Classical geometry to quantum behavior correspondence in a Virtual Extra Dimension
aa r X i v : . [ h e p - t h ] J u l Classical geometry to quantum behaviorcorrespondence in a
Virtual
Extra Dimension
Donatello Dolce
Centre of Excellence for Particle Physics (CoEPP),The University of Melbourne,Parkville 3010 VIC,Australia.
Abstract
In the Lorentz invariant formalism of compact space-time dimensions theassumption of periodic boundary conditions represents a consistent semi-classical quantization condition for relativistic fields. In [1] we have shown,for instance, that the ordinary Feynman path integral is obtained from theinterference between the classical paths with different winding numbers asso-ciated with the cyclic dynamics of the field solutions. By means of the bound-ary conditions, the kinematics information of interactions can be encoded onthe relativistic geometrodynamics of the boundary [2]. Furthermore, sucha purely four-dimensional theory is manifestly dual to an extra-dimensionalfield theory. The resulting correspondence between extra-dimensional geo-metrodynamics and ordinary quantum behavior can be interpreted in termsof AdS/CFT correspondence. By applying this approach to a simple Quark-Gluon-Plasma freeze-out model we obtain fundamental analogies with basicaspects of AdS/QCD phenomenology.
Introduction
Originally proposed by T. Kaluza [3] and O. Klein [4], field theory in com-pact eXtra Dimension (XD) represents one of the most investigated candi-dates for new physics beyond the Standard Model (SM), providing an elegantexplanation for the hierarchy problem [5, 6, 7, 8].
Email address: [email protected] (Donatello Dolce)
Preprint submitted to Elsevier March 8, 2018 evertheless, Kaluza, following Nordström’s idea [9], originally intro-duced the XD formalism as a “mathematical trick” to unify gravity and elec-tromagnetism [3]. This mathematical property of XD theories has inspiredthe purely 4D, semi-classical, geometrodynamical formulation of scalar QEDrecently published in [2], in which gauge symmetries are related to corre-sponding space-time symmetries.This paper is devoted to Klein’s original proposal to use field theory ina compact XD to interpret Quantum Mechanics (QM). In his famous paper
Quantentheorie und fünfdimensionale Relativitätstheorie [4] Klein noted thatPeriodic Boundary Conditions (PBCs) at the ends of a compact XD yieldan analogy to the Bohr-Sommerfeld quantization condition — the cyclic XDwas introduced to interpret the quantization of the electric charge. As wewill see in detail, the formalism of XD theory shares interesting analogieswith the quantum formalism. For instance, in XD theories the KK massspectrum is fixed by consistent Boundary Conditions (BCs) similarly to theenergy spectrum in semi-classical theories, whereas the evolution along theXD is described by bulk Equations of Motion (EoMs) playing the role of theSchrödinger equation of QM, [10, 11]. In general the essential requirement forthe BCs is that they must minimize the action at the boundaries, fulfilling theclassical variational principle [15, 8]. It is well known from XD field theoryor string theory that the variational principle allows PBCs (or anti-PBCs)in relativistic bosonic actions — or in an Orbifold description combinationsof BCs of Neumann (N-BCs) and Dirichlet (D-BCs) type. These BCs havethe same formal validity of the usual Synchronous ( a.k.a.
Stationary) BCs(SBCs) of ordinary field theory, in which the fields have fixed values at theboundaries.The dimensional (de)construction mechanism [16, 17] has shown anotherremarkable application of XD theories, that is to say a dualism between theclassical configuration of XD fields, in particular in warped XD (AdS), andthe quantum behavior of Strongly Coupled Field Theories (SCFTs), such asQCD [18, 19]. This so-called AdS/QCD correspondence is a phenomenolog-ical application of the more fundamental (and not yet proven) AdS/CFTcorrespondence [20], whose validity seems to be more general than its origi-nal formulation, as confirmed by many bottom-up investigations. Accordingto the words used by Witten in the abstract of his famous paper [21], inAdS/CFT “ quantum phenomena [...] are encoded in classical geometry ”.This aspect can be regarded as a “classical to quantum” correspondencewhich, however, does not involve any explicit quantization condition.2n this paper we will seek the origin of this kind of “classical to quan-tum” correspondence of XD theories in term of the formalism of bosonicfield theory in compact space-time dimensions (compact 4D). The founda-tional aspects of this novel theory have been defined in recent papers [1, 2],see also [22, 23, 24, 25, 27, 26], and will be summarized in sec.(1-4). Thebasic assumption is that every (free) elementary relativistic particle has as-sociated an intrinsic (persistent) periodicity . This hypothesis was originallyintroduced by de Broglie in 1923, in terms of intrinsic “ periodic phenomenonallied with every parcel of energy ” whose “ character is yet to be determined ”[28], i.e. a so-called “internal clock” associated to every elementary particle.In other words we will enforce the undulatory nature of elementary particle(wave-particle duality) at the base of the modern description of QM, implic-itly tested in 80 years of QFT and indirectly observed in a recent experiment[30]. As is well known, through the Planck constant, the space-time period-icity of a de Broglie “periodic phenomenon” (whose minimal topology is S )describes the energy-momentum of a corresponding particle. In order to seethe is full consistency with relativity of the assumption of intrinsic period-icity, it is therefore sufficient to note that the retarded and local variationsof energy-momentum characterizing relativistic interaction and causality canbe equivalently described by corresponding retarded and local modulationsof space-time periodicity.Essentially, by considering that in classical-relativistic mechanics the kine-matics of a particle is described by its 4-momentum, that in ordinary undu-latory mechanics the energy-momentum is encoded in temporal and spatialperiodicity of a corresponding intrinsically “periodic phenomenon” (wave-particle duality), and that in the atomistic description of modern physicselementary particles are the basic constituents of every physical system, wecan figure out the possibility of a consistent description of physics in termsof intrinsically cyclic elementary systems (with locally modulated periodicityin the interacting case). In addition to this, we must consider that masslessfields (such as the mediators of the EM and gravitational interactions) canhave infinite periodicity, providing the long space-time reference scales for arelational description of the elementary particles. Field theory in compact4D, is a natural realization of the de Broglie “periodic phenomenon”, [1, 2].In fact, it is essentially a relativistic bosonic theory in which, in agreementwith the variational principle, PBCs are imposed at the ends of compact 4D.As a consequence, the resulting field solution is constrained to have intrinsicspace-time periodicity. In turn, the de Broglie spacial and temporal peri-3dicities are encoded in the spatial and temporal compactification lengthsrespectively, i.e. on the boundary of the theory through the PBCs. Hence,according to undulatory mechanics, this will allow us to describe relativisticinteractions in terms of corresponding retarded and local deformations of thecompact 4D of the theory [2].The assumption of intrinsic periodicity represents a semi-classical quanti-zation condition for relativistic particles. Field theory in compact 4D can beregarded as the full relativistic generalization of the quantization of a “par-ticle in a box” or of a “vibrating string”, or more in general of sound theory.Moreover it has fundamental analogies with the Matsubara theory [32]. Ourperiodic field, owing to PBCs, is nothing but a relativistic string vibratingin a compact 4D. Through discrete Fourier transform and the Planck con-stant, space-time periodicity directly implies a discretized (quantized) energy-momentum spectrum. Transformations of reference frame imply relativisticmodulation of space-time periodicity ( e.g. as in the relativistic Doppler ef-fect). In this way we find out that a periodic field solution has the sameenergy-momentum spectrum as an ordinary second quantized bosonic field(after normal ordering).As proven in [1, 2, 22], without any explicit quantization condition ex-cept intrinsic periodicity, field theory in compact 4D reproduces formallythe fundamental aspects of QM such as Hilbert space, Schrödinger equa-tion, commutation relations, Heisenberg relation and so on. In particularin sec.(2) we will show that the Feynman Path Integral (FPI) formulationarises in a semi-classical and intuitive way in terms of interference betweenclassical paths of different winding numbers associated with the underlyingcyclic geometry S of the “periodic phenomenon”, [1, 22]. Remarkably, in [2]we have extended such a semi-classical formulation of quantum-relativisticmechanics to scalar QED.In this paper, we will explore another interesting property of field theoryin compact 4D. This is a remarkable mathematical analogy of our theorywith a KK theory, as already pointed out in [1]. In sec.(3) we will see, forinstance, that the parameterization of the intrinsic periodicity of an “internalclock” enters into the equations in perfect analogy with a cyclic XD of corre-sponding periodicity. In other words, at a mathematical level, the intrinsicperiodicity of a de Broglie “periodic phenomenon”, whose topology S is thesame topology of a KK theory, appears to be parameterized by cyclic XD,though the theory is purely 4D. To show this we will start with an ordinaryKK theory and use the cyclic XD to parameterize the cyclic world-line of a4periodic phenomenon”. To address this parameterization we say that the XDis Virtual (VXD). In particular, as already described in [1], we will find outthat under this identification, the KK field theory turns out to be equivalentto field theory in compact 4D, i.e. the two theories are dual . Also, with theterm virtual we want to emphasize the fact, owing to this duality, that fieldtheory in compact 4D inherits important mathematical properties from theXD theories without involving any “real” XD. For instance, in the ordinaryXD the KK modes describe independent particles. In the formalism of com-pact 4D the KK mode will correspond to the collective energy eigenmodes ofthe same 4-periodic field (similarly to the thermal modes of the Matsubaratheory). According to the correspondence to QM, this energy eigenmodescan be interpreted as quantum excitations or, loosely speaking, as virtual particles, so that we will for instance say that, in field theory in compact 4D,the KK modes are virtual . We will note that a similar description of the KKmodes is implicitly obtained through the Holography prescription.In sec.(4) we will generalize the theory to curved 4D in order to achievea geometrodynamical description of interaction, similar to GR, see [2]. Thelocal variations of 4-momentum of a given interaction scheme can be equiva-lently described as the corresponding local modulation of space-time periodic-ity, according to undulatory mechanics. As already noticed in [2], through thePBCs of our theory, this modulation can be equivalently formalized as corre-sponding local stretching of the compact 4D, that is to say in correspondinglocal deformations of the underlying space-time metric. We will see that thisformulation of interaction mimics the ordinary geometrodynamic descriptionof linearized gravity. For instance, it is well known that in General Relativity(GR) the deformations of space-time associated to gravitational interactionencode the modulations of space-time periodicity of reference lengths andclocks, [44]. Moreover, [2], in this formalism the kinematical informationof the particle is encoded in the geometrodynamics of the boundary in themanner of the holographic principle [45, 46].Combining the correspondence of the theory in compact 4D with ordinaryQFT, described in [1, 2] and summarized in sec.(2), and the dualism to XDtheory, described in [1] and in sec.(3), together with the geometrodynami-cal formulation of interactions, described in [2] and summarized in sec.(4),we will infer that the classical configuration of fields in a deformed VXDbackground encodes the quantum behavior of the corresponding interactionscheme . In sec.(5) we will apply this idea to a simple Bjorken Hydrodynam-ical Model (BHM) for Quark-Gluon-Plasma (QGP) exponential freeze-out533], and—considering also the analogies of QCD with a thermodynamicsystem [34]—we will find a parallelism with basic aspects of AdS/QCD phe-nomenology. In particular, during the exponential free-out the 4-momentumof the QGP fields decays exponentially—in terms of thermal QCD this cor-responds to Newton’s law of cooling. Thus the de Broglie “internal clocks”space-time periodicity during the freeze-out has an exponential dilatation(modulation). This in turn can be equivalently encoded in a virtual
AdSmetric. By solving the classical propagation of the field in this VXD ge-ometry we find that: gauge coupling has a logarithmic running typical ofthe asymptotic freedom; the energy excitations ( i.e. virtual
KK modes) ofthe de Broglie “periodic phenomena” can be interpreted as hadrons; and thetime periodicity turns out to be the conformal parameter of AdS/CFT, sothat it naturally parameterizes the inverse of the energy of the QGP. Weconclude that the classical geometry to quantum behavior correspondence offield theory in compact 4D has important justifications in basic aspects ofAdS/CFT phenomenology, providing at the same time an unconventional in-terpretation of Maldacena’s conjecture in terms of the wave-particle dualityand space-time geometrodynamics.Every section of this paper is concluded by comments and outlooks, pro-viding physical interpretations of the formal results.
