Four-dimensional Reality and its Coherence

An Outline of Rietdijk's Theory on Physics


C.W. Rietdijk, D.Sc.


Truth can be recognized by its

simplicity and its beauty.

Richard Feynman




  1. We prove from relativity theory that the world is realistically four-dimensional; the future is, and it is determined. It even contributes backwards in time to what happens now: natural law too acts in four dimensions.
  2. A new, four-dimensional, concept of distance is introduced in which "occurring" (action) rather than objects (measuring rods) is essential. It explains the quantum-mechanical nonlocality paradoxes, such as the wavelike manifestation of particles and the intriguing paradox of Einstein, Podolsky and Rosen.
  3. From the above, Planck's "quantum of action" h can be conceived as a realistic four-dimensional "atom of occurring". The existence in time of particles and all other processes are constituted by such quanta.
  4. A clear, realistic and imaginable model of the quantum of action is constructed, from which various new results can be derived, such as three of the four fundamental forces – strong, weak and electromagnetic –, the fractional electric charges  e and  e of quarks, and the size of the proton.
  5. Particles in the corpuscular state and those in the wave state are considered to carry physical information in two different but isomorphic data codes. Think of how DNA "helices" encode human characteristics in a different language. This contributes much to making imaginable models of quantum processes.
  6. Our theory derives some ratios of the masses of elementary particles, and of fundamental forces.
  7. We investigate how compound particles originate from simpler components, using detailed properties of action quanta. Within this scope we show how protons or other compounds manage to produce waves with  = h/mv in which m is the mass of the compound.
  8. A "Mendelejev system" of elementary particles – electrons, neutrinos, quarks, protons, muons, W and Z bosons, virtual and normal photons – is constructed that coherently shows their existence in time to be built up as various structures from the realistic action quanta of 3. and 4. above. In such system, neither virtual photons nor heavy bosons need "borrow energy from God" any longer.
  9. An explanation of Einstein's "hidden variables" is introduced that also calls on the four-dimensional character of natural law and the action metric of 2.
  10. The philosophical consequences of the above theory appear to be radical. Inter alia, it surprisingly reconciles the standpoints of Einstein and Bohr in their famous controversy: "God does not play dice" (Einstein) and "A microprocess acts as a whole and is often only defined by an act of observation" (Bohr) both appear to be correct. Determinism and local "uncertainty" appear to go together.



1. Special relativity implies that the world is realistically four-dimensional and, therefore, deterministic


In Ref. 1 a first rigorous demonstration has been given that the universe cannot but be realistically four-dimensional. That is, both future and past have the intrinsic (ontological) state of real existence, be it beyond our present human capacity of experience. The proof is based on the Special Theory of Relativity and holds on the (obvious) condition that reality at a distance for various observers does not violate natural laws such as the constancy of the velocity of light c.

            Some more demonstrations can be found in Refs. 2–5. The gist of one among them follows below.


Figure 1. The spacetime coordinate systems (x,O,t) and (x',O,t') of two observers S and T, respectively, according to relativity theory. S and T pass each other at O, at t = t' = 0. Their respective coordinate systems imply that T moves to the right with respect to S.


Consider Fig. 1, roughly explained in the caption. It sketches the mutual situation of two observers S and T, both at O. S and T have worldlines COD and POQ, respectively, and a mirror M has worldline m. A worldline is the path of some entity through space-time.

            S sent a light signal L from C to mirror M. It is reflected at A and recaptured by S at D later. Because according to relativity the velocity of L – just as that of all light signals in vacuum – is c, L takes equal times for its movement from S at C to M at A and its way back to S at D from the reflection at A. Hence CO = OD. This implies that S at his passing T at O can legitimately say: "At this very moment, now, my signal L is reflected at A, which is a realistic phenomenon at a distance for me". If S were not entitled to conclude this at O, the thesis that L takes equal times for its travels CA and AD and that CO = OD would not hold. Conclusion: the reflection of L at A actually appears and exists now for S at O.

            However, T at O near S experiences the world relativistically too. If he sent a light signal L' from P to M it is reflected at B and recaptured by T at Q. The circumstance that L' has the same velocity c on its paths PB and BQ is reflected in our figure by PO = OQ: L' takes equal times for the way to B and the one back to Q.

            But then T at O is fully entitled to conclude and say to S: "At this very moment my L' is realistically reflected at B, which is still future for your present A, and hence such future is real and determined". Again, B's reality for T at O can only be denied on pain of denying that L' has the same velocity on its paths PB and BQ.

            We can easily see that the thought experiment of Fig. 1 can be varied so as to make any arbitrary event in my future to be past or present for some well-chosen other observer in my present.

            It is a remarkable fact that the demonstration of Ref. 1 evoked a mere one opposing reaction, as far as I know, viz. the one of Ref. 6. The other demonstrations, of Refs. 2–5, remained unchallenged. In Ref. 7 I discuss the argument of Ref. 6.

            The realistic four-dimensionality of the universe has important consequences for physics in general. For it is obvious that if nature is four-dimensional, natural law too is probably four-dimensional. That is, to some extent nature can be expected also to function four-dimensionally, past, present and future possibly being mutually related in more comprehensive and coherent ways than has been thought up to now. In section 2. below we will see a first specimen of this. And, of course, things like coincidence, free will and possibly evolution are put in a different light. In what follows we will see many more aspects of realistic four-dimensionality.

Remark. Some physicists used to deny that we may start from the idea that, in going there and going back of a light signal (such as L and L' in Fig. 1), its velocity is the same in both cases: we should merely be permitted to conclude from experience that ½(c1 + c2) = c, in obvious notation. However, in Ref. 8 I demonstrated that c1 = c2 = c can be experimentally established. This refutes a possible objection to the above argument.



2. Retroaction: the future can influence the present – within uncertainty margins


In Fig. 2 we vary Young's well-known experiment. Remind that in the usual set-up the photon or particle waves interfere between screens S and T after having passed the slits A and B. This results in interference fringes on T as to the distribution of the "corpuscular" impacts of the momentum carriers. If we remove T, such carriers will land on one of the plates. We manage the produced parts of all of these to pass through C between A and B. This results in the "interference" pattern to change because now the waves from A can only hit the upper sides of some plate, whereas waves from B only hit some lower side. That is: A and B waves now act mutually independently, without interfering (and land on upper and lower sides, respectively). But then we will see on both sides a "one-slit pattern", which roughly amounts to an even distribution of both A and B impacts rather than the fringe one.


Figure 2. A variant of Young's double-slit interference experiment. We consider it on a very large scale. If one removes screen T, the momentum carriers land on the plates of system P.


            Now consider this (thought) experiment on such scale that the momentum carriers need an hour to reach the T region from S, whereas an experimenter with T decides 59 minutes after they passed S whether he will remove T or not, one minute before the carriers hit either T or the P system. Then the conclusion is remarkable. For the momentum carriers cannot change their momentum in the vacuum between S and T, but at the latest can do so in their interactions with S. But then such interactions should decide whether the carriers are attuned to a fringe pattern of impacts or a more even distribution! Note that the relevant decision – about a removal of T – will be made 59 minutes after the carriers pass S. Hence we see here such decision casting its shadows 59 minutes before in interfering with the momentum exchange in A and B. Or rather, the consequences of the decision (viz. the impact-events) will have influence 60 minutes earlier. This is a clear instance of retroaction: physical influence backwards in time. It is easily seen that such influence has only the latitute of quantum uncertainties (the range of the waves) and therefore never violates causality.

            The above proof is from Ref. 9, and alternative demonstrations of retroaction have been given in Refs. 4, 10 and 11. These and the above one remained unchallenged too.

            Note that retroaction is not merely a striking example of the realistically four-dimensional character of natural law spoken of in Sect. 1., but also constitutes an additional proof of the realistic nature of the future: if at some occasions it has influence, such future should somehow be defined. (In Refs. 12 and 13 we discussed explanations of retroaction.)



3. Logically, the stuff of the four-dimensional world is events – occurring or action – rather than objects. We integrate spacetime and energy-momentum into action. This is in line with – a continuation of – the integration of space and time by Special Relativity


Special Relativity integrated space and time into spacetime. Also it integrated energy and momentum into energy-momentum. (This is a four-vector in Minkowski-space.) Such integration, inter alia, solved the problem of the constancy of the velocity of light. A paradox with respect to this originated from our mistakenly conceiving space and time as separate, independent "rigid" entities or characteristics of the universe. What we now intend to do is further integrating space, time, energy and momentum – that is, spacetime and energy-momentum – jointly into action. Just as space and time follow from spacetime (becoming less rigid in the process), we now intend to derive distance, time, energy and momentum – and even many things more – from action. The latter then becomes the basic stuff of the four-dimensional universe. In deriving distance etcetera from action, they become less independent, less absolute and less "rigid". Again, such new integration and relativation produce the solution of some fundamental problems and paradoxes. We will see in what follows that the nonlocality paradoxes of quantum mechanics (QM) are among them, just as those about wave-particle "duality" and uncertainty. They originate from similar "absolutistic" assumptions about the "components" of action (such as spacetime) as the prerelativistic prejudices were about separate space and time. (Especially compare Refs. 4 and 12.)

            Concretely, space and time were integrated by the relation s2 = c2t2 - x2 - y2 - z2, in which the "relative" x, y, z, t produced the invariant or more "objective" spacetime distance s. In the further integration we now propose, the "objective" action W = Et -  is produced by the more relative spacetime four-vector  = (ict, x, y, z) and the energy-momentum four-vector  = (i/c E, px, py, pz) of which W is the inner product.

            It is one of the consequences of our further "de-absolutizing" space and time (viz. relativizing even spacetime) that, e.g., there is no more any inherent physical difference between, say, "3 meters of vacuum" and "5 meters of vacuum". Neither does it make sense to say that 3 rather than 5 seconds elapsed in a certain vacuum domain. Such amounts of space and time can only be derived from real physical processes. That is, from action. Sects. 4 and 5 elaborate.

            In connection with our intended reduction of everything physical to action mind that this fits in with various other indications of the essential role of action. Think of the Principle of least action, of Noether's Theorem about conservation, the derivation of all other quantizations (and deviations from Classical Theory) from the quantization of action, and the predominant role of the Lagrangian (action density).



