of complex individual processes is far more extended than these examples, but these examples demonstrate that the EPT applies to various nuclear reactions. 3.2.2 Theory of the Planck era of the universe The theory of the Planck era of the universe will be given below in Table 2 in the form of a sequence of formulas, describing events that follow one another in the direction of evolution. Technically this is a deduction within the axiomatic system containing the EPT: every formula of the sequence is either an assumption of the theory of the Planck era of the universe, or a corollary of a previous assumption and the EPT. The last corollary identifies a condensed matter field in a nonempty vacuum. Table 2 Theory of the Planck era of the universe; a superscript i in the first column refers to a Remark 3.2.2.i below. # Formula (i)1 (ii)2 (iii) 3 (iv) (v) 4 (vi) (vii)5 (viii)6 (ix) (x) 7 (xi) (xii) NP ?01 NP 0 ??1 Justification ? ME ? NP ?01 NP 0 ??1 = NP ?01 NP 0 ??1 assumption ?01 0 = LW 0 0 ??1 NP 0 EP 0 ?1 ?1 0 ? : NP 0 EP 0 0 ? ??1 ??1 EP 0 ?1 ? ME EP 0 ??1 EP 0 EP 0 ?1 ?1 = EP 0 EP 0 ??1 ??1 EP 0 NW 0 ?1 ?1 ? 0 : EP 0 NW 0 ? 0 ??1 ??1 NW 0 ?1 ? ME NW 0 ??1 EP 0 NW 0 ?1 ?1 ?10 = EP 0 + N W 0 ??1 ??10 ??1 NW 0 EP 0 NP 1 ?1 ?1 ?1 ? : EP 0 NW 0 NP 1 ? ??1 ??1 ??1 NP 1 NP 1 ?1 ?2 + . . . + N P ?12K+1 = NP 1 for some K ? Z + NP 1 ??2 + . . . + N P ??12K+1 ??1 NP 1 ?2 + . . . + N P ?12K+1 ? ME NP 1 ??2 + . . . + N P ??12K+1 NP 1 LW 1 0 ?2 + . . . + N P ?12K+1 ?1 ? : NP 1 LW 1 0 ? ??2 + . . . + N P ??12K+1 ??1 LW www.ann-phys.org assumption (i), (ii), 3.1.3.12 (i), (iii), 3.1.3.21 (iv), 3.1.3.15 (v), 3.1.3.6 (iv), (vi), 3.1.3.21 (v), 3.1.3.18 (v), 3.1.3.8 assumption (iv), (ix), 3.1.3.21 (x), 3.1.3.10 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 726 M. J. T. F. Cabbolet: Elementary Process Theory Table 2 (Continued). # Formula (xiii) 8 (xiv) 9 (xv) LW ?11 LW 1 ??1 LW ?11 LW 1 ??1 EP Justification ? ME : NP ?11 NP 1 ??1 ?12 + . . . + EP 1 ??2 + . . . + (xi), (xii), 3.1.3.21 ? ? EP ?12K+1 EP 1 ??2K+1 EP ?12 + . . . + EP 1 ??2 + . . . + ? ME EP ?12K+1 EP 1 ??2K+1 (x), 3.1.3.12 (x), (xi), (xiv), 3.1.3.21 EP 1 ?12 + . . . + EP ?12K+1 ?1 (xvi) = EP 1 assumption EP 1 ??1 ??2 + . . . + EP ??12K+1 EP 1 NW 1 ?2 + . . . + EP ?12K+1 ?1 ? 0 (xvi), 3.1.3.6 (xvii) : EP 1 NW 1 ? 0 ??2 + . . . + EP ??12K+1 ??1 NW 1 ?1 (xviii) (xv), (xvii) , 3.1.3.21 ? ME NW 1 ??1 S 2 ?1 11 (xiii), 3.1.3.17 (xix) ? ME S 2 ??1 NW 1 EP 1 NP 2 ?1 ?2 + . . . + EP ?12K+1 ?1 ? (xx) : EP 1 (xvi), 3.1.3.8 NW 1 NP 2 ? ??2 + . . . + EP ??12K+1 ??1 ??1 NP 2 NP 2 ?1 ?2K+2 + . . . + N P ?22K+L+1 12 (xxi) = NP 2 assumption NP 2 ??2K+2 + . . . + N P ??22K+L+1 ??1 NP 2 ?2K+2 + . . . + N P ?22K+L+1 (xv), (xx), 3.1.3.21 ? ME (xxii) NP 2 ??2K+2 + . . . + N P ??22K+L+1 NP 2 LW 2 0 ?2K+2 + . . . + N P ?22K+L+1 ?1 ? (xxiii) : NP 2 (xxi), 3.1.3.10 LW 2 0 ? ??2K+2 + . . . + N P ??22K+L+1 ??1 LW 2 ?1 (xxiv) (xxii), (xxiii), 3.1.3.21 ? ME LW 2 ??1 LW 2 NP 2 EP 2 ?1 ?1 ?2K+2 + . . . + EP ?22K+L+1 ? (xxv) : NP 2 (xxi), 3.1.3.12 LW 2 EP 2 ??2K+2 + . . . + EP ??22K+L+1 ??1 ??1 ? EP 2 EP 2 ?1 + . . . + EP ?2?(2) ?2K+2 + . . . + EP ?22K+L+1 13 (xxvi) = EP 2 3.1.3.15, 3.1.2.2 EP 2 ??1 + . . . + EP ??2?(2) ??2K+2 + . . . + EP ??22K+L+1 EP 2 NW 2 0 ?k ?k ? (xxvii) : EP 2 3.1.3.6 for every k ? S?(2) NW 2 0 ? ??k ??k EP 2 NW 2 S 2 LW 2 ?1 + . . . +EP ?2?(2) ?1 + . . . +N W ?2?(2) ?1 ?1 14 + S 2 + LW 2 + N W 2 ? ME (xxviii) EP 2 ??1 + . . . +EP ??2?(2) ??1 ??1 ??1 + . . . +N W ??2?(2) 10 EP c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 727 Remark 3.2.2.1. The interpretation of assumption (i) is straightforward from Table 1: initially, the NP 0 ?1 component N P 0 exists, while the nonextended particlelike phase quantum N P ?01 is simple, and ??1 composed of the nonextended particlelike matter quantum N P ?01 that precedes the first monad. Following the convention of the interpretation rule strictly, N P ?01 is the nonextended particlelike phase quantum occurring in the world in the first (and only) process from the (N ? 1)th to the 0th degree of evolution. NP 0 ?1 This one process thus consists of the events succeeding the existence of the component N P 0 ? to be ??1 discussed ? and events at a higher degree of evolution that lead back to the initial com in this paragraph NP 0 ?1 ponent N P 0 : these events at the higher degree of evolution will then take place at a later time than ??1 N W N ?1 EP N ?1 NP 0 ? ?1 ?1 ?1 : EP N ?1 the initial events. In particular, the nonlocal mediation N W N ?1 NP 0 ??1 ??1 ??1 ? will take place at a later time. The degree of abstractness of the EPT thus enables one to state that it is certain that this mediation will happen, although it is now not known what the actual constitution of the EP N ?1 ? component EP 1N ?1 will be: the entries of the matrices are designators, not representations. ??1 Remark 3.2.2.2. In this initial individual process at the 0th degree of evolution, no local equilibrium takes place; the corresponding elementary principle of local equilibrium 3.1.3.10 is trivially true. It can be derived from this elementary principle that then a discrete transition N P ?01 ? 0 takes place in the world ? that is, nothing is emitted from the initial nonextended particlelike matter quantum N P ?01 . Remark 3.2.2.3. The of corollary (iii) is that the local equilibrium between the compo interpretation NP 0 EP 0 ? ? nents N P 10 and EP 10 is not mediated by any physical object (that is, by any nonzero compo??1 ??1 nent), but occurs spontaneously. Thus, in the world spontaneously a discrete transition N P ?01 ? EP ?01 takes place, and this is accompanied by a discrete transition EP ??01 ? N P ??01 in the antiworld. Energy is conserved: E(N P ?01 ) = E(EP ?01 ) (34) This equation can not be generalized to other processes: general laws of conservation of energy will be discussed in Sect. 4.2.2. Remark 3.2.2.4. At this point a new individual process begins in the numbering of individual processes, so it has to be taken that ?(0) = 1, so that S?(0) = {1}, cf. Definition 3.1.1.1. This means that there is precisely one individual process from the 0th to the 1st degree of evolution. Remark 3.2.2.5. The vacuum system in the world at the 0th degree of evolution is solely composed of the nonlocal wavelike phase quantum N W ?01 . In other words: no energy can be absorbed from the surroundings, so that E(N W ?01 ) = E(EP ?01 ) = E(EP ?01 ) (35) Compare (10) for the general case. The suggested duration of the life-time of N W ?01 is a Planck-time, so from here on time comes into existence. There is no such thing as a metric in this early vacuum system. Remark 3.2.2.6. From corollaries (iv), (vii), (viii), and Postulate 3.1.3.5 it follows that the following holds: EP 0 NW 0 ?1 ?1 ?10 (36) ? ME ? EP 0 ? ME ? N W 0 ? ME ??1 ??10 ??1 www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 728 M. J. T. F. Cabbolet: Elementary Process Theory The principle of particle/wave duality, Theorem 3.1.3.20, thus holds already at the 0th degree of evolution. Remark 3.2.2.7. Substituting assumption (x) in corollary (ix) yields the formula NW 0 EP 0 NP 1 ? ?1 ?1 ?2 + N P ?13 + . . . + N P ?12K+1 : EP 0 (37) NW 0 NP 1 ? ??1 ??2 + N P ??13 + . . . + N P ??12K+1 ??1 NW 0 EP 0 ?1 ?1 According to (37) the component N W 0 mediates an equilibrium between the components EP 0 ??1 ??1 NP 1 NP 1 NP 1 ? + ?3 + . . . + ?2K+1 . Following Interpretation Rule 3.1.3.9, this entails that the and N P 21 NP 1 NP 1 ??2 + ??3 + . . . + ??2K+1 nonlocal wavelike phase quantum N W ?01 collapses into the superposition N P ?12 + N P ?13 +. . .+ N P ?12K+1 of 2K nonextended particlelike matter quanta N P ?1i , each at a different position. Here K is a large integer, estimated in the order of magnitude of 1075 . In accordance with the general case (11), energy is conserved in this collapse: E(N W ?01 ) = E(N P ?12 + NP ?13 + . . . + NP ?12K+1 ) (38) The 2K monads numbered 2, 3, 4, . . . , 2K + 1 concern K pre-protons and K pre-electrons, see Remark 3.2.2.10 for further elaboration. Remark 3.2.2.8. The emitted local wavelike phase quantum LW ?11 is nonzero, so that the following inequality holds: E(LW ?11 ) ? 0 (39) Thus, energy is emitted into the surroundings. At this point in the early universe, spatial phase quanta, that together with nonlocal wavelike phase quanta are to function as ?carrier? for local wavelike phase quanta, do not yet exist: this first local wavelike phase quantum in the universe, LW ?11 , therefore has its spatial extension immediately. Furthermore, because there is not yet such a thing as momentum, the local wavelike phase quantum LW ?11 contains no photons. Remark 3.2.2.9. Substituting Axiom (x) in corollary (xiv) gives LW 1 NP 1 EP 1 ?1 ?2 + . . . + N P ?12K+1 ?2 + . . . + EP ?12K+1 ? : NP 1 (40) EP 1 LW 1 ??2 + . . . + N P ??12K+1 ??2 + . . . + EP ??12K+1 ??1 ? The right superscript of matter quanta is thus always conserved in a local mediation: there can be no exception. Loosely speaking, an object EP ?1j is the phenomenon that appears at a definite position when the j th monad exists in particlelike form, and N P ?1j is the energy needed at that position to get the j th monad existing there in particlelike form. Remark 3.2.2.10. Although all 2K extended particlelike matter quanta EP ?1j , formed in the one process from the 0th to the 1st degree of evolution, have a different position, there exists no real distance between the different matter quanta because the early vacuum has no metric. The 2K extended particlelike matter quanta EP ?1j thus constitute a single extended particlelike phase quantum EP ?11 . Thus, by no means, the 2K monads numbered 2 to 2K + 1 concern free protons and free electrons: together with the wavelike phase quanta LW ?11 and N W ?11 these matter quanta EP ?1j form what can be called a ?primordial soup?; the monads 2 to 2K + 1 are therefore to be called pre-electrons and pre-protons. The extended particlelike phase quantum formed, EP ?11 , is the starting point of the one process from the 1st to the 2nd degree of evolution. Thus speaking, it has to be taken that ?(1) = 1, so that S?(1) = {1}, cf. Definition 3.1.1.1. Using the general case (12) and inequality (39) it can then be derived from the foregoing that E(EP ?11 ) < E(EP ?01 ) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (41) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 729 The rest mass of the 1st monad at the 0th degree of evolution is thus larger than the sum of rest masses of the K pre-protons and K pre-electrons at the 1st degree of evolution; the one process from the 1st to the 2nd degree of evolution thus begins with less energy than the one process from the 0th to the 1st degree of evolution. Furthermore, in the one process from the 0th to the 1st degree of evolution the composite particlelike phase quantum EP ?11 has been formed out of the simple particlelike phase quantum EP ?01 , but that initial phase quantum EP ?01 existed in an empty space with no surroundings: it is therefore best to label the chain of events in the world as a decay reaction (nuclear disintegration) due to intrinsic instability, rather than as a weak interaction. Remark 3.2.2.11. Due to the fact that in the process from the 0th to the 1st degree of evolution energy has been emitted in the form of the local wavelike phase quantum LW ?11 , at the 2nd degree of evolution space is formed. Thus, while time already existed, cf. Remark 3.2.2.5, now also the three dimensional space exists. Remark 3.2.2.12. Substituting Axiom (xxi) in corollary (xx) yields the formula NW 1 EP 1 NP 2 ? ?1 ?2 + . . . + EP ?12K+1 ?2K+2 + . . . + N P ?22K+L+1 : EP 1 (42) NP 2 NW 1 ? ??2 + . . . + EP ??12K+1 ??2K+2 + . . . + N P ??22K+L+1 ??1 Following Interpretation Rule 3.1.3.9 this entails that the nonlocal wavelike phase quantum N W ?11 , which emerged from the superposition EP ?12 + . . . + EP ?12K+1 of 2K extended particlelike matter quanta, has collapsed into the superposition N P ?22K+2 + . . . + N P ?22K+L+1 of L nonextended particlelike matter quanta. While the 2K monads 2, 3, . . . , 2K + 1 concerned K pre-protons and K pre-electrons, the newly arisen matter quanta precede the L monads 2K + 2, 2K + 3, . . . , 2K + L + 1 that concern p protons, p electrons, and q neutrons with 2p + 2q = 2K. These protons, electrons, and neutrons will thus appear in particleform at the positions of these nonextended particlelike matter quanta N P ?2j in the superposition NP 2 ?2K+2 + . . . + N P ?22K+L+1 . Remark 3.2.2.13. Due to the existence of space at the 2nd degree of evolution, cf. Remark 3.2.2.11, the superposition EP ?22K+2 + . . . + EP ?22K+L+1 of L extended particlelike matter quanta no longer form a single phase quantum: instead, these L matter quanta form ?