1. Intrinsic periodicity
In this section we define the basic aspects of field theory in compact 4D,see [1, 2]. The results are presented for scalar bosonic fields, so that they canbe easily generalized to vector bosons — the generalization to fermions willbe given in future papers.The undulatory nature of elementary particles was introduced by deBroglie in 1923 in these terms: “ we proceed in this work from the assumptionof the existence of a certain periodic phenomenon of a yet to be determinedcharacter, which is to be attributed to each and every isolated energy parcel[elementary particle] ”, [28]. This “ yet to be determined ” de Broglie “periodicphenomenon” has been tested by 80 years of successes of QFT and observedindirectly in a recent experiment [29, 30]. According to de Broglie, a particlewith mass ¯ M can be characterized by a proper-time periodicity T τ definedby the relation T τ ≡ h/ ¯ M c , i.e. by a “periodic phenomenon” with minimaltopology S . Depending on the reference frame, such a proper-time period-icity T τ induces corresponding spatial and temporal periodicities ~λ x and T t cT τ = cγT t − γ ~β · ~λ x . (1)It is convenient to write the resulting de Broglie space-time periodicity in acovariant notation T µ = { T t , − ~λ x ˆn /c } . (2)As is well known from undulatory mechanics, T µ can be used to describethe 4-momentum ¯ p µ = { ¯ E/c, − ¯p } of an elementary particle with mass ¯ M .In fact, in the new reference frame denoted by ¯p , the resulting energy andmomentum are ¯ E ( ¯p ) = γ ¯ M c and ¯p = γ ~β ¯ M c respectively, so that the deBroglie relation ( a.k.a. de Broglie phase harmony) can be in general rewrittenas T τ ¯ M c ≡ h → T µ ¯ p µ c ≡ h . (3)In particular we have the familiar relation ¯ p µ = ~ ¯ ω µ /c where ¯ ω µ is the fun-damental 4-angular-frequency associated with T µ .The periodicity T τ of the proper-time τ can be equivalently parameterizedby a world-line parameter s = cτ of corresponding periodicity λ s = cT τ = h ¯ M c . (4)We note that λ s is nothing other than the Compton wavelength of the cor-responding particle of mass ¯ M . Therefore we find out that massive particles(except neutrinos) have typically extremely fast periodic dynamics if com-pared with the characteristic periodicity of the Cs atom T Cs ∼ − s , orwith the present experimental resolution in time ∆ T exp ∼ − s . For in-stance a hypothetical light boson with the mass of an electron has intrinsicproper-time periodicity T τ ∼ − s , whereas an hypothetical TeV boson hasintrinsic proper-time periodicity T τ ∼ − s — a possible generalization ofthe de Broglie “periodic phenomenon” to fermionic fields is naturally repre-sented by the Zitterbewegung model ideated by Schrödinger. In sec.(2) wewill show that these extremely small time scales are relevant for interpretingthe quantum behavior of elementary particles [1, 35, 36].
We want to formalize the de Broglie assumption of a “periodic phe-nomenon” by describing an isolated elementary bosonic particle of classical7-momentum ¯ p µ as a bosonic field solution Φ( x ) of persistent de Brogliespace-time periodicity T µ imposed as a constraint, according to (3). Thatis, suppressing the Lorentz index, such a de Broglie “periodic phenomenon”is represented as the bosonic solution Φ( x ) = Φ( x + T c ) . (5)Furthermore we note that every isolated elementary particle, i.e. constant4-momentum, has associated persistent periodicity, i.e. constant T µ . There-fore it can be regarded as a reference clock, a.k.a. the “de Broglie internalclock”. In fact, according to Einstein, " by a [relativistic] clock we under-stand anything characterized by a phenomenon passing periodically throughidentical 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 inan arbitrary period ", [31]. That is, as in Einstein’s relativistic clock, in a deBroglie "periodic phenomenon" the whole physical information is containedin a single space-time period. Hence we can represent a de Broglie “periodicphenomenon” as the bosonic field (5) of intrinsic periodicity T µ , i.e. as thesolution of the following bosonic action in compact 4D and PBCs S λ s = I T µ d x L ( ∂ µ Φ( x ) , Φ( x )) . (6)In this notation the PBCs are represented by the circle in the integral symbol H . As is well known from XD theory, PBCs (5)—or combinations of N BCsor D BCs—are fully consistent with the above bosonic action in compactdimensions, since they minimize the action at the boundary fulfilling thevariational principle. Note that in ordinary field theory this requirement isthe motivation for the assumption of the stationary field (fixed value) at theends of a time interval, i.e. SBCs. Both SBCs and PBCs are equivalentlyallowed by the variational principle. In particular they both preserve therelativistic symmetries of a bosonic theory, see par.(4.2). This is the funda-mental reason why field theory in compact 4D is a fully consistent relativistictheory.Here it is sufficient to note that, by using a global Lorentz transformation x µ → x ′ µ = Λ νµ x ν , (7)as a linear transformation of variables in the action (6), we get the trans-formed action S λ s = I Λ µν T ν d x ′ L ( ∂ ′ µ Φ ′ ( x ′ ) , Φ ′ ( x ′ )) . (8)8esides the transformation of the integrand, a related transformation of theboundary of the theory must be considered. Thus, [38], the compactificationlength T µ of the action transforms in contravariant way with respect to (7), T µ → T ′ µ = Λ µν T ν . (9)This also means that the field solution Φ ′ ( x ′ ) minimizing the transformed ac-tion (8) has intrinsic persistent periodicity T ′ µ . That is, in the new referenceframe the resulting space-time periodicity T ′ µ of the field Φ ′ ( x ′ ) solution of(8) is transformed in a contravariant way. Indeed, according to the de Brogliephase harmony (3), it actually describes the transformed 4-momentum of theparticle of mass ¯ M in the transformation of reference frame (7). In fact, ¯ p µ → ¯ p ′ µ = Λ νµ ¯ p ν . (10)In this formalism the covariance of the theory in compact 4D is manifest[1]. As a double check we can also note that, according to (3), the phase ofa field is invariant under translations of de Broglie 4-periods T µ and it mustalso be a scalar under Lorentz transformations [1, 28, 29].The consistency of this covariant formalism can also be seen by noticingthat, as well known from undulatory mechanics, see (3), the local and re-tarded variations of energy-momentum ¯ p µ occurring during interaction canbe equivalently interpreted as local and retarded modulations of de Brogliespace-time periodicity T µ . Indeed in par.(4) we will describe interactionsin terms of modulation of periodicity. More precisely, it is possible to showthat the de Broglie space-time periodicity T µ transforms as is a covariant“tangent” ¯ p µ c = h/T µ , so that the relativisticdispersion relation ¯ M c = ¯ p µ ¯ p µ , (11) It must be noticed that the local (“instantaneous” in the modulated wave terminology)4-periodicity of a field is in general different from the compactification 4-length of thelocal action. Under a generic local interaction scheme (52), the local periodicity of thefield T µ ( x ) transforms as a “tangent”, [2, 37] 4-vector (53) ( i.e T µ ∝ ∂x µ = e µa ( x ) ∂x a ).In fact, ¯ p µ ∝ ∂/∂x µ and the periodicity are fixed by (3). The boundary of the theorytransforms as an ordinary 4-vector x ′ µ ( X ) ≃ Λ µa ( x ) x = X x a where e aµ ( x ) ≃ ∂x a /∂x ′ µ , see[2]. In this paper we investigate the case of “smooth” interactions, i.e. the approximation Λ aµ ( x ) ∼ e aµ ( x ) . T τ = 1 T µ T µ . (12)In the massless case, ¯ M ≡ , i.e. infinite proper-time periodicity T τ ≡ ∞ or “frozen” de Broglie “internal clock”, we say that the spatial and temporalperiodicities are conformally modulated, c T t ( ¯p ) = λ x ( ¯p ) .Another important property of the formalism of field theory in compact4D is that the de Broglie periodicity is encoded on the boundary of the theorythrough PBCs. In the cases investigated in this paper the compactification 4-length can be approximated with the de Broglie 4-periodicity, but this is nottrue in general, see [2] and footnote.(1). Thus the compactification 4-length T µ of (6) is dynamically fixed by the 4-momentum (3). As a consequence,in sec.