4. A new metric based on action: distances between events, measured in amounts of "occurring" (action). It explains nonlocality in QM: the appearance of matter waves, particles interacting with a whole grating, and the paradox of Einstein, Podolsky and Rosen (EPR)


Figure 3. Four-dimensional picture of the wave pattern of a uniformly moving particle. The action distance between P, Q and R is zero. Hence the particle is at P, Q and R "at the same time". The slices are wavelike four-dimensional action quanta.


Consider Fig. 3 that shows a uniformly moving particle S with worldline OE. E.g., counted from O, point-events P, Q and R all have an action 3½ h if we take periods such as BC to correspond with one de Broglie clock tick in the existence of S. That is, BC corresponds with the duration of one action quantum in the existence in time of S. We consider such existence to be nothing but a series of action quanta succeeding each other in time. This is within the scope of our constructing the entire four-dimensional world from mere action and its quanta.

            In Fig. 3, we consider S to be in the wavelike state. Slices such as between m and n [which are hyperplanes in four-dimensional (Minkowski) space] represent one quantum of action, whereas OA is a wavelength as (indirectly) observed in his tree-dimensional "now" hyperplane by an observer whose coordinate system is (x,O,t). The rest system of S is (x',O,t').

            Now we introduce action distance – as relevant to two point-events in a process F – as follows: The action distance between point-events G and H in F is the amount of action (of "occurring") needed to get from G to H (or conversely). Hence in Fig. 3 the action distance between O and C is 4h and that between P, Q and R is zero because all correspond to the same action 3½ h as we saw. We can also say: The action distance between subprocesses G and H of the process F (say, constituted by the moving particle S) is the amount of action needed to transform sub-event G into sub-event H. This is quite natural a measure or gauge for "distance" in a four-dimensional world in which processes or "occurring" are central, just as objects are so in three-dimensional space. Therefore, we use standard objects such as measuring rods there to gauge distances. Our gauge in the realistic four-dimensional space of events is action, in units h.

            Now it is clear from the figure how matter waves can appear at all. For if P, Q and R all have the same action 3½ h and the amount of action needed to get from P to R (or: to transform situation P into situation R, or into Q) is zero (from 3½ h to 3½ h), there is no real physical difference between situations P, Q and R, at least as far as the internal behaviour of our process F (the moving S) is at stake. This means that the action-quantal slices – such as the one wherein we see P, Q and R – are indeed "stretched" from our point of view (in both Euclidean and Minkowski metric), but are not at all so in the internal action metric of our process F. For the latter, P, Q and R are mutually contiguous, form a "compact whole". We see something similar as with the discrepancy between Euclidean and relativistic metric.

            The above explains why "extended", wavelike particles can interact with a grating as a whole, and also Bohr's "A microprocess constitutes a whole" at all.

            For instance, consider Young's double-slit experiment of Fig. 2: if the plane waves approach S and the slits A and B, the action distance between wave parts corresponding to A and B, respectively, is zero, just as that between P and Q in Fig. 3. Hence, for a relevant process of a moving momentum carrier internally, there is no difference between such wave parts at all, measured in the essential action metric. For this reason, the momentum carrier "passes through A as well as B".

            It has indeed to be noted that action metric each time refers to a specific process, for which nature actually appears "to consider it relevant". E.g., for other processes than moving particle S of Fig. 3, P, Q and R may have quite different action values. Still, the various action situations in the many processes that appear, jointly build up the macro Minkowski metric as a rough scheme, as is elucidated in Sect. 5.


Figure 4. Four-dimensional EPR situation. The two correlated systems are separated at E and measured at A and B, respectively.


Figure 5. A spherical-waveslice picture of the same EPR situation. Via C the action distance between the two measurement events at A and B is zero; hence these can show feedback communication.


The paradox of Einstein, Podolsky and Rosen is one more instance of nonlocality that can be solved by our new concept of action metric. First consider Fig. 4, where, say, two photons are emitted from E, and later are measured as regards their polarization directions. In some special cases the two photons are "correlated", which not only means that their polarization directions happen to be orthogonal "as separate conditions", but also shows a more "deep" feature. Viz. that, each time we set up two analysers at A and B, respectively, whose directions (in which photons can pass) are orthogonal too, both photons pass or both are absorbed. This phenomenon excludes the possibility that each of them merely locally interacts with one of the analysers (see Refs. 14 and 15. Also in Ref. 16 it is proved that quantum mechanics in any case implies nonlocality).

            Now compare Fig. 5. If we consider the spherical waves emitted from E we see action metric again becoming relevant. For all point-events, say, on ACB have the same action (  h) in the relevant process the two photons implement. Hence ACB is zero in action metric and measurements at A and B are action-physically mutually contiguous, which explains their "nonlocal feedback". Just as in the case of Fig. 3, we see action metric to be vital in "distorting" our conventional conceptions about physically relevant distances. We already see such "distortion" in comparing Euclidean and relativistic metric, but an even more striking one manifests itself in the above examples, viz. one between on the one side action metric and on the other both Euclidean and relativistic metric. Actually, both the Euclidean and the Minkowski vacuum space (distance) are as much a theoretical construction as the old-time aether.

            It has to be emphasized that action metric – as to its nonlocality consequences – is relevant each time with respect to only one separate microprocess. The Minkowski scheme is a "rough" macro coordinate system that can be derived from the various process-bound action metrics via an argument considered in Ref. 12; especially see the discussion of its Fig. 3. Also compare Sect. 5 below.


5. The quantum of action as the realistic four-dimensional elementary process: as "atom of occurring". We abandon as a redundant hypothesis the idea that more than action (quanta) is needed to construct everything physical: particles, processes, distances, ...


Action is of the dimension energy times time, that is, "physical existence during time". This is the elementary stuff of the four-dimensional world. Now we put the hypothesis that the quantum of action, in such world, is simply a universal "atom of occurring", the most elementary process from which all other processes are built up. Actually, we already saw an example in Sect. 4, where we considered the existence in time of a particle S to be a series of (this time wave- or slicelike) action quanta. We hypothesize the existence in time of all microparticles – compound or not – to consist of a (sometimes more complicated) series of action quanta. We go further into this in later sections.

            As long as we do not need anything else than realistic four-dimensional action quanta, and series or more complicated structures ("lattices") of them, in order to construct processes and whatever is relevant in physics at all (such as metric, forces,...), we abandon as redundant the hypothesis that there is more in the physical world than such quanta and their four-dimensional "lattices". In this context remind that in the second paragraph of Sect. 4 we showed a three-dimensional matter wave like OA to be a section of our now-hyperplane with a four-dimensional action-metrically stretched quantal wave slice. The wave system  then actually describes the internal structure of an action-quantal series representing, say, a particle's existence in time. (Note that we added the dimension t, restoring four-dimensionality.) Such  is actually a realistic model of the stretched quantal system. Small wonder that it contains all "local" physical information of the system, because there is nothing but the action-quantal series, so that, describing the – stretched – action quanta,  describes the physical system completely, with one exception. Viz. it leaves out the hidden variables implemented by nonlocal influences from beyond the relevant process (or: coherence domain), such as retroaction (compare Sect. 11) that, e.g., may define an impact location. Note that the location "uncertainty"  is accounted for in our model by the very slice extension. (For an explanation of the momentum uncertainty  see Refs. 4 and 12, where it is elucidated how  is constructed out of many Fourier components.) In spite of such uncertainties, our model as a whole will appear to be imaginable and deterministic. (Apart from Sect. 1, see for this Sect. 11.)

            Realize that only the realistically four-dimensionl model of the universe implied by Sects. 1 and 2 constitutes a natural starting point for introducing the quantum of action as an (all-important) realistic entity at all. Within this scope it also becomes clearer why Ref. 17 could say that "the failure of Classical Theory seems to have as sole origin the atomism of action". For this quotation is more obvious if such atomism refers to a realistic elementary physical process than if an action quantum would be a mere "mathematical" quantity of action that for some unknown reason always happens to be h.

            In all, we will try to construct a model of physics – its metric, the existence in time of various elementary particles, three of the fundamental forces (strong, weak and electromagnetic),... – by taking serious the quantum of action as four-dimensional "building block" of processes in Minkowski space.

            In Sect. 4 we introduced action metric in the wavelike process of a uniformly moving particle. We saw that it could diverge from relativistic Minkowski metric. We now sketch how the macro ordering scheme the latter is in our model, can be constructed from action quanta. See Fig. 6. (For a more extensive treatment see Ref. 12, the discussion of its Fig. 3.)

            Consider Fig. 6, in which we first imagine that objects A and B are two electric charges. Virtual photons evidently gauge the distance AB and attune their energy to it via Coulomb's law . This relates the Coulomb force to the distance AB. In Sect. 10 we will see that such virtual photons (their existence in time) are separate quanta of action. Also note that in Fig. 6 the distance of B from A is proportional to the number of de Broglie clock ticks – that is, action quanta – elapsed in A between the emission from A to B and its recapture on A of some virtual or normal photon L.


Figure 6. The distance between two electric charges A and B is in principle defined by the number of quanta (or de Broglie clock ticks) in A between its emission of a light signal L to B and L's  recapture by A. 


We generally see that the mutual distance of A and B is actually defined by action-quantal series and their numbers of quanta, without our in principle needing the vacuum as a gauge and "absolute reference frame". We can do with the relevant action-quantal structures or lattices, in which relevant ratios of the lengths (in quanta or de Broglie clock ticks) define everyting as to spacetime metric. (A simple case is that time in a micro system is defined by action quanta via , where  defines action-quantal based de Broglie clock ticks every  sec.) This means that we can abandon the vacuum as objective reference frame jointly with old-time aether: the mere action quanta and their lattices suffice here for defining everything about metric. (This, in fact, amounts to a quantization of spacetime. That is, the adjustment of our space and time concepts to their dependence on action and its quantization.) Note that this ultimately ensues from our integrating into action of spacetime and energy-momentum in Sect. 3: the vacuum follows from the integration result (action) rather than conversely, and becomes less objective and absolute with that. In Fig. 6 and the above discussion we see this happen concretely in distances originating from action quanta and their lattices rather than such distances and the vacuum existing independently. In all, the four-dimensional structure or lattice S of all action quanta and series of them constitutes the "frame" of the macro Minkowski scheme M. Our position is that S is primary and M secondary. Also, e.g., the action-quantal frequency  in  is primary – an entity in terms of action quanta – and the energy E is derived from it, secondary. More generally: distance, energy, time, momentum, charge,... are properties of processes, i.e., of four-dimensional lattices of action quanta and/or of the internal construction of the quanta. (As to the latter – charge and internal structure – see Sect. 6, and in particular the discussion of Fig. 8.)