(2) extended particlelike phase quanta EP ?2i that are now spatially separated. If ?(2) = L, then all L extended particlelike matter quanta EP ?2j are spatially separated from each other, and each such matter quantum EP ?2j then on its own forms a phase quantum EP ?2i . If ?(2) < L, then there is at least one phase quantum EP ?2i composed of more then one matter quantum EP ?2j ? but still all phase quanta EP ?2k are spatially separated. The case ?(2) > L is physically impossible. In this one process from the 1st to the 2nd degree of evolution, in the world the phase quantum EP ?11 , which was composed of 2K subconstituents, has been transformed into ?(2) spatially separated extended particlelike phase quanta EP ?2i . The nonlocal wavelike phase quantum N W ?11 , which essentially effected the separation according to (xx) and (xxi), could ?see? the emitted local wavelike phase quantum LW ?11 ; the chain of events in the world is therefore best labeled an electroweak interaction. In addition, this yields a new view to the so-called ?horizon problem?, cf. [30], different from Guth?s inflation, cf. [31]. In a nutshell, this horizon problem is the following: taking into account the estimated age of the universe, then the currently most distant galaxies could never have originated from one point, even if they would have traveled close to the speed of light: at t = 0, they still were lightyears apart, see Fig. 6 for an illustration. The point of view that the EPT offers is that there is no such thing as the speed of light at the 0th and 1st degree of evolution, and thus no principle such as ?no object can travel faster than light? is valid in this early universe: for the nonlocal mediation in the one process from the 1st to the 2nd degree of evolution, cf. corollary (xx) and assumption (xxi), there is thus no restriction on the spatial positions at which the L nonextended particlelike matter quanta N P ?2j in the superposition N P ?22K+2 + . . . + N P ?22K+L+1 arise. The ?(2) extended particlelike phase quanta EP ?2i that are formed from there can thus be located at the www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 730 M. J. T. F. Cabbolet: Elementary Process Theory Fig. 6 Illustration of the horizon problem. In a space vs. time diagram, the lines (1) and (2) depict the path travelled by the objects, that are now furthest apart in the universe, assuming a speed close to the speed of light; the arrows (3) and (5) indicate the distance between the objects now and at t = 0, respectively. Fig. 7 Illustration in a space vs. time diagram of the approach to the horizon problem based on the EPT. The interpretations of the lines (1) and (2) and the arrow (3) are identical to those for Fig. 6. The array of dots schematically indicates the positions of the phase quanta in the superposition EP 2 ?1 + . . . +EP ?2?(2) ; the lowest dot indicates the position of the phase quantum EP ?11 ; the dotted arrows illustrate the action of the nonlocal wavelike phase quantum NW ?11 . 2nd degree of evolution at positions in a way that can not be reconciled with Einstein?s relativity. See Fig. 7 for an illustration of this approach to the horizon problem. Remark 3.2.2.14. The superposition S ?21 + LW ?21 + N W ?21 +. . .+ N W ?2?(2) + EP ?21 +. . .+ EP ?2?(2) concerns a condensed matter field in a heterogeneous vacuum system. The condensed matter field is the superposition EP ?21 + . . . + EP ?2?(2) which is composed of spatially separated phase quanta EP ?2j . The vacuum system is the superposition S ?21 + LW ?21 + N W ?21 + . . . + N W ?2?(2) , in which the superposition S 2 ?1 + N W ?21 + . . . + N W ?2?(2) is a homogenous phase, and the one local wavelike phase quantum LW 2 ?1 is a homogenous phase. The homogenous phase S ?21 + N W ?21 + . . . + N W ?2?(2) is observable as a space with non-Euclidean geometry: the constituent S ?21 in itself has a homogenous energy density, but is transcended by the constituent N W ?21 + . . . + N W ?2?(2) , which is the source of curvature, that is, of differences in energy density. If this homogenous phase could be modeled by a four-dimensional manifold with a metric g, then the metric would depend on the phase quanta: g = g(S ?21 , N W ?21 , . . . , N W ?2?(2) ). This homogenous phase acts as a carrier for the second homogenous phase, at the 2nd degree of evolution formed by the one local wavelike phase quantum LW ?21 , which spreads out with the speed of light. 4 Discussion 4.1 Relation of the EPT with established theories 4.1.1 Incompatibility of the EPT and QM Proposition 4.1.1.1. The theory, obtained by extending the EPT with the following translation of the orthodox quantum-mechanical view on nonzero rest mass entities in the language of the EPT, is inconsistent: EP x ?k ?kx ? ME ? EP x ? ME (43) ?x ? ZN ?k ? S?(x) ??kx ??k P r o o f. Formula (43) is a contraposition of Theorem 3.1.3.20. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 731 The orthodox position of QM implies that observable nonzero rest mass entities, such as electrons, have no definite position in absence of observation. Translated in the language of the EPT, this would imply that (43) would be true. Obviously, this view yields a contradiction with the EPT. To spell it out, QM is incompatible with the EPT. One of the fundamental differences, expressed by the fact that Theorem 3.1.3.20 and formula (43) contradict each other, is that according to QM the wave function of a microsystem does not collapse in absence of observation, while according to the EPT nonlocal wavelike phase quanta spontaneously collapse regardless whether someone is watching or not. 4.1.2 Incompatibility of the EPT and GR Proposition 4.1.2.1. The theory, obtained by extending the EPT with the following translation of the classical view on nonzero rest mass entities in the language of the EPT, is inconsistent: NW x ?kx ?k ?x ? ZN ?k ? S?(x) ? ME ? N W x ? ME (44) ??kx ??k P r o o f. Formula (44) is a contraposition of Theorem 3.1.3.20. The classical view, which is also incorporated in GR (General Relativity), implies that an observable nonzero rest mass entity is in a particlelike state at every point of its worldline. Translated in the language of the EPT, this would imply that (44) would be true. Obviously, this view yields a contradiction with the EPT. To spell it out, GR is incompatible with the EPT. One of the fundamental differences, expressed by the fact that Theorem 3.1.3.20 and formula (44) contradict each other, is that according to GR a nonzero rest mass particle does not spontaneously transform into a wavelike state, while according to the EPT extended particlelike phase quanta spontaneously transform into nonlocal wavelike phase quanta. Given that both cornerstones of modern physics, QM and GR, are incompatible with the EPT, the EPT the EPT should thus not be mistaken for an attempt to unify GR and QM: instead, it has to be seen as (a proposal for) the fundamental laws of physics governing the supersmall scale. 4.1.3 Relation with Special Relativity Arriving at the relation between the EPT and Special Relativity (SR), it should be noted that the latter entails a rejection of the idea of an aether, while in the context of the former spatial phase quanta occur as energetic constituents of the vacuum system. Although it is widely believed that the idea of an aether has been disproven by the Michelson-Morley experiment, the implications of the outcome of this experiment should be reviewed: what has been falsified, namely, is the idea of an aether such as mathematically represented by classical theory. In particular, what has been falsified is that the Galilean law of velocity transformation is universally applicable. Proceeding, the first major point is that the Galilean law of relativity, that it is not possible to determine the absolute velocity of a nonzero rest mass entity, cf. [33], has no meaning in the context of the EPT: at the supersmall scale, where the concept of stepwise motion applies, there is no such thing as the ?velocity? of an electron, of a proton, etc. To put that in other words: in the context of the EPT, velocity is a secondary property as meant by John Locke ? it is present in the observation of the object but not in the object itself. Thus, one can perform measurements on microsystems and use the obtained results to calculate a value that can be called ?velocity?, but the idea, that the calculated velocity then corresponds to a really existing property of the material object that was subjected to measurement (e.g. an electron), is purely classical, and is connected to the concept of continuous motion of classical mechanics. In the context of the EPT, particlelike phase quanta do not move at all, and thus simply have no velocity. In other words, the Galilean principle of special relativity has no meaning in the context of the EPT. The second of the two postulates of SR, Einstein?s principle of universality of the speed of light is to be retained in the framework of the EPT. It should be noted that nonextended particlelike phase quanta, www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 732 M. J. T. F. Cabbolet: Elementary Process Theory from which local wavelike phase quanta (and thus light ? photons occur in such phase quanta, cf. Remark 3.2.1.6) are emitted, do not move at all: therefore, there is no such thing as ?the motion of the light?s source? in the framework of the EPT, nor does the Galilean law of velocity transformation, falsified by the Michelson-Morley experiment, apply to light in the framework of the EPT. The speed of change of the changing spatial extension of the local wavelike phase quanta is identical to the speed of light, and this speed of light is then a property of the vacuum system, having at every point of position space the same value for all observers. Einstein?s principle of universality of the speed of light thus remains valid in the framework of the EPT; it should be noted that the principle is then analytic in the context of the EPT. A quantitative formulation of the principle is to be incorporated in a mathematical model of the EPT. 4.2 Relation of the EPT with established principles 4.2.1 The EPT vs. the general principle of relativity Proposition 4.2.1.1. The EPT is in agreement with the general principle of relativity. P r o o f. The following four statements hold: (i) a degree of evolution is the same for all observers; (ii) the kind of a phase quantum is the same for all observers; (iii) the elementary principles, laid down in the EPT, do not depend on the numbering of the individual processes; (iv) the elementary principles, laid down in the EPT, do not depend on the numbering of the matter quanta. From this it follows that the elementary principles of the EPT are the same for all observers, and the EPT hence satisfies the general principle of relativity. This general principle of relativity is not formulated exactly, that is, word for word, the same as the general principle of relativity formulated by Einstein: the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion [34]. But because of the similarity in intended meaning, the same name was given to the present principle. 