(4.1) we will describe relativistic interactions in terms of local andretarded deformations of the compact 4D of the theory, i.e. in terms ofboundary geometrodynamics with interesting analogies with the holographicprinciple [45, 46].As we will discuss in more detail, the compact world-line parameter ofmassive fields in our theory plays a role similar to the compact world-sheetparameter of string theory, so that (6) can be regarded as the action of asimple bosonic string and the de Broglie “periodic phenomenon” as a stringvibrating in compact 4D. In the first part of the paper we will only consider the free case, i.e. persis-tent 4-periodicity T µ . An isolated “periodic phenomenon” in a given referenceframe is in fact represented by the action (6) with persistent compactificationlength T µ . The bosonic field solution (5) with persistent periodicity T µ canbe expanded in the harmonics modes of a “string” vibrating in compact 4Dor in analogy with a “particle in a box”. In fact, in a given reference framedenoted by ¯ p , if we consider only time periodicity T t (¯ p ) = 2 π/ ¯ ω ( ¯p ) , by dis-crete Fourier transform we find that the field solution has quantized harmonicangular-frequency spectrum ω n ( ¯p ) = n ¯ ω ( ¯p ) = n/T t (¯ p ) . Multiplying by thePlanck constant, this corresponds to the quantized energy spectrum E n (¯ p ) = n ¯ E (¯ p ) = n hT t (¯ p ) . (13)10hus the periodic field (5), solution of the action in compact 4D (6), can bewritten as a quantized tower (wave packet) of 3D energy eigenmodes Φ n ( x ) , Φ( x ) ≡ X n Φ n ( x ) = X n e − i ~ n ¯ E (¯ p ) t Φ n ( x ) . (14)We will address the quantities related to the fundamental mode n = 1 with the bar sign, e.g. ¯Φ ( x ) = ¯Φ( x ) , E (¯ p ) = ¯ E (¯ p ) , and so on. As is wellknown from the relativistic Doppler effect the time periodicity T t (¯ p ) varieswith the reference frame ¯ p . The variation of time periodicity is describedby (9) and must fulfill (12). Thus, from (11) we see that the dispersionrelation of fundamental mode ¯Φ( x ) is actually the dispersion relation ¯ E (¯ p ) =¯ p c + ¯ M c of a classical particle with mass ¯ M . By plugging this in (13), itis easy to see that the dispersion relation of the quantized harmonic energyspectrum of a free periodic field Φ( x ) is E n (¯ p ) = n ¯ E (¯ p ) = n p ¯ p c + ¯ M c . (15)It is interesting to note that this is nothing other than the energy spectrumof an ordinary second quantized bosonic fields (after normal ordering). Thisis the first element of a series of remarkable correspondences between fieldtheory in compact 4D and ordinary QFT, fully discussed in [1, 2]. They willbe summarized in sec.(2) for the free case and in sec.(6) for the interactingcase. By using common terminology of QFT we simply say that ¯ E (¯ p ) and ¯ p are the energy and the momentum of the 4D field Φ( x ) , respectively.Besides the intrinsic time periodicity we must also consider the induced deBroglie spatial periodicity of our field solution Φ( x ) of topology S . Denotingby λ x modulo of the spatial wavelength, the resulting harmonic quantizationof the momentum spectrum is | p n | = n | ¯p | = n hλ x (¯ p ) . (16)Through the Planck constant we can say that the energy and the momen-tum are the “physical conjugate” variables of the temporal and of the spatialcoordinate, respectively. Therefore, through discrete Fourier transformation,the intrinsic de Broglie 4-periodicity of Φ( x ) implies a discretization (quan-tization) of the energy-momentum spectrum: p nµ = n ¯ p µ . Such a “periodicphenomenon” has fundamental topology S , so that its harmonic spectrumturns out to be written in terms of the single (quantum) number n .11he simple topology S investigated so far, however, must be extendedwith two other spherical coordinates in isotropic or topological equivalentsystems. In case of spherical symmetry in fact the “periodic phenomenon”describing the particle can be parameterized in terms of the angles θ → θ + π and ϕ → ϕ + 2 π , [22]. As a consequence of the resulting topology S ⊗ S ,in addition to the quantization of the 4-momentum labeled by n , we obtainthe ordinary quantization of the angular momentum described in terms oftwo additional quantum numbers, typically denoted by { l, m } . For the sakeof simplicity in this paper we will not consider this further expansion of thefield Φ( x ) in spherical harmonics or their deformations.In the rest frame, the quantized energy spectrum (15) of a periodic field Φ( x ) , or equivalently to the energy spectrum of a second quantized KG field,yields a quantized mass spectrum M n ≡ E n (0) /c such that M n = n ¯ M = n hcλ s . (17)According to (4), λ s is nothing other but the Compton wavelength of the fieldof mass ¯ M . The mass is the “physical conjugate” of the world-line parameter.As for the semi-classical quantization of the energy-momentum describedabove, in this case the discretized mass spectrum is a direct consequence ofthe intrinsic proper-time periodicity of the field Φ( x ) . In sec.(3) we will seethat this result, formally equivalent to the KK mass tower, is part of a moregeneral dualism between periodic fields and XD fields.In ordinary QFT we can imagine to fix the value of the field at the bound-aries ( e.g. through SBCs) in order to select a particular single mode Φ KG ( x ) ,solution of the Klein-Gordon (KG) equation, and then impose commutationrelations. The difference with the ordinary approach to field theory (whereBCs have however a marginal role in practical computations) is that in (8) wefix a particular solution of the corresponding relativistic differential systemby imposing PBCs rather than SBCs. As shown in [2] the formalism of fieldin compact 4D is useful to study classical ( i.e. non-quantum) relativisticfields. In fact, the fundamental mode ¯Φ( x ) ∝ e − i ~ ¯ p µ x µ of the periodic fieldsolution Φ( x ) can be locally matched to a corresponding single KG mode Φ KG ( x ) = ¯Φ( x ) of an ordinary non-quantized KG field of mass M KG = ¯ M and energy E KG (¯ p ) = ¯ E (¯ p ) ( i.e. to a single de Broglie matter wave). Thus,the fundamental mode ¯Φ( x ) , as Φ KG ( x ) , describes the relativistic behaviorof a classical particle. This means that ordinary QFT can be retrieved by12he second quantization of the fundamental mode ¯Φ , neglecting all the highermodes (for instance by imposing SBCs instead of PBCs), [2]Nevertheless, as we will see in the next section, the assumption of PBCscan be used as a semi-classical quantization condition. In fact, if all theharmonics are considered, a de Broglie “periodic phenomenon” is describedas an harmonic wave-packet whose evolution has remarkable formal overlapswith an ordinary second quantized field, [1]. In case of interaction howeverthe periodicity of the corresponding harmonic wave-packet must be locallymodulated to encode the corresponding variations of kinematical state. Insec.(4), the strategy to describe local interactions will be that of modulatinglocally the spatial and temporal intrinsic periodicities of the field Φ( x ) (in-stead of using creation and annihilation operators). This will be realized byassociating to every local space-time point X an action with locally deformedcompactification length T µ ( X ) . Comments and Outlooks
In defense of the scientific method, de Broglie said “ this hypothesis [ofperiodic phenomenon] is at the base of our theory: it is worth as much, likeall hypothesis, as can be deduced from its consequences ”, [28]. In this spirit,similar to recent papers [1, 2], our study wants to explore the validity andthe consistency, in applications of modern physics, of such a description ofelementary particles as intrinsically periodic phenomena.In our formalism the intrinsic periodicity is realized in terms of field theoryin compact 4D through PBCs. We have already seen elements of the formalconsistency of this theory with relativity, such as its covariance. Furtherfundamental elements of its consistency with QM and GR will be discussedin sec.(2) and sec.(4), respectively. Essentially we find out that, as long aswe consider the relativistic modulations of periodicity, our formulation turnsout to be a consistent quantum-relativistic theory. Moreover, it is importantto bear in mind that, from a formal point of view, SR—and GR as we willsee later—defines the differential properties of space-time without giving anyparticular restriction about the BCs whose only requirement is to fulfill thevariational principle. Indeed, [1, 2], such a dynamical description of “periodicphenomena” is achieved by applying consistently the variational principle atthe geometrodynamical boundary of a relativistic wave theory. On the otherhand, it is well known that BCs play a fundamental role in QM as we willdiscuss in the next section. 