            Note in the above context that our action-based metric still needs the timelike extension of the quanta of objects like A and B of Fig. 6 because the quantal series corresponding, say, to protons and electrons differ as to their periods because of the formula  that actually defines mass m by the frequency . (Again we see a natural phenomenon, i.e. mass, being defined by action quanta.)

            The above does not detract anything from our essential result, viz. that action quanta and their extended "lattices" define space, distances, mass (and many more phenomena) rather than the latter being independent, "absolute" concepts. The analogy with spacetime defining space and time – relativizing them in the process – forces itself upon us. We can abandon as redundant all "absolute" quantities like "old-fashioned distance" (as "an amount of vacuum") that are indeed already inherent to or implied by action-quantal lattices or structures as sketched in Fig. 6.



6. Spherical rotation as a model of action quanta. In-principle explanation by this of the strong and EM forces. Also explaining the  e and  e charges of quarks, and why there is no e2/r infinity paradox


As soon as we consider action quanta to be realistic elementary processes the question arises what kind of concrete model we can make of such process.

            Figure 7 refers to such model. In Ref. 18 spherical rotation is shown to be a periodical (rotation) process having the peculiarity that, though connected with its environment by "strings" like the tracers of Fig. 7, it can continue indefinitely without the appearance of ruptures or entanglements. Ref. 18 also demonstrates that the configurations corresponding to the phases of an ongoing spherical rotation are isomorphic with those of the spinor waves of a Dirac particle – of all things. The paper identifies matter waves with the strings figuring in spherical rotation, in its endeavour to conceive spherical rotation as the basic model of action quanta.


Figure 7. A model of spherical rotation. The first half-turn of S evokes a twist in the two tracers (see dotted lines in the second figure), which is undone by S's next half-turn. After two complete rotations of S the tracers resume their initial positions and one spherical rotation has been completed. (This figure is essentially derived from Ref. 18.)


Though inspired by this, we chose another approach, i.e. one in which the strings – that is, the connections of the rotating core S with its environment – are seen as in principle representing the fundamental forces particles (as timelike series of action quanta) can exert on each other. The above-mentioned isomorphy demonstrated in Ref. 18 will then be used by us to found a theory according to which matter waves contain and process the physical information of the corpuscular particles they represent in another, isomorphically coded way, in "another language". (See Sect. 8.)

            We hypothesize spherical rotation as considered above to be the basic process embodying the action quanta that, as a series, constitute the existence in time of corpuscular particles. Then, it appears that we can make correspond various details of spherical rotation on the one side and several properties of "corpuscular" action quanta on the other (for a further elaboration we refer to Refs. 5, 19 and 11):

1) We vary the model of Fig. 7 in the sense that sphere S is substituted by a rotating (pointlike) entity E periodically covering the equator of S in Fig. 7. The strings "guide" E that now actually performs the Zitterbewegung Dirac derived for fermions. The strings witness neither ruptures nor entanglements in the ongoing process.

2) We assume three "strings" – corresponding to the three "colors" red, green and blue – to appear in the spherical-rotation or zitter-process and also that these are in principle the implementers or "embodiers" of strong forces. That is, strongly interacting particles do so by making mutual contacts via three, two or one of the strings of their "constituting" action quanta.


Figure 8. In (a), (b) and (c) models are given of the spherical rotations, that is, constituting action quanta, referring to electrons, u-quarks and d-quarks, respectively. In (c) the strings of the  component of a d-quark have not yet been integrated.


3) In our model, we can integrate the EM force too. Compare Fig. 8 for this. Our basic hypothesis is here that, whereas all separate strings represent or transmit strong forces between relevant particles, three strings of different "colors" may also be bundled to one cluster that is in a position of interacting with virtual photons. This is how the EM force, just as the strong one, is embedded in the elementary process action quanta are. Note that "electric charge" is then the capacity of a particle to interact with virtual photons rather than some special "stuff" on a charged system. We further assume that, in forming a "three-bundle" defining electric charge, the three relevant strings may originate with more than one particle in a compound. As an example we below consider a proton and the strong and electric forces inherent to it. (See Fig. 11.)

            As to Fig. 8 we add that for the u-quarks in (b) and (c) we take a (positive) positron (anti-electron) as a basis. In the model of Ref. 18 the spherical rotation of an anti-particle is considered to be the mirror image of the one of the corresponding "normal" particle.

4) A proton p is constituted by two u-quarks and one d-quark. As we see in Fig. 8, the u kind provides one "strong" colored string (which defines the color of the quark too) and two bundled EM ones, while the d kind corresponds with one colored string and five bundled ones corresponding to the charge  e. (The strings of the anti-neutrino  are discussed later.) Realize that the two "bundled" strings of u-quarks represent  of what is needed for a standard charge e which corresponds to three bundled strings that can absorb and emit virtual photons. (We assume virtual photons not to be in a position to interact with one or two EM strings, but only with bundled triads of them that are a "white" combination of a red, green and blue string.) Hence u-quarks' fractional charge  e. In a proton uud the role of those strings is that the colored ones see to the mutual strong mutual binding of the three quarks whereas the bundled ones in an imaginable way result in a combined "charge" of  via the bundling of in sum 2 + 2 + 5 = 9 strings (three d-electron strings compensate three "positive" ones).


Figure 9. The common Zitterbewegung of the three proton quarks makes it possible that a proton is both a compound particle and satisfies the demand of Dirac's equation that also a compound fermion experiences a Zitterbewegung. We assume all three quarks to cover the same path. This amounts to integrating their respective action quanta into "common" proton quanta that we call trunk quanta. They are typical of compound particles.


5) We can also explain now why a compound like a proton at the same time can perform something like a "pointlike" zittering and be constituted by three component quarks. It is clear from Fig. 9 and its caption: the three quarks' "zitter cores" all cover the same zitter path, at periods equal to the de Broglie one for the proton as a whole. (In such way component quanta integrate into compound trunk ones.) Correspondingly, our model explains how, in interference experiments, compounds such as a proton in the wave shape indeed show wavelengths and frequencies corresponding to de Broglie periods of  where m is the total compound mass. That is, apparently the action quanta and their timelike periods of the component particles are integrated in ("trunk") action quanta of the compound system. We see this roughly illustrated by Fig. 9. (Mind that the transition of corpuscular quanta into wavelike ones has still to be added to this model; compare here Sect. 4 and the discussion of Fig. 3.)

6) The above model also solves the  infinity paradox of electromagnetism. If electrons and quarks are pointlike and charge would be some stuff on the points, we would be confronted with infinite energies if, say, an electron and a quark would approach each other nearer and nearer.


Figure 10. Mutual approach of a proton and an electron. They interact electromagnetically via their strings, not "pointlike". Also, the interaction implies periodic movements of such strings which are bound to a minimum period by the de Broglie periods of the respective action quanta, which are finite. (This figure is derived from Ref. 20.)


Figure 10 solves the paradox by illustrating that strings and their periodic movements rather than pointlike particles are responsible for EM interaction. In Ref. 20 it is explained in detail how, at their minimum distance, the (bundled) strings Se and Sp of an electron and a proton experience a common periodic process having as its frequency that of the proton's de Broglie period. The role in it of a virtual photon traveling, e.g., via DCFCD is also elucidated. (B, G and R are zittering proton quarks; compare Fig. 11 below.) The above also limits the minimum distance of the charges to about the proton's radius. All of this shows that charge is not some "stuff" carried by a pointlike particle but that it is the capability of whole action quanta to exchange virtual photons and interact with them.



7. The "atomic" model of compound particles, the range of the strong force and the size of protons; the nature of virtual photons and more about EM interaction


Figure 11. Prototypical model of a proton or comparable (but unstable) micro-compound, such as a neutron or muon. Besides their implementing the trunk-quantal Zitterbewegung (A, B, C) we assume the quark-components to also cover paths like electrons in atoms (A1, B1, C1). The mutual binding of the quarks (or other components) is hypothesized to be implemented by strings between zitter cores and the code-quantal cores on the atomic-electronlike paths. a, b and c are the EM-bundled strings.


There is an analogy between the spectra of elementary compound particles such as protons and those of atoms. This suggests an analogy between the models of both categories too. We imagine that of a proton to be roughly as sketched in Fig. 11. The quarks of such proton cover both zitter paths and paths like the electronic ones in an atom [compare 2) below], whereas the strings embody centripetal forces on both the "zittercores" on the zitterpath and the quark representatives on the "atomic" outer path(s). (We hypothesize this to be inherent to trunk and elementary action quanta.) In Ref. 21 the model of Fig. 11 is elaborated as a minimized proton model in which the quarks adopted a simplified form that conserves all observables, and therefore is physically allowed. In particular two assumptions are made there:

1) The relevant internally binding colored strings correspond to a periodic virtual-photonlike process between both kinds of cores, a process whose period equals the de Broglie quantal period of the proton as a whole,  Because the velocity going with the relevant virtual-photonlike proces is c, this actually defines the size of the proton to be of the order  fermi; it roughly agrees with experiment;

2) We hypothesize a qubitlike relation between on the one side the pointlike quark "representatives" (cores) on the zitterpath and on the other side the code-quantal ones on the outer path(s) analogous to those of atomic electrons. (Compare the discussion, inter alia of qubits, below Fig. 16.) Such qubit nature makes it possible that – in a non-interacting compound – the quarks are substantially "invested" in the zittercores, whereas in an interaction their physical substance is (partly) transferred to the "code-quantal" cores on the outer path(s). This, e.g., appears when a probe to some extent interacts with separate quarks in a hadron. Note that in the former case the component quarks are wholly integrated in the compound; this precisely corresponds with one wavelength  and one de Broglie frequency  for the compound as a whole. Before the introduction of the common-Zitterbewegung model, the wavelength and frequency of whole-compound waves were unexplained. Note again that the translation of the corpuscular state of a system into the wavelike one is relevant here too. Compare Sect. 4 and also Sect. 8 below.  

