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Ph.D. Donatello Dolce picture

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School of Physics - CoEPP
The University of Melbourne
Parkville, VIC 3010
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donatello.dolce@coepp.org.au


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Slides: Ferrara, TM2012, DICE2012, ICHEP2012, London2012


"Io stimo piu' un vero, benche' di cosa leggiera, che al disputar lungamente delle massime questioni senza conseguir verita' nissuna", (Galileo Galilei) 

[I deem it of more value to find out a truth about however simple thing than to engage in long disputes about the greatest questions without achieving any truth] Galileo Galilei


I wish, my dear Kepler, that we could have a good laugh together at the extraordinary stupidity of the mob. What do you think of the foremost philosophers of this University? In spite of my oft-repeated efforts and invitations, they have refused, with the obstinacy of a glutted adder, to look at the planets or Moon or my telescope.” (Galileo Galilei)


"Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction" (A. Einstein)

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The cyclic nature of elementary particles

The Forgotten Lesson from Quantum World

Donatello Dolce


The assumption of an intrinsically cyclic nature of elementary systems represents a fundamental principle to address in a natural way important open (and sometime “forgotten over the decades”) questions in physics and beyond. In fact intrinsic periodicity for elementary systems provides a concrete formalization of the undulatory nature of elementary particles at the base of the ordinary wave-particle duality. This also means that every elementary particle can be represented as a reference clock and its interaction described as modulation of periodicity, similarly to general relativity.
The resulting theory is extremely intuitive and natural, nonetheless it must be noticed that it has been formulated in a rigorous mathematically way. For instance, in recent publication in leading reviewed journals of theoretical physics [1,4], it has been proved that the theory formally reproduces the most fundamental aspects of the modern Quantum Field Theory. Here we will presents some basic aspects of this innovative interpretation of the quantum realm.  


            I.     The ‘incomplete revolution’ of quantum physics.



Quantum mechanics (QM) has been defined the “incomplete revolution”. In fact it is an “axiomatic” theory, i.e. based on purely mathematical axioms whose physical meaning, after more than a century, is still debated. From a formal point of view QM works in surprising agreement with our experimental observations. This means that its mathematical formulation must be considered to be correct. But over the decades it has left unsolved a consistent number of paradoxes and questions. The elusive nature of the quantum world has been pointed out by the most eminent minds of quantum physics. Einstein addressed the indeterministic nature of QM by saying  “God doesn’t play dice?”. Feynman, who is the father of the most powerful and modern formulation of QM, said “I think I can safely say that nobody understands QM”.

The most problematic aspects arise when QM is combined to relativity. In this case the difficulties are also computational and mathematical. In fact observables in particle physics are calculated through techniques that involves cancellation, a.k.a. renormalization, of infinities (e.g. loop diagrams). These calculation are so complex that two of the fathers of renormalization theory, Prof. G. ‘t Hooft and Prof. F. Wilzcek, both important Nobel laureates, are actively working on foundations of QM  --- in models that for some aspects are related to the present proposal. To add a single digit in the precision of these calculations nowadays are necessary brute-force calculations with long simulations in extremely expensive supercomputers.  Indeed, besides the purely conceptual interest, the open questions of QM affect directly most of the ongoing research areas of modern physics. For instance the idea of the Higgs boson (a.k.a. God’s particle) originates from superconductivity (Cooper pair), that is a (not completely understood after a century) quantum phenomenon. The problematics of the Higgs sector are essentially of quantum nature (e.g. vacuum stability, radiative corrections, hierarchy problem, unitarization).

Another example is that, in General Relativity (GR), gravitational interaction has a well-established geometrical space-time meaning.  On the other hand gauge interactions, investigated by LHC and representing the other three fundamental interactions of nature, are based on quantum symmetries. Despite Weyl’s [43] and Wheeler’s original attempts to interpret these so-called gauge symmetries, their geometrical space-time meaning remains obscure and their existence must be postulate mathematically.

Furthermore the situation is even worse when QM is combined to GR. In fact, theories like quantum gravity or string theory are becoming more and more speculative and unnatural. One of the most important outcomes of these theories, the so-called gauge/gravity duality or AdS/CFT correspondence [42], remains after more than a decade an unproved conjecture, a.k.a. Maldacena’s conjecture.  A similar speculative tendency can also be observed in High Energy Physics (HEP). Data coming from the new experiments are in fact excluding or strongly constraining the most popular and investigated models of new physics beyond the standard model, including minimal models of supersymmetry and extra dimensions. Of course it is always possible to extend these models introducing new sectors [7] (i.e. more untested hypothesis), but this inevitably involves new parameters. As a result the constraints on the model can be relaxed but at the same time this decreases the predictivity, falsifiability and naturalness of the models. LHC has only left a tiny allowed  region for the Higgs boson (mostly between 122-128 GeV, for comparison the original theoretical allowed region was about 0-500 GeV, whereas the indirect from indirect observation was about 40-170 GeV). Nevertheless this remaining region is problematic even for the SM itself. For instance this implies either that gauge interactions are practically decoupled from gravity or that the vacuum of the universe is unstable.  On the other hand, if the Higgs boson will not be observed, radically new quantum mechanisms for the generation of the mass of ordinary matter are required, see for instance Higgsless/Composite-Higgs models [6,5] or superconductivity [1,32,33].  Whether the Higgs boson will be discover or not, radically new ideas in particle physics will be required (I would like to mention that even WIMPs, the main candidate for dark-matter, is almost excluded by recent experiments).