13esides the purely mathematical aspects of the theory, in this section, aswell as in similar sections throughout the whole paper, we want to provideconceptual arguments to interpret the formal consistency of our results. Inparticular here we want to justify the formalism of compact 4D in the contextof SR and undulatory mechanics.Neglecting for a moment the mode expansion of a “periodic phenomenon”,to figure out the possibility of a description of elementary particles in termsof angular variables (such as the real coordinates appearing in the phase ofwaves) we may, for instance, follow few simple logical steps: i ) According to classical-relativistic mechanics, every elementary particle,in a given local reference frame, is characterized by an energy ¯ E (¯ p ) andmomentum ¯ p . In the free case the particle is in an inertial frame ¯ p andits 4-momentum ¯ p µ is constant ( i.e. Newton’s law of inertia). ii ) In undulatory mechanics every elementary isolated particle has asso-ciated intrinsic temporal and spatial de Broglie periodicity T t (¯ p ) = h/ ¯ E (¯ p ) and λ i = h/p i (de Broglie hypothesis or wave-particle dual-ity). The resulting local 4-periodicity T µ depends on the local referenceframe and it is persistent for isolated particles. iii ) In the atomistic description characterizing modern physics, elementaryparticles are the fundamental constituents of every system in nature.These elementary particles can interact exchanging 4-momentum ¯ p µ and thus modulating their 4-periodicity T µ , according to the relativisticlaws. Hence, the logical combination of these scientific truths naturally implies thepossibility of a formulation of physical systems in terms of intrinsically “pe-riodic phenomena” representing the elementary particles . This justifies thegood properties of our formalism. In few words, the consistency of thisdescription is already implicit in the ordinary undulary description of ele-mentary particle (wave-particle duality) at the base of modern QFT.In particular, we may imagine a system composed by a single free elemen-tary particle. Since in the undulatory description such an isolated particlehas persistent periodicity, it can be imagined as a pendulum in the vac-uum. Indeed, the periodic oscillations of this system can be parameterizedby an angular variable, e.g. by a phasor or a wave. For instance, as wellknown, a free classical-relativistic particle can be described by a correspond-ing KG mode Φ KG . Similarly, adding more isolated particles, the system can14e described by a set of angular variables parameterizing every elementary“periodic phenomenon”.For the sake of simplicity in this conceptual digression we consider onlythe time variable. In the international system (SI) the unit of time “second” isdefined as the duration of 9,192,631,770 characteristic cycles of the Cs atom.Indeed, time can be only defined by counting the cycles of a phenomenonwhich is supposed to be periodic in order to guarantee the invariance of theunit of time. We may for instance think to Galileo’s experiment of the pendu-lum isochronism in the Pisa dome or to Einstein’s definition of a relativisticclock in which “all that happens in a given period is identical with all thathappens in an arbitrary period”, see sec.(1.1). Thus, an isolated elementaryparticle, having persistent periodicity as a pendulum in the vacuum, canbe regarded as a reference clock, the so-called de Broglie “internal clock”,[29, 30].For an isolated system (universe) composed by a single “periodic phe-nomenon”, as for an isolated pendulum in the vacuum, time can be regardedas a cyclic variable since “all that happens in a given period is identical withall that happens in an arbitrary period”. Being that the system is isolated,there are no external variations of energy distinguishing a given period fromany other. Since the whole physical information is contained in a single pe-riod we say, by using a terminology typical of XD theories, that the time ofsuch an isolated system is compactified on a circle S (cyclic universe). Ifwe parameterize its oscillation with an external time axis t ∈ R , for instancedefined by the “ticks” of the Cs clock, we see that the time parameter entersinto the equations of motion as an angular variable with periodicity fixed bythe energy of the system as in a wave function. The external time axis canbe also regarded as the time of a reference system of infinite periodicity orwhose periodicity is many orders of magnitude bigger than the time scalesunder investigation. In this way we find out that the angular variables ofdifferent “periodic phenomena” can be parameterized by the same parameter t . Thus, in a generic interval t ∈ ( t i , t f ) on such an external axis, a “peri-odic phenomenon” Φ( x , t ) assumes particular initial and final values Φ( x , t i ) and Φ( x , t f ) during its evolution. This means that a “periodic phenomenon”is not localized in a particular temporal region. However, the evolution is For some aspects this can be regarded as the beginning of modern physics since itallowed a sufficient accuracy in the measurement of time to test theory of dynamics. t i , the energy propagates according to the retardedrelativistic potential (the theory is in fact based upon relativistic differentialequations). After a given delay, say at time t f , this induces a variation of pe-riodic regime to the particle, depending on the amount of energy exchanged.As a result, the system passes from a periodic regime to another periodicregime so that it is possible to establish a before and an after with respectto this retarded event in time (in this case all that happens in a given pe-riod is not identical with all that happens in an arbitrary period). Hence wecan intuitively understand relativistic causality and time ordering of eventsin terms of modulations of periodicity of the elementary particles’ internalclocks [1].In a system composed by more non-interacting particles, the same exter-nal time parameter appears in the EoMs of every “periodic phenomenon” asan angular variable with different periodicity, depending on its kinematicalstate. However, such a non-elementary system in general has no periodicdynamics. As the ratio of periodicities does not necessarily form a rationalnumber, the resulting evolution is indeed an ergodic evolution .In particular, every isolated elementary particle can be regarded as areference clock so that, in principle, its “ticks” can be used to define theexternal time axis similarly to the Cs clock . If we also consider relativisticeffects, every external observer or elementary particle in the system describesa different combination of the phases of the de Broglie internal clocks and thusa different “present”, depending on their relativistic kinematic state. This isnothing other than the ordinary relativistic description of simultaneity.Every value (instant in time) of an external relativistic time axis t ∈ R can be characterized by a unique combination of the “ticks” of the particle“internal clocks”, as in a stopwatch or in a calendar . Thus, as noted in A system can be regarded as elementary as long as our resolution does not allows usto resolve its possible egodics dynamics. For instance, this would yield a remarkable improvement in the experimental resolutionof time, if only it were possible to count the “ticks”, say, of an electron without perturbingits periodicity. In everyday life we fix events in time in terms of reference periods of years, months,days, hours, minutes, seconds. These reference cycles are conventionally rational each T τ = 8 . ± × − s . On the other hand, the time peri-odicity of a massless particle such a photon or graviton can vary from zeroto infinity. Since the world-line compactification length, say, of photons isinfinite we say that light has a “frozen proper-time internal clock” T τ ≡ ∞ .In a relational description of the elementary particles, the long temporal andspatial compactification lengths of light provide the long space-time scalestructure of the system. They can be regarded as the reference temporal(and spatial) axis of ordinary relativity, allowing a common parameteriza-tion of the elementary cyclic phenomena; such long space-time scales withrespect to the typical periodicities of the matter fields can be used as a ref-erence upon which the ordinary relativistic structure of space-time can bebuilt. In particular the “frozen” clock of light or of gravity plays the role ofthe emphatically non-cyclic world-line in relativity. We will come back tothis point in the next sections.An interacting particle will be formally described by modulating the deBroglie 4-periodicity. This description can also be extended to a single KGmode Φ KG , as pointed out in [2]. In addition to this, if we imagine to switchon interactions among particles, the modulations of periodicity associatedwith the exchange of energy-momentum yield very chaotic evolutions of thesystem — thus the theory does not necessarily imply ordinary cyclic cosmol-ogy.Summarizing, this description in which every elementary particle can beregarded as a Einstein relativistic reference clock, not only is fully consistentwith the special relativity, but the local nature of relativistic time turns outto be enforced. This provides interesting new elements to address the notionof time in physics. We will not discuss further the problem of the flow of time other, in particular in a sexagesimal base. But they need regular adjustments since theymimic natural cycles ( e.g. Moon and Earth rotations) which are not rational to each other.