            Because of the nature of spherical rotation (see above and also Ref. 18) two zitter rotations correspond with the de Broglie period of  In such period of time a string – its end point, in our model, going once to and fro between the centre and the "outer electron path", at velocity c – has a range of ½ , whereas the zitter radius is ½ . (Mind again that in one de Broglie period two zitter rotations appear.) This means that the radius of the outer path has a diameter roughly  times that of the Zitterbewegung. In this way we find the proton radius, which is roughly that of the outer code-quantal path.

            In Ref. 21 we discussed a qubitlike relation between the zittercores and the "code-quantal" ones on the outer path. At the same time we see from the above model that the range of the strong force – which is the one of the strings – is about that of the proton size, in agreement with experiment. Generally, the above sketches the relation between the trunk quanta of the compound – represented by the "common Zitterbewegung" – and the separate quarks, represented by their "atomic-electron behaviour" on the outer path. We associate the latter behaviour (referring to some quark independence) with "code quanta" representing such components' periodic processes.

            Anticipating Sect. 10 (to which we refer for details), where we go into a "Mendelejev" system that orders elementary particles according to their action-quantal lattices – the specific construction of their existence in time from action quanta -, we already introduce the idea that virtual photons are separate action quanta that, in three-dimensional language, travel to and fro between the ends of bundled triads of charged-particle strings (compare Sect. 6).

            Viz. in Ref. 22 we give a theory alternative to the current "borrowing energy from God" explanation of Coulomb interaction. The latter explanation starts from the idea that a virtual photon is on its way during  sec, where r is the distance between the charges. Because the virtual photon should be unobservable, its energy  should satisfy  if it represents a quantum of action, we should take the equality sign. Then, however, the energy  is hc/r. On the other hand, the Coulomb energy satisfies  A being about 861. This means that "orthodox" theory works with virtual photons whose energy is much larger than the Coulomb one. This also requires the assumption that a virtual photon is on its way between the charges only during  th part of the time.

            Our alternative theory has three mutually related advantages:

a) Our virtual photon embodies the Coulomb energy, no "borrowing from God" being necessary;

b) Virtual photons continuously succeed each other in being on their way between the charges (those of the standard theory do not; most of the time none is on its way);

c) We can demonstrate that momentum and frequency of collisions of the  photons with the charges precisely suffices for transmitting the Coulomb force of  

            Point b) is implied by a) because excluding borrowing from God makes conservation of (Coulomb) energy inevitable: the Coulomb energy does not cease to exist intermittently; hence the virtual photons do not.

            Point c) can be demonstrated as follows:

Its existence in time (duration  ) corresponding to one action quantum, a virtual photon satisfies  This means that it travels A (about 861) times to and fro (on the average) between two charges C and D before being absorbed (and re-emitted) by one of them. With each reflection on a charge a momentum




is transferred, which has to account for the Coulomb force on, say, C during a time  between two successive reflections on C. Because p2 - p1 = F(t2 - t1), where p2 - p1 is C's (and D's) change of momentum during the period from t1 to t2, we have that during the period  the average force F on C will be




which is the Coulomb force. The same thing holds for the other periods of  We see in an imaginable way the pushes and recoils of the virtual photons and charged particles, respectively, cause the Coulomb force. In the standard theory, things are less imaginable. In Ref. 22 we also discuss a mechanism that – via negative-energy photons – can implement Coulomb attraction.


Figure 12. In (a) we see the dotted lines show an action-quantal slice p of a virtual photon implementing Coulomb repulsion. (b) sketches a negative-energy photon n going with attraction. In both figures, a and b are the worldlines of the two charges. We did not sketch the many reflections of the photons but assumed one crossing.


In Fig. 12 a characteristic difference between positive-energy virtual photon p and a negative-energy one n is in their contacts with the charges. p is emitted in P of its 0-phase hyperplane PS and absorbed in Q of its  -phase hyperplane TQ. In (b) we see a difference: absorption Q is in the earlier 0-phase and emission P in the later  -phase of the photon. This somehow corresponds to the traveling backwards in time that has been associated by Feynman and others with negative energy which, note, is inherent to Coulomb attraction. Hence we see again the power of the imaginable action-quantal model in its making Coulomb interaction understandable.

            As to one more point our model tends to serve consistency. I.e., Ref. 22 also demonstrates that the above 861 crossings precisely correspond to a situation in which, in an H-atom, a virtual photon is completed every rotation of the relevant electron e. Because such rotation, on account of  for the e waves, also goes with one action quantum of e, this situation precisely allows a virtual photon p and e to complete an action quantum simultaneously, so that after each rotation they can interact in their  -phases. (The interaction is the absorption and re-emission of p by e.) In more complicated atoms the mutual "fittings" appear to be similar.

            The wavelength of our one-quantal virtual photons is  This corresponds with the photon traveling A times to and fro between C and D in order to complete "its" action-quantal "slice" (compare Sect. 4).

            In all, virtual photons as discussed above once more illustrate the construction of the four-dimensional world from action quanta; here one quantum separately figures as transmitter of the Coulomb force. It links up with the bundled triads of strings of the action quanta of (elementary) charges.



8. The coded-information theory about the relation between corpuscular and wavelike particles and their action-quantal series


In our theory, matter waves neither guide particles hidden within them, nor are mere mathematically expedient means for predicting statistically experimental outcomes. Actually, as discussed in Sect. 4, four-dimensional wave slices are an alternative manifestation of action quanta in series that constitute particles existing in time.

            We saw in Sect. 6 that Dirac matter waves were proved in Ref. 18 to be a process that is isomorphic with that of spherical rotation. That is, an isomorphism exists between the group of configurations of a spherical-rotation period and the phases of a four-dimensional spinor-wave slice, as two variants of an action-quantal process. This simply amounts to matter waves being an alternative state of spherical-rotational "corpuscular" action quanta; they carry essentially the same information. A state appearing if the action metric of Sect. 4 is relevant. Generally, this will be so if the particle in question is so little localized by interactions that it moves sufficiently freely for action metric (and a coherence domain in which it applies) to be actualized. It may even be that the "free state" of such particle ensues intrinsic variations of the functioning of its action quanta on top of their being smeared out in the slicelike form of Fig. 3. In any case, the isomorphism result of Ref. 18 suggests our hypothesis that wavelike particles simply carry the physical information contained in their corresponding, spherical-rotational, corpuscular shape in another data code. Here the action quanta embodying the corpuscular and the wavelike manifestation of a particle still retain the same structure, because of the isomorphism proof in Ref. 18. We indeed see that matter waves neither guide particles, nor are of a merely immaterial, mathematical nature.

            Our coded-information idea means a considerable step in the direction of understandable models in quantum mechanics, as we can start now from the hitherto "unimaginable" formalism as a data-processing instrument referring to information carriers that as such are imaginable and whose relevance does not in essence violate imaginable models. The abstract formalism actually does no more make the microworld unimaginable than the DNA code of reproductive cells makes man unimaginable, or than tv waves that transmit pictures make such pictures unimaginable. In Ref. 19 we show in detail how matrices, representation spaces and various other mathematical processing methods of waves simply refer to the information code inherent to waves that are isomorphic with spherical rotation, and not less realistic. Two simple examples of the wave information code are  encoding the momentum of a system as a wavelength, and  encoding its energy as a wave frequency.

            From a general point of view we can complete the second paragraph of Sect. 5, about  as follows:

1. There is a simpler way to formulate the equations of motion, viz. the Principle of least action; it "translates" the behaviour of particles or waves in four-dimensional terms of action.

2. In Sect. 5 we also considered the wave function  in terms of action (quanta).

3. Well, what is done by the coded-information idea of this section is generalizing this. That is, we consider waves, wave equations, matrices and other formalism no longer primarily as mere mathematical instruments for producing correct predictions, but also as referring to realistic "stretched" action-quantal series (that is, quanta in their wave shape). Hence, they refer to realistic four-dimensional action-quantal processes rather than our three-dimensional model of objects, forces and Minkowski metric.

4. We saw earlier that the nonlocality paradoxes of QM can be solved by the introduction of action metric, after our abandoning the "aether-like" absoluteness of the vacuum space and our integrating it into action. Now the coded-information idea shows that in principle the other QM paradoxes – that is, the other instances violating coherent imaginable models of micro phenomena – can be solved too. Viz. by abandoning the assumption that nature invariably stores and transmits the information associated with physical systems in the same way, i.e., the corpuscular one. The two abandonments are mutually related: both have to do with the primacy of action, over conventional spacetime and metric as well as over conventional objects, or mass concepts. Quantum nonlocality and "unimaginable" formalism both refer to "distortions" stemming from our prejudical assumptions about spacetime, information, particles and action quanta. These caused such nonlocality and formalism – and many quantum phenomena – to seem paradoxical and to defy understandable models of a coherent reality. To a high degree we can say that the quantum formalism is a translation algorithm for expressing action laws and data codes in our familiar concepts and ways of thinking that are concomitant with assumptions as the above. Only after re-translation in action-quantal terms imaginable coherent realism reappears. Natural laws attuned to action and its (two) information codes can seem very "convoluted" if adjusted to our "ordinary-life world". (For a thourough discussion see Ref. 19.)



9. More on the nature of the fundamental forces and their ratios


We earlier associated the strong force with three action-quantal strings of different colors, that somehow share the periodic action-quantal process and connect its "zitter core" with interacting systems. In Fig. 11 we sketched something of this for the strong internal binding between the quarks of a proton. It also shows the origin of the proton's charge e: the "bundled" triad of differently colored strings resulting in "white" or neutral, and with which virtual photons can interact (reflect or being absorbed and re-emitted). The colored "strong" strings need no "intermediary" momentum carriers such as virtual photons.

            As discussed in Sect. 7, the colored strings themselves act as a kind of virtual photons by periodically covering the distance to an attracted particle. In the proton model of Fig. 11 we see two "strong" strings centripetally enforcing rotational movements of the relevant quarks, both on the zitter path and the outer path (remind the qubit relation). In this "minimized" proton model we imagine two colored strings per quark implementing the strong force and one being bundled electromagnetically.

            We saw in Sect. 7 that a virtual photon travels 861 times to and fro between relevant charges before the action quantum it embodies in time is completed. If virtual photons had a wavelength  rather than  and they and their successors still were active continuously, the Coulomb force would be 861 times stronger. (The photons' energy and momentum would be 861 times larger.) Now the colored strings indeed only once cover the distance to a "colleague quark", which strongly suggests the force they transmit to be 861 times stronger than the Coulomb force. At the same time we see from our model in Fig. 11 that per zitter core two colored strings are active. This can be hypothesized to produce a centripetal force  times stronger than if it had to be implemented by virtual photons in a similar role.