4.2.2 The EPT vs. the law of conservation of energy The laws of conservation of energy for any individual process are given by Eqs. (11) and (12). Equation (11) expresses that no energy is lost in the collapse of the nonlocal wavelike phase quantum N W ?xk . Equation (12) expresses that no energy is lost in the decay of the nonextended particlelike phase quantum N P ?x+1 . k A corollary of (11) and (12) is the following law: E(EP ?x+1 ? (1) + . . . + EP EP x ?x+1 ?k ) = E(N W ?xk ) ? E(EP ?xk ) ? E(LW ?x+1 ) k ? (q) ) ? E( (45) Concerning the right-hand term, it is always the case that E(N W ?xk ) ? E(EP ?xk ), cf. (10). That is, the amount of energy distributed in the nonlocal wavelike phase quantum N W ?xk is more than or equal to the amount of energy distributed in the extended particlelike phase quantum EP ?xk from which it originated. The difference E(N W ?xk ) ? E(EP ?xk ) ? 0 is absorbed from the vacuum (the surrounding system) in the first phase of the individual process. Keeping in mind that the local wavelike phase quantum LW ?x+1 is k emitted, the right hand term of (45) thus designates the net amount of energy exchanged with the vacuum. The left hand side of (45) is merely the difference in energy between ?output? and ?input? of the process: this is, thus, exactly equal to the net amount of energy exchanged with the vacuum. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 733 Laws of conservation of energy thus have been formulated for the EPT, but it is emphasized that ?energy? is a primitive notion here. In further research it still has to be proven that this relates to the macroscopic notion of energy defined as the ability to do work. Such a proof is best given on the basis of a mathematical model of the EPT, satisfying the conditions set forth in Sect. 1.3. 4.3 Relation of the EPT with the hypothesis on gravitational repulsion 4.3.1 Extension of the language for the EPT Definition 4.3.1.1. Let the for language for the EPT, given by Definition 3.1.1.1, be extended with the following individual constants: (i) the subset {?1, 0, 1} of the set Z of integers, elements of which are to be denoted by a symbol cn : cn ? {?1, 0, 1} ? Z (ii) the set RZN of functions from ZN to the real numbers R, elements of which are to be denoted by a symbol s: s = {[0 m0 ], [1 m1 ], . . . , [N ? 1 mN ?1 ]} Here the 2 О 1 matrices [a b] represent two-tuples a, b. Interpretation Rule 4.3.1.2. An element cn ? {?1, 0, 1} denotes a characteristic number of normality; this is an essential property of a monad according to the following guidelines: (i) cn = 1 for all normal matter such as protons, electrons, neutrons; (ii) cn = ?1 for all abnormal matter such as antiprotons, positrons, antineutrons; (iii) cn = 0 for all annihilating monads, cf. Example 3.2.1.8. Interpretation Rule 4.3.1.3. An element s ? RZN denotes a rest mass spectrum; this is an essential property of a monad. If the j th monad has the rest mass spectrum s, and if [x mx ] ? s, then the amount of energy, distributed in the extended particlelike matter quantum EP ?xj is identical to mx , that is, if the extended particlelike matter quantum EP ?xj indeed exists. In other words, for every monad, the rest mass is predetermined at every degree of evolution. 2 Thus, using Remarks 3.1.2.4 and 3.1.2.5, if the j th monad is, for example, a free antiproton with rest mass spectrum sp? , then in accordance with (6) and (7) one would get: EP x EP x ?j ?k (46) = EP x ? ME ? EP ?xj = sp? (x) EP x ??j ??k All electrons and all positrons have one and the same rest mass spectrum. This is also the case for all free protons and all free antiprotons, and also for all free neutrons and all free antineutrons. More general: every nonzero rest mass matter entity and its antimatter counterpart have the same rest mass spectrum. Interpretation rule 4.3.1.3 thus covers the observation that matter particles and their antimatter counterparts have the same rest mass. 4.3.2 Explanation of gravitational repulsion of matter and antimatter Given that the EPT contradicts the accepted theory of gravitation (GR), cf. Proposition 4.1.2.1, it has to be redefined what ?to gravitate? is in order to explain the original hypothesis on gravitational repulsion. Defining ?to gravitate? now as ?to move under the influence of a long-distance interaction with predominantly gravitational aspects?, then in the universe governed by the EPT nonzero rest mass entities like www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 734 M. J. T. F. Cabbolet: Elementary Process Theory electrons, positrons, (anti-)neutrons and (anti-)protons gravitate in a wave-state, while the actual displacement is determined by the reaction of these wave-states to the state of the vacuum system. The difference in the assigned characteristic number of normality, cf. Interpretation Rule 4.3.1.2, can now be used to explain how matter and antimatter can behave differently under the influence of a long-distance interaction with predominantly gravitational aspects. For the sake of simplicity, the case is narrowed down to the simplest processes (Definition 3.2.1.2), that is, to electrons, positrons, (anti-)neutrons and (anti-)protons. Let the k th individual process from the xth to the (x + 1)th degree of evolution be a simplest individual process for which EP x EP x ?? (1) ?k = EP x for some ? ? P (47) EP x ??? (1) ??k Then one of the two following cases hold: (i) if cn = 1 for the ? (1)th monad, then during the nonlocal mediation in this simplest process the nonlocal wavelike phase quantum N W ?xk has the tendency to effect a transition EP ?x?(1) ?N P ?x+1 ? (1) in the world towards a stronger gravitational field (higher energy density); (ii) if cn = ?1 for the ? (1)th monad, then during the nonlocal mediation in this simplest process the nonlocal wavelike phase quantum N W ?xk has the tendency to effect a transition EP ?x?(1) ?N P ?x+1 ? (1) in the world towards a weaker gravitational field (lower energy density); The n consecutive spatiotemporal positions Xx , Xx+1 , . . . , Xx+n?1 attained by n consecutive matter quanta EP ?x?(1) , . . . ,EP ?x+n?1 arising in a sequence of simplest processes in which gravitation is the ? (1) dominant factor thus depend on the characteristic number of normality cn ? {?1, 0, 1}, which yields Xj = X(cn )j . See Fig. 8 for an illustration. To see the symmetry, let the k th individual process from the xth to the (x+ 1)th degree of evolution be a simplest individual process involving the ? (1)th monad, for which cn = 1 (e.g. a proton). Let the lth individual process from the y th to the (y + 1)th degree of evolution be a simplest individual process involving the ? (1)th monad, for which cn = ?1 (e.g. an antiproton). Then the following nonlocal mediations happen in these two individual processes: N P x+1 EP x NW x ?? (1) ?? (1) ?k ? : EP x (48) NW x N P x+1 ??? (1) ??? (1) ??k ? EP y N P y+1 NW y ?? (2) ?? (2) ?l ? : EP y (49) NW y N P y+1 ??? (2) ??? (2) ??l ? Focussing on the gravitational aspects, then the nonlocal wavelike phase quantum N W ??xk in (48) has the EP x tendency to effect a transition N P ??x+1 ??? (1) towards a weaker gravitational field, a?nd the nonlocal ? (1) ? wavelike phase quantum N W ?yl in (49) has the tendency to effect a transition EP ?y? (2) ? N P ?y+1 ? (2) That is, the behavior of abnormal forms of matter (cn = ?1: positrons, antiprotons, antineutrons, etc.) in the world resembles the behavior of normal forms of matter (cn = 1: electrons, protons, neutrons) in the antiworld in opposite time-direction. Furthermore, while rest mass is connected with Newton?s theory of gravitation and all energy is connected with Einstein?s theory of gravitation, for the idea of gravitation in the context of the EPT the following relation is suggested for the gravitational mass mg of the the ? (1)th monad (e.g. a proton) occurring as a binad ?kx = EP ?x?(1) + N W ?xk in the k th individual process from the xth to the (x + 1)th degree of evolution: mg = cn и E(N W ?xk ) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (50) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 735 Fig. 8 Illustration of how matter and antimatter can behave differently in the gravitational field of the earth according to the EPT. The upper grey area depicts the gravitational field of the earth: the arrow on the right indicates the direction of increasing height (h) above the earth?s surface, and an increasing darker taint indicates a stronger gravitational field. The lower hatched area indicates the earth?s surface. The left dot indicates the position of the extended particlelike matter quantum EP ?x?(1) before the gravitational interaction. The two dots on the right indicate the position where its successor, the extended particlelike matter quantum EP ?x+1 ? (1) , arises a?fter the gravitational interaction: the upper right dot applies to the case that the ? (1)th monad concerns antimatter, and the lower right dot applies when it concerns matter. The upper arrow indicates the action of the intermediate nonlocal wavelike phase quantum for a constituent of antimatter (e.g. when the ? (1)th monad is an antineutron) with cn = ?1; the lower arrow indicates the action of the intermediate nonlocal wavelike phase quantum for a constituent of matter (e.g. when the ? (1)th monad is a neutron) with cn = ?1. This results in negative gravitational mass for antimatter components such as positrons, antineutrons, antiprotons, etc. The negative sign is thus not meant to indicate that the energy quantum, distributed in the corresponding nonlocal wavelike phase quantum is negative (which is not the case!): it merely indicates that the action of the corresponding nonlocal wavelike phase quantum is opposite. In the universe of the EPT, gravitational mass is thus a secondary property as meant by John Locke. From the inequality (10) it follows that |mg | ? m0 , cf. (5), where m0 = E(EP ?x?(1) ) > 0 is the rest mass of the ? (1)th monad at the xth degree of evolution as determined by the rest mass spectrum, cf. Definition 4.3.1.1 and Interpretation Rule 4.3.1.3. Concluding, gravitational repulsion of matter and antimatter can be descibed by the EPT; laws of conservation of energy have been given. Remark 4.3.2.1. Using the notion of a rest mass spectrum, consider the case that electrons have an increasing rest mass spectrum se and protons have a decreasing rest mass spectrum sp , that is, consider that the rest mass spectra se and sp of electrons and protons, respectively, satisfy the following conditions for all x ? ZN \{N ? 1}: se (x + 1) > se (x) (51) sp (x) = se (N ? 