2. Correspondence to free QFT
We now want to study in more in detail the mechanics of the periodicsolution of our field theory in compact 4D. We assume persistent 4-periodicity T µ , so that the results of this section concern essentially free particles. Wewill find that the mechanics of our periodic fields have remarkable formalcorrespondences with ordinary QM [1]. In this paper, as exemplification, wemainly consider the correspondence with the FPI formulation of QM; moreevidence is given in [1, 2]. For the sake of simplicity here we assume a singlespatial dimension denoted by x , avoiding expansion in spherical harmonics.The periodic scalar field (14) with dispersion relation E n ( ¯p ) in (15) canbe written with the following notation [39] Φ(x , t ) = X n A n a n φ n (x , t ) = X n A n a n e − i ~ ( E n t − p n x) . (18)The normalization factors A n are fixed for instance by choosing the innerproduct induced by the conservation of the “charge density” h f | g i Q = Z λ x d x f ∗ (x , t ) i ↔ ∂ t ~ c g (x , t ) , (19) The purely translational zero mode n = 0 can be regarded as unphysical. A ↔ ∂ t B = A ( ∂ t B ) − ( ∂ t A ) B , so that A n = q ~ c ω n λ x . Similarly toordinary field theory, the coefficients a n of the Fourier expansion are givenby a n = h f n (x , t ) | Φ n (x , t ) i Q , where f n (x , t ) = A n φ (x , t ) .A periodic field is essentially a sum over the harmonic modes of a vibratingclassical string. This actually is the typical classical system which can bedescribed in a Hilbert space. In fact, the energy eigenmodes φ n (x) form acomplete set with respect to the inner product (“non-relativistic limit” of h | i Q ) h φ | χ i ≡ Z λ x d x λ x φ ∗ (x) χ (x) . (20)The energy eigenmodes can be used to define Hilbert eigenstates h x | φ n i ≡ φ n (x) √ λ x . (21)Note that the integral over a single spatial period λ x can be extended to anarbitrary large (or infinite) number N T ∈ N of periods: R λ x d x λ x → R V x d x V x with V x = N T λ x .The evolution along the compact time, described by the so-called bulkEoMs ( ∂ t + ω n ) φ n (x , t ) = 0 , can be reduced to first order differential equations[11], so that i ~ ∂ t φ n (x , t ) = E n φ n (x , t ) . (22)This set of differential equations corresponds to the ordinary Schrödingerequation of QM. In fact, [40], from the Hilbert eigenstates (21), it is possibleto formally define a Hamiltonian operator H such that H | φ n i ≡ E n | φ n i . (23)Similarly, the momentum operator P can be defined as P | φ n i ≡ − p n | φ n i . (24)A Hilbert state | φ i = | φ ( t ′ ) i describing a generic periodic field at time t ′ , isa generic superposition of Hilbert eigenstates | φ i ≡ X n a n | φ n i . (25)19herefore, from the EoMs (22), we find that the time evolution is actuallydescribed by the familiar Schrödinger equation of ordinary QM, i ~ ∂ t | φ ( t ) i = H| φ ( t ) i . (26)From (18) we see that the time evolution of the generic Hilbert state | φ i is also described by the exponential operator U ( t ′ ; t ) = e − i ~ H ( t − t ′ ) . (27)Moreover this time evolution is Markovian (unitary), and can be written asproduct of N infinitesimal evolutions, U ( t ′′ ; t ′ ) = N − Y m =0 U ( t ′ + t m +1 ; t ′ + t m − ∆ t ) , (28)where N ∆ t = t ′′ − t ′ .Without any further assumption than intrinsic periodicity we have ob-tained all the elements necessary to build the ordinary Feynman Path Inte-gral (FPI) formulation of QM. In fact, if we plug the completeness relationof the energy eigenmodes in between the elementary Markovian evolutions,we obtain the finite evolution Z = Z V x N − Y m =1 d x m V x ! U (x ′′ , t ”; x N − , t N − ) × . . .. . . × U (x , t ; x , t ) U (x , t ; x ′ , t ′ ) . (29)From the notation introduced so far the elementary 4D evolutions can bewritten in the following way U (x m +1 , t m +1 ; x m , t m ) = X n m e − i ~ ( E nm ∆ t m − p nm ∆x m ) = h φ | e − i ~ ( H ∆ t m −P ∆x m ) | φ i , (30)where ∆x m = x m +1 − x m and ∆ t m = t m +1 − t m . As in the usual FPI formula-tion we are assuming on-shell elementary 4D evolutions [41]. Therefore, [22],we have obtained that the classical evolution of our system with intrinsic pe-riodicity is formally described by the ordinary FPI, which in the phase-spaceformulation is Z = lim N →∞ Z V x N − Y m =1 d x m V x ! N − Y m =0 h φ | e − i ~ ( H ∆ t m −P ∆x m ) | φ i . (31)20his remarkable result has been obtained in a semi-classical way, withoutany further assumption than intrinsic 4-periodicity.In complete analogy with the ordinary Feynman formulation, (31) can beexpressed in configuration space by means of the action S cl ( t b , t a ) ≡ Z t b t a dtL cl . (32)This is nothing other than the action of the free classical particle associatedwith our “periodic phenomenon”. In fact it is defined by the Lagrangian L cl ≡ P ˙x m − H , (33)Thus , the infinitesimal evolutions can be written as h φ | e − i ~ ( H ∆ t m −P ∆x m ) | φ i = e i ~ S cl ( t m +1 ,t m ) . (34)Finally, by using the notation of the functional measure , the FPI (31) inconfiguration space describing the classical evolution of a classical “periodicphenomenon” is Z = Z V x D x e i ~ S cl ( t f ,t i ) . (35)It is important to note that this demonstration, being based on thecomposition of elementary evolutions, can be formally generalized to a non-homogeneous Hamiltonian H ′ through the formal substitution H → H ′ . Asshown in [2] and as we will summarize in sec.(6), in case of interaction thefree action S cl of the FPI (35) will be substituted with the action of thecorresponding interacting classical particle S cl → S ′ cl .We also note that, [1], by evaluating the expectation value associated tothe Hilbert space (20-21) of the observable ∂ x F (x) , and by integrating byparts, we find h φ f | ∂ x F (x) | φ i i = i ~ h φ f |PF (x) − F (x) P| φ i i . (36) As in the ordinary formulation of the FPI, to pass from the phase-space formulationto the configuration space description we have performed the sum over the momentumspectrum of the Hilbert space [22]. R V x D x = lim N →∞ R V x Q N − m =1 d x m V x .
21n fact the boundary terms vanish because of the periodicity of the spatialcoordinate. By assuming that the observable is such that F (x) ≡ x [41],the above equation for generic initial and final Hilbert states | φ i i and | φ f i ,is nothing other than the commutation relation [x , P ] = i ~ of ordinary QM— more in general [ F (x) , P ] = i ~ ∂ x F (x) . The commutation relations canbe regarded as implicit in this theory. With this result we have checkedthe correspondence with canonical QM. Note that a similar demonstrationwas used by Feynman in [41] to show that the FPI formulation of QM isequivalent to the canonical formulation. Comments and Outlooks
The correspondence to ordinary relativistic mechanics obtained above isone of the main motivations for our formalism based on compact 4D andPBCs. The FPI obtained in (35) can be intuitively, as well as formally andgraphically [1, 2, 22, 26], interpreted in a semi-classical way as the conse-quence of the topology S associated to the de Broglie “periodic phenomenon”.In fact in such a cylindrical geometry, because of the invariance under 4-periodic translations (5), the initial and final configurations of the field Φ( x i ) and Φ( x f ) can be reached by an infinite set of periodic classical paths char-acterized by different winding numbers. In other words, because of the PBCsa field in compact 4D can self-interfere, and the resulting evolution turns outto be the one prescribed by ordinary QM.The reader interested in more correspondences between fields in com-pact 4D and ordinary QFT may refer to [1, 2]. For instance, the assump-tion of intrinsic periodicity in the free case leads to the harmonic spectrum p nµ T µ = n ¯ p µ T µ = nh , see (15). More in general, as shown in [2], in case ofinteraction the quantized spectrum associated with the corresponding modu-lation of periodicity turns out to be given by the Bohr-Sommerfeld condition H p nµ ( x ) dx µ = h ( n + v ) (see also [43], the so-called Morse index v can beretrieved by assuming a twist factor in the PBCs). Similarly to the Bohrhydrogen atom or to a particle in a box, the allowed orbits and energy levelsare those with an integer number of cycles [1, 22, 43]. This approach hasbeen tested in the description of several quantum phenomena [22] such asblack-body radiation, double slit experiment, wave particle duality, etc, aswell as in applications of non-relativistic quantum mechanics and condensedmatter such as superconductivity, Schrödinger problems, atomic physics, etc,[22, 42, 43]. 22t is instructive to interpret the black-body radiation in terms of “peri-odic phenomena”. In this case it is natural to assume that the population a n of the n th energy levels is fixed by the Boltzmann distribution. For theIR components of the radiation, i.e. massless periodic phenomena with longperiodicity, the PBCs can be neglected and the energy spectrum can beapproximated to a continuum (classical limit) since many energy levels arepopulated (we may say that the thermal noise destroys the intrinsic peri-odicity in a sort of decoherence). For the UV components, however, thefundamental energy ¯ E is big with respect to the thermal energy so that onlya few energy levels can be populated. That is to say, these modes have veryshort periodicity (not destroyed by the thermal noise), so that the PBCsare important and there is a manifest quantization of the energy spectrum(quantum limit). The UV catastrophe is avoided according to Planck.For massive particles, in the non-relativistic limit ¯ p ≪ ¯ M c , only the fun-damental energy level of a periodic phenomenon is largely populated as thegap between the energy levels goes to infinity, ¯ M → ∞ . The cyclic fieldcan therefore be approximated as Φ( x ) ∼ exp[ − i ¯ Mc ~ t + i ¯ M ~ x t ] , see [1]. Ne-glecting the proper de Broglie “internal clock” (first term), it is possible tosee by plotting the modulo square, that the wavefunction of a massive “peri-odic phenomenon” is centered along the path of the corresponding classicalparticle and its width is smaller than the Compton wavelength λ s . Thus, inthe non-relativistic limit, the Dirac delta distribution is reproduced as in theusual Feynman description. Similarly, the spatial compactification lengthtends to infinity whereas the time compatification tends to zero, so that inthe non-relativistic limit we have a point-like distribution in R , i.e. theordinary three-dimensional description of a classical particle. Indeed, thecorpuscular description therefore arises at high frequencies. On the otherhand, in the relativistic limit the non-local nature of a massive periodic phe-nomena can not be neglected (the distribution width is of the order of theCompton wavelength). This gives an intuitive interpretation of the wave-particle duality and of the double slit experiment. In particular, if probedwith high energy or observed with good resolution, more and more energylevels (in general with either positive or negative frequencies) turn out to be In the non-relativistic limit or when stopped in a detector, we say that the wavefunc-tion collapses to the ground state . In this limit the inner product (19) describing therelativistic probability to find a charged particle must be replaced by (20) (Born rule), [1]. i.e. more and more harmonics can be resolved. In this way we canfigure out that the energy excitations play the role of the quantum excita-tions of the same fundamental elementary system, so that we have an analogywith the virtual particles of ordinary relativistic QM. This aspect will moti-vate the terminology “ virtual