            With an electron – whose existence in time is a series of simple action quanta, in contrast with protons in which quarks "compose" trunk-quanta – the three strings are bundled in the EM way. That is, their centripetal force on the zitter core is a factor 1722 smaller than the comparable one in protons. This, then, in our theory is the basic explanation why the ratio of the proton mass  and the electron mass  is roughly 1800. For if the centripetal force is 1722 times stronger in the proton case, the zitter radius in such case is also 1722 times smaller, and the de Broglie frequency  is 1722 times higher. This implies mp to be 1722 times larger than me. For we assume all zitter cores (of electrons, protons,...) to be equal, so that our conclusion follows from  Mind in the above context that we consider the core rotation and the string action, as two aspects of spherical rotation, to be inherently mutually connected in such way that such action works centripetally on the core – as an aspect of the construction of action quanta, whose frame is spherical rotation.

            We generally hypothesize that the radius of the zitter path of a fermion is inversely proportional to the centripetal force on the zitter core. Some may object that  is problematic "inside" an action quantum, which sounds plausible. However, if we take all action quantal cores, and their velocities c on the zitter paths, to be equal,  will derive its variation in different particles only from F, that defines r and the mass.

            We also see that it is the centripetal forces by strings or other instances that define ratios such as  we roughly estimated above, because they determine the mass via de Broglie clock tick frequencies  while  is defined by the time a zitter core covers two zitter paths of radii ½ , that is, by r.

            The conclusion is that generally the variation of microparticles – simple like an electron, or compounds – highly stems from their zitter radii defined by centripetal forces, from the number of cores on the zitter path, and from the nature of such forces (defining their strength): strong, EM or weak.


Figure 13. The one-quantal helical slice representing the movement of an H-atomic electron e originates from successive slices covering each other in being bent into helices: an optimum of simplicity. Before e's capture, its worldline was OQ. Compare Fig. 3.



As to the latter forces, our theory introduces some new ideas. In order to explain them we first consider a four-dimensional action-quantal picture of the movement of H-atomic electrons. Consider Fig. 13 that, in the first instance, represents the slicelike quanta of a uniformly moving electron e. Now suppose the electron is captured by a proton so that it starts covering a circular path. How can this be "reconstructed" in four-dimensional action-quantal terms? Well, the circumference  of the electronic path is  in Fig. 13. (We here consider the case of an H-atom, but the argument can be generalized to other atoms.) If we now adjust the quantal slices to such path, we see a remarkable simplification appear. For in bending the slices in a way corresponding to the circular path, we see D and C cover O, while B and E cover A, and M and F cover G, etc. The net result is that all successive action quanta of e (when it moved linearly) mutually overlap and merge into one helical quantal slice that represents e's "entire history". This means an enormous simplification as compared with the non-quantal three-dimensional picture. Further, the most consistent structure or action-quantal lattice appears if we imagine one new e-quantum to appear after every two e-rotations. (See Ref. 22, pp. 356-7, where it is demonstrated that in an H-atom a virtual photon P that mutually attracts electron e and proton p "lives" a time equal to the rotation period of e. If, then, a photon is alternately absorbed by e and p, there should be a photon-absorption-prone  -phase end of an e-quantum every two rotations.) This and the above in general will appear to be relevant to the weak-force discussion below because if a new e-quantum appears every two rotation periods, the energy "invested" in the sub-process of the atom the e movement is, is very small, much smaller than the rest energy of an electron. (This state of matters contributes to the trunk quanta of an atom embodying virtually all its energy.)


Figure 14. The weak bond between action quanta of particles and antiparticles; the quantal lattices of weak bosons. (a) refers to interactional W-bosons and (b) to free ones with plane (non-helical) wave slices. In (a) slices 1, 2, 3,... are alternately of the e and  kind. In (b) trunk-quantal slices (e,  ) appear.



            Now we consider something analogous in Fig. 14 (a), where we see a series of quantal slices that intermittently go with an electron and an anti-neutrino. We then construct a model of a wavelike weak boson W = e +  by hypothesizing

1) quantal slices and anti-quantal slices can be mutually bound like, say, the successive action quanta of an electron, or the trunk-quanta of a proton;

2) bound, and again slicelike, combinations of an e and a  quantal slice (1 and 2, 3 and 4,...) can be bent like the slices of an electron in Fig. 13. [Now, in Fig. 14 (a), A will cover O and D will not.] Depending on after how many concomitant rotations we suppose a new quantum to start, we may see wavelike W-bosons of arbitrarily small masses. [Another "option" is imagining that all slices separately are bent and cover each other's helices (then both A and D will cover O); the difference is not relevant as to what follows.] The helical case we consider with Fig. 14 (a) refers to weakly interacting W-bosons. For the free W's we refer to sect. 10, point 6) and Fig. 14 (b).

            The consequence of the above is that in weak interactions such as  we no more have to "borrow energy from God" than in the case of vitual photons as discussed in Sect. 7. The mass of the interactional W adjusts itself to conservation of energy via the frequency with which W quanta succeed each other in the helical slices, in a mechanism comparable to what we see with the atomic electrons of Fig. 13.

            In essence, we found two things in connection with Fig. 14:

a) a new kind of binding between two particles, viz. in the weak way as appearing with electrons and anti-neutrinos that are weakly bound into W bosons (and into a Z boson if we similarly combine an electron and an anti-electron);

b) a mechanism in a position of adjusting W or Z mass to conservation of energy in interactions. For, say, in the above interaction of muon decay the mass of the W boson can adjust to about the muon mass by means of the helical-slice formation discussed, that allows adjustment of energies according to the frequency with which helical-slice-shaped quanta succeed each other, in atomic electrons as well as in interacting W- and Z-bosons.

            We take the "weak bond" as a third one besides the strong and EM one, and assume it to "mimic" the bond between successive action quanta such as the one between those of the series in Fig. 3. But in the weak case it specifically appears between a quantum and an anti-quantum. Moreover it only takes part in forming the unstable bosons W and Z.

            The above gives some insight into what the proper weak force is: a binding instrument like the strong one, but now being derived from the basic bond between successive quanta in series.

            Note further that with strong and EM forces we took the strings of "corpuscular" spherical rotation (quanta) as starting point, but with the weak "force" (binding instrument) the wave variant is better imaginable. Still, in both cases the corpuscular and the wave state are mutually isomorphic, as demonstrated in Ref. 18 by proving the isomorphism of spherical rotation and Dirac waves at all.

            The weak bond and associated helical-slice-like action quanta may also solve the paradoxical behaviour of so-called sea-particles. These are supposed to arise (from virtually nothing) in the environment of "normal" ones. E.g., sea pairs e+e do so with electrons. Still, the pairs do not seem to correspond to any measurable mass effects. Well, we may assume then their existence in time to be of a similar helical nature as with weakly interacting W- and Z-bosons: the helices can correspond to arbitrarily tiny masses whereas, say, the  nature survives.



10. A "Mendelejev" system of elementary particles from the details and various "lattices" of their constituent action quanta, by particularly using zitter cores, strings, the weak bond and the qubit concept. Coherent models in terms of action quanta of electrons, quarks, protons, (virtual) photons, neutrinos, muons and weak bosons


In Ref. 21, we especially considered models of the main elementary particles in terms of how the action quanta constituting their existence in time act as to their cores and strings, and of how they are integrated into trunk quanta in compound systems such as protons. The arguments are often subtle and elaborate, and we only summarize below a number of results amounting to a kind of Mendelejev system for elementry particles. It became only possible from a four-dimensional point of view that focusses on action-quantal processes that embody the particles' existence in time. We not merely produced models of the main elementary particles but also of their capacity of strong, weak and EM interactions, by varying the relevant action quanta's core, string and other behaviour and the "lattices" they can form. An outline:

1) An electron existing in time fits in the system as a series of simple indivisible (non-trunk) action quanta, whose three strings are bundled in the EM way so that they carry a unit charge by their mere capacity of interacting with virtual photons.

2) An electron neutrino  as to its existence in time, is considered to be a mere separate electron action quantum characterized by two properties: first, its position in the comprehensive action-quantal structure S that physically embodies Minkowski space corresponds to a velocity c (the quantal slice makes an angle of ¼  with the time axis of an arbitrary observer); second, its strings are not in a position to interact with virtual photons because of the neutrino's velocity c. Instead, they participate in the neutrino's emission and absorption processes.

3) A muon neutrino is a compound  of two electron neutrinos and one anti-electron neutrino. See Fig. 15 where we see that the three components can be mutually bound in the weak way because on hyperplanes 1 and 2 each time a "normal" and an anti-action quantum make contact, which we took to be characteristic of the weak bond.


Figure 15. The muon neutrino as a compound of three weakly bound simple (anti-)action quanta corresponding to a velocity c. The internal weak bonds are via hyperplanes 1 and 2, whereas O and Q represent emission and absorption "linkup", respectively.


4) Quarks can be conceived to exist in time as variants of series of "simple electron action quanta". Reconsidering Fig. 8, we see a u-quark differ from an electron by its action quanta having only two of their strings bundled electromagnetically, while one colored string is disposable for strong interactions. Since each string has one of the three colors red, green or blue, a quark too can have, according to its "free" string.

            Further, we assume the "basis" of a u-quark action quantum to be an anti-electronic, that is, positronic one. (In Ref. 18 "anti-quanta" are taken to be characterized by spherical rotation that is a mirror image of "standard" spherical rotation.)

            As to d-quarks, sketched in Fig. 8 as compounds, we refer to Ref. 21 for details, especially referring to their role in compounds like a proton. If we reconsider Fig. 11 above it has to be added that our theory in Ref. 21 also starts from the idea that compound systems, as to their action quanta and cores and strings of these, tend to optimum simplicity. That is, cores and "anti-cores", strings and "anti-strings", may mutually compensate or annihilate on the condition of the conservation of all observables. Via such simplicity Ref. 21 conduced to the proton model of Fig. 11, in which we see two rather than one colored strings start from each of the three quark-co-constituting zitter cores.

5) Hence, for a model of the proton see Fig. 11. We found in Sect. 9 that its very component-binding by colored strings caused its mass to be of the order of 1722 times that of the eletron. We also inferred the diameter of the proton [Sect. 7 point 1)]. In all corpuscular fermions we take the radius of the zitter path to be ½  and two rotations to complete one (trunk or elementary) action quantum. This appears to correspond with the proton spin and size. (Compare Sect. 7, fourth paragraph.)

