aa r X i v : . [ qu a n t - ph ] A p r Intrinsic periodicity:the forgotten lesson of quantum mechanics Donatello Dolce the University of Melbourne, Parkville 3010 VIC, Australia.the University of Camerino, Piazza Cavour 19F, 62032 Camerino, Italy.E-mail: [email protected]
Abstract.
Wave-particle duality, together with the concept of elementary particles, wasintroduced by de Broglie in terms of intrinsically periodic phenomena. However, after nearly90 years, the physical origin of such undulatory mechanics remains unrevealed. We proposea natural realization of the de Broglie periodic phenomenon in terms of harmonic vibrationalmodes associated to space-time periodicities. In this way we find that, similarly to a vibratingstring or a particle in a box, the intrinsic recurrence imposed as a constraint to elementaryparticles represents a fully consistent quantization condition. The resulting classical cyclicdynamics formally match ordinary relativistic Quantum Mechanics in both the canonical andFeynman formulations. Interactions are introduced in a geometrodynamical way, similarly togeneral relativity, by simply considering that variations of kinematical state can be equivalentlydescribed in terms of modulations of space-time recurrences, as known from undulatorymechanics. We present this novel quantization prescription from an historical prospective.
1. Introduction
We give some historical motivation for a formulation of Quantum Mechanics (QM) in termsof elementary space-time cycles [1, 2, 3]. Since its earliest days, QM was characterized byan assumption intrinsic periodicity. We may think to the wave-particle duality, introducedby de Broglie in this terms: “we proceed with the assumption of the existence of a certainperiodic phenomenon of a yet to be determined character, which is to be attributed to each andevery isolated energy parcel” [4]. That is to say, every elementary particle is characterized byan intrinsic recurrence in time and space. As known from undulatory mechanics, every localretarded variation of the kinematical state of a particles (i.e. the local four-momentum) can beequivalently expressed, by means of the Planck constant, in terms of the corresponding localretarded modulation of space-time recurrence. According to the Standard Model, every systemin physics is described by a set of elementary particles and their relativistic interactions, thusthe wave-particle duality implicitly says that every system in physics can be described in termsof elementary modulated cycles.The intrinsic periodicity of elementary particles can be used as a quantization condition,similarly to the walls of a “particle in a box”. This is intuitive if we think to Bohr’s hydrogenatom in which, similarly to the harmonic spectrum of a vibrating ‘string’, the energy levelsare determined by requiring that the only possible orbits are those with an integer number of Dedicated to Ignaz Philipp Semmelweis, the “savior of mothers” (Buda 1818 — D¨obling 1865). avelengths. Indeed this is the idea at the base of the Bohr-Sommerfeld quantization: in asemi-classical description of QM, the energy spectrum can be determined by requiring closedorbits associated to a given potential. It has been shown recently that such a semi-classicaldescription can be used to solve complex quantum systems such as atoms with more electronsor the Zeeman effect, considered as some of the main limitations of this approach.More generally, as mathematically proven in recent papers [1, 2, 3], we will show how thesemi-classical description based on the assumption of intrinsic periodicity of elementary systems,when correctly implemented in a relativistic theory, can be used to formally derive the moderndescriptions of relativistic QM.