1 ? x) (52) This allows an answer to the question ?what makes the universe expand?? by de Sitter in [35]: the gradually disintegrating protons (i.e. the gradual decrease of rest mass of protons). Namely, in every individual process involving a proton, the positive amount of energy corresponding with the decrease in rest mass is www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 736 M. J. T. F. Cabbolet: Elementary Process Theory emitted in a local equilibrium according to elementary principle 3.1.3.10, and is contained in a local wavelike phase quantum in the form of a matter quantum ?kx+1 according to (23); the latter then form space in accordance with the elementary principle of formation of space 3.1.3.17. By this mechanism, the decrease in rest mass of protons leads to an increase in distance between extended particlelike phase quanta in the world ? the universe thus expands, as long as this effect exceeds the opposite effect caused by the increase in rest mass of electrons, that is, as long as sp (x + 1) + se (x + 1) < sp (x) + se (x) (53) The currently established rest mass ratio between protons and electrons, the estimated age of the universe and the expansion of the universe should yield the rest mass spectra of electrons and protons. Given that neutrons decay in electrons and protons, for the rest mass spectra sn of neutrons one gets the approximation sn (x) ? sn (x) + se (x). A simple idea for the rest mass spectra sn of neutrons is then sn (x) = sp (x) + se (x) + C (54) where C is a (small) real constant. As mentioned earlier, the rest mass spectra se? , sp? and sn? of positrons, antiprotons, and antineutrons, respectively, satisfy se = se? , sp = sp? and sn = sn? . This concludes this exposition on the EPT. Acknowledgements The author wishes to thank Sergey Sannikov, Institute of Theoretical Physics, Kharkov Institute for Physics and Technology (deceased the 25th of March, 2007) for his contribution to the development of the EPT. The author also wishes to thank Harrie de Swart, Group of Logic, Department of Philosophy, University of Tilburg, for his contribution to the formalization of the EPT. This work was financially supported by PlusPunt Eindhoven B.V., entry #823353. Additional financial support was given by the Foundation Liberalitas. Appendix A On set matrix theory In the mathematical-logical framework of set matrix theory, all terms are sets or set matrices. A set is an object made up of elements. For example, {1, 2} is the set made up of the numbers 1 and 2. Because a set is completely determined by its elements, the set {1, 1, 2} is thus identical to the set {1, 2}, and the set {1, 2} is identical to the set {2, 1}. A formula 1 ? {1, 2} means that 1 is an element of the set {1, 2}, and a formula 3 ? {1, 2} means that 3 is not an element of the set {1, 2}. A set x is a subset of a set y, notation: x ? y, if and only if all elements of x are also elements of y. Thus speaking, {1, 2} ? {1, 2, 3}. For any positive integers m and n, a m О n matrix is a rectangular object where m ? ? и n entries tij are ordered ? in m rows and n columns between square brackets, as in ? ? t11 .. . tm1 1 3 2 4 иии t1n .. ? ? . ?. For example, the object и и и tmn is a 2 О 2 matrix with integer-valued entries 1, 2, 3 and 4. In set matrix theory, the entries tij of ? t11 и и и t1n ? . .. ? ? a matrix ? . ? are allowed to be pij О qij matrices themselves, but in the end every matrix ? .. ? tm1 и и и tmn has to consist of a finitenumber of simple entries (sets); such a matrix is called a set matrix. For example, ? ? ? the object is a 2 О 3 set matrix where all entries are identical to the empty set ?. Moreover, ? ? ? c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 737 constant terms in this language do not necessarily have to be specific sets, such as the empty set ? or the set R of real numbers: one can introduce a set S as an individual constant, without specifying the set S any further. This feature, which is also a feature of the framework of ZF set theory, allows the application of the concept of a designator, widely used in logic, to physics. The mathematical axioms for the terms and the ?-symbol are laid down in set matrix theory: this is the axiom scheme, that provides a rigorous mathematical basis for the use of sets and sets matrices in one and the same framework. The main features of this framework are the following: (i) set matrices of any other dimension than 1 О 1 are not sets, and do not have elements in the sense of the ?-relation; (ii) set matrices of dimension 1 О 1, that is, set matrices of the type [x], are identical to their sole entry: [x] = x; (iii) set matrices of any dimension can be elements of sets; (iv) new sets can be constructed from existing ones using the familiar set-theoretical operations (union, intersection, powerset, etc.). ? ? ? on account of (i), but ? ? [{?}] on account of (ii). It should be noted Thus, for example, ? ? ? ? ? that in general any set is a 1 О 1 set matrix on account of (ii), so that the adage ?everything is a matrix? x x holds in the framework of set matrix theory. Furthermore, for any sets x and y, ? on y y x , has no sets as elements, because the 2 О 1 account of (iii). It should be noted that this latter set, y x set matrix is not a set on account of (i). So in this framework the empty set ? is the only set that y has no elements, but it is notthe only set that has no sets as elements. Looking at the language of the EP x ? EPT, the set matrix EP kx itself is thus not a set, and has no elements in the sense of the ?-relation. ??k Proceeding with the set-theoretical operations, the reader is assumed to be familiar with the usual ones like etc. So, for example, in the framework of set union, intersection, matrix theory the union of the sets x ? ? ? x ? ? ? and is the set , on account of (iv). Likewise, all y ? ? ? y ? ? ? other set theoretical operations that can be applied in usual set-theoretical framework, can also be applied in the framework of set matrix theory. For a rigorous introduction to the foundations of this mathematicallogical framework, the reader is referred to the literature, cf. [25]. References [1] [2] [3] [4] [5] [6] [7] [8] M. M. Nieto and T. Goldman, Phys. Rep. 205, 221?281 (1991). M. M. Nieto and T. Goldman, Phys. Rep. 216, 343 (Erratum) (1991) G. Chardin and J.-M. Rax, Phys. Lett. B 282, 256?262 (1992). M. Fischler, J. Lykken, and T. Roberts, Fermilab Report FERMILAB-FN-0822-CD-T, 13 pp., arXiv: 0808.3929v1 [hep-tp], (2008). P. Morrison, Am. J. Phys. 26, 358?368 (1958). L. I. Schiff, Phys. Rev. Lett. 1, 254?255 (1958). L. I. Schiff, Proc. Natl. Acad. Sci. 93, 69?80 (1959). M. L. Good, Phys. Rev. 121, 311?313 (1961). www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 738 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] M. J. T. F. Cabbolet: Elementary Process Theory M. Sachs, Einstein versus Bohr: the Continuing Controversies in Physics (Open Court, New York, 1988). C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, Cambridge, 1993). C. M. Will, Living Rev. Relativ. 4(4) (2001). C. A. Fuchs and A. Peres, Phys. Today 53(3), 70 (2000). P. Pe?rez and A. Rosowsky, Nucl. Instrum. Methods A 545, 20?30 (2005). W. Heisenberg, Physics and Philosophy, Modern Classics Edition (HarperCollins Publishers, New York, 2007) p. 67. P. R. Feynman, Phys. Rev. 76, 749?759 (1949). J. R. Shoenfield, Mathematical Logic (AK Peters Ltd., Natick, 2001), p. 22. R. M. Santilli, Int. J. Mod. Phys. A 14(10), 2205?2238 (1999). D. van Dantzig, Erkenntnis 7(1), 142?146 (1937). J. Bell, Physics 1(3), 195?200 (1964). D. Bohm, Phys. Rev. 85, 166?179 (1952). D. Bohm, Phys. Rev. 85, 180?193 (1952). F. W. Dyson, A. S. Eddington, and C. R. Davidson, Philos. Trans. R. Soc. Lond. A 220, 291?333 (1920). E. Hubble, Proc. Natl. Acad. Sci. (USA) 15(3), 168?173 (1929). G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34(2), 470?491 (1986). M. J. T. F. Cabbolet and H. C. M. de Swart, Set Matrix Theory as a Physically Motivated Generalization of Zermelo-Fraenkel Set Theory, submitted for publication. J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton Universitry Press, Princeton, 1955), pp. 3?4. S. S. Sannikov, Russ. Phys. J. 39(9), 767?775 (1996). H. von Helmholtz, Handbuch der physiologischen Optik (Hamburg, Leipzig, 1896); excerpt translated in Dutch language in W. M. Weber, De Meningen van the Filosofen, Deel I, (Uitgeverij Konstapel, Groningen, 1981), pp. 244?245. S. S. Sannikov, Theor. Math. Phys. 34, 21?29 (1978). C. Misner, Phys. Rev. Lett. 22, 1071?1074 (1969). A. Guth, Phys. Rev. D 23, 347 (1981). W. E. Pauli, Address to Group on Radioactivity, Tu?bingen, Germany, 1930 (unpublished letter). G. Galileo, Dialogo sopra i due massimi sistemi del mondo (Florence, Italy, 1632); discussed in B. F. Schutz, A First Course in General Relativity (Cambridge University Press, Cambridge, UK, 1990), p. 2. A. Einstein, Ann. Phys. (Berlin) 49, 769?822 (1916). W. de Sitter, Algemeen Handelsblad, July the 9th (1930); discussed in P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, 1993), p. 81. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org nal time-direction in the direction of evolution. Assuming the elementary principle of choice means that the ideas of a deterministic universe and of a probabilistic universe have to be rejected. Example 3.1.3.25. To illustrate the notion of parallel possible nonextended particlelike phase quanta, let the following be given for the k th process from the xth to the (x + 1)th degree of evolution: EP x EP x EP x ?? (1) ?k ?k (14) = EP x ? EP x ? ME EP x ??? (1) ??k ??k At the instant when the discrete transition EP ?x?(1) ? N W ?xk takes place ? and this transition will certainly take place because of the elementary principle of nonlocal equilibrium 3.1.3.6 ? the nonlocal wavelike phase quantum in the world N W ?xk has not yet collapsed. Thus, at said instant the collapse can still happen at every point of the spatial extension of the nonlocal wavelike phase quantum N W ?xk : every a priori possible point of collapse now corresponds with a parallel possible nonextended particlelike phase quantum N P x+1 NW x ?k . At the instant when the nonlocal wavelike phase quantum ?k collapses, the nonextended , and thus the component particlelike phase quantum N P ?x+1 k www.ann-phys.org NP NP ?x+1 k ??x+1 k , is chosen. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 720 M. J. T. F. Cabbolet: Elementary Process Theory Remark 3.1.3.26. (causal laws): The five principles of action 3.1.3.6, 3.1.3.8, 3.1.3.10, 3.1.3.12, and 3.1.3.17 are causal laws: all other causal laws are hereby rejected as invalid at the supersmall scale. The elementary principle of identity of binads 3.1.3.18 and the elementary principle of choice 3.1.3.22 are not causal laws, but are still synthetic propositions. In addition, the elementary principles of the EPT describe every individual process in the universe governed by the EPT, and by that the creation of every fundamental building block of this universe: the EPT is therefore physically complete. 3.2 Applications of the Elementary Process Theory 3.2.1 Formalization of some observed processes in the framework of the EPT In this section, first the notions of simple, simplest, and complex individual processes will be formalized in the framework of the EPT, and the concept of stepwise motion will be discussed. Next, a variety of observed process will be formalized in the framework of the EPT: currently, there is no other framework in physics in which all these processes can be formalized. Definition 3.2.1.1. The k th individual process from the xth to the (x + 1)th degree of evolution is simple if and only if in this k th individual process the following expressions are true: EP x EP x ?? (1) + . . . + EP ?x?