XD” in describing the cyclic proper-time of anelementary “periodic phenomenon”.The assumption of intrinsic periodicity implicitly contains the Heisenberguncertain relation of QM. Briefly, the phase of a “ de Broglie clock” can notbe determined and is defined modulo factor πn since only the square of thefield has physical meaning, according to (20)— this also brings the factor / in the resulting uncertain relation. To determine the frequency, and thusthe energy ¯ E ( ¯p ) = ~ ¯ ω ( ¯p ) , with good accuracy ∆ ¯ E ( ¯p ) we must count a largenumber of cycles. That is to say we must observe the system for a long time ∆ t ( ¯p ) according to the relation ∆ ¯ E ( ¯p )∆ t ( ¯p ) & ~ / . The simple mathematicdemonstration of this relation is given in [1].As already pointed out in previous publications, because of the extremelyfast periodic dynamics typically associated to matter particles, QM can be re-garded as an emerging phenomenon. The situation is therefore similar to theone considered in recent attempts to interpret QM as an emerging theory,such as in the ’t Hooft determinism and in the stroboscopic quantization.According to ’t Hooft, there is a “close relationship between the quantumharmonic oscillator and a particle moving on a circle” if their periodicity T t is faster than our resolution in time [35]. This correspondence can be seen inour theory from the fact that the “periodic phenomenon” with periodicity T t describes the quantum behavior of the corresponding KG mode, which cor-responds to a quantum harmonic oscillator of the same periodicity. A similardescription is given by the “stroboscopic quantization” in which QM and, asalready mentioned, the arrow of time are interpreted as emerging phenomenaassociated with ergodic dynamics resulting from XDs compactified on a torus[36].The periodicity upper limit of typical quantum systems, characterized byelectrodynamics, can be regarded as fixed by the electron (the lighter massiveparticle except neutrinos) proper-time periodicity T τ ∼ − s . Thus, usinga Cs clock, T Cs ∼ − s , to determine the dynamics of such a system islike using a time scale of the order of the age of the universe ( ∼ y ) toinvestigate annual dynamics. For every “tick” of the Cs clock the electron,in its evolution, does an enormous number of cycles, i.e. about windings . The actual experimental time resolution, about ∆ T exp ∼ − , is still24oo low to resolve such small time scales, though the internal clock of anelectron has been observed in a recent interference experiment [30]. Indeedthe observation of such a fast de Broglie internal clock is similar to theobservation of a “clock under a stroboscopic light”. The results is that atevery observation the particle appears to be in an aleatoric phase of its cyclicevolution. As for a dice rolling too fast with respect to our resolution in time,the outcomes can be described only in a statistical way. Loosely speaking,an observer with infinite resolution in time can in principle resolve exactlythe underling deterministic dynamics, and would have no fun playing dice(“God doesn’t play dice”, Einstein). In [25] we have named a dice rollingwith de Broglie frequency as “de Broglie deterministic dice”. Similarly tothe deterministic models mentioned above, the formal results of the previoussection show that the statistics associated with these extremely fast cyclicbehaviors have formal correspondence with ordinary QM. This also suggeststhat a direct exploration of such microscopical time scales could be of primaryinterest in understanding the inner nature of the quantum world – the deBroglie “periodic phenomenon” has been addressed as “the missing link”, [29].BCs have played a fundamental role since the earliest days of QM. Indeed,another advantage to using BCs as quantization conditions is the remarkableproperty that QM is obtained without involving any (local) hidden variable.Since Bell’s theorem is based on the fundamental hypothesis of local hiddenvariables, it can not be applied to our theory. On the other hand the as-sumption of intrinsic periodicity introduces an element of non-locality, whichis however consistent with SR since the periodicity is modulated relativisti-cally. This could be used for a novel interpretation of quantum phenomenasuch as entanglement. Thus the theory can in principle violate Bell’s inequal-ity (in particular, as we will explicitly check in future studies, if we try toadapt Bell’s theorem to our theory, i.e. to evaluate the expectation values ofan observable in the Hilbert space described above, we expect to find againa formal parallelism with ordinary QM). For this reason we speak about—mathematical—determinism [1, 35]. We conclude that the manifestation ofour description of elementary particles in terms of compact 4D and PBCs isa remarkable correspondence with relativistic QM . Virtual
Extra Dimension (VXD)
A de Broglie “periodic phenomenon” with topology S can be equivalentlydescribed by either assuming intrinsic time periodicity T t (¯ p ) in a generic25eference frame as in [1], or by assuming intrinsic periodicity of the proper-time T τ and using Lorentz transformations as shown in sec.(1). In this sectionwe want to show that, from a formal point of view, the same purely 4D fieldtheory in compact 4D described so far can be equivalently derived from aKK theory with flat cyclic XD. This approach, already described in [1] andrepeated in this section, will show us that our pure 4D theory is manifestly dual to a corresponding XD theory. As we will discuss in more in detail at theend of this section, in this way our description of “periodic phenomenon” willinherit, and for some aspects justify, interesting properties of XD theories,without actually introducing any (so far unobserved experimentally) XD.From a mathematical point of view this can be seen by using the cyclic XD(relativistic invariant) to parameterize the intrinsic proper-time periodicityof the de Broglie “internal clock”. That is, the cyclic XD will be used asa “mathematical trick” to describe the undulatory behavior of elementaryparticles. For this reason we will say that field theory in compact 4D has a“virtual” XD (VXD), to distinguish it from the “real” XD of the KK theory.To introduce the idea of VXD we consider a non-interacting 5D masslessfield with “real” XD denoted by s in the flat metric dS = dx µ dx µ − ds ≡ . (37)Since an XD field with zero 5D mass is constrained on the 5D light-cone( dS ≡ ), the modulo of the 4D components and the XD component areproportional ds = dx µ dx µ . (38)As already described in detail in [1] and summarized in par.(3.2), we willidentify the XD with the world-line parameter of a purely 4D field theory.That is, we will impose s ≡ cτ , (39)where τ is the proper-time. To address the identification (39) we say that theXD, s , is virtual , [1]. Upon this identification the original XD theory will bereduced to a purely 4D theory with ordinary Minkowskian metric (38) (wedo not consider the case of 5D massive fields since dS = 0 and this wouldimply tachyonic modes in the purely 4D theory).Moreover, as in the ordinary KK theory, we will also assume that the XDis compact with compactification length λ s and PBCs, i.e. cyclic XD, [4].Thus, with the identification (39), the cyclic XD will describe a cyclic proper-time. From this our description of a de Broglie “periodic phenomenon” can be26erived as in sec.(1) through the Lorentz transformation (1). In other words,the original 5D theory will be reduced to a purely 4D theory; the KK modesof the original 5D field, which in the “purely” XD theory are independent 4Dfields, will correspond to the harmonic modes Φ n ( x ) of the 4D periodic field Φ( x ) described in (14). Here we give a short review of the formalism of the ordinary KK theory.A purely 5D scalar field with zero 5D mass ( dS ≡ ) in a flat XD (37)denoted by s , and compactification length λ s , is described by the 5D action S λ s = 12 Z λ s dsλ s Z d x (cid:2) ∂ M Φ ∗ ( x, s ) ∂ M Φ( x, s ) (cid:3) . (40)By varying this action with respect to the 5D field Φ( x, s ) and integratingby parts, we obtain δ S λ s = 12 λ s Z d x n [ δ Φ( x, s ) ∂ Φ( x, s )] s = λ s s =0 + Z T s dsδ Φ( x, s ) ∂ M ∂ M Φ( x, s ) o . (41)The bulk term of this variation yields the 5D EoMs ∂ M ∂ M Φ( x, s ) = 0 , ∀ s ∈ (0 , λ s ) . (42)The generic solution Φ( x, s ) , through a discrete Fourier transform, can bewritten as a sum over 4D normal modes Φ n ( x ) , Φ( x, s ) = X n Φ n ( x, s ) = X n e ic ~ M n s Φ n ( x ) . (43)Through the EoMs (42), i.e. in an indirect way, these can be interpreted as4D massive fields, a.k.a. the KK mass eigenmodes. In fact (cid:18) ∂ µ ∂ µ + M n c ~ (cid:19) Φ n ( x, s ) = 0 . (44)The fundamental requirement is, as usual, that the BCs must minimizethe action (40) at the boundary, i.e. that the boundary term of (41) mustvanish. In the KK theory, this requirement is fulfilled by assuming PBCs(though anti-PBCs or, more in general, N and D BCs are equally allowed).27he resulting theory with cyclic XD has topology S . We may already notethat the PBCs at the ends of the XD Φ( x, s ) ≡ Φ( x, s + λ s ) , (45)lead indirectly (see comments at the end of this section) to a KK mass tower M n . This turns out to be formally the same mass spectrum of field theoryin compact 4D given in (17), i.e. M n = n ¯ M = nh/λ s c .Integrating over the XD, the KK action (40) can be equivalently writtenas a sum ( n ∈ Z ) over the 4D Lagrangian densities of the KK mass modes Φ n ( x ) : S λ s = X n Z d x h ∂ µ Φ ∗ n ( x ) ∂ µ Φ n ( x ) − n ¯ M c ~ Φ n ( x ) i . (46)Next we will show the dualism between field theory compact 4D and theKK theory described here, by explicitly assuming that the XD is virtual ,(39). The 4D modes Φ n ( x ) which in the KK theory are independent fieldswill be the energy excitations of the same fundamental 4D field (14). Here we explicitly assume that the cyclic XD of the KK theory intro-duced above is a virtual