As noticed by G. ‘t Hooft and many other well-regarded physicists, the new experimental constraints will eventually give rise to a renewed interest on foundations of physics.  In particle physics we typically assume that there is something that we don’t see for some experimental reason. Nevertheless data and anomalies coming from new experiments suggest us that the correct attitude to solve the problems of modern physics should be rather to think that there is something that we do not fully understand about the quantum realm.

        II.   Digging into the history of physics

  As in a Pandora box, the simple single assumption of intrinsic periodicity of elementary systems, introduced by de Broglie in terms of “periodic phenomena”, unlocks in a natural way the important unsolved (and sometimes “forgotten over the decades”) problems of physics mentioned above, according to [1-4,8-9,30-33].  The approach is so innovative, original and exciting but, at the same time it has deep roots the history of physics. It worth to use the report of the Ann. Phys. referee of my recent paper [1] to introduce the issue in the most objective way.

The referee, in Feb 2012 wrote: “In this paper the author revisits an old question due to de Broglie about the physics of the "periodic phenomenon" which is implicit in the standard quantum mechanical treatment of (relativistic) particles. Experiment tells us that a particle of mass M, at rest, is to be associated with a temporal period of size T=h/M c^2 [h is the Planck constant]. When the particle moves [with momentum p], M gets replaced by the relativistic energy, and a corresponding spatial period l=h/p emerges.  The natural question, unexplained by standard quantum mechanics, is about the physical nature of the "something" which actually carries these periodicities. While in the ordinary theory free particle wave functions are designed by hand so as to realize these periodicities, no physical system is associated to them, in the way a pressure field is associated to a sound wave, for instance; thus questions like "what is the clock that ticks with period h/M " remained unanswered. In several earlier papers, reviewed and extended in the present one, the author [D. Dolce] has made a concrete proposal for a physical system exhibiting de Broglie's periodic phenomenon, namely a classical Klein-Gordon field (for spinless particles) which is subject to periodic boundary conditions in space and time. He then argues that standard first quantized semiclassical quantum theory is recovered in this setting, see sect. 5 of the present paper for those arguments.  The paper is certainly important in that it correctly pinpoints an open issue of the standard theory which is of central importance. To some extent it had been forgotten over the decades, but it is by no means clear that, equipped with modern concepts and technology, it is necessarily impossible to make progress here. The solution proposed by the author appears natural and plausible as the Klein-Gordon field he employs also makes its appearance in standard SECOND quantization, while the formalism presented is basically a substitute for semiclassical first quantization. This approach is certainly very inspiring and intriguing. The new results in the present paper concern the interpretation of the electromagnetic interaction (and non-abelian generalizations) in this framework. The author proposes to interpret the vector potential A_{\mu}(x)  as defining an infinitesimal local change of the periods T and l  [space-time periodicities]; this is motivated by the rule of minimal substitution p_\mu -> p_\mu +e A_\mu [gauge interaction], implying T^\mu \rightarrow T^\mu +e A^\mu /h for the reciprocal periods [modulation of periodicity]. He describes a reinterpretation of gauge transformations as geometrical transformations acting on the space of periodic Klein-Gordon fields with all possible periods”.


fig

In the left side of the (Fig.1a) shows the ordinary description of a system of elementary particles (e.g. a photon, an electron and three muons) in terms of waves. Such an description must be generalized to QFT as relativistic corrections are considered. In the right side (Fig.2a) the same system of particles in terms of original de Broglie “periodic phenomena” which in 1924 gave rise the wave-particle duality and thus to the modern undulatory description of QM. The de Broglie “periodic phenomena” can be represented as clocks, the so-called de Broglie “internal clocks” of the elementary particles. In fact “By a clock we understand anything characterized by a  phenomenon passing periodically through identical phases” [Einstein,1910].  The periodicity of these clock can vary from 0 to infinity in the case of the photon. Matter particles have tyoically an incredibly fast time periodicity. For the electron it is faster that 10^-21 s whereas for the muon is faster than 10^-23 s. In both figures it is also  represented the Cs-133 atominc clock whose periodicity is of the order of 10^-10 s. The “ticks” of the Cs-133 clock is used to defined the external time arrow. Every generic instant of time t_0 can be characterized by an combinations of the ticks of the de Briglie “internal clocks” of the particles constituting the system as in a calendar or in a stopwatch.  The scale difference between the “ticks” of a Cs-133 clock and of the internal clock of an electron is of the order of the difference between the age of the universe and a solar year.