6) As to W and Z bosons we have to distinguish two different variants: First, the bosons as participating in weak interactions, with which we saw in the discussion of Fig. 14 (a) that their quantal structure is helical, which allowed them to adjust their energy to what conservation demands. Second, short-lived free and very massive "non-helical" ("normal") weak bosons appear, viz. with mass of the order of 80 GeV, a multiple of the proton's. They fit in our "Mendelejev" system by our assuming them to be coumpounds (trunk-quantal systems) of, in the W case, an e and an anti-neutrino particle and, in the Z case, an e and an anti-e particle, which are weakly bound in one of two possible ways. Viz. in such weak way in which the strings of the two components participate in the weak bond in the colored or strong way. [Compare 7) below.] This appears to produce the strongest bond of all, which results in powerful centripetal bending of the zitter paths (not of corresponding slices!), a tiny zitter radius and the exceptional mass of weak bosons not participating as "transmitters" in interactions. See Fig. 14 (b) for the non-bent trunk-quantal slices of a free W-boson.

7) Muons can be compared with protons except that their zitter cores (and anti-cores) are centripetally forced in their common zitter path by the weak rather than the strong force. Guided assumptions in Ref. 21 result in our hypothesizing two variants of the weak force: one in which colored strings take an additional part and another in which bundled basically EM ones do. We assume their ratio to be again 861, just as for "normal" strong and EM forces. This leads us to the mass ratio  via the hypothesis that in W and  particles the centripetal forces on the zitter cores are of the "strongly assisted" and the "EM assisted" weak kind, respectively. On account of the interaction  we conceive the muon trunk quantum basically to consist of the three quantal components of the muon neutrino in addition to those of a W particle, which results in five cores on the zitter path of the muon as a compound (and their qubit counterparts on the "outskirts" of the system.

In Ref. 21 the role of the qubit concept as playing a part in compound particles is discussed. We can summarize this by considering Fig. 16 or, similarly, the analogue Fig. 11 referring to a proton.


Figure 16. Sketch of the muon and its five zitter cores and anti-cores, and their qubit counterparts.


 In the latter, we consider, say, A and A1 – a zitter core and a code-quantal one – as similarly related to each other as states A and B in Fig. 17: they are two superposed alternative pure states in a mixture m = a|0> + b|l> that has the remarkable specific quantum property that the system m can indeed be partly in the state |0> and partly in the state |1>. In Fig. 11 these corresponded to the situations A and A1, respectively. We assume that if the system (proton, muon,...) is isolated, a = 1 and b = 0 in the above formula. That is, in an isolated proton the zitter cores get all "weight", whereas in interactions the code-quantal (or "atomic-electron") cores play more or less radical parts. In general, they represent sub-processes in a micro-compound, just as rotating electrons in an atom. The code-quanta get an increasing weight as the compound is more involved in interactions. This corresponds to b > 0 in a|0> + b|1>. The trunk-quantal formation gets incomplete or perturbed. In the case a = 0, b = 1 we see the emission of a sub-particle from the compound.

            We see too that, in all, our "Mendelejev" system shows some similarity to that of atoms.


Figure 17. In an example of a qubit, two states A and B are each "partly realized". S is a semi-transparent mirror and l is a light ray. Note that A and B have an action distance 0, and that this is also the case with respect to, inter alia, A and A1 in Fig. 11 if we consider a specific phase of the relevant quantum of action.


8) Photons - virtual or normal – fit in our scheme if considered as separate Z boson action quanta also corresponding to a velocity c (which actually makes them a weakly bound electron neutrino and anti-neutrino). This means that in our model normal photons have a similar relation to Z bosons as electron neutrinos have to electrons. Viz. they are constituted by one action quantum and one anti-action quantum, mutually bound as in Z bosons. That is, weakly (as in muon neutrinos), but this time in the "colored-strings-assisted way", as in Z bosons. The strongly and electromagnetically inactive strings of the two weakly bound action quanta are also involved with the photons' emission and absorption processes, as we suppose in neutrinos too.

            With virtual photons we hypothesize it to be the essential difference that – in contrast with normal photons – their strings are attuned to interaction with the bundled triads of strings characterizing charged particles. That is, whereas normal photons, and neutrinos, make contact with their emitter and absorber in their 0- and  -phases, respectively ("point-events" P and Q in Fig. 18), it is different with virtual ones. Viz. consider the strings, in all phases of the system, to make contact in (varying) points A and C, with both relevant charges (i.e., with their bundled, EM, strings). Such combined or double contact is possible by virtue of the action metric in which the distance ABC is zero. At the same time, this nonlocal aspect allows the virtual photon to attune its energy e2/r to the mutual distance of the two charges. The relevant feedback also causes virtual potons to travel 861 times to and fro, contrary to normal ones. The very above "mechanism" explains why only charged particles, with bundled EM strings, can exchange Coulomb photons.

            Also, it appeared to be possible in Ref. 21 to derive the ratio  by arguments about centripetal forces on the relevant zitter cores that are similar to those indicated above. The strong, EM and weak forces, by their acting on the cores, define the zitter radii and by them the masses of the various microparticles.


Figure 18. Action-quantal slice(s) of a photon. With normal photons, in P the 0-phase makes contact with the emitter, and in Q the  -phase does so with the absorber. In virtual photons the "binding-assisting strings" can make contact with bundled strings of charged particles too. In a certain phase, this occurs in A and C. In other phases these shift accordingly. a and b are the worldlines of emitter and absorber (normal photons) or of the two charges (virtual photons).


In all, the gist of the above is that

a) the properties of the action-quantal process, such as of the relevant strings, and

b) the ways the quanta can form trunk-quantal systems,

define what kind of particles can appear and what their differences are.

            That is, systematizing elementary particles and a "unification" of fundamental forces has to depart from the proper elementary "particle" and its details as a process, viz. the quantum of action, its string processes and the ways quanta can make mutual contact, by means of strings or otherwise, such as in the weak ways discussed.

            In a final section, Ref. 21 concludes to one more kind of nature's simplicity. Viz. that Noether's formula does not merely describe all observable densities and dimensions in a unifying comprehensive way, but, in the coded-information mode, also describes physical reality completely as far as it is defined by causal factors (i.e., by the past), in spite of the highly "smeared-out" ("fuzzy") wavelike nature of the Noether substrate. Two mutually related factors explain this:

1. The circumstance that in the wavelike picture Noether refers to, much of the physical information is stored nonlocally, so that it need not be there at all point-events separately where a measurement might produce a relevant observable value. (In essence this "local inarticulateness" represents a radically simplifying, "economical", feature of nature!)

2. The Noether formula leaves out hidden variable influences, such as retroactive ones. Inter alia, this causes many quantizations to be left "undecided". That is, mixed rather than pure states dominate the picture. Before any measurement takes place, local articulation is minimum. This point is related to Wheeler's: "No phenomenon is a phenomenon until it is an observed phenomenon". For, only by a measurement (interference from outside) some nonlocal and/or hidden-variable information is evoked also locally, e.g., by retroactive influences. (Compare Sect. 11; by local information we mean here information manifesting itself in the local physical "articulation" or "construction".)



11. The role of hidden variables: their origin, their nonlocal character and their function in natural law; Optimum Simplicity


In Ref. 14 it has been proved that, if hidden variables (hv) exist, they have to be nonlocal; in Ref. 15 it was demonstrated that they appear indeed, as nonlocal entities. Ref. 16 proved that, if QM predictions are correct, nonlocal hv should exist.


Figure 19. Hidden variables as retroactive influences or feedback results co-defined by C, D, E, P, Q,... and influencing, say, a moving particle F and its impact location A on screen S. B is an alternative impact location, equally probable if we do not count hv influences.


What natural influence do they represent? On account of our knowing from Sect. 2 that retroactive influences appear, and are precisely operative within Heisenberg's uncertainty margins within which causal laws leave things undecided (such as with respect to the direction in which the momentum carriers move between S and T in Fig. 2), it is rather obvious to assume retroactive influences to be responsible for defining details within such margins. That is, just as in Figs. 2 and 4 (think of the feedback "causal chains" EA and EB in the latter), we see feedback in various "physical trajectories" such as AC, AE, PE etc. in Fig. 19, or between the moving system F and physically connected events such as C, E and P. We now simply assume the retroactive influences from the four-dimensionally pre-existing future of F and its impact process to embody the hidden variable defining impact location A. This actually amounts to not merely seeing the physical world of events as four-dimensional, but physical laws too: they define processes not merely locally and causally from the past, but also partly from the existing future. Note that in a four-dimensional model we can better formulate: natural law does not "move" or influence things or processes, but corresponds to certain four-dimensional symmetries and coherences that we, from our three-dimensional point of view, experience as "influences". E.g., think of the relation between four-dimensional Least action and three-dimensional equations of motion.

            In comparing alternative worldline point-events such as G and H in Fig. 19, or alternative impact locations A and B, realize that their action distances are zero. From a causal point of view, there is no difference between producing them. We can also say: the amount of action (or "occurring") needed, say, to transform situation B into A, is zero. That is, such two situations constitute a touch and go case: even an infinitesimal retroactive influence could tilt over the scales to A rather than B, as causal and (local) action considerations are neutral. That is, if any retroaction exists, it can solve the hv problem, i.e. decide on action-physically contiguous alternatives. It also does so, as we see in Ref. 23, on alternative eigenvalues of observables in general. This explains too the collapse or contraction of wave packets into a pure state: some retroactive hv decide in cases that causally are of a touch and go nature.

            Up to now, we assume the many retroactive or feedback contributors (C, D, E, P,...) to the hv, that jointly define A within the relevant unertainty margins, to act chaotically, as mere "noise". It may be that nature acts indeed in this way, but it may equally hold that there is as much coherence in retroactive influences as nature apparently implements in causal ones. In Ref. 23 we discuss three alternatives of the above noise variant, in which we assume more coherence which, by the way, is nonlocal on account of the earlier-mentioned results of Refs. 14-16. In the most coherent variant of the three, we consider a situation in which the latitude the uncertainty margins provide allows natural law to enforce nonlocal coherence on a macro scale. So much so that "guided evolution" can appear, and also the mutual attuning of natural constants so that life became possible and, maybe, even "coincidences" that imply "directions" which make the world meaningful. (Think of Einstein: "The most surprising of all is that the world almost certainly has a meaning.")