(p) ?k (15) = EP x for some ? ? P and p ? Z + EP x ??? (1) + . . . + EP ??x?(p) ??k Furthermore, NP NP EP ?x?(1) + . . . + ?x+1 ? (1) + . . . + ??x+1 ? (1) + . . . + NP NP EP ?x+1 ? (p) ??x+1 ? (p) ?x?(p) = ?xk : EP x NW x ??? (1) + . . . + EP ??x?(p) ??k N P x+1 EP x NW x ?? (1) + . . . + N P ?x+1 ?? (1) + . . . + EP ?x?(p) ?k ? ? (p) : EP x NW x N P x+1 ??k ? ??? (1) + . . . + EP ??x?(p) ??? (1) + . . . + N P ??x+1 ? (p) N P x+1 N P x+1 LW x+1 ? (1) + . . . + ?? (p) ?k ? 0 : N P ?x+1 LW x+1 N P x+1 ??? (1) + . . . + ??? (p) ??k ? 0 N P x+1 EP x+1 LW x+1 ?? (1) + . . . + N P ?x+1 ?? (1) + . . . + EP ?x+1 ? ?k ? (p) ? (p) : N P x+1 LW x+1 EP x+1 ? ??? (1) + . . . + N P ??x+1 ??? (1) + . . . + EP ??x+1 ??k ? (p) ? (p) 0 0 ? ? ?x+1 k N P x+1 ??k NP NW (16) (17) (18) (19) (20) in expression (17). 2 In a simple individual process no nuclear reactions take place. This property is captured in expression (17) by conserving the right subscript index of the matter quanta; if a nuclear reaction occurs, these right subscript indices are not conserved (see below for examples). All motion under the influence of longdistance interactions is due to simple processes, regardless whether gravitational or electromagnetic aspects are predominant. A simple individual process can, of course, be succeeded by a new simple individual EP x+1 EP x+1 ? (1) + . . . + ?? (p) of (19) satisfies for some integer process. In that case, the superposition EP ?x+1 EP x+1 ??? (1) + . . . + ??? (p) EP x+1 EP x+1 ? (1) + . . . + EP ?x+1 ?l ? (p) l ? S?(x+1) the identity EP ?x+1 = while the expressions (15)/(19) x+1 EP x+1 ??? (1) + . . . + EP ??? (p) ??l then hold for the lth individual process from the (x + 1)th to the (x + 2)th degree of evolution. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 721 Definition 3.2.1.2. The k th individual process from the xth to the (x + 1)th degree of evolution is a simplest individual process if and only if this k th individual process is a simple individual process for which the following expression is true: EP x EP x ?? (1) ?k = EP x for some ? ? P (21) EP x ??? (1) ??k This is a special case of (15), meaning that in a simplest individual process the extended particlelike phase quantum in the world is simple, that is, is composed of only one extended particlelike matter quantum. 2 To sketch what happens in the world during a simplest process, let?s assume that the k th individual process from the xth tothe (x +1)th degree of evolution is a simplest individual process; the starting EP x EP x ?? (1) ? = , cf. formula (22). The extended particlelike matter point is the component EP kx EP x ??? (1) ??k quantum has an observable spatiotemporal position, say, X. In this individual process, in the world first spontaneously a discrete transition EP ?x?(1) ? N W ?xk takes place, cf. formula (16) with p = 1, whereby the nonlocal wavelike phase quantum N W ?xk comes into existence. In the antiworld, then the discrete transition N W ??xk ? EP ??x?(1) has happened. Next, in this individual process in the world the nonlocal wavelike phase quantum N W ?xk collapses after a finite amount of time into a nonextended particlelike matter EP x quantum N P ?x+1 ?? (1) to N P ?x+1 ? (1) , thus in effect bringing about a transition from ? (1) ; the right subscript index ? (1) is thus preserved in this latter transition. In the antiworld then the opposite has happened in accordance with the elementary principle 3.1.3.8. Finally, in this individual process in the world the local wavelike phase quantum LW ?x+1 is emitted from the nonextended particlelike matter quantum N P ?x+1 k ? (1) , N P x+1 EP x+1 ?? (1) ? ?? (1) . The extended particlelike which immediately brings about the discrete transition EP x+1 ?? (1) , formed in this last phase of the individual process, has then an observable spamatter quantum tiotemporal position Y . In the antiworld then the opposite has happened in accordance with the principles 3.1.3.10 and 3.1.3.12. This individual process has then brought about the stepwise motion from the extended particlelike matter quantum EP ?x?(1) at position X to the extended particlelike matter quantum EP ?x+1 ? (1) at position Y . For an observer, who himself is subjected to the same individual processes, this is observed as the motion of, say, an electron from X to Y . Two factors have then contributed to the change in observable position ?X = Y ? X: (i) the displacement effected by the nonlocal wavelike phase quantum N W ?xk ; (ii) an intermediate change of the state of the vacuum by the formation of spatial phase quanta. Concerning (i), the larger the displacement ?X effected, the larger E(N W ?xk ) in (10). That is, the larger the displacement ?X effected, the more energy is absorbed from the surroundings of the matter quantum EP ?x?(1) . These surroundings are a two-phased heterogeneous vacuum system, consisting of a curved space (the first phase composed of nonlocal wavelike phase quanta and spatial phase quanta) and a second phase, carried by the curved phase, composed of local wavelike phase quanta. In such a simplest process a long-distance interaction having gravitational and electromagnetic aspects take place: the principle of the gravitational aspect is that the nonlocal wavelike phase quantum N W ?xk interacts with the curved space in the sense that the displacement ?X depends on the metric of the curved space (the nonlocal wavelike phase quantum N W ?xk ?sees? the metric) and simultaneously the metric of the curved space depends on the nonlocal wavelike phase quantum N W ?xk ; the principle of the electromagnetic aspect is that the displacement ?X depends on the state of the second phase (the nonlocal wavelike phase quantum N W ?xk ?sees? the field of nonlocal wavelike phase quanta) and on the intermediate change of state of the vacuum. Definition 3.2.1.3. The k th individual process from the xth to the (x + 1)th degree of evolution is complex if and only if it is not simple. 2 www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 722 M. J. T. F. Cabbolet: Elementary Process Theory th Suppose process from the xth tothe (x + 1)th degree of evolution starts with a com the k individual EP x EP x ? + . . . + EP ?x?(p) ? ponent EP kx = EP ?x(1) , and suppose that in this k th process the nonlocal ??? (1) + . . . + EP ??x?(p) ??k N P x+1 N P x+1 ??(1) + . . . + N P ?x+1 ?k ?(q) mediation yields a component N P x+1 = N P x+1 for some ? ? P . If the ???(1) + . . . + N P ??x+1 ??k ?(q) EP x ?? (i) individual process is complex, then the number (p) of subcomponents EP x of the component ??? (i) N P x+1 EP x ??(j) ?k is not necessarily the same as the number (q) of subcomponents N P x+1 of the compoEP x ?? ???(j) k N P x+1 ? nent N P kx+1 . In any complex individual process, the following holds for the right subscript indices ??k of these components: {? (1), . . . , ? (p)} ? {?(1), . . . , ?(q)} = ? (22) That is, the extended particlelike matter quanta EP ?x?(i) , entering a complex individual process, and the extended particlelike matter quanta N P ?x+1 ?(j) , formed in a complex individual process, can not be attributed to the same monads. Example 3.2.1.4. (electron in an electron shell): A sequence of the simplest processes applies to any electron orbiting any atomic nucleus in the world. At n consecutive degrees of evolution the electron is x+n?1 x , . . . , ?k(n) , for which the identity 3.1.3.18 is valid. then designated by the n corresponding binads ?k(1) Here the right subscript index k(j) has not necessarily the same value for every value of j, because the successor of the k th individual process from the xth to the (x + 1)th degree of evolution is not necessarily the k th individual process from the (x + 1)th to the (x + 2)th degree of evolution. However, because of Definition 3.2.1.2, for these n binads the following identities do hold for some i ? Z + : x ? ?k(1) = EP ?xi + NW x+1 ? ?k(2) = EP ?x+1 + i ?xk(1) NW ?x+1 k(2) and so forth. Thus, to indicate that the n binads are to be attributed to the same electron in itself, the right subscript i of EP ?xi is conserved in a sequence of simple processes. x = EP ?xi + N W ?xk(1) then reads: the binad occurring in the world in the k(1)th The equation ?k(1) individual process from the xth to the (x + 1)th degree of evolution consists of the extended particlelike matter quantum of the ith monad at the xth degree of evolution and the nonlocal wavelike phase quantum occurring in the world at the xth degree of evolution in that individual process. Thus speaking, in this sequence of simplest processes the electron orbiting an atomic nucleus has consecutively n definite spatiotemporal positions Xx , Xx+1 , . . . , Xx+n?1 : these are the positions where the n extended particlelike matter quanta EP ?ix+j for j = 0 to n ? 1 happen to find themselves. Example 3.2.1.5. (free neutron gravitating towards earth): A sequence of the simplest processes also y y+p?1 applies to any free neutron gravitating towards earth. A sequence of p consecutive binads ?l(1) , . . . , ?l(p) then designates the neutron at p consecutive degrees of evolution y, y + 1, . . . , y + p ? 1. To these p binads the following equations apply for some j ? Z + : y ? ?l(1) = y+1 ? ?l(2) = EP ?yj + EP NW ?y+1 + j ?yl(1) NW ?y+1 l(2) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 723 and so forth. In the process of gravitating towards earth, the neutron then has p consecutive, definite spatiotemporal positions Yy , Yy+1 , . . . , Yy+p?1 corresponding with the p extended particlelike matter quanta EP y ?j , . . . ,EP ?y+p?1 . 2 j x+n?1 x , . . . , ?k(n) will be very different from the p Comparing the last two examples, the n binads ?k(1) y y+p?1 binads ?l(1) , . . . , ?l(p) , and the n ? 1 leaps between the n consecutive positions Xx , . . . , Xx+n?1 will be very different from the p ? 1 leaps between the p consecutive positions Yy , . . . , Yy+p?1 . The point is, however, that the elementary principles of action are exactly the same for both processes. That is, at the degree of abstractness of the EPT, there is absolutely no difference between an electron orbiting an atom and a neutron gravitating towards earth: all simplest processes are the same. Remark 3.2.1.6. (photons): In any sequence of simplest individual processes, the nonzero rest mass entity has a spatial momentum in the world when leaping from the j th to the (j + 1)th position. Being motionless, the nonextended particlelike matter quantum at the (j + 1)th position has no spatial momentum. The aforementioned spatial momentum is then conserved by a photon: photons are local wavelike matter quanta, that occur in the world in local wavelike phase quanta. Assuming that the k th individual process fromthe xth to the (x + 1)th degree of evolution is a simplest individual process, the component LW x+1 ?k is then a superposition LW x+1 ??k ?x+1 k LW x+1 ??k LW ?kx+1 ??kx+1 = + ?kx+1 ??kx+1 (23) where ?kx+1 is the photon emitted in the k th individual process from the xth to the (x + 1)th degree of evolution, and ?kx+1 is another local wavelike matter quantum emitted in this process (cf. Remark 4.3.2.1 for some elaboration). Example 3.2.1.7. (decay of a neutron): For some ? ? P , let the ? (1)th monad be a neutron, the ? (2)th a proton, and the ? (3)th an electron. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP ?xk EP x ??k NW ?xk NW x ??k = EP EP ?x+1 k LW x+1 ??k LW : EP EP : ?x?(1) (24) ??x?(1) ?x?(1) ??x?