2. The model
We introduce the formalism by following the same arguments used by de Broglie to derivethe wave-particle duality [4]. In his paper he pointed out that every elementary particle ischaracterized by recurrences in time and space. These are fully determined, though Lorentztransformations, by the recurrence along the proper time, fixed by the mass of the particle ¯ M .The periodicity of this rest recurrence is the Compton time of the particle T τ = h ¯ M c . Equivalently, every particle has a quantum recurrence λ s of along its worldline s = cτ , where λ s is the Compton wavelength λ s = cT τ . When observed in a generic reference frame thiscorresponds to instantaneous temporal and spatial periodicities, T t and ~λ x respectively, as canbe seen by using the Lorentz transformation cT τ = cγT t − γ ~β · ~λ x . They can be written in a contravariant tangent four-vector T µ = { T t , ~λ x } . In fact, from the relativistic definition of four-momentum ¯ p µ = { ¯ E, − ¯p } = { γ ¯ M c , − γ ~β ¯ M c } , itis easy to derive the following relativistic invariant (de Broglie phase harmony) T τ ¯ M c ≡ h ←→ T µ ¯ p µ ≡ h. In other words, the four-momentum of a particle can be equivalently encoded in correspondingmodulations of the instantaneous space-time periodicity T µ . According to undulatory mechanics,particles can be described in terms of periodic phenomena (phasors or waves) in which thespace-time coordinates enter as angular variables; their periodicities define their kinematicalstate. Thus, every system in nature can be consistently described in terms of modulations ofelementary space-time cycles. Since they are completely characterized by proper time periodicity,we can consider a minimal topologies S for elementary bosonic cycles .It must be noted that these recurrences are typically extremely fast with respect to the Cs atomic clock, whose time scale is (by definition of “second”) about 10 − s , or to the presentresolution in time, about 10 − s (using laser pulses). For instance, simple electrodynamicssystem are characterized by the the proper time periodicity of the electron T τ ( e ) = h/m ( e ) c =8 . × − s (Compton time), corresponding to the zitterbewegung . The heavier themass or the energy of the particles, the faster the corresponding temporal periodicities. For We neglect the spin which however can be regarded as another manifestation of the intrinsic periodicity ofelementary particles. See Schr¨odinger’s zitterbewegung and related semiclassical derivation of the Dirac equation. nstance, LHC is exploring “indirectly” time cycles of the order of T t ∼ − s (correspondingto energies scales of the TeV).We want to impose the intrinsic periodicity T µ of elementary particles as a constraint. Thisrepresents a semiclassical quantization condition. A particle with intrinsic periodicity is similarto a “particle in a box”. Through discrete Fourier transform the periodicity T µ directly impliesa quantization of the conjugate spectrum p µn = n ¯ p µ ; n is the single quantum number associatedto the topology S . For instance the quantization of the energy spectrum associated to the timeperiodicity T t is the harmonic spectrum E n = n ¯ E = nh/T t . A bosonic particle can be therefore represented as a one dimensional bosonic ‘string’ Φ( x )vibrating in compact space-time dimensions of length T µ and Periodicity Boundary Conditions(PBCs — denoted by the circle in H ): S λ s = I T µ d x L ( ∂ µ Φ( x ) , Φ( x )) . (1)As known from string theory or extra-dimensional theories, PBCs (or combinations ofDirichlet/Neumann BCs or anti-PBCs) minimizes the action at the boundary so that all therelativistic properties of (1) are preserved. This is a consequence of the fact that relativity fixesthe differential structure of space-time whereas the only requirement for the BCs is to fulfill thevariational principle. The expansion in harmonics (discrete Fourier expansion with coefficients a n and normalization A n ) of a field/string vibrating with persistent periodicity isΦ( x ) = X n φ n ( x ) = X n A n a n (¯ p µ ) e − i ~ p nµ x µ . Such a description in terms of elementary space-time cycles is covariant as prescribed byrelativity. This can be easily seen by using Lorentz transformations. The action (1) describing aparticle in a given reference frame, after the generic transformation of variables x µ → x ′ µ = Λ νµ x ν ,transforms to the action S λ s = I T ′ µ =Λ µν T ν d x ′ L ( ∂ ′ µ Φ ′ ( x ′ ) , Φ ′ ( x ′ )) . (2)This turns out to have a transformed boundary, such that the resulting solution has transformedperiodicity T µ → T ′ µ = Λ µν T ν . According to ¯ p ′ µ cT ′ µ = h , this is the periodicity associated to the four-momentum of the particlein the new frame ¯ p µ → ¯ p ′ µ = Λ νµ ¯ p ν . Indeed T µ is a contravariant tangent four-vector satisfying the relativistic constraint1 T τ = 1 T µ T µ , With “direct” observation we mean to compare these fast dynamics with a clock of resolution higher than thistime scale. For instance, with the modern clocks, a direct observation of these quantum recurrences are possiblefor neutrinos and other coherent quantum phenomena in condensed matter such as lasers, superconductivity orcarbon nanotubes. hich is the geometric counterpart of the relativistic dispersion relation of the particle ¯
M c =¯ p µ ¯ p µ . By considering the relativistic modulations of temporal periodicity (relativistic Dopplereffect) associated to variations of reference frames, the resulting energy spectrum is E n (¯ p ) = n hT t (¯ p ) = n q ¯ p c + ¯ M c . In other words we have obtained semi-classically the energy spectrum prescribed by ordinarysecond quantization (after normal ordering) for bosonic particles .