            Realize that even if we reject macro-coherence it could be a function of the uncertainty margins that they create the latitude for nonlocal (e.g., retroactive) influences and laws to produce or enforce more-than-local coherence: feedback between emissions and absorptions, EPR-correlation etc. In other words, the "indeterminism" seemingly appearing on the local level could precisely create the margins within which nonlocal rather than local factors (laws) define and coordinate the course of events even more coherently than local causality could ever do.

            All of this could happen in conjunction with action metric and its nonlocal purport. And it again puts Bohr's "A microprocess constitutes a whole" in a more concrete light, just as the "instantaneous collapse" of a wave packet. As to the latter, note that in Fig. 19 the "collapse" of a stretched wave packet into impact point-event A is explained within the above scope of action metric and stretched quanta, retroaction and the nonlocality they imply.

            The nonlocal macro coherence spoken of above would amount to a radical "extrapolation" of the one we see in the EPR case of Figs. 4 and 5. Also note that EPR nonlocal coherence is not at all an exotic exception in nature's functioning. For it essentially appears in any emission-recoil situation: an emitted system and its recoiling emitter have a relation similar to that of the two correlated EPR particles. Actually, nonlocality – that in any case exists and is inherent to QM (see Refs. 15 and 16) – obeying four-dimensional laws (which correspond to four-dimensional symmetries in the world's events) and of which retroaction is part and parcel could make the universe more coherent than 19th-century causal determinism assumed, rather than less.

            The role of hv and their nonlocal nature induces us to suggest a Principle of optimum simplicity, joining with both action-metrical stretching and the coded-information idea. (See also Ref. 13, pp. 726-7.) It contains:

Nonlocal and coded information storage, inherent to the nature and primacy of action, tends to be so much "economical" that nature will not implement local "articulation" (that is, local information) to any extent exceeding what is needed for making a situation consistent. E.g., if an atom interacts as a whole, the details of its electronic sub-processes are not locally defined but could nonlocally be evoked according to what various possible alternative experimental situations may require, viz. via nonlocal information contributed by hv.

            The essential simplifying factor in Optimum simplicity is that nonlocality-inducing action metric and retroaction remove the necessity for nature of having to store locally all information needed in some point-event or process, with respect to all potential alternative relevant experiments, in order to keep the world coherent in all such cases. Actually, such information can be called on nonlocally as various experiments require. "Undetermined" observables such as  if x is measured precisely merely correspond to information nonlocally stored in quantal structures; information that would be called on if in a subsequent measurement  should become precise.

            The above puts Bohr, "Copenhagen" and neo-positivism in the right as to the thesis: Quantum mechanics is complete with respect to local "articulation" of micro-physical situations. Still, the last word is to Einstein: if nonlocal information and the future (e.g., intended experiments included) are counted in, "God does not play dice" and everything – the complete four-dimensional world – is defined, all local situations being optimally simple as can be harmonized with experiment. ("Economical" information distribution.) Einstein is put in the right on the condition of his abandoning pure locality (separability).

            The above can also be formulated as: The four-dimensional action-quantal lattice S embodying the physical world, and the feedback channels implementing hv action, have such symmetries (obey such natural laws) that no experimentally and logically redundant information (redundant as to observable facts and their logical coherence) is stored locally. I.e., local situations adjust to various actions from outside (such as the initiation of an experiment) by invoking relevant articulation or information nonlocally, e.g., by retroaction. The Principle of least action can be seen as a special case of the simplicity of the four-dimensional point of view.

            One more reformulation of Optimum simplicity: all action-quantal lattices of micro-processes are as simple as can be harmonized with their interactions (outside connections) according to physical law. E.g., they should obey conservation. That is, only observation defines many details (articulations). Locally, QM is complete; still, deterministic and imaginable nonlocal models are possible if we count in hv.

            Generally, EPR as regards the two systems of Figs. 4 and 5 illustrates the above: neither the A, nor the B measurement outcome is locally defined. For with each measurement the lacking hv information is nonlocally invoked, inter alia, from the correlated system! This examples the above, also as to feedback channels EA and EB in Fig. 4 and the nonlocal communication via ACB in Fig. 5.

            Further, a system is "inarticulate" – as regards some observables, not as to the above quantal lattice S – as far as incompatible observables would correspond with different, incompatible action-quantal lattices. Hence only some obervables are determined. Those that are, are often selected by experiments, which retroactively define the relevant lattice. All of this explains why it remained elusive to construct objective local models definite as to all observables at the same time.



12. Some special items: experimental verification of retroaction and of the physical existence of action quanta; explaining the uncertainty margins and the  probability rule


1) An experiment on retroaction. In Ref. 24 we propose an experiment that could show retroaction in a direct and measurable way.


Figure 20. Our choice to put up, in the M region of T, either a light revolving plate L or a nicol prism N, influences the angular momentum carried by the polarized photons coming from A and B.


 In Fig. 20, z-polarized photons pass screen S. Instrument R rotates the polarization of the A-waves from the z- to the y-direction. In the region M of T, where the A and B waves differ ½  in phase, quantum mechanics (QM) requires that, in "interfering", both kinds of waves produce photons that each carry an angular momentum of  This can cause a light revolving plate L installed in M to start rotating. If, however, we substitute L by the nicol prism N that discriminates between A and B waves, the latter will not interfere at all and will remain either z- or y-polarized, according to QM. Then, much less angular momentum will arrive at our instrument (nicol N plus absorbers P and Q that capture y- and z-photons, respectively). If n photons are measured, the total angular momentum carried by them will be – as is measurable in principle –  and  respectively.

The gist is that an observer in M, by deciding to install either L or N + P + Q, appears to have influence on the amounts of angular momentum that the photons supply from the S region where they were before reaching M.

2) Experimental demonstration of the existence of action quanta. In Ref. 25 we discussed some possible experimental verifications of the indivisibility of action quanta. With the aid of Fig. 21 we sketch one of them.


Figure 21. An electron comes from S. We measure very precisely time and place of its hitting Ai  and, later, of its arrival at Bk. Our aim is veryfying whether an integer number of electron action quanta appears between Ai and Bk.


Consider plates P and Q. On both of them, a great number of measuring points, A1,..., An and B1,..., Bm, respectively, are in a position of indicating very precisely time and place of their possible interaction with an electron e coming from S. Note that this does not violate any uncertainty relation such as  or  because location and time are no complementary observables. They allow simultaneous precise measurements.

            As soon as we know the two relevant interaction times and locations, referring to, say, Ai and Bk, respectively, we in principle can arbitrarily precisely calculate

a) the velocity and momentum of e between Ai and Bk;

b) its energy between both points and how long it was on its way.

            These data define two variables: the total action  corresponding with e's movement between Ai and Bk, and also, via e's calculated momentum, its wavelength in covering its path. The number of waves on AiBk is that of the action-quantal slices coresponding to e's existence in time between Ai and Bk. Well, we now only have to verify whether such number is integer and whether, dividing W by it, we indeed get h. Then it is corroborated that an integer number of action quanta appears between Ai and Bk.

            Also note that some retroaction cannot but be active here: either the direction or the absolute value of e's momentum from Ai on should have been attuned to e's hitting one of the Bk's so that the action of its movement is an integer multiple of h.

            Some might object that our experiment violates an uncertainty relation: it would allow to define both e's momentum and its location at a certain time as it moved between Ai and Bk. However, realize that the uncertainty relations only limit simultaneous predictions about, e.g., momentum and location; they do not exclude relevant precise knowledge afterwards, as is demonstrated in Ref. 26.

            In Ref. 25 we also propose four other experiments attuned to demonstrating the indivisibility of action quanta in an imaginable way, and considering them as realistic entities. Some of the experiments are indeed practical.

3) At the end of Ref. 21 we propose an experiment that could discriminate between our idea of virtual photons traveling 861 times to and fro, and the standard theory with its "borrow-from-God" aspect.

4) The nature and size of "uncertainty". In Ref. 27 we gave an explanation of the uncertainty relations which can be summarized as follows:

First consider an emission process. Say, a photon is emitted and the duration of the emission process is  (whatever its precise "model" may be). Now one of our starting points is that all natural processes consist of action-quantal lattices whose "branches" (series of quanta) consist of integer numbers of quanta. Suppose that in the first instance the photon energy is E. Now it would be a mere coincidence if  would be an integer number of action quanta, as this emission process, too, should be. Hence E should adjust in such way that its new (ultimately measured) value E1 satisfies  n being integer. Without adjustment,  could have been any broken number of action quanta, such as 100½ h or 99¼ h. The adjustment amounts to  ½h, ¼h,... That is, all alternatives for  of the order h, or maybe ½h. So far as to spontaneous emission.

            In the case of "measurement disturbances" being the cause of "uncertainty", we can argue similarly. Viz. suppose that in the measurement process of an amount of energy some action-quantal series has a time duration  (it may be the duration of the measurement as such). Then we are confronted with a similar adjustment of E as above in order to complete  to an integer number of action quanta. (Such adjustment is the very "measurement disturbance".) In the case that E is rather "rigid" and cannot adjust, the relevant period  has to. In both cases the basic cause of the "disturbance" or adjustment is the necessity of completing a relevant action-quantal series - in a spontaneous emission or measurement process – to an integer number of action quanta.

            Note that via  and  we easily find from  that also

5) Explaining the  probability rule micro-realistically.

            In Sects. 3–6 we considered matter waves (four-dimensional slices) as realistic but action-metrically "stretched" (distorted) variants of "corpuscular" action-quanta, whose inner nature ("bowels") manifests itself in the Dirac-spinorial "construction" of the waves in a way that makes the latter isomorphic with spherical rotation (Ref. 18), they still being realistic. This is a starting point of our explanation of the  rule. It is an inherent aspect of the isomorphism that the realistic waves  and  - where S is the action  - are mutually coupled in the specific "complex" way described by Eq (1) below. This does not detract anything from the fact that the two realistic sinusoidal waves carry the conserved energy and momentum of what in other circumstances is a corpuscule. The complex coupling is irrelevant to this. The realistic waves still carry energy in the normal  way. Analogously to how this is the case with classical waves, deriving the energy content for one Fourier component implies finding it for an entire wave packet.