As well known, in the so-called “old” formulation of QM, to every elementary particle is associated a wave. This famous wave-particle duality is shown in fig.1a. A wave is a phenomenon which has recurrences in time (periodicity) and space (wavelength). Nearly 90 years ago de Broglie noticed that, in the microscopic world, periodicity and wavelength, T  and l respectively, of a relativistic wave can be used to describe the energy momentum, E and p, of the particle through the Planck constant h. That is, E= h / T and p = h/ l (these relations are often written in terms of time frequency and spatial wavenumber). The undulatory nature of elementary particles is at the base of the modern description of  relativistic QM and it is confirmed by  many experiments. However,  if we want to include Special Relativity in QM it is necessary to introduce Quantum Field Theory (QFT), i.e. an evolution of the wave-particle duality based on more complicated and axiomatic quantizations prescriptions, such as second quantization and Feynman Path Integral (FPI).

      Nevertheless in physics the most groundbreaking ideas are the simple ones.  To see how the simple assumption of intrinsic periodicity can address the central questions listed above we must reconsider the concept of elementary particle and of wave-particle duality as originally introduced by de Broglie in 1922: “we proceed with the assumption of the existence of a certain “periodic phenomenon” of a yet to be determined character, which is to be attributed to each and every isolated energy parcel [elementary particle]” [de Broglie:1924,1924]. As the referee noticed, the theory that I am going to introduce is a (literal) realization of this yet to be determined de Broglie “periodic phenomenon”, a.k.a. de Broglie “internal clock”. Similarly to the wave-particle duality, the de Broglie temporal-spatial periodicity describes the energy-momentum of the particle through the Planck constant h. For the sake of simplicity here we will consider only time periodicity T and energy E. Thus to the time periodicity T there is associated a fundamental energy E=h/T. The only assumption that we need is that every elementary isolated particle has intrinsic de Broglie  periodicity. This assumption arises naturally as we combine the wave-particle duality to Newton’s law of inertia. For instance, in classical-relativistic mechanics an isolated elementary particle has constant energy E, so that the corresponding temporal periodicities of the de Broglie periodic phenomenon are persistent, i.e. T is constant.  We can for instance imagine a pendulum (to avoid the use of the gravitational external force we may consider a system of two masses and a spring) in the vacuum. If no interaction or friction affect the pendulum, the physics laws says that it will continue to oscillate forever with the same periodicity fixed by the energy initially given to it. It is important to bear in mind that, according to de Broglie, the energy E and the periodicity T are “two faces of the same coin”.  This means that an assumption of intrinsic periodicity is fully consistent with relativity as long as the relativistic modulations of periodicity associated to relativistic variations of kinematical state are considered. In fact we will describe interaction as modulation of periodicity as for gravitational interaction. Since our world is composed by elementary particles, and elementary particles are intrinsically periodic phenomena, it is natural to assume that reality can be described iin terms of elementary cycles. As we will see below this is the teaching of QM.

 As Galileo taught us with the pendulum experiment in the Pisa dome, time can only be defined by counting the number of periods of a phenomenon which is supposed to be periodic. The persistence of periodicity guaranties that the unit of time does not vary. The modern definition of time is based on the same principle: “A second is the duration of 9,192,631,770 periods of the radiation corresponding to [...] the Cs 133 atom.”. The importance of the assumption of intrinsic time periodicity is also present in Einstein’s definition of relativistic clock: “By a clock we understand anything characterized by a  phenomenon passing periodically through identical phases so that we must  assume, by the principle of sufficient reason, that all that happens in a  given period is identical with all that happens in an arbitrary period” [Einstein, 1910]. Therefore, under the assumption of intrinsic periodic periodicity, every elementary isolated particle can be regarded as a reference clock, the so-called de Broglie “internal clock” [38,39] (an isolated particle has constant energy E and therefore it has persistent time periodicity T, similarly to a inertial pendulum in the vacuum).  In fig.1b, in fact, we show the same systems of elementary particles of fig.1 (e.g. one photon, one electron a three muons) described in terms of de Broglie “internal clocks”.  By assuming that an elementary particle is an intrinsically periodic phenomenon we are enforcing of the undulatory nature of elementary particles of ordinary QM, and at the same time we are also enforcing the local nature of relativistic time. In fact fig.1b also shows the arrow of time defined in terms of the periods of the Cs atomic clock (green). Every instant in time is characterized by a different combination of the phases of the internal clocks constituting the system under observation. This is what we do every day when we fix events in time by combining phases of time periods that we call years, months, days, hours, minutes, seconds, and so on.   