3. Interactions
Interactions can be introduced in this formalism by considering that, as known in undulatorymechanics, relativistic variations of four-momentum associated to a given interaction schemecan be equivalently described in terms of relativistic modulations of periodicities. That is, inevery point x = X , a relativistic interaction can be characterized by the relativistic variationsof four-momenta of a particle with respect to the free case¯ p µ → ¯ p ′ µ ( X ) = e aµ ( x ) | x = X ¯ p a . Through ~ , this generic interaction can be equivalently encoded by corresponding relativisticmodulations of quantum recurrence T µ → T ′ µ ( X ) ∼ e µa ( x ) | x = X T a . According to our formalism, this corresponds to local and retarded “stretching” the compactifiedspace-time dimensions of (1), and thus by corresponding local and retarded deformations of themetric η µν → g µν ( X ) = [ e aµ ( x ) e bν ( x )] | x = X η ab . This geometrodynamical description of interaction can be easily checked by imposing the localtransformation of reference frame dx µ → dx ′ µ ( X ) = e aµ ( x ) | x = X dx a as substitution of variables in the free action (1), see [2, 3]. Indeed, the resulting action turnsout to have locally deformed metric g µν ( X ) and local transformations of boundary. In this way,the solution of this transformed action correctly describes the local modulations of periodicity T ′ µ ( X ) associated to the interaction scheme. That is, we pass from a free solution of persistenttype φ ( x ) ∝ e − i ~ p µ x µ to the interacting solution with modulated periodicity of type φ ′ ( x ) ∝ e − i ~ R xµ dx ′ µ p µ . (3)Note also that in our formalism the (integral of the) kinematics associated to the given interactionscheme turns out to be encoded on the boundary of the theory. The holographic principle hasthus a natural realization in this theory. In fact, second quantization prescribes that every mode with angular frequency ¯ ω (¯ p ) = p ¯ p c + ¯ M c / ~ of aKlein Gordon fields has a quantized energy spectrum E n (¯ p ) = nh ¯ ω (¯ p ), after normal ordering. By equivalentlyassuming in (1) anti-PBCs we get E n (¯ p ) = ( n + ) h ¯ ω (¯ p ), avoiding normal ordering. uch a geometrodynamical description of generic interactions is of the same type of the oneprescribed by General Relativity (GR). In a weak Newtonian interaction, the correspondingvariation energy ¯ E → ¯ E ′ ∼ (cid:0) GM ⊙ / | x | c (cid:1) ¯ E implies, through ~ , a modulation of timeperiodicity T t → T ′ t ∼ (cid:0) − GM ⊙ / | x | c (cid:1) T t , i.e. we have redshift and time dilatation. If we alsoconsider the variation of momentum and the corresponding modulation of spatial periodicity,the resulting metric encoding the Newtonian interaction is actually the linearized Schwarzschildmetric. Thus we have obtained linearized gravity, from which GR follows by assuming self-interaction.We also mention another remarkable consequence of our description of elementary cycles,which can be regarded as a realization of Weyl’s, Kaluza’s, Wheeler’s original proposals. In[2, 3] we have found that gauge interaction has a geometrodynamical description similar to thatof gravitational interaction in GR. This possibility is provided by the fact that transformationsof flat reference frames, which in ordinary QFT do not vary the solution of the theory , in ourformulation induce local transformations of the boundary, and thus local transformations of thesolution of the theory. The resulting transformation of the field solution is formally equivalentto the internal transformation of ordinary gauge theories. For instance, it is possible to showthat the local transformation of flat reference frame dx µ ( x ) → dx ′ µ ∼ dx µ − edx a ω µa ( x )can be parametrized parametrizing by introducing a vectorial field¯ A µ ( x ) ≡ ω aµ ( x )¯ p a , so that the interaction scheme is actually given by the minimal substitution¯ p ′ µ ( x ) ∼ ¯ p µ − e ¯ A µ ( x ) . Thus the transformed solution associated to this interaction is of the type φ ′ ( x ) ∝ e − i ~ R xµ dx ′ µ A µ e − i ~ p µ x µ . The gauge connection, which must be postulated in ordinary gauge theory, turns out to describethe modulations of periodicity associated to the local transformations of flat reference framewith respect to the free solution (we say that the gauge field “tunes” the periodicity). Fromthis, ordinary Maxwell dynamics (and more in general Yang-Mills theories) can be derived, see[2] for more details.
4. Quantization
From an historical point of view, the modern mathematical formulation of QM is inspired tothe theory of sound and Reyleigh studies. Indeed the theory described so far can be regardedas a fully relativistic generalization of the theory of sound (we assume vibrations in compacttime, and not only in compact space as in ordinary classical sound sources). Therefore the theoryinherits fundamental aspects of the ordinary quantum formalism. The assumption of periodicityat the base of our formulation is a quantization condition (similarly to the quantization of aparticle in a box). In particular, when the recurrences of elementary particles are imposed as aconstraint, the resulting (classical) cyclic dynamics reproduce formally ordinary relativistic QM. In ordinary QFT the BCs have a marginal role: the KG field used in computation is the most general solutionof the KG equation. Intuitively we want to describe the trembling motion of a particle interacting electromagnetically as localtransformation of reference frame, similarly to the equivalence principle (applied to the boundary) of GR. n this section we summarize basic aspects of this remarkable correspondence. It is knownthat a vibrating ‘string’ is the typical classical system that can be described locally in a Hilbertspace. Even if we consider modulations of periodicities, the harmonics of such an interacting‘string’ form locally a complete set with respect to the corresponding local inner product h φ | χ i ≡ Z λ x ( X )0 dxλ x ( X ) φ ∗ ( x ) χ ( x ) . (4)These harmonics define locally a Hilbert base (cid:10) x | φ ′ n (cid:11) = φ ′ n ( x ) . Thus a modulated vibrating ‘string’, generic superposition of harmonics, is represented by thegeneric Hilbert state (cid:12)(cid:12) φ ′ (cid:11) = X a n | φ n i . The non-homogeneous Hamiltonian H ′ and momentum P ′ i operator are introduced as theoperators associated to the four-momentum spectrum of the locally modulate ‘string’: P ′ µ (cid:12)(cid:12) φ ′ n (cid:11) = p ′ nµ (cid:12)(cid:12) φ ′ n (cid:11) , where P ′ µ = {H ′ , −P ′ i } . From the modulated wave equation, the temporal and spatial evolution of every modulatedharmonics satisfies i ~ ∂ µ φ ′ n ( x ) = p ′ n φ ′ n ( x ) , thus the time evolution of our modulated ‘string’ | φ ′ i is given by the ordinary Schr¨odingerequation i ~ ∂ t (cid:12)(cid:12) φ ′ (cid:11) = H ′ | φ i . (5)Moreover, since we are assuming intrinsic periodicity, this classical-relativistic theory implicitlycontains the ordinary commutation relations of QM. This can be seen by evaluating theexpectation value of a total derivative ∂ x F ( x ), and considering that the boundary terms ofthe integration by parts cancel each other owing the assumption of intrinsic periodicity. Forgeneric Hilbert states we obtain [ F ( x ) , P ′ i ] = i ~ ∂ x F ( x ) and thus, from F ( x ) = x j ,[ x j , P ′ i ] = i ~ δ j,i . The formal correspondence to ordinary relativistic QM is confirmed by the fact that, remarkably,the classical evolution of such a classical vibrating ‘string’ with all its modulated harmonics isdescribed by the ordinary Feynman Path Integral (we are integrating over a sufficiently largenumber N of spatial periods so that the V x = N λ x is bigger than the interaction region) Z = Z V x D x e i ~ S ′ ( t f ,t i ) . (6)The action S ′ is, by construction, the classical action of the corresponding interaction scheme,with lagrangian L ′ = P ′ x − H ′ . This result has a very intuitive justification in the fact that in acyclic geometry such as