XD rather than a “real” XD. This means to imposethe condition (39). The resulting purely 4D theory with cyclic world-lineparameter will be nothing other than the field theory in compact 4D withMinkowskian metric (38) described in sec.(1,2). That is, Φ( x, s ) in (43) willdescribe the same de Broglie “periodic phenomenon” as the purely 4-periodicfield Φ( x ) .The condition of VXD, (39), means that the parameter s must be identi-fied with a cyclic world-line parameter (implicit function of the 4-coordinate x µ = { ct, − x } ) with periodicity λ s . This corresponds to imposing that theproper-time τ has intrinsic periodicity T τ = λ s /c , according to (4). From thisit is easy to see that a bosonic field theory with such a proper-time period-icity is nothing other than our field theory in compact 4D. By following theline of par.(1), it is sufficient to note that, if we move a de Broglie “periodicphenomenon” of intrinsic proper-time periodicity T τ from its rest frame to ageneric frame denoted by the spatial momentum ¯p , we get a resulting timeperiodicity T t ( ¯p ) , [28, 29]. That is, in a generic reference frame, according28o the Lorentz transformation (1), the proper-time periodicity (45) inducesthe following time periodicity Φ( x , t, s ) ≡ Φ( x , t + T t ( ¯p ) , s ) . (47)From this we can for instance repeat the same demonstration used in [1]obtaining back field theory in compact 4D of sec.(1). In fact, the intrinsictime periodicity (47) implies that the 4D field Φ( x, s ) in (43) can be furtherexpanded as three-dimensional energy eigenmodes Φ n ( x )Φ( x, s ) = X n e − in ~ ( ¯ E ( ¯p ) t − ¯ Mcs ) Φ n ( x ) . (48)The resulting quantization of the energy spectrum associated to the timeperiodicity T t ( ¯p ) of the field (47) is the same harmonic energy spectrum(13) of field theory in compact 4D, i.e. E n ( ¯p ) = n ¯ E ( ¯p ) = nh/T t ( ¯p ) . Inparticular, the expansion in energy and mass eigenmodes of Φ( x, s ) is labeledby the same (quantum) number n because the corresponding time or world-line periodicities are relativistic projections of the same de Broglie “clock” offundamental topology S . Note that in the rest frame ( ¯p ≡ ) we have thematching condition T t (0) = T τ .After decompactification of the time coordinate, the action (46) turns outto be a single sum over 3D Lagrangians of the energy eigenmodes Φ n ( x ) , S λ s = T t Z R d x X n (cid:20) n c ~ (cid:0) ¯ E (¯ p ) − ¯ M c (cid:1) Φ n ( x ) − | ∂ i Φ n ( x ) | (cid:21) . (49)In order to find the dispersion relation of the 4D eigenmodes Φ n ( x ) wemust resolve the EoMs of (49), so that n ¯ E (¯ p ) = p n c + n ¯ M c . From thisit is easy to see that these eigenmodes are described in a collective or coherentway and that the quantization of the momentum spectrum is p n = n p . Infact, as noticed for instance with the Lorentz transformation (1), in a genericreference frame the intrinsic proper-time periodicity (47), or equivalentlythe periodicity of the VXD (45), together with the time periodicity T t ( ¯p ) ,also induces a periodicity λ x (wave-length) on the modulo of the spatialdimensions. The quantization of the momentum spectrum in terms of thisspatial periodicity is given by (16). In the rest frame ( i.e. dτ ≡ dt ) wehave p n = 0 , ∀ n ( i.e. zero separation d x ≡ ) and the matching condition ¯ E (0) = ¯ M c . Thus, the fundamental mode has dispersion relation of a29elativistic particle (11), and the resulting space-time periodicity T µ is theusual de Broglie 4-periodicity (9) with geometrical constraint (12). From thisfollows that the resulting dispersion relation of the quantized energy spectrumis again that of a “periodic phenomenon” (15), which in turn matches withthe energy spectrum of a second quantized KG field with mass ¯ M (afternormal ordering).The action (40) or (46) can be regarded as purely 4D under the assump-tion of VXD. In fact, in (46) the explicit dependence on the world-line param-eter of (40), i.e. the VXD, has been integrated out. Since the whole infor-mation of the decompactified action (46) is contained in a single space-timeperiod, we can limit the integration region of the action to T µ — analogousarguments apply to the spatial integration region of (49).In conclusion, the field solution Φ( x, s ) of a KK theory with VXD isactually the periodic solution Φ( x ) , (5), of field theory in compact 4D (8).The KK modes, under the assumption of VXD, turn out to be the energyexcitations of the de Broglie “periodic phenomenon”. They can be named virtual KK modes, for reasons that will be described in the comments below.To pass from the VXD field Φ( x, s ) satisfying (40) and (45) to the cor-responding 4D field Φ( x ) , (14), satisfying (6) we integrate out the explicitdependence on the world-line parameter of the field as follows: Φ( x ) = X n ′ Z λ s dsλ s e + iM n ′ cs Φ( x, s ) . (50)Notice that, as long as PBCs are assumed and s is identified with theworld-line parameter, (40), (46) and (49) are equivalent formulations of thesame purely 4D action (6) describing our de Broglie “periodic phenomenon”.By imposing the bulk EoMs (42) as a constraint, the action in a VXD (40)can also be rewritten as a 4D boundary action S λ s = 12 Z T t (¯ p ) dt Z R d x [Φ( x, s ) ∂ Φ( x, s )] s = λ s s =0 . (51)In this formulation we say that the bulk physics of the VXD has been pro-jected on the boundaries. This reduction is at the base of the holographicprescription that we will discuss later. In (51) we have not yet explicitlyimposed any BCs at the ends of the VXD. If we impose PBCs and the bulkEoM as constraints, (51) describes the same physics of (6) as well.30 .3. Comments and Outlooks Field theory in XD has remarkable mathematical properties such as Kaluza’sunification of gravity and electromagnetism [3]; an elegant explanation for thehierarchy of about 17 orders of magnitude between the electroweak scale andthe Planck scale [5]; and more recently, with the warped models, interestinginterpretations of the fermion mass hierarchy [48], of the electroweak gaugesymmetry and unitarization of the elastic scattering without involving anyadditional fundamental scalar ( e.g.
Higgsless [8], Gauge-Higgs-Unificationor composite-Higgs models). On the other hand, no experimental evidenceof the existence of an XD has been obtained so far. This has strongly con-strained some of the above XD theories, which in turn are sometimes regardedas of purely mathematical interest.The idea emerging from our analysis of KK theory, and its reductionto a purely 4D theory in compact dimensions, is that some of the goodproperties of XD theories can be justified without actually introducing any“real” (unobserved) XD. This is because, as we have seen, the role played bythe proper-time of a de Broglie “periodic phenomena” is, from a mathematicalpoint of view, exactly analogous to the one played by an XD in the KK theory.We want now give conceptual elements to interpret the dualism between afree field in compact 4D and a 5D field in flat compact XD formally obtainedabove. To see this we must consider that, in sec.(1), according to undulatorymechanics, we have described an isolated elementary particle as an intrin-sically periodic phenomenon in which the proper-time can be regarded ascyclic. In this section we have seen that the same undulatory description canbe retrieved from a simple 5D field in flat cyclic XD simply by identifyingthe XD as the cyclic proper-time. Applying this recipe, it is easy to figureout that the resulting theory inherits fundamental mathematical properties ofthe initial XD theory, though remaining a purely 4D theory .In doing this identification we say that we impose that the XD is virtual ,or equivalently that the cyclic world-line parameter plays the role of a virtual
XD. The term virtual
XD is loosely used to highlight the fact that in eithercase the resulting field theory in compact 4D is purely 4D, though it sharessome good behavior with the XD theory. Another reason for using the term virtual has been already mentioned during the conceptual interpretation ofthe quantum behavior associated to a “periodic phenomenon”. In fact, theenergy excitations of a periodic field can be regarded as quantum excitationsor, roughly speaking, as virtual particles, according to the correspondence torelativistic QM described in the previous section. But we have also seen here31hat, under the assumption of VXD, these energy excitations correspond tothe KK modes of the original XD, so that they can be addressed as virtual
KK modes. We will come back to this point when we will discuss aboutAdS/CFT and hadrons.Note that in an XD theory the mass spectrum arises indirectly in (44)through the EoMs (42). That is, the quantized conjugate variable of the XDcan be identified with a mass spectrum because dS ≡ (this is analogousto the fact that, through the EoMs, the modulo of the spatial momentum isproportional to the energy in massless particles). On the other hand, in thecase of field theory in compact 4D, the mass spectrum arises directly througha discrete Fourier transform: it is the quantized “physical conjugate” variableof the compact world-line parameter. Thus it can be named a virtual KKtower. In the ordinary XD theory the KK modes are independent 4D fields(with unrelated 4-momenta) whereas under the assumption of VXD theyturn out to be the excitations of the same 4D field. Indeed the virtual
KKtower of the 4D field Φ( x ) is a coherent sum of energy eigenmodes Φ n ( x ) , i.e. the virtual KK modes have a collective description since they all satisfy thesame 4D PBCs. Such a collective description of the energy eigenmodes (14)as virtual
KK modes shows also an interesting aspect of the analogy of fieldtheory in compact 4D with the Matsubara theory. As is well known, statis-tical systems are quantized by using the “mathematical trick” of an intrinsicEuclidean time periodicity, so that the 4D Euclidean field solution can bedecomposed into a coherent sum of thermal energy eigenmodes. We will seethat a collective description of the KK modes typical of a VXD is implicit inthe usual holographic description of an ordinary KK theory. The compact-ification length of ordinary XD theories is a free parameter, which howevermust be extremely small since no XD has been experimentally observed sofar. Current theory sets this value to be smaller than − m , i.e. to anenergy scale above TeV. With the assumption of VXD, every elementary sys-tem has associated a different compactification length, uniquely fixed by theinverse of the mass. This assumption is completely falsifiable, and accordingto the previous section has remarkable correspondences to ordinary relativis-tic QM. We will see later that our theory also has fundamental analogieswith string theory. In this way it will be natural to interpret the hadrons asquantum excitations of the same elementary string.32 . Interactions An exhaustive formalization of interacting periodic fields, and thus theexact transition between different periodic regimes, is beyond the scope ofthis paper. Here we want to give a qualitative description by consideringthe deformations of the boundary induced by particularly simple interactionschemes. A more rigorous formalization of the problem is given in [2]. Fromthis section on, we adopt natural units ( ~ ≡ c ≡ ).A simple starting point to introduce our formalization of interactions is toconsider the Compton scattering e + γ → e ′ + γ ′ . According to the de Broglie(3), the conservation