(1) NP NP ? ? ?x+1 ? (2) + ??x+1 ? (2) + NP NP NP NP ?x+1 ? (3) ??x+1 ? (3) ?x+1 ? (2) + ??x+1 ? (2) + ? ? NP NP ?x+1 ? (3) (25) ??x+1 ? (3) EP EP ?x+1 ? (2) + ??x+1 ? (2) + EP EP ?x+1 ? (3) ??x+1 ? (3) (26) In this process, in the world the extended particlelike matter quantum EP ?x?(1) , that can be attributed to a EP x+1 neutron, has decayed in a superposition EP ?x+1 ?? (3) of two extended particlelike matter quanta, ? (2) + that can be attributed to a proton and an electron. This process can then by two simplest be succeeded EP x+1 EP x+1 ?? (2) ?? (3) individual processes, the starting points of which are the components EP x+1 and EP x+1 in ??? (2) ??? (3) accordance with (22). In the individual process described above in the world a decay of a neutron occurs according to the decay reaction n ? p+ + e? + ? as originally proposed by Pauli in [32]. The neutrino ? is a local wavelike www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 724 M. J. T. F. Cabbolet: Elementary Process Theory matter quantum, that exists in the local wavelike phase quantum LW ?x+1 emitted in this process: k LW x+1 ?k ?kx+1 ?kx+1 ?kx+1 = + + LW x+1 ??kx+1 ??kx+1 ??k ?»kx+1 (27) Here ?kx+1 designates the neutrino in the world. Example 3.2.1.8. (annihilation of a proton/antiproton pair): For some ? ? P , let the ?(1)th monad be a proton, and the ?(2)th an antiproton. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP x EP x ??(1) + EP ?x?(2) ?k = EP x (28) EP x ??k ???(1) + EP ??x?(2) N P x+1 EP x NW x ??(3) ??(1) + EP ?x?(2) ? ?k : EP x (29) N P x+1 NW x ? ???(1) + EP ??x?(2) ???(3) ??k N P x+1 LW x+1 ??(3) ?k ? 0 : N P x+1 (30) LW x+1 ???(3) ??k ? 0 In this individual process, in the world a proton and an antiproton are annihilated: such an annihilation process applies to any other pair of nonzero rest mass entities (such as electron/positron, neutron/antineutron, etc.). The right subscript index ?(3) in the designator N P ?x+1 ?(3) refers to a monad, which has the property that it immediately decays completely into a local wavelike phase quantum, cf. Definition 3.1.3.14 with N P x+1 N P x+1 ? ?k = N P ?(3) . The annihilating mediation (30) is thus a special case of a local mediation N P x+1 ??x+1 ??k ?(3) EP x+1 EP x+1 EP x+1 ?? (1) ?? (1) ??(3) 0 3.1.3.12 with q = 1 and EP x+1 = (and EP x+1 = EP x+1 ). 0 ??? (1) ??? (1) ???(3) Example 3.2.1.9. (formation of Deuterium):For some ? ? P , let the ? (1)th monad be a proton, the ? (2)th a neutron, the ? (3)th a proton of a Deuterium nucleus, and the ? (4)th a neutron of a Deuterium nucleus. Let the k th individual process from the xth to the (x + 1)th degree of evolution be determined by the following: EP x EP x ?? (1) + EP ?x?(2) ?k = EP x (31) EP x ??? (1) + EP ??x?(2) ??k N P x+1 EP x NW x ?? (3) + N P ?x+1 ?? (1) + EP ?x?(2) ?k ? ? (4) : EP x (32) N P x+1 NW x ? ??? (1) + EP ??x?(2) ??? (3) + N P ??x+1 ??k ? (4) N P x+1 EP x+1 EP x+1 LW x+1 ?? (3) + N P ?x+1 ? + ? ? ?k ? (4) ? (3) ? (4) : N P x+1 (33) LW x+1 EP x+1 ? ??? (3) + N P ??x+1 ??? (3) + EP ??x+1 ??k ? (4) ? (4) In this individual process, in the world a proton and a neutron have formed a deuterium nucleus. This complex individual process is then succeeded by a simple individualprocess, say the lth individual process from EP x+1 EP x+1 EP x+1 ? + ? ?l ? (3) ? (4) th th the (x + 1) to the (x + 2) degree of evolution, for which EP x+1 = EP x+1 . ??? (3) + EP ??x+1 ??l ? (4) The free neutrons and protons are different from (that is: have to be attributed to other monads than) the neutrons and protons bound in deuterium nuclei, because their rest masses add up differently: the rest mass of a deuterium nucleus is different from the sum of the rest masses of a free proton and a free neutron. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 725 Other examples of complex processes will be given in the next section. The variety of complex individual processes is far more extended than these examples, but these examples demonstrate that the EPT applies to various nuclear reactions. 3.2.2 Theory of the Planck era of the universe The theory of the Planck era of the universe will be given below in Table 2 in the form of a sequence of formulas, describing events that follow one another in the direction of evolution. Technically this is a deduction within the axiomatic system containing the EPT: every formula of the sequence is either an assumption of the theory of the Planck era of the universe, or a corollary of a previous assumption and the EPT. The last corollary identifies a condensed matter field in a nonempty vacuum. Table 2 Theory of the Planck era of the universe; a superscript i in the first column refers to a Remark 3.2.2.i below. # Formula (i)1 (ii)2 (iii) 3 (iv) (v) 4 (vi) (vii)5 (viii)6 (ix) (x) 7 (xi) (xii) NP ?01 NP 0 ??1 Justification ? ME ? NP ?01 NP 0 ??1 = NP ?01 NP 0 ??1 assumption ?01 0 = LW 0 0 ??1 NP 0 EP 0 ?1 ?1 0 ? : NP 0 EP 0 0 ? ??1 ??1 EP 0 ?1 ? ME EP 0 ??1 EP 0 EP 0 ?1 ?1 = EP 0 EP 0 ??1 ??1 EP 0 NW 0 ?1 ?1 ? 0 : EP 0 NW 0 ? 0 ??1 ??1 NW 0 ?1 ? ME NW 0 ??1 EP 0 NW 0 ?1 ?1 ?10 = EP 0 + N W 0 ??1 ??10 ??1 NW 0 EP 0 NP 1 ?1 ?1 ?1 ? : EP 0 NW 0 NP 1 ? ??1 ??1 ??1 NP 1 NP 1 ?1 ?2 + . . . + N P ?12K+1 = NP 1 for some K ? Z + NP 1 ??2 + . . . + N P ??12K+1 ??1 NP 1 ?2 + . . . + N P ?12K+1 ? ME NP 1 ??2 + . . . + N P ??12K+1 NP 1 LW 1 0 ?2 + . . . + N P ?12K+1 ?1 ? : NP 1 LW 1 0 ? ??2 + . . . + N P ??12K+1 ??1 LW www.ann-phys.org assumption (i), (ii), 3.1.3.12 (i), (iii), 3.1.3.21 (iv), 3.1.3.15 (v), 3.1.3.6 (iv), (vi), 3.1.3.21 (v), 3.1.3.18 (v), 3.1.3.8 assumption (iv), (ix), 3.1.3.21 (x), 3.1.3.10 c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 726 M. J. T. F. Cabbolet: Elementary Process Theory Table 2 (Continued). # Formula (xiii) 8 (xiv) 9 (xv) LW ?11 LW 1 ??1 LW ?11 LW 1 ??1 EP Justification ? ME : NP ?11 NP 1 ??1 ?12 + . . . + EP 1 ??2 + . . . + (xi), (xii), 3.1.3.21 ? ? EP ?12K+1 EP 1 ??2K+1 EP ?12 + . . . + EP 1 ??2 + . . . + ? ME EP ?12K+1 EP 1 ??2K+1 (x), 3.1.3.12 (x), (xi), (xiv), 3.1.3.21 EP 1 ?12 + . . . + EP ?12K+1 ?1 (xvi) = EP 1 assumption EP 1 ??1 ??2 + . . . + EP ??12K+1 EP 1 NW 1 ?2 + . . . + EP ?12K+1 ?1 ? 0 (xvi), 3.1.3.6 (xvii) : EP 1 NW 1 ? 0 ??2 + . . . + EP ??12K+1 ??1 NW 1 ?1 (xviii) (xv), (xvii) , 3.1.3.21 ? ME NW 1 ??1 S 2 ?1 11 (xiii), 3.1.3.17 (xix) ? ME S 2 ??1 NW 1 EP 1 NP 2 ?1 ?2 + . . . + EP ?12K+1 ?1 ? (xx) : EP 1 (xvi), 3.1.3.8 NW 1 NP 2 ? ??2 + . . . + EP ??12K+1 ??1 ??1 NP 2 NP 2 ?1 ?2K+2 + . . . + N P ?22K+L+1 12 (xxi) = NP 2 assumption NP 2 ??2K+2 + . . . + N P ??22K+L+1 ??1 NP 2 ?2K+2 + . . . + N P ?22K+L+1 (xv), (xx), 3.1.3.21 ? ME (xxii) NP 2 ??2K+2 + . . . + N P ??22K+L+1 NP 2 LW 2 0 ?2K+2 + . . . + N P ?22K+L+1 ?1 ? (xxiii) : NP 2 (xxi), 3.1.3.10 LW 2 0 ? ??2K+2 + . . . + N P ??22K+L+1 ??1 LW 2 ?1 (xxiv) (xxii), (xxiii), 3.1.3.21 ? ME LW 2 ??1 LW 2 NP 2 EP 2 ?1 ?1 ?2K+2 + . . . + EP ?22K+L+1 ? (xxv) : NP 2 (xxi), 3.1.3.12 LW 2 EP 2 ??2K+2 + . . . + EP ??22K+L+1 ??1 ??1 ? EP 2 EP 2 ?1 + . . . + EP ?2?(2) ?2K+2 + . . . + EP ?22K+L+1 13 (xxvi) = EP 2 3.1.3.15, 3.1.2.2 EP 2 ??1 + . . . + EP ??2?(2) ??2K+2 + . . . + EP ??22K+L+1 EP 2 NW 2 0 ?k ?k ? (xxvii) : EP 2 3.1.3.6 for every k ? S?(2) NW 2 0 ? ??k ??k EP 2 NW 2 S 2 LW 2 ?1 + . . . +EP ?2?(2) ?1 + . . . +N W ?2?(2) ?1 ?1 14 + S 2 + LW 2 + N W 2 ? ME (xxviii) EP 2 ??1 + . . . +EP ??2?(2) ??1 ??1 ??1 + . . . +N W ??2?(2) 10 EP c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 727 Remark 3.2.2.1. The interpretation of assumption (i) is straightforward from Table 1: initially, the NP 0 ?1 component N P 0 exists, while the nonextended particlelike phase quantum N P ?01 is simple, and ??1 composed of the nonextended particlelike matter quantum N P ?01 that precedes the first monad. Following the convention of the interpretation rule strictly, N P ?01 is the nonextended particlelike phase quantum occurring in the world in the first (and only) process from the (N ? 1)th to the 0th degree of evolution. NP 0 ?1 This one process thus consists of the events succeeding the existence of the component N P 0 ? to be ??1 discussed ? and events at a higher degree of evolution that lead back to the initial com in this paragraph NP 0 ?1 ponent N P 0 : these events at the higher degree of evolution will then take place at a later time than ??1 N W N ?1 EP N ?1 NP 0 ? ?1 ?1 ?1 : EP N ?1 the initial events. In particular, the nonlocal mediation N W N ?1 NP 0 ??1 ??1 ??1 ? will take place at a later time. The degree of abstractness of the EPT thus enables one to state that it is certain that this mediation will happen, although it is now not known what the actual constitution of the EP N ?1 ? component EP 1N ?1 will be: the entries of the matrices are designators, not representations. ??1 Remark 3.2.2.2. In this initial individual process at the 0th degree of evolution, no local equilibrium takes place; the corresponding elementary principle of local equilibrium 3.1.3.10 is trivially true. It can be derived from this elementary principle that then a discrete transition N P ?01 ? 0 takes place in the world ? that is, nothing is emitted from the initial nonextended particlelike matter quantum N P ?01 . Remark 3.2.2.3. The of corollary (iii) is that the local equilibrium between the compo interpretation NP 0 EP 0 ? ? nents N P 10 and EP 10 is not mediated by any physical object (that is, by any nonzero compo??1 ??1 nent), but occurs spontaneously. Thus, in the world spontaneously a discrete transition N P ?01 ? EP ?01 takes place, and this is accompanied by a discrete transition EP ??01 ? N P ??01 in the antiworld. Energy is conserved: E(N P ?01 ) = E(EP ?01 ) (34) This equation can not be generalized to other processes: general laws of conservation of energy will be discussed in Sect. 4.2.2. Remark 3.2.2.4. At this point a new individual process begins in the numbering of individual processes, so it has to be taken that ?(0) = 1, so that S?(0) = {1}, cf. Definition 3.1.1.1. This means that there is precisely one individual process from the 0th to the 1st degree of evolution. Remark 3.2.2.5. The vacuum system in the world at the 0th degree of evolution is solely composed of the nonlocal wavelike phase quantum N W ?01 . In other words: no energy can be absorbed from the surroundings, so that E(N W ?01 ) = E(EP ?01 ) = E(EP ?01 ) (35) Compare (10) for the general case. The suggested duration of the life-time of N W ?01 is a Planck-time, so from here on time comes into existence. There is no such thing as a metric in this early vacuum system. Remark 3.2.2.6. From corollaries (iv), (vii), (viii), and Postulate 3.1.3.5 it follows that the following holds: EP 0 NW 0 ?1 ?1 ?10 (36) ? ME ? EP 0 ? ME ? N W 0 ? ME ??1 ??10 ??1 www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 728 M. J. T. F. Cabbolet: Elementary Process Theory The principle of particle/wave duality, Theorem 3.1.3.20, thus holds already at the 0th degree of evolution. Remark 3.2.2.7. Substituting assumption (x) in corollary (ix) yields the formula NW 0 EP 0 NP 1 ? ?1 ?1 ?2 + N P ?13 + . . . + N P ?12K+1 : EP 0 (37) NW 0 NP 1 ? ??1 ??2 + N P ??13 + . . . + N P ??12K+1 ??1 NW 0 EP 0 ?1 ?1 According to (37) the component N W 0 mediates an equilibrium between the components EP 0 ??1 ??1 NP 1 NP 1 NP 1 ? + ?3 + . . . + ?2K+1 . Following Interpretation Rule 3.1.3.9, this entails that the and N P 21 NP 1 NP 1 ??2 + ??3 + . . . + ??2K+1 nonlocal wavelike phase quantum N W ?01 collapses into the superposition N P ?12 + N P ?13 +. . .