that associated to the topology S , the classical evolution of φ ( x ) from aninitial configuration to a final configuration is given by the interference of all the possible classicalpaths with different windings numbers; without relaxing the classical variational principle.Thus the harmonics of the vibrating string/field φ ( x ) are interpreted as quantum excitations.We have also proven that this classical to quantum correspondence pinpoints a fundamentalaspect of Maldacena’s duality [2]. Therefore, the assumption of intrinsic periodicity whichas characterized quantum theory in its early days has a renewed interest in modern physics.This correspondence to relativistic QM can be interpreted in analogy with ’t Hooft determinism:“there is a deep relationship between a particle moving very fast in a circle [of periodicity T t ] anda quantum harmonic oscillator [with the same periodicity]” [5]; or the stroboscopic quantization[6]). QM emerges as a statistical description associated to the extremely fast recurrences of theelementary systems in nature: this is the meaning of the Hilbert notation above. Our actualexperimental time resolution, about ∆ T exp ∼ − , is too low to resolve the small time scalesof ordinary quantum systems, though the internal clock of an electron has been observed in arecent interference experiment [7]. As for a dice rolling too fast with respect to our resolutionin time, the outcomes can be described only in a statistical way. Loosely speaking, an observerwith infinite resolution in time can in principle resolve exactly the underling deterministic cyclicdynamics: it would have no fun playing dice (“God doesn’t play dice”, Einstein).It is instructive to interpret the black-body radiation in terms of “periodic phenomena”. Inthis case it is natural to assume that the population a n of the n th energy levels is fixed bythe Boltzmann distribution. For the IR components of the radiation, i.e. massless periodicphenomena with long periodicity, the PBCs can be neglected and the energy spectrum canbe approximated to a continuum (classical limit) since many energy levels are populated (thethermal noise, i.e. the euclidean time periodicity, destroys the intrinsic periodicity in a sort ofdecoherence). For the UV components, however, the fundamental energy ¯ E is big with respectto the thermal energy so that only a few energy levels can be populated. That is to say, thesemodes have very short periodicity (not destroyed by the thermal noise), so that the PBCs areimportant and there is a manifest quantization of the energy spectrum (quantum limit). TheUV catastrophe is avoided according to Planck.We have a consistent interpretation of quantum to classical transition as ~ →
0. For massiveparticles, in the non-relativistic limit ¯ p ≪ ¯ M c , only the fundamental energy level is largelypopulated as the gap between the energy levels goes to infinity, ¯ M → ∞ . The cyclic field cantherefore be approximated as Φ( x ) ∼ exp[ − i ¯ Mc ~ t + i ¯ M ~ x t ], see [1]. Neglecting the proper timrecurrence, it is possible to see by plotting the || that the wavefunction of a massive periodicphenomenon is centered along the path of the corresponding classical particle and its width isof the order of the Compton wavelength λ s . Thus, in the non-relativistic limit, the Dirac deltadistribution describing a classical particle is reproduced as in the usual Feynman description.Similarly, the spatial compactification length tends to infinity whereas the time compactificationtends to zero, so that in the non-relativistic limit we have a point-like distribution in R , i.e. theordinary three-dimensional description of a classical particle. Indeed, the corpuscular descriptiontherefore arises at high frequencies. On the other hand, in the relativistic limit the non-localnature of a massive periodic phenomena can not be neglected (the distribution width is of theorder of the Compton wavelength). This gives an intuitive interpretation of the wave-particleduality