As well known, a system of periodic phenomena is ergodic (i.e. quasi-periodic evolution, such as that of the combination of the rotation and the revolution of the earth, which causes the addition of a day every four months and further fine tuning of the calendar, or with the rotation of the moon so that every years the phase of the moon correspond to different days). If we also consider that the clocks can vary time periodicity through exchange of energy (i.e. interaction), we see that the evolution of a simple system of interacting periodic phenomena is in general very chaotic [e.g. the orbits of three gravitational bodies]. Similar to an ordinary calendar or stopwatch the arrow of time in the quantum world can be therefore described in terms of the “ticks” of the internal clocks of the particles, and in principle the external time axis can be dropped. This description of reality terms of elementary cycles also provides a better understanding of problems related to the notion of time in physics [2,4,31].  In particular the de Broglie internal clock associated to an elementary particle, as our ordinary clocks, can be conventionally chosen to be clockwise or anticlockwise. Only the reciprocal combinations of the ticks of these clocks are important. However if invert a single clock with respect to the others we obtain a different physical system. This corresponds to pass from a particle to an antiparticle and vice versa. 

   The periodicity of the de Broglie internal clocks of the elementary particles is tipically incredibly fast with respect to our human time scale.  In fact their periodicity is always faster that the proper time periodicity, which is fixed by the rest mass M of the particle T_0 = h / M c^2 - in fact the rest energy is related to the mass (E=Mc^2). The proper time periodicity is the is equal to the time that takes light to pass through the Compton wavelength of a particle, which for the electron is 10^-12 m.  In this way it is possible to see that the order of the intrinsic time  periodicity of an electron is  10^-21 s. To have an idea, the periodicity of the Cs-133 clock is, by definition, of the order of 10^-10 s.  This means that for every “tick” of the Cs-133 clock, an electron (the lighter particle except neutrinos) does a huge number of  “ticks”, comparables with the age of the universe expressed in solar years (i.e. it is like comparing a solar year with the age of the universe). The proper time periodicity is the is equal to the time that takes light to pass through the Compton wavelength of a particle, which for the electron is 10^-12 m.   The heavier the mass, the faster the periodicity (the mass scale explore by LHC corresponds to the time scale 10^-27 s). The modern resolution in time, nearly 10^-17 s, is still far from these scales. However the internal clock of the electron has been indirectly observed in a recent interference experiment [38]. This intrinsic periodicity allows a semi-classical formulation of the spin and the intrinsic magnetic momentum of the electron as already noticed in 1932  by Schrödinger in his Zitterbewegung model. Remarkably in my papers I have rigorously mathematically proven that the assumption of intrinsic periodicity is sufficient to completely derive the fundamental quantum behavior of elementary particles.

 
      III.     The harmony of elementary particles

 

 An elementary system constrained to have intrinsic periodicity can be typically represented as a string vibrating in compact dimensions. Therefore, as shown in fig.3, we will represent elementary particles as tiny vibrating strings. Their quantum behavior can be understood in terms of harmonic vibrations along the space-time dimensions [4].

 


pithagoras
Fig.2

A representation of Pitagoras playing strings. A vibrating string can be considered one of the most fundamental physical systems in nature. It is in fact at the base of many theories in physics.  The expansion in harmonics modes is also at the origin of mathematics, as well as of the concept of harmony in art.


 

As known since Pythagoras, see fig.3a, a system (e.g. a string) constrained in a compact or periodic space can only vibrate with discretized (quantized) frequencies.  That is a homogeneous string, which in our case corresponds to a free particle, can be represented as a homogeneous string vibrating with fundamental periodicity T=1 / v , so that the resulting harmonic frequency spectrum is v_n = n v.  According to the de Broglie, the frequency v multiplied by the Planck constant h (which can be imagined to be the “compression coefficient” of the string) corresponds to an energy E = h v. As a consequence of the assumption of intrinsic periodicity every elementary isolated particle exhibits a harmonic quantized energy spectrum E_n = n E = n h v.  This is in perfect analogy with the semi-classical quantization of a “particle in a box” fid.3a.


fig.3
Fig.3

A de Broglie “periodic phenomenon” describing an elementary particle can be either imagined as a “de Broglie internal clock” or as a vibrating string with de Broglie periodicity T=1/v. As well know, a vibrating string can be expanded in harmonics modes whose quantized harmonic spectrum v_n = n v is fixed by its length. In this way it is easy to see that to the intrinsic periodicity of an elementary particle there is naturally  associated a quantized energy spectrum E_n = n E = n h v. It is possible to show that, analogously to the quantization of a particle in a box, this se-classical quantization reproduces the basics aspects of ordinary quantum relativistic mechanics.


    To understand the physical meaning of this quantization we may consider the case of the Black-Body radiation, which historically was one of the first arenas of QM. In the so-called Black-Body the electromagnetic radiation is continuously emitted and absorbed by the walls of a cavity kept at fixed temperature. We must also notice that the photons are massless particle, M=0, so that thy have infinite proper time periodicity,  i.e. the internal clocks of the photons are frozen.  This means that their time periodicity can span from infinite to zero. Thus, for those components of the electromagnetic radiation with long time periodicity the spectrum can be approximated to a continuous. That is the effect of the periodicity can be neglected like in a very long string that can vibrate with all the possible frequencies, like a string constrained to vibrate within very small dimension. On the other hand, for those components with very small periodicities (the component with very high frequencies are called UV component), the effect of the periodicity cannot be neglected and the quantization of the energy spectrum E_n  = n E becomes evident. This is nothing but the Planck description of the Black-Body radiation, introduced to  avoid the so called UV catastrophe. We can imagine that for photons with small energy w.r.t. the thermal noise their time periodicities is destroyed (decoherence), whereas if the photons are sufficiently energetic their periodicity can not be neglect and we can observe the quantization of the energy spectrum.

  At this point we may consider the relativistic Doppler effect, so that the frequency, and thus the periodicity, of a periodic phenomenon varies in a relativistic way. If we consider the relativistic modulation of periodicity associated to variations of reference frame it is easy to find out that the energy spectrum of an intrinsically periodic phenomenon is nothing but the energy spectrum prescribed in ordinary QFT by the famous “second quantization”.

This is the first element of a long chain of exact correspondences with ordinary QFT. For instance, a vibrating string is the typical classical system which can be described in a Hilbert space, the relativistic wave describing our string (Klein-Gordon equation) is the “square” of the Schrödinger equation, the evolution is Markovian and unitary (i.e. it can be cut in infinitesimal evolutions), it is possible to define an Hamiltonian and Momentum operator, and so on. Remarkably, by integration by parts, from this intrinsic cyclic behavior it is possible to derive the ordinary commutation relations of QM as well as the Heisenberg uncertainty relation. Intuitively, to establish with good accuracy the frequency Dv, or equivalently the energy DE=h Dv of a periodic phenomenon it is necessary to count its ticks for a very long time with respect to its fundamental periodicity (i.e. to count a large number of periods), according to the relation DEDt ≥ h.

Even the most powerful mathematical tool of QFT, i.e. the Feynman Path Integral (FPI), can be derived in a extremely intuitive way from the assumption of intrinsic periodicity, [4,32]. The FPI prescribes that the quantum evolution of a particle is affected by self-interference of all its possible paths starting from A to B. But in classical mechanics it is only possible a single “straight” path between two point, so that in the ordinary formulation the FPI can be only achieved by relaxing one of the pillars of classical mechanics, the least action principle. But  this is not the case of  a classical cyclic geometry. If we imagine to drawn all the possible “straight” lines between two points in a cylinder, see fig.4, we immediately see that an infinite set of classical paths with different windings number are possible. In few words a classical periodic phenomenon can self-interfere and, as proven in [1-4], its classical evolution is naturally described by the ordinary FPI of QM. From this also follows a similarly natural interpretation of double slit experiment [32] [another possible way to see the wave-particle duality in this formulation is that an elementary system is in fact represented as a string which is not a localized object, but if its time periodicity is small it can be approximated to a particle. In fact in the classical limit a massive string turns out to have nearly zero time compactification and infinite spatial compactification. If we also notice that such a string is mainly always localized inside its Compton length, we find that in the classical limit such a string is a point particle living in an infinite three dimensional space and effective time evolution]. 

fog.5
Fig.4

In an intrinsically periodically phenomenon, such as that associated to an elementary particle, the evolution from a give initial configuration to a final configuration is given by the interference of all the possible paths with different winding number. It is possible to show that this sum over such classical paths associated to a cylindrical geometry reproduce the Feynman Path Integral.



Indeed, in this theory the quantized energy spectrum of a particle is described by the vibrations of a “periodic phenomenon” along the time dimension. Similarly, the vibrations along the modulo of spatial dimension and of the angular dimensions describe the quantization of the momentum and of the angular momentum, respectively. It is interesting to note that the theory is the full relativistic generalization of sound theory. Sound theory, developed by the Nobel laureate 1904 Reyleigh, is actually at the origin of the modern quantum formalism. A sound source is an object vibrating along spatial dimensions in a classical wave framework. A quantum system turns out to be an object vibrating along space-time dimensions in a relativistic wave framework [4]. The idea that quantum mechanics could be related to space-time vibrational modes has recently inspired the Nobel laureate F. Wilczek which has recently published two papers on a similar idea that he named “time-crystal”, [37] (a crystal is characterized by a periodic structure, and it can be described by considering a single period, see Brillouin zone).          Indeed this description  also yields the Bohr-Sommerfeld quantization which is historically was one of the first quantization prescription.  Its most famous application is in the description of quantized orbits of the Hydrogen atom which can be obtained by considering only closed orbits, i.e. a periodicity conditions (modulo a twist factor) for the electrons orbitating around the nucleus. It can be shown, for instance, that the harmonics expansion of the temporal and angular vibrations of a periodic phenomenon in a Coulomb potential gives the correct atomic orbitals of the hydrogen (or helium) atom, [2-4] (similarly it is possible to describes Zeeman effect ).


 

bohr      orbital1     membrane

         Fig.5a                                                           Fig.5b                                                   Fig.5c
In the hydrogen atom the allowed orbits are those in which the periodic phenomenon associated to an electron does an integer number of de Broglie time periods, i.e. closed orbits along the time direction (modulo twist factors). The winding number along the time dimension describes the principal quantum number.  This is nothing that a generalization of the original description given by Bohr, Fig.5a.   The azimuthal quantum number and the magnetic quantum number can be obtained by considering the quantization of the vibrational modes associated to the spherical spatial periodicity of the orbits, Fig.5b.  The resulting description is therefore the generalization of the vibrational modes of spherical membrane allowed to vibrate in the time direction, fig.b.





     IV.     An unexpected unified scenario

As shown with deep mathematical detail in recent publications [1-4,8,9], the axioms of ordinary QM can be inferred directly from the simple and natural assumption of intrinsic periodicity of elementary systems. This promotes intrinsic periodicity as the physical hypothesis (the “missing link”) to complete the “incomplete revolution” of QM, see also [39]. In fact intrinsic periodicity is also a way out to the Bell’s theorem (it states that theory with local-hidden variables cannot consistently reproduce ordinary QM). In fact, the quantization condition in the theory is represented by the assumption of intrinsic periodicity, as for the semi-classical quantization of a particle in a box.  This also means the QM is retrieved without introducing any local-hidden-variable in the theory (time is a physical variable that can not be integrated out and the assumption of periodicity is an element of non-locality in the theory). As noticed in [1], if we try to reformulate the Bell’s theorem within our theory we find again a formal correspondence with the inequalities of ordinary QM.  Therefore we may speak about determinism.

We may thing in the following way. A typical periodic dynamics of a typical quantum system (governed by the so-call quantum electrodynamics) is so intrinsically fast  (the intrinsic periodicities associated to electrons are always faster than 10^-21 s) w.r.t. our resolution in time that at every observation the system appears to be in an aleatoric phase of its cyclic evolution.   Intuitively, a phenomenon with such a fast periodicity is like a dice (“de Broglie deterministic Dice”, see [3,31]) rolling too fast w.r.t. our resolution in time, so that its outcomes can only be described statistically. As already noticed by G. ‘t Hooft, the statistical laws associated to a periodic phenomena formally corresponds to the ordinary quantum harmonic oscillator [35], which in turns is the basic ingredient of ordinary QFT. According to this mathematical formulation, QM can emerge as a statistical description of the intrinsically deterministic periodic dynamics of such fast vibrating strings, i.e. de Broglie dices.  Thus an imaginary observer with infinite time resolution would be able to resolve the deterministic dynamics of the de Broglie deterministic dice and in principle to predict the outcomes, as for an ordinary dice. That is such an observer would not have fun playing dices.

            This interpretation of QM not only is interesting for the advance on the conceptual and philosophical knowledge of the quantum world and the concept of time, it is also interesting to improve computational methods and to address open questions in HEP. To see this we must introduce interaction in our theory.  Since temporal-spatial periodicity and energy-momentum are two faces of the same coin, we can equivalently describe the retarded and local variations of energy-momentum occurring during relativistic interactions in terms of corresponding modulations of temporal-spatial periodicity of the elementary particles. In particular we may note that Einstein derived his description of gravitational interaction of general relativity by considering the modulations of temporal-spatial periodicity associated to a Newtonian gravitational potential. In general, local modulations of periodicity of reference clocks can be equivalently encoded in corresponding local deformations of the underlying space-time metric. Fig.5, show  the modulation of periodicity in linearize gravity  and the corresponding  deformation of the metric, a.k.a. Schwarzschild metric. As well known, reference clocks in a gravitational well go slower w.r.t. those outside. Therefore the assumption of intrinsic periodicity is fully consistent with special and general relativity as long as modulations are considered. In fact relativity sets the differential structure of space-time without giving any particular prescription about the boundary conditions. The intrinsic periodicity of the elementary particle can be formalized through periodic boundary conditions which are absolutely consistent with th variational principle of relativistic theories.

   The geometrodynamical description of gravity and its relation to clock modulation is well known.  As mathematically proven in [1], by considering the undulatory nature of elementary particles, the geometrodynamical description of gravitational interaction in GR  can be extended to gauge interactions. This represents an important and conceptual step towards a unified geometrodynamical description of all the fundamental interactions of nature and constitutes an historical, exceptional further success of the theory. Basically, a gauge field, through a formalism very similar to the one of modulated signals, turns out to describe modulations of periodicity associated to local transformation of reference frame. This can be easily understood if  we consider how an electromagnetic field is generated by the trembling motion of a charged particle in an antenna.  Such a trembling motion can be described as local transformation of reference frame of the charged particle, and the resulting modulation of periodicity, similarly to the Doppler effect, turns out to be described by the Maxwell equations. In fig.6 this is shown together with the corresponding local “rotation” of reference frame (e.g. zitterbewegung). Remarkably, gauge symmetries, which in ordinary QFT are internal symmetries whose existence must be postulated, turn out to be related to space-time symmetries through the assumption of intrinsic periodicity, and the Equivalence Principle of General Relativity can be extended to gauge interaction. Remarkably this was Weyl’s original proposal when he introduced the idea of gauge invariance, but similar attempts can also be found in Kaluza’s and Wheeler’s works.   Moreover, when the modulation of periodicity of all the harmonic modes is considered, the classical evolution of our vibrating string turns out to be described by ordinary quantum electrodynamics (QED). This means the quantum behavior of gauge interactions can be derived directly from cyclic dynamics, so that, in principle, Feynman diagrams can be expanded in harmonics and calculated semi-classically. That is, the explicit quantized spectrum of a periodic phenomenon could regularize the infinities of the loop diagrams (this possibility is also confirmed by Light-Front-Quantization, twistor theory, holography, AdS/CFT; all these theories have important formal and conceptual analogies with the theory described here). 

figFig.6a                                                                                          fig.6b                                                                 Fig.6c

Fig.6b describes the ordinary relativistic description of gravitational interaction in terms of space-time deformation or equivalently in terms of modulation of space-time periodicity of reference clocks which therefore run slower, i.e. are redshifted, with respect to clocks outside the gravitational well. Under the assumption of intrinsic periodicity the same geometrodynamical description can be applied to the other fundamental interactions in nature. In fig.6a shows  the modulation of periodicity of a de Broglie internal clock interacting electromagnetically together with the corresponding local polarized rotation of space-time metric [1]. In fig.6c it is represented the logarithmic freeze-out of the quark-gluon-plasma which, analogously to the Newton’s law of cooling, passes logarithmically from an hot regime characterized by small periodicities to a cold regime with long time periodicity. The corresponding deformation of space-time is a warped metric and resulting dynamics are those prescribed by ordinary quantum chromodynamics [4,32].


    Periodicity conditions impose that the gauge field can vary only by finite amounts in unit h/e (being e the electric charge). Thus the magnetic flux is quantized and the electric current can not vary continuously. Therefore the theory explain superconductivity in a immediate and material-independent way. This also means that intrinsic periodicity points towards a gauge symmetry breaking condition alternative to the Higgs mechanism. According to [40], from these quantization condition of the magnetic flux (a.k.a. Dirac quantization) it is possible to derive realistic models for the electroweak symmetry breaking in HEP. Another relevant result is that, according to the Bjorken Hydrodynamical model [42], in an accelerator experiment the energy of the fields constituting the Quark-Gluon-Plasma (QGP) decay exponentially during the freeze-out. In thermal QCD this corresponds to the Newton Law of Cooling, in fact the QGP can be represented as a volume of hot fluid.  Therefore, as shown in Fig.7, the periodicity of the fields has an exponential modulation. As shown in my recent e-print [30], this can consistently describe the asymptotic freedom of Quantum-Chromo-Dynamics (the third gauge interaction).  Least but not last, form the formalism of extra-dimensional theory it is possible to note that the extra-dimension acts in surprising mathematical analogy with the intrinsic proper time periodicity of a de Broglie internal clock.  This indicates a correspondence between periodic phenomenon and extra-dimensional theory [30], similarly to Klein’s original proposal, and justifies the good behavior of extra-dimensional theory without introducing an (unobserved) extra-dimension. Considering this dualism, the theory also provides an elegant explanation for the quantum to classical correspondence of at the base the AdS/CFT correspondence, which is one of the greatest open questions of the last decade of theoretical physics [42]. The quantization of elementary particles trough the assumption of intrinsic periodicity has important applications in condensed matter where actually semi-classical are widely used. In [33] we have for instance pointed out that such an approach can be used for a formulation of superconductivity based on fundamental principle of QM mechanics rather than on microscopical properties of the material. Similarly in [33] we have applied the same idea to interpret the behavior of electron in graphene and carbon-nanotubes, or to describe laser and Black-Body radiations. This just to mention few potential aspects explored in this research project.  But for the general nature of the assumption of intrinsic periodicity the applications of this method can go from cosmology (see cyclic model of the universe, or quasi normal modes in black-holes) [31,32] or to quantum gravity [1,31], to biology or econophysics (see cyclic models).

        V.     The origin of masses

Such a theory potentially opens the way to new challenging new scenarios in phenomenological aspects of high energy physics.  For instance, by working in analogy with current extra-dimensional Higgsless/composite-Higgs models [9,10] it can be used to understand the fundamental nature of the electroweak symmetry breaking mechanisms in analogies. Roughlty speaking this means to investigate the problem of the origin of the masses in the more fundamental and original approach allowed by my theory. We have seen that the masses of elementary particles have precise geometrical interpretation in terms of proper time periodicity of the elementary particles, i.e. in terms of the compactification lengths of the vibrating strings [1-4,30,31]. The question of the origin of the masses that LHC is trying to solve can be therefore reformulated in the following way:  why is nature playing that particular string chord? In this theory such a problem can be addressed by using the analogies with superconductivity [1,32] and the Dirac string description of monopoles [40]. Another possibility allowed by the theory is a semi-classical computation of loop diagrams in QED in terms of the harmonics of a vibrating string.  The feasibility of this point is indicated by the recent successes of similar recent calculations in empirical integrable models, e.g. Light-Front-Quantization [44], Twistors, AdS/CFT, Holography [42].

    With the publication of [1-4],  the first critical, foundational stage of this research project can be considered successfully accomplished. As in a Pandora box, the simple assumption of intrinsic periodicity, if well implemented, has showed to unlock most of the historical (and sometimes forgotten) open questions of modern physics in an extremely natural way and with an amazing mathematical beauty.  Now it is time for brave physicists to really face challenging, feasible, natural issues of nature. 

References

[1] Gauge Interaction as Periodicity Modulation. Donatello Dolce, (Melbourne U., CoEPP). Nov 2011. 53pp. Published in Annals of Physics, 327 (6) June 2012, pp. 1562-1592, DOI: 10.1016/j.aop.2012.02.007

[2] On the Intrinsically Cyclic Nature of Space-Time in Elementary Particles. Donatello Dolce, (Melbourne U., CoEPP). Sep 2011. 13pp. Published in J. Phys.: Conf. Ser. 343 012031, 2012.

[3] de Broglie Deterministic Dice and emerging Relativistic Quantum Mechanics. Donatello Dolce, (Melbourne U., CoEPP). Jan 2011. 10pp. Published in J. Phys.: Conf. Ser. 306 012049, 2011.

[4] Compact Time and Determinism for Bosons: foundations. Donatello Dolce, (Mainz U., THEP) . MZ-TH-09-06, 2010. 43pp. Published in Found.Phys. 41:178-203,2011.

[5] Holographic approach to Higgsless models. D. Dolce (Universitat Autonoma de Barcelona and IFAE),R. Casal- buoni, S. De Curtis, D. Dominici (Florence U. and INFN, Florence) . Feb 2007. 23pp. Published in JHEP 0708:053,2007

[6] Playing with fermion couplings in Higgsless models. R. Casalbuoni, S. De Curtis, D. Dolce, D. Dominici (Florence U. and INFN, Florence) . Feb 2005. 20pp. Published in Phys.Rev.D71:075015,2005,

[7] The Scalar sector in an extended electroweak gauge symmetry model. Stefania De Curtis (INFN, Florence U.) , Donatello Dolce (Florence U.) , Daniele Dominici (INFN, Florence U.) . Jul 2003. 12pp.

Published in Phys.Lett.B 579:140-148,2004

[8] Deterministic Quantization by Dynamical Boundary Conditions.

Donatello Dolce, (Mainz U., THEP) . MZ-TH-10-24, Jun 2010. 4pp. Published in AIP Conf.Proc.1246:178-181,2010.

[9] Quantum Mechanics from Periodic Dynamics: The Bosonic Case.

Donatello Dolce, (Mainz U., THEP) . MZ-TH-09-23, Jan 2010. 6pp. Published in AIP Conf.Proc.1232:222-227,2010.

[30] AdS/CFT interpretation of a Virtual Extra Dimension. Donatello Dolce, (Melbourne U., CoEPP). Mov 2012. 40pp. Submitted to Annals of Physics e-Print: arXiv:1110.0316

[31] Clockwork Quantum Universe. Donatello Dolce, (Melbourne U., CoEPP). Feb 2011. 8pp. Forth Price, FQXi contest 2011 “Is reality Digital or Analog?” web page: http://www.fqxi.org/community/essay/winners/2011.1#dolce

[32] Compact Time and Determinism for Bosons Donatello Dolce, (Melbourne U., CoEPP). Mar 2009. e-Print: arXiv:0903.3680v1 - arXiv:0903.3680v4.

[33] Gauge Symmetry Breaking without the VEV and other considerations about superconductivity Donatello Dolce, (Melbourne U., CoEPP). Feb 2012. 16pp. Submitted to Annals of Physics

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