+ N P ?12K+1 of 2K nonextended particlelike matter quanta N P ?1i , each at a different position. Here K is a large integer, estimated in the order of magnitude of 1075 . In accordance with the general case (11), energy is conserved in this collapse: E(N W ?01 ) = E(N P ?12 + NP ?13 + . . . + NP ?12K+1 ) (38) The 2K monads numbered 2, 3, 4, . . . , 2K + 1 concern K pre-protons and K pre-electrons, see Remark 3.2.2.10 for further elaboration. Remark 3.2.2.8. The emitted local wavelike phase quantum LW ?11 is nonzero, so that the following inequality holds: E(LW ?11 ) ? 0 (39) Thus, energy is emitted into the surroundings. At this point in the early universe, spatial phase quanta, that together with nonlocal wavelike phase quanta are to function as ?carrier? for local wavelike phase quanta, do not yet exist: this first local wavelike phase quantum in the universe, LW ?11 , therefore has its spatial extension immediately. Furthermore, because there is not yet such a thing as momentum, the local wavelike phase quantum LW ?11 contains no photons. Remark 3.2.2.9. Substituting Axiom (x) in corollary (xiv) gives LW 1 NP 1 EP 1 ?1 ?2 + . . . + N P ?12K+1 ?2 + . . . + EP ?12K+1 ? : NP 1 (40) EP 1 LW 1 ??2 + . . . + N P ??12K+1 ??2 + . . . + EP ??12K+1 ??1 ? The right superscript of matter quanta is thus always conserved in a local mediation: there can be no exception. Loosely speaking, an object EP ?1j is the phenomenon that appears at a definite position when the j th monad exists in particlelike form, and N P ?1j is the energy needed at that position to get the j th monad existing there in particlelike form. Remark 3.2.2.10. Although all 2K extended particlelike matter quanta EP ?1j , formed in the one process from the 0th to the 1st degree of evolution, have a different position, there exists no real distance between the different matter quanta because the early vacuum has no metric. The 2K extended particlelike matter quanta EP ?1j thus constitute a single extended particlelike phase quantum EP ?11 . Thus, by no means, the 2K monads numbered 2 to 2K + 1 concern free protons and free electrons: together with the wavelike phase quanta LW ?11 and N W ?11 these matter quanta EP ?1j form what can be called a ?primordial soup?; the monads 2 to 2K + 1 are therefore to be called pre-electrons and pre-protons. The extended particlelike phase quantum formed, EP ?11 , is the starting point of the one process from the 1st to the 2nd degree of evolution. Thus speaking, it has to be taken that ?(1) = 1, so that S?(1) = {1}, cf. Definition 3.1.1.1. Using the general case (12) and inequality (39) it can then be derived from the foregoing that E(EP ?11 ) < E(EP ?01 ) c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (41) www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 729 The rest mass of the 1st monad at the 0th degree of evolution is thus larger than the sum of rest masses of the K pre-protons and K pre-electrons at the 1st degree of evolution; the one process from the 1st to the 2nd degree of evolution thus begins with less energy than the one process from the 0th to the 1st degree of evolution. Furthermore, in the one process from the 0th to the 1st degree of evolution the composite particlelike phase quantum EP ?11 has been formed out of the simple particlelike phase quantum EP ?01 , but that initial phase quantum EP ?01 existed in an empty space with no surroundings: it is therefore best to label the chain of events in the world as a decay reaction (nuclear disintegration) due to intrinsic instability, rather than as a weak interaction. Remark 3.2.2.11. Due to the fact that in the process from the 0th to the 1st degree of evolution energy has been emitted in the form of the local wavelike phase quantum LW ?11 , at the 2nd degree of evolution space is formed. Thus, while time already existed, cf. Remark 3.2.2.5, now also the three dimensional space exists. Remark 3.2.2.12. Substituting Axiom (xxi) in corollary (xx) yields the formula NW 1 EP 1 NP 2 ? ?1 ?2 + . . . + EP ?12K+1 ?2K+2 + . . . + N P ?22K+L+1 : EP 1 (42) NP 2 NW 1 ? ??2 + . . . + EP ??12K+1 ??2K+2 + . . . + N P ??22K+L+1 ??1 Following Interpretation Rule 3.1.3.9 this entails that the nonlocal wavelike phase quantum N W ?11 , which emerged from the superposition EP ?12 + . . . + EP ?12K+1 of 2K extended particlelike matter quanta, has collapsed into the superposition N P ?22K+2 + . . . + N P ?22K+L+1 of L nonextended particlelike matter quanta. While the 2K monads 2, 3, . . . , 2K + 1 concerned K pre-protons and K pre-electrons, the newly arisen matter quanta precede the L monads 2K + 2, 2K + 3, . . . , 2K + L + 1 that concern p protons, p electrons, and q neutrons with 2p + 2q = 2K. These protons, electrons, and neutrons will thus appear in particleform at the positions of these nonextended particlelike matter quanta N P ?2j in the superposition NP 2 ?2K+2 + . . . + N P ?22K+L+1 . Remark 3.2.2.13. Due to the existence of space at the 2nd degree of evolution, cf. Remark 3.2.2.11, the superposition EP ?22K+2 + . . . + EP ?22K+L+1 of L extended particlelike matter quanta no longer form a single phase quantum: instead, these L matter quanta form ?(2) extended particlelike phase quanta EP ?2i that are now spatially separated. If ?(2) = L, then all L extended particlelike matter quanta EP ?2j are spatially separated from each other, and each such matter quantum EP ?2j then on its own forms a phase quantum EP ?2i . If ?(2) < L, then there is at least one phase quantum EP ?2i composed of more then one matter quantum EP ?2j ? but still all phase quanta EP ?2k are spatially separated. The case ?(2) > L is physically impossible. In this one process from the 1st to the 2nd degree of evolution, in the world the phase quantum EP ?11 , which was composed of 2K subconstituents, has been transformed into ?(2) spatially separated extended particlelike phase quanta EP ?2i . The nonlocal wavelike phase quantum N W ?11 , which essentially effected the separation according to (xx) and (xxi), could ?see? the emitted local wavelike phase quantum LW ?11 ; the chain of events in the world is therefore best labeled an electroweak interaction. In addition, this yields a new view to the so-called ?horizon problem?, cf. [30], different from Guth?s inflation, cf. [31]. In a nutshell, this horizon problem is the following: taking into account the estimated age of the universe, then the currently most distant galaxies could never have originated from one point, even if they would have traveled close to the speed of light: at t = 0, they still were lightyears apart, see Fig. 6 for an illustration. The point of view that the EPT offers is that there is no such thing as the speed of light at the 0th and 1st degree of evolution, and thus no principle such as ?no object can travel faster than light? is valid in this early universe: for the nonlocal mediation in the one process from the 1st to the 2nd degree of evolution, cf. corollary (xx) and assumption (xxi), there is thus no restriction on the spatial positions at which the L nonextended particlelike matter quanta N P ?2j in the superposition N P ?22K+2 + . . . + N P ?22K+L+1 arise. The ?(2) extended particlelike phase quanta EP ?2i that are formed from there can thus be located at the www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 730 M. J. T. F. Cabbolet: Elementary Process Theory Fig. 6 Illustration of the horizon problem. In a space vs. time diagram, the lines (1) and (2) depict the path travelled by the objects, that are now furthest apart in the universe, assuming a speed close to the speed of light; the arrows (3) and (5) indicate the distance between the objects now and at t = 0, respectively. Fig. 7 Illustration in a space vs. time diagram of the approach to the horizon problem based on the EPT. The interpretations of the lines (1) and (2) and the arrow (3) are identical to those for Fig. 6. The array of dots schematically indicates the positions of the phase quanta in the superposition EP 2 ?1 + . . . +EP ?2?(2) ; the lowest dot indicates the position of the phase quantum EP ?11 ; the dotted arrows illustrate the action of the nonlocal wavelike phase quantum NW ?11 . 2nd degree of evolution at positions in a way that can not be reconciled with Einstein?s relativity. See Fig. 7 for an illustration of this approach to the horizon problem. Remark 3.2.2.14. The superposition S ?21 + LW ?21 + N W ?21 +. . .+ N W ?2?(2) + EP ?21 +. . .+ EP ?2?(2) concerns a condensed matter field in a heterogeneous vacuum system. The condensed matter field is the superposition EP ?21 + . . . + EP ?2?(2) which is composed of spatially separated phase quanta EP ?2j . The vacuum system is the superposition S ?21 + LW ?21 + N W ?21 + . . . + N W ?2?(2) , in which the superposition S 2 ?1 + N W ?21 + . . . + N W ?2?(2) is a homogenous phase, and the one local wavelike phase quantum LW 2 ?1 is a homogenous phase. The homogenous phase S ?21 + N W ?21 + . . . + N W ?2?(2) is observable as a space with non-Euclidean geometry: the constituent S ?21 in itself has a homogenous energy density, but is transcended by the constituent N W ?21 + . . . + N W ?2?(2) , which is the source of curvature, that is, of differences in energy density. If this homogenous phase could be modeled by a four-dimensional manifold with a metric g, then the metric would depend on the phase quanta: g = g(S ?21 , N W ?21 , . . . , N W ?2?(2) ). This homogenous phase acts as a carrier for the second homogenous phase, at the 2nd degree of evolution formed by the one local wavelike phase quantum LW ?21 , which spreads out with the speed of light. 4 Discussion 4.1 Relation of the EPT with established theories 4.1.1 Incompatibility of the EPT and QM Proposition 4.1.1.1. The theory, obtained by extending the EPT with the following translation of the orthodox quantum-mechanical view on nonzero rest mass entities in the language of the EPT, is inconsistent: EP x ?k ?kx ? ME ? EP x ? ME (43) ?x ? ZN ?k ? S?(x) ??kx ??k P r o o f. Formula (43) is a contraposition of Theorem 3.1.3.20. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 731 The orthodox position of QM implies that observable nonzero rest mass entities, such as electrons, have no definite position in absence of observation. Translated in the language of the EPT, this would imply that (43) would be true. Obviously, this view yields a contradiction with the EPT. To spell it out, QM is incompatible with the EPT. One of the fundamental differences, expressed by the fact that Theorem 3.1.3.20 and formula (43) contradict each other, is that according to QM the wave function of a microsystem does not collapse in absence of observation, while according to the EPT nonlocal wavelike phase quanta spontaneously collapse regardless whether someone is watching or not. 4.1.2 Incompatibility of the EPT and GR Proposition 4.1.2.1. The theory, obtained by extending the EPT with the following translation of the classical view on nonzero rest mass entities in the language of the EPT, is inconsistent: NW x ?kx ?k ?x ? ZN ?k ? S?(x) ? ME ? N W x ? ME (44) ??kx ??k P r o o f. Formula (44) is a contraposition of Theorem 3.1.3.20. The classical view, which is also incorporated in GR (General Relativity), implies that an observable nonzero rest mass entity is in a particlelike state at every point of its worldline. Translated in the language of the EPT, this would imply that (44) would be true. Obviously, this view yields a contradiction with the EPT. To spell it out, GR is incompatible with the EPT. One of the fundamental differences, expressed by the fact that Theorem 3.1.3.20 and formula (44) contradict each other, is that according to GR a nonzero rest mass particle does not spontaneously transform into a wavelike state, while according to the EPT extended particlelike phase quanta spontaneously transform into nonlocal wavelike phase quanta. Given that both cornerstones of modern physics, QM and GR, are incompatible with the EPT, the EPT the EPT should thus not be mistaken for an attempt to unify GR and QM: instead, it has to be seen as (a proposal for) the fundamental laws of physics governing the supersmall scale. 4.1.3 Relation with Special Relativity Arriving at the relation between the EPT and Special Relativity (SR), it should be noted that the latter entails a rejection of the idea of an aether, while in the context of the former spatial phase quanta occur as energetic constituents of the vacuum system. Although it is widely believed that the idea of an aether has been disproven by the Michelson-Morley experiment, the implications of the outcome of this experiment should be reviewed: what has been falsified, namely, is the idea of an aether such as mathematically represented by classical theory. In particular, what has been falsified is that the Galilean law of velocity transformation is universally applicable. Proceeding, the first major point is that the Galilean law of relativity, that it is not possible to determine the absolute velocity of a nonzero rest mass entity, cf. [33], has no meaning in the context of the EPT: at the supersmall scale, where the concept of stepwise motion applies, there is no such thing as the ?velocity? of an electron, of a proton, etc. To put that in other words: in the context of the EPT, velocity is a secondary property as meant by John Locke ? it is present in the observation of the object but not in the object itself. Thus, one can perform measurements on microsystems and use the obtained results to calculate a value that can be called ?velocity?, but the idea, that the calculated velocity then corresponds to a really existing property of the material object that was subjected to measurement (e.g. an electron), is purely classical, and is connected to the concept of continuous motion of classical mechanics. In the context of the EPT, particlelike phase quanta do not move at all, and thus simply have no velocity. In other words, the Galilean principle of special relativity has no meaning in the context of the EPT. The second of the two postulates of SR, Einstein?s principle of universality of the speed of light is to be retained in the framework of the EPT. It should be noted that nonextended particlelike phase quanta, www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 732 M. J. T. F. Cabbolet: Elementary Process Theory from which local wavelike phase quanta (and thus light ? photons occur in such phase quanta, cf. Remark 3.2.1.6) are emitted, do not move at all: therefore, there is no such thing as ?the motion of the light?s source? in the framework of the EPT, nor does the Galilean law of velocity transformation, falsified by the Michelson-Morley experiment, apply to light in the framework of the EPT. The speed of change of the changing spatial extension of the local wavelike phase quanta is identical to the speed of light, and this speed of light is then a property of the vacuum system, having at every point of position space the same value for all observers. Einstein?s principle of universality of the speed of light thus remains valid in the framework of the EPT; it should be noted that the principle is then analytic in the context of the EPT. A quantitative formulation of the principle is to be incorporated in a mathematical model of the EPT. 4.2 Relation of the EPT with established principles 4.2.1 The EPT vs. the general principle of relativity Proposition 4.2.1.1. The EPT is in agreement with the general principle of relativity. P r o o f. The following four statements hold: (i) a degree of evolution is the same for all observers; (ii) the kind of a phase quantum is the same for all observers; (iii) the elementary principles, laid down in the EPT, do not depend on the numbering of the individual processes; (iv) the elementary principles, laid down in the EPT, do not depend on the numbering of the matter quanta. From this it follows that the elementary principles of the EPT are the same for all observers, and the EPT hence satisfies the general principle of relativity. This general principle of relativity is not formulated exactly, that is, word for word, the same as the general principle of relativity formulated by Einstein: the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion [34]. But because of the similarity in intended meaning, the same name was given to the present principle. 4.2.2 The EPT vs. the law of conservation of energy The laws of conservation of energy for any individual process are given by Eqs. (11) and (12). Equation (11) expresses that no energy is lost in the collapse of the nonlocal wavelike phase quantum N W ?xk . Equation (12) expresses that no energy is lost in the decay of the nonextended particlelike phase quantum N P ?x+1 . k A corollary of (11) and (12) is the following law: E(EP ?x+1 ? (1) + . . . + EP EP x ?x+1 ?k ) = E(N W ?xk ) ? E(EP ?xk ) ? E(LW ?x+1 ) k ? (q) ) ? E( (45) Concerning the right-hand term, it is always the case that E(N W ?xk ) ? E(EP ?xk ), cf. (10). That is, the amount of energy distributed in the nonlocal wavelike phase quantum N W ?xk is more than or equal to the amount of energy distributed in the extended particlelike phase quantum EP ?xk from which it originated. The difference E(N W ?xk ) ? E(EP ?xk ) ? 0 is absorbed from the vacuum (the surrounding system) in the first phase of the individual process. Keeping in mind that the local wavelike phase quantum LW ?x+1 is k emitted, the right hand term of (45) thus designates the net amount of energy exchanged with the vacuum. The left hand side of (45) is merely the difference in energy between ?output? and ?input? of the process: this is, thus, exactly equal to the net amount of energy exchanged with the vacuum. c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 522, No. 10 (2010) 733 Laws of conservation of energy thus have been formulated for the EPT, but it is emphasized that ?energy? is a primitive notion here. In further research it still has to be proven that this relates to the macroscopic notion of energy defined as the ability to do work. Such a proof is best given on the basis of a mathematical model of the EPT, satisfying the conditions set forth in Sect. 1.3. 4.3 Relation of the EPT with the hypothesis on gravitational repulsion 4.3.1 Extension of the language for the EPT Definition 4.3.1.1. Let the for language for the EPT, given by Definition 3.1.1.1, be extended with the following individual constants: (i) the subset {?1, 0, 1} of the set Z of integers, elements of which are to be denoted by a symbol cn : cn ? {?1, 0, 1} ? Z (ii) the set RZN of functions from ZN to the real numbers R, elements of which are to be denoted by a symbol s: s = {[0 m0 ], [1 m1 ], . . . , [N ? 1 mN ?1 ]} Here the 2 О 1 matrices [a b] represent two-tuples a, b. Interpretation Rule 4.3.1.2. An element cn ? {?1, 0, 1} denotes a characteristic number of normality; this is an essential property of a monad according to the following guidelines: (i) cn = 1 for all normal matter such as protons, electrons, neutrons; (ii) cn = ?1 for all abnormal matter such as antiprotons, positrons, antineutrons; (iii) cn = 0 for all annihilating monads, cf. Example 3.2.1.8. Interpretation Rule 4.3.1.3. An element s ? RZN denotes a rest mass spectrum; this is an essential property of a monad. If the j th monad has the rest mass spectrum s, and if [x mx ] ? s, then the amount of energy, distributed in the extended particlelike matter quantum EP ?xj is identical to mx , that is, if the extended particlelike matter quantum EP ?xj indeed exists. In other words, for every monad, the rest mass is predetermined at every degree of evolution. 2 Thus, using Remarks 3.1.2.4 and 3.1.2.5, if the j th monad is, for example, a free antiproton with rest mass spectrum sp? , then in accordance with (6) and (7) one would get: EP x EP x ?j ?k (46) = EP x ? ME ? EP ?xj = sp? (x) EP x ??j ??k All electrons and all positrons have one and the same rest mass spectrum. This is also the case for all free protons and all free antiprotons, and also for all free neutrons and all free antineutrons. More general: every nonzero rest mass matter entity and its antimatter counterpart have the same rest mass spectrum. Interpretation rule 4.3.1.3 thus covers the observation that matter particles and their antimatter counterparts have the same rest mass. 4.3.2 Explanation of gravitational repulsion of matter and antimatter Given that the EPT contradicts the accepted theory of gravitation (GR), cf. Proposition 4.1.2.1, it has to be redefined what ?to gravitate? is in order to explain the original hypothesis on gravitational repulsion. Defining ?to gravitate? now as ?to move under the influence of a long-distance interaction with predominantly gravitational aspects?, then in the universe governed by the EPT nonzero rest mass entities like www.ann-phys.org c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 734 M. J. T. F. Cabbolet: Elementary Process Theory electrons, positrons, (anti-)neutrons and (anti-)protons gravitate in a wave-state, while the actual displacement is determined by the reaction of these wave-states to the state of the vacuum system. The difference in the assigned characteristic number of normality, cf. Interpretation Rule 4.3.1.2, can now be used to explain how matter and antimatter can behave differently under the influence of a long-distance interaction with predominantly gravitational aspects. For the sake of simplicity, the case is narrowed down to the simplest processes (Definition 3.2.1.2), that is, to electrons, positrons, (anti-)neutrons and (anti-)protons. Let the k th individual process from the xth to the (x + 1)th degree of evolution be a simplest individual process for which EP x EP x ?? (1) ?k = EP x for some ? ? P (47) EP x ??? (1) ??k Then one of the two following cases hold: (i) if cn = 1 for the ? (1)th monad, then during the nonlocal mediation in this simplest process the nonlocal wavelike phase quantum N W ?xk has the tendency to effect a transition EP ?x?(1) ?N P ?x+1 ? (1) in the world towards a stronger gravitational field (higher energy density); (ii) if cn = ?1 for the ? (1)th monad, then during the nonlocal mediation in this simplest process the nonlocal wavelike phase quantum N W ?xk has the tendency to effect a transition EP ?x?(1) ?N P ?x+1 ? (1) in the world towards a weaker gravitational field (lower energy density); The n consecutive spatiotemporal positions Xx , Xx+1 , . . . , Xx+n?1 attained by n consecutive matter quanta EP ?x?(1) , . . . ,EP ?x+n?1 arising in a sequence of simplest processes in which gravitation is the ? (1) dominant factor thus depend on the characteristic number of normality cn ? {?1, 0, 1}, which yields Xj = X(cn )j . See Fig. 8 for an illustration. To see the symmetry, let the k th individual process from the xth to the (x+ 1)th degree of evolution be a simplest individual process involving the ? (1)th monad, for which cn = 1 (e.g. a proton). Let the lth individual process from the y th to the (y + 1)th degree of evolution be a simplest individual process involving the ? (1)th monad, for which cn = ?1 (e.g. an antiproton). Then the following nonlocal mediations happen in these two individual processes: N P x+1 EP x NW x ?? (1) ?? (1) ?k ? : EP x (48) NW x N P x+1 ??? (1) ??? (1) ??k ? EP y N P y+1 NW y ?? (2) ?? (2) ?l ? : EP y (49) NW y N P y+1 ??? (2) ??? (2) ??l ? Focussing on the gravitational

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