and of the double slit experiment. In particular, if probed with high energy or observedwith good resolution, more and more energy levels turn out to be excited, i.e. more and moreharmonics can be resolved. In this way we can figure out that the energy excitations playthe role of the quantum excitations of the same fundamental elementary system, so that wehave an analogy with the virtual particles of ordinary relativistic QM. The modes can have ingeneral with either positive or negative frequencies: this means that the relativistic theory hasHamiltonia operator positively defined (contrarily to ’t Hooft model).We conclude this section by noting that the assumption of intrinsic periodicity implicitlycontains the Heisenberg uncertain relation of ordinary QM. Briefly, in the Hilbert notation(statistical description of a “periodic phenomenon”) the phase of the periodicity can not bedetermined — in (4) only the square of the field has physical meaning . To determine thefrequency and thus the energy ¯ E ( ¯p ) = ~ ¯ ω ( ¯p ) with good accuracy ∆ ¯ E ( ¯p ) we must count alarge number of cycles. That is to say we must observe the system for a long time ∆ t ( ¯p ). Thehase is defined modulo factors π which can be written as a energy or a temporal uncertaintyexp[ − i ¯ Et ~ + π ] = exp[ − i ( ¯ E +∆ ¯ E ) t ~ ] = exp[ − i ¯ E ( t +∆ t ) ~ ]. Considering that every cycles is such that t < T t (for time longer than a cycle we must consider the general phase invariance nπ ) we findthat this uncertain is described by the famous Heisenberg relation ∆ ¯ E ( ¯p )∆ t ( ¯p ) & ~ /
2, [1].
5. Schr¨odinger problems
We continue with the historical overview of our assumption of intrinsic periodicity — aspectsconcerning modern physics are discussed in [1, 2, 3]. In particular we what to show how to solvesimple non-relativistic quantum problems in terns of the vibrational modes associated to thequantum recurrences .First of all we must consider that the assumption of intrinsic periodicity is equivalent tothe Bohr-Sommerfeld condition: it in fact prescribes that the only possible modes are thosewith a finite number of cycles as in a vibrating ‘string’. For instance, the assumption ofintrinsic periodicity in the free case (homogeneous periodicity) leads to the harmonic spectrum p nµ T µ = n ¯ p µ T µ = nh . More in general, as shown in [2], in case of interaction the quantizedspectrum associated with the corresponding modulation of periodicity is given by the condition H p nµ ( x ) dx µ = h ( n + v ) (the so-called Morse index v interpreted as a twist factor in the PBCs,see also [1, 2, 9]). This can be seen by imposing periodicity to the modulated solution of the type(3); note also the correspondences with the WKB method. To obtain the the Bohr-Sommerfeldwe quantized only the spatial momentum p n ( x ); the corresponding energy spectrum is obtainedthrough the equations of motion of the interaction. The quantization condition is therefore I p n ( x ) d x = h ( n + v ) . To this spatial momentum we associate a potential, which in the non-relativistic limit can bedefined such that E n = p n ( x )2 m + V ( x ). The Hamiltonian operator is so that the Schr¨odingerequation (5), mode by mode, turns out to be given by − ~ m ∂ x φ n ( x ) + V ( x ) φ n ( x ) = E n φ n ( x ) . (7)Taking into account all previous arguments, the quantized harmonic oscillator turns out tohave an immediate solution with respect to the usual formulation. It is sufficient to considerthe (Galileo) isochronism of the pendulum: every orbit of an harmonic oscillator has the sametime (homogeneous) periodicity. We have just to impose that the solution is a wave with thecharacteristic periodicity T t = 2 π/ ¯ ω for every energy level. That is the quantization conditionof the energy spectrum is simply H E nµ dt = E n T t = h ( n + v ). Therefore the solution has anharmonic energy spectrum E n = hT t ( n + ) and can be decomposed asΦ( x , t ) = X n φ n ( x , t ) = X n e i ( n +1 / ωt φ n ( x ) . (8)where we have assumed a twist of half a period (anti-periodicity) in the BCs in order to reproducethe Morse factor v = . The Schr¨odinger differential equations of the problem can be written More in general, by quantum recurrence we mean that quantum system can be described semi-classically byimposing appropriate BCs to relativistic wave equations. Trivial examples are the quantization of a particle ina box, a particle with Dirac delta potentials, and similar problems. However this approach can be extended tonon-obvious problems such as the Casimir effect [8], the tunnel effect or atomic physics [9] This avoides to the complication of the Fock space. In this approach the creation and annihilation operatorscan be regarded as describing the excitation or de-excitation of the different levels of the harmonic system. Intuitively the assumption of intrinsic antiperiodicity is natural for fermion in order to reproduce the correctspin statistics y defining x = ρ y with ρ = q ~ m ¯ ω and h n ( y ) = φ n ( y ) e y . Thus the solution of the n -thharmonic mode of the vibration associated to the harmonic potential is ∂ y h n ( y ) − y ∂ y h n ( y ) = (cid:20) (cid:18) n + 12 (cid:19) − (cid:21) h n ( y ) . (9)We have solved the quantum harmonic oscillator by simply assuming intrinsic periodicity. Thisexample is extremely important since the harmonic oscillator is the basic ingredient of ordinary(bosonic) quantum field theory. That is, a second quantized Klein Gordon field is the integralover the quantum harmonic oscillators with all the possible periodicities, from zero to infinity.The creation and annihilation operators in this approach describe the excitation of the different n -th mode of this harmonic ‘string’. This suggests that our correspondence to QM can be fullyextended to ordinary quantum field theory.In a similar way it is easy to solve other Schr¨odinger problems. For instance here we reportthe energy spectrum of simple one-dimensional potentials. This method applied to the linearpotential V ( x ) = mgx actually gives the correct quantized energy spectrum E n = 12 [3 π ( n + 1 / / ( ~ mg ) / where we have assumed v = 1 /
4. Another example is the solution of the quantum anharmonicoscillator, i.e. we assume a perturbation ǫx /l of the harmonics potential, with l = p ~ /mω .In this way we obtain (modulo Morse factors) the deviation for the harmonic energy spectrum E n = ~ ¯ ωn of the same quantity prescribed by the ordinary methods:∆ E n = ǫ
32 ( n + n ) . In this way it is easy to see that the vibrational modes associated to these one-dimensionalpotentials reproduce the correct solutions of the Schr¨odinger problems.From a historical point of view this approach to quantum problems is the same adoptedby Bohr to solve the energy spectrum of the hydrogen atom. Indeed, according to the aboveprescription, the harmonic modes of our ‘string’ with the modulated periodicities allowed by theCoulomb potential yields the resulting energy spectrum (modulo Morse factors) E n = − . n . This famous results is obtained without necessarily requiring circular orbits. The single space-time periodicity of topology S associated to the orbits describes correctly the principal quantumnumber n . Furthermore, in spherical problems we must also consider the quantization associatedto the spherical periodicity, i.e. the quantized vibrational modes associated to the additionalperiodicities in the spherical angles θ ∈ [0 , π ) and ψ ∈ [0 , π ). As known from ordinary QMthese lead to the ordinary quantization of angular momentum. In fact, the spherical vibrationalmodes of solution are described in terms of the two additional angular and magnetic quantumnumbers { m, l } . In other words, the atomic orbitals can be represented as the vibrational modesof an harmonic system of topology S ⊗ S in a Coulomb potential. This harmonic descriptionof atoms has been recently re-proposed by eminent scientists such as the Nobel laureate Prof.F. Wilczek. It has been successfully generalized to achieve classical descriptions of quantumproperties of atoms with more electrons (e.g. Helium atom), the Zeeman effect [9]; these wereconsidered the major limitation of the approach based on the assumption of intrinsic periodicity.The example of the Hydrogen atom provides us the last element of the formal correspondenceto QM: different harmonic sets characterized by different fundamental periodicity are describedy the tensor products of their different Hilbert spaces. For instance, in the Hilbert notation,the orbitals φ n,l,m , composition of the harmonics associated to the space-time vibration | φ n i , ormore simply | n i , and the spherical harmonics | l, m i , can be written in a Hilbert space with base | n i ⊗ | l, m i = | n, l, m i associated to the topology S ⊗ S . This property plays an importantrole in the demonstration of the Bells’ theorem. In particular, as it will be shown explicitlyin a dedicated paper, the formal correspondence of our approach with canonical QM stronglysuggests the statistical description of elementary space-time cycles satisfies the same inequalitiesof ordinary QM. In particular it must be noted that the theory has not local-hidden-variables;the assumption of intrinsic periodicity is a element of non-locality (though this type non-localityis consistent with relativity, as the periodicity varies according to the retarded potentials).
6. Conclusions
We have shown how ordinary relativistic QM can be formally derived from an effective statisticaldescription of the classical mechanics associated to the space-time recurrences of elementarysystems [1, 2, 3]. In this paper we have given some historical motivations of this approach. It hasits origin in Bohr’s description of hydrogen atom, Bohr-Sommerfeld quantization, de Broglie’swave particle-duality, Schr¨odinger’s zitterbewegung , and in other foundational contributions tomodern QM (not mentioned here for brevity). These ideas are the base of modern quantumfield theory, but the attempt of a semi-classical description of QM has been abandoned afterHeisenberg and Bell. Nevertheless, we have seen that these limitations can be avoided bythinking in terms of elementary cycles. The formal correspondence to ordinary QM obtainedby imposing intrinsic periodicity to elementary particles strongly suggests that relativistic QMcan be interpreted as an effective (statistical) description of semi-classical mechanics associatedto the fast cyclic dynamics of elementary particles, as also suggested for instance by ’t Hooft— and not the contrary. We have proven that fundamental aspects of QM can be consistentlydescribed in terms of the harmonic modes of four-dimensional vibrating ‘strings’. The resultingtheory is a relativistic generalization of the theory of sound (where sources can vibrate along thetime dimension). This aspect anticipates the study of QM to Pythagoras, who first investigatedthe “quantized” spectrum of harmonic system, as also recently suggested by Wilczek. Theresulting theory conciliates together important aspects of modern physics: it can be regardedas a (purely 4D) string theory (defined on the single compact word-parameter s ) similarly tooriginal Veneziano’s proposal; it provides a geometrodynamical description of gauge interaction,similarly to original Weyl’s proposal; it pinpoints fundamental aspects (e.g. classical to quantumcorrespondence) of Maldacena’s duality, originally pointed out by Witten; it justifies themathematical beauty of extra-dimensional theories as original proposed by N¨ordstrom, Kaluzaand Klein; and so on. We conclude that physical origin (the “missing link” [10]) of QM must befound in the assumption of intrinsic periodicity, as originally proposed by the father’s of QM andthen forgotten. The resulting formulation of physics in terms of elementary space-time cyclesbrings important new elements to face the open problems of modern physics [1, 2, 3]. References [1] Dolce D 2011
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