            In all, matter waves which are mathematically written as the complex expression  are actually two kinds of "very imaginable" waves that are mutually coupled in a particular way [see Eq (1)] rather than, say, mutually interfering or appearing independently. It is such particular coupling that apparently is part and parcel of their representing the spherical-rotation process isomorphically.

            The above means that we can assume the two realistic "components" of the "complex" wave packet




to be normal carriers of energy, just as light waves are. This leads us to an energy, carried by the above matter waves, that is proportional to




Because of the above relation E = ½Cu2, this makes the expression  proportional to how the energy of the system(s) in question is distributed over space at a time t. [Note that  represents u.]

            Then, the argument explaining the  probability rule can be completed by our realizing:

a) If the physical system considered consists of many identical particles, the probability P of finding one at a certain place and time is proportional to the local energy density at that time;

b) If we gradually reduce the relevant system to one particle, the distribution of the probability of producing it at a certain location and time remains proportional to the one in the many-particles case. This demonstrates the probability rule even for the one-particle case.

            Some might have a problem with harmonizing 1. the wavelike "spread" of the energy and 2. the "instantaneous collapse" of a wave packet. The solution is retroaction that enforces conservation, just as it does, e.g., in the case of Fig. 2. It is an influence additional to what has been elucidated above and amounts to a retroactively enforced proper path within "pre-collapsing" wave packets. Since retroaction never violates causality and its laws, such as  it can solve the problem mentioned without refuting the above derivation of such causal law. The latter indeed gives an incomplete picture of reality, as QM does at all, though both the  rule and QM in general are in agreement with experiment in predicting relevant outcomes statistically.

            The above is treated more completely in Ref. 28.


13. General and philosophical implications of the above theory


The most general characteristics of such theory are

1. Microrealism, in contrast with formalism, the Copenhagen interpretation and neopositivism: understandable models are considered as the basis of the Aha-Erlebnis, that is, of real understanding and physical insight;

2. Determinism, in contrast with uncertainty, probabilism and, again, the Copenhagen and also subjectivistic interpretations;

3. Nonlocal coherence, against the entire spirit of indeterminism and neopositivism.


Still, our "super"-deterministic model in a way reconciles Einstein and Bohr. In the first place, "God does not play dice" (Einstein) in our theory. On account of our explanation of hidden variables, He does not even do so on a nonlocal level because natural law is even coherent in the super-local sphere. (That is, to begin with EPR, and even on a larger and subtler scale if the least "chaotic" of our hidden-variable explanation hypotheses would be corroborated.) In the second place, Bohr was right in considering microprocesses as wholes and in rejecting local determinism. Also, he was right in saying that micro-systems co-depend on their environment, such as a variable measurement situation. A link with Einstein is that the latter's hidden variables appear to be precisely the retroactive or otherwise nonlocal influences from the environment we discussed with Fig. 19 and that cause a micro-system not to be "intrinsically" defined objectively. But the whole combination of system plus environment is four-dimensionally determined according to Sects. 1 and 2.

            Quantum mechanics is only incomplete as far as it leaves out of consideration retroaction (e.g., emanating from measurement attempts) and four-dimensional feedback (compare Sect. 11). Bohr was wrong in feeling "indeterminate" to be an intrinsic state and Einstein was in that he rejected nonlocality (inseparability of mutually distant systems) that (in EPR and elsewhere) precisely restores determinism.

            One essential impact of the forgoing theory is that natural law is four-dimensional, defining many (all?) processes even so much coherently that the idea of a four-dimensional organism might be applicable to the universe. However, a less macro-coherent model of the world can just as well harmonize with our theory. For example, if hidden variables would indeed be mere "retroactive noise"; additionally realize that determinism and four-dimensional realism need not mean macro-coherent four-dimensional reality.

            Besides action quanta as detailedly structured realistic four-dimensional atoms of occurring – a consequence of the four-dimensional realism we demonstrated – the most important concept introduced above may be action metric. This, in coherence with realistic four-dimensionality, not only allows us to explain nonlocality but also to make a new kind of understandable models of microprocesses which start from realistic action quanta as building blocks. For such elementary processes remain imaginable, defined and fitting in coherent models in spite of their frequent "nonlocal" stretching into their wavelike shape.

            Most of the above, however, has little to do with philosophy or interpretation but simply shows that four-dimensional realism is in a position to explain various otherwise mysterious or paradoxical phenomena.

            Still, it has to be observed that the more "paradoxical" phenomena refused to coherently fit in our traditional three-dimensional picture and corpusular information code, the more many got accustomed to evade the concepts of coherence, explanation, objective reality and definite laws at all, reverting to "fundamental uncertainty", neopositivism (that no longer is interested in coherent explanations) and mere formalism. Actually, reality allows a definite and precise blueprint that happens not to conform to our traditional concepts and pictures about it. But still, science is ordering coherently sense data and Aha-Erlebnisse. Not a few, however, repress the idea of a detailed and coherent reality in order to evade the necessity of abandoning some of our pre-conceptions if we want to solve paradoxes and indeed construct coherent realistic models.





1. C.W. Rietdijk: A rigorous proof of determinism derived from the Special Theory of Relativity, Philosophy of Science, Vol. 33, No. 4 (1966), pp. 341-4.

2. C.W. Rietdijk: Thought experiments relevant to Special and General Relativity – and some theses, in M.C. Duffy (Ed.), Physical Interpretations of Relativity Theory (Brit. Soc. Phil. Sci., London 1990), pp. 246-55. See also C.W. Rietdijk: Towards a consistent four-dimensional action physics, in C. Cormier-Delanoue et al. (Eds.), Courants, Amers, Écueils en Microphysique (Fond. L. de Broglie, Paris 1993), pp. 361-78.

3. C.W. Rietdijk: Special Relativity and determinism (Discussion), Philosophy of Science, Vol. 43, No. 4 (1976), pp. 598-609.

4. C.W. Rietdijk: On the four-dimensional character of micro-physical phenomena, in S. Diner et al. (Eds.), The Wave-Particle Dualism (D. Reidel, Dordrecht 1984), pp. 433-56.

5. C.W. Rietdijk: Four-dimensional realism and waves carrying physical data codedly as keys to understandable models, in A. van der Merwe et al. (Eds.), Microphysical Reality and Quantum Formalism (Kluwer Acad. Publ., Dordrecht 1988), pp. 245-66.

6. H. Stein: On Einstein-Minkowski Space-Time, The Journal of Philosophy, Vol. LXV, No. 1 (1968), pp. 5-23.

7. C.W. Rietdijk: Four-dimensional reality and determinism; a reply to Stein, to be published in V. Petkov and A. Wayne (Eds.), The Ontology of Spacetime.

8. C.W. Rietdijk: On the velocity of light in opposite directions, and on the structure of spacetime, in M.C. Duffy (Ed.), Physical Interpretations of Relativity Theory, Late Papers (Brit. Soc. Phil. Sci., London 1996), pp. 243-6.

9. C.W. Rietdijk: Another proof that the future can influence the present, Foundations of Physics, Vol. 11, Nos. 9/10 (1981), pp. 783-90.

10. C.W. Rietdijk: Proof of a retroactive influence, Foundations of Physics, Vol. 8, Nos. 7/8 (1978), pp. 615-28.

11. C.W. Rietdijk: A largely classical experiment demonstrating retroaction, Ann. Fond. L. de Broglie, Vol. 16, No. 1 (1991), pp. 91-6.

12. C.W. Rietdijk: A microrealistic explanation of fundamental quantum phenomena, Foundations of Physics, Vol. 10, Nos. 5/6 (1980), pp. 403-57.

13. C.W. Rietdijk: The Zitterbewegung embodies understandable models of the action quantum and retroactive influences, in L. Kostro et al. (Eds.), Problems in Quantum Physics; Gda?sk '87, (World Scientific, Singapore 1988), pp. 713-37.

14. J.S. Bell: On the Einstein Podolsky Rosen paradox, Physics, Vol. 1, No. 3 (1964), pp. 195-200.

15. C.W. Rietdijk: On nonlocal influences, in G. Tarozzi et al. (Eds.), Open Questions in Quantum Physics (D. Reidel, Dordrecht 1985), pp. 129-51.

16. C.W. Rietdijk and F. Selleri: Proof of a quantum mechanical nonlocal influence, Foundations of Physics, Vol. 15, No. 3 (1985), pp. 303-17.

17. A. Messiah: Quantum Mechanics (North-Holland Publ. Comp., Amsterdam 1961), p.42.

18. E.P. Battey-Pratt and T.J. Racey: Geometric model for fundamental particles, Intern. J. Theoret. Phys., Vol. 19, No. 6 (1980), pp. 437-75.

19. C.W. Rietdijk: The world is realistically four-dimensional, waves contain information embodied by particles codedly, and microphysics allows understandable models, Parts I and II, Ann. Fond. L. de Broglie, Vol. 13, Nos. 2 and 3 (1988), pp. 141-82 and 299-336.

20. C.W. Rietdijk: A rough derivation of  by abandoning an assumption about space and time, Physics Essays, Vol. 11, No. 2 (1998), pp. 222-34.

21. C.W. Rietdijk: Towards understandable models in microphysics, Physics Essays, Vol. 15, No. 3 (2002).

22. C.W. Rietdijk: How do "virtual" photons and mesons transmit forces between charged particles and nucleons?, Foundations of Physics, Vol. 7, Nos. 5/6 (1977), pp. 351-74.

23. C.W. Rietdijk: How do hidden variables fit in natural law?, Physics Essays, Vol. 16, No. 1 (2003).

24. C.W. Rietdijk: A possible experimental verification of retroactivity, Physica B, Vol. 151 (1988), pp. 358-61.

25. C.W. Rietdijk: Suggestions for experiments on action quanta, Il Nuovo Cimento, Vol. 63 B, No. 2 (1981), pp. 541-64.

26. J. Terletzky et L. de Broglie: Sur la possibilité de mesurer simultanément les coordinées et l'impulsion d'une particule pour un événement passé, Comptes Rendues de l'Academie des Sciences, Vol. 272, Serie B (1972), pp. 1161-3.

27. See Ref. 2, second part (Courants, Amers,...), pp. 374-6.

28. C.W. Rietdijk: A micro-realistic explanation of the  probability rule, Ann. Fond. L. de Broglie, Vol. 20, No. 2 (1995), pp. 187-206.



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