of the 4-momentum ¯ p eµ + ¯ p γµ = ¯ p e ′ µ + ¯ p γ ′ µ in the interactionpoint x = X , can be equivalently expressed in terms of conservation of theinverse of the 4-periodicity, ( T µe ) − + ( T µγ ) − = ( T µe ′ ) − + ( T µγ ′ ) − .With this example we see that local and retarded variations of the 4-momentum occurring in the interaction point can be equivalently interpretedas local and retarded modulations of the 4-periodicity of the fields. In the for-malism of field theory in compact 4D, modulation of periodicity correspondsto a local stretching of the compactification 4-length, and thus to a localdeformation of the metric. Therefore, to describe the interaction we will usethe formalism field theory in curved 4D [38]. In particular, as is well knowfrom the geometrodynamical description of gravitational interaction of GR,the local modulations of periodicity of reference clocks can be equivalentlyencoded in local deformations of the underlying space-time metric. In classical-relativistic mechanics a particular interaction scheme can bedescribed in terms of corresponding local variations of 4-momentum withrespect to the non-interacting case. That it, in a generic interaction point x = X , it can be described by the following local variation of 4-momentum ¯ p µ → ¯ p ′ µ ( X ) = e aµ ( x ) | x = X ¯ p a . (52)With this notation we mean that, as the interaction is switched on, the per-sistent 4-momentum ¯ p a of a free elementary system is forced to vary frompoint to point. The specific interaction scheme is therefore encoded in thetetrad e aµ ( x ) . We now generalize the analysis given for the Lorentz invari-ance (7) to the more general case of invariance of the theory under localtransformations of variables, see also [2].33ecause of the relation between the 4-momentum and the compactifi-cation 4-length (3), the interaction (52) can be equivalently encoded in acorresponding local contravariant modulation of 4-periodicity with respectto the persistent periodic case: T µ → T ′ µ ( X ) ∼ e µa ( x ) | x = X T a . (53)In the approximations of this paper the 4-periodicity can be identified withthe compactification 4-length of the theory [2]. Thus, during interaction wehave a corresponding stretching of the compact 4D and in turn a deformationof the flat 4D metric, η µν → g µν ( x ) = e aµ ( x ) e bν ( x ) η ab . (54)To check this geometrodynamical description it is sufficient to consider thetransformation of variables associated with the interaction scheme (52-54), dx µ → dx ′ µ ( X ) = e aµ ( x ) | x = X dx a , (55)as a substitution of variables in the free action (6). The determinant of theJacobian is p − g ( x ) = det[ e aµ ( x )] . In this way we actually obtain [38] S λ s ∼ I e µa ( X ) T a d x p − g ( x ) L ( e µa ( x ) ∂ µ Φ ′ ( x ) , Φ ′ ( x )) . (56)This transformed action has compactification 4-length T ′ µ ( X ) varying lo-cally. Hence, its field solution has modulated periodicities at different in-teraction points X . This actually describes the modulation of de Broglieperiodicity of a particle under the interaction scheme (52). Note that thegeometrical description given in this paper is limited to the cases in whichthe compactification 4-length of the theory can be approximated with thelocal 4-periodicity; for a more general formalism see [2].According to our analysis, interactions can be formalized in terms of localdiffeomorphisms (55), i.e. in terms of field theory in curved 4D (56). Thefree case is given by the classical field solution with persistent periodicity T µ associated to the bosonic action in flat 4D and persistent boundary (6). Theinteraction scheme (52) is described by 4-periodicity local modulations of thede Broglie “internal clocks” of the particles and encoded in the correspondingdeformed metric (54). As we will discuss below, this mimics GR. In fact, the34ravitational interaction can be interpreted as modulations of spatial andtemporal periodicity of reference lengths and clocks, respectively, encoded incorresponding deformed space-time metric.The conditions at the transformed boundary T ′ µ must be of the sametype as the ones assumed for the free fields of (6) at the persistent boundary T µ . In our case, to reproduce the quantum behavior, we assume PBCs(or conditions describing the same physics at an effective level, as in theholographic formulation). In this case the matching between the fundamentalmode of a cyclic field and the corresponding KG mode is local, [2].To extend this geometrodynamical description of interactions to the VXDformalism, the flat VXD metric describing a free cyclic field must be deformedin the following way η MN → g MN ( s ) = e AM ( s ) e BN ( s ) η AB . (57)For the scope of this paper it is sufficient to assume only deformations of the4D components of the virtual metric g MN ( s ) ∼ (cid:18) g µν ( x ( s )) 00 1 (cid:19) , (58)where the capital letters label the 5D Lorentz indices. In fact, in this paper,such a formalism will be exclusively used to describe massless fields (a con-sistent generalization to massive fields may involve, for instance, dilatons inthe deformed VXD metric). Thus, to describe interactions of massless fieldsin the VXD formalism, (40) must be transformed to an action in the corre-sponding curved 5D (58). The resulting theory in curved 4D (56) is obtainedby imposing that the XD is virtual , (39), i.e. by integrating out the VXD asprescribed in sec.(3). As already noticed for instance in [2], the analogy with GR arises natu-rally if we think of the de Broglie internal clocks of elementary particles asreference clocks. In fact gravitational interaction can be described as modu-lations of 4-periodicity and thus encoded in corresponding local deformationsof the underlying 4D metric.More explicitly we consider the case of weak Newtonian interaction in-stead of the generic interaction (52). In a weak gravitational potential V ( x ) = − GM ⊙ | x | ≪ , (59)35ccording to (52), the variation of the energy with respect to the non-interacting case is [44] ¯ E → ¯ E ′ ∼ (cid:18) GM ⊙ | x | (cid:19) ¯ E . (60)Thus, according to (53) and to the de Broglie phase harmony (3), the corre-sponding modulation of time periodicity is T t → T ′ t ∼ (cid:18) − GM ⊙ | x | (cid:19) T t . (61)Indeed, see [44], this simple description is already sufficient to retrieve two im-portant predictions of GR: gravitational redshift and the usual gravitationaldilatation of the Minkowskian time (as arises explicitly from the Hafele andKeating experiment this aspect is described in terms of variation of the pe-riodicity of reference clocks). Essentially, this is a direct consequence of themodulation with the energy of the de Broglie internal clocks, which thereforerun slower in a gravitational well with respect to the non-interacting case,according to (3). Thus, our formalism mimics linearized gravity g µν ( x ) ≃ η µν + κh µν ( x ) . In fact, from (54) we explicitly obtain the con-tribution h = − κ GM | x | to the deformed metric. Besides the energy variationand time modulation we must consider the variation of the spatial momentum(52) induced by the gravitational interaction and the consequent modulationof the spatial periodicity (53). In this way it is possible to find out that (54)is nothing other than the usual Schwarzschild metric in the linear approxima-tion (59), see for instance [44]. Furthermore, it is well known that GR canbe derived from the linearized formulation by considering self-interactions[44, 25] — for instance by relaxing the assumption of smooth interactions. Comments and Outlooks
In this section we have shown that the geometrodynamics encoding inter-actions in the compact 4D formalism mimic the relativistic geometrodynam-ics of linear gravity. Though the exact formalism must be explicitly workedout and checked, this provides evidence of the consistency of our formalismof compact 4D with GR.To understand this description it is important to point out that “ what isfixed at the boundary of the action principle of GR ” is not uniquely defined[47]. SR and GR fix the differential framework of the 4D without giving any36articular prescription about the BCs. The only requirement for the BCs isthat to minimize a relativistic action at the boundary. For this aspect bothSBCs and PBCs have the same formal validity and consistency with relativity.Thus
SR and GR allow us the freedom to play with BCs , as long as we fulfillthe variational principle; and the BCs have played a fundamental role sincethe first days of QM, according to de Broglie, Planck, Sommerfeld, Bohr,Klein, etc. Nevertheless, the BCs play a very marginal role in ordinary QFT.This has motivated the implementation of our boundary methods in fieldtheory, showing at the same time the possibility that quantum phenomenacan be interpreted semi-classically by playing with BCs in relativistic theory .These considerations may also offer novel elements to address the problemof the quantization of gravity.It is interesting to point out that, from a historical point of view, Einsteinformulated GR by thinking of the modulations of periodicity of referenceclocks in a gravitational potential. Indeed, in general relativity, the localdeformation of the space-time metric encodes the modulations of periodicityof reference clocks. In perfect analogy with this description, the formalismof field theory in compact 4D, associated to every point a local 4-vector T µ ( X ) describing the modulations of periodicity of the internal clocks ofinteracting elementary particles, which in turn can be equivalently encodedto deformations of the underlying flat metric to locally curved metrics (54).The local modulations of de Broglie 4-periodicity describe the local variationsof 4-momentum associated to the interaction scheme (52). This also meansthat in this formulation the kinematics of the interaction is encoded on therelativistic geometrodynamics of the boundary of the theory (56). Hence wehave an interesting analogy with the holographic principle [45, 46].To figure out the conceptual consistency of our description, we must re-member the considerations given in sec.(1.1), when we noticed that masslessfields have infinite world-line compactification length (frozen proper “internalclock”), so that they can be used as reference for a relational description ofthe massive particles constituting the system under investigation. This de-scription can be extended to interactions as the modulation of periodicity isconsidered. In particular, gravitational interaction, as EM interaction, beingmediated by massless fields, naturally sets the reference space-time coordi-nates to describe the reciprocal dynamics of the gravitating particles. Thatis, the gravitational field can have infinite proper-time periodicity, so that asystem of particles interacting gravitationally can be described, as in usualnon-cyclic cosmological theory, by a non-compact world-line parameter.37emarkably, in [2] we have argued that the same approach can be usedto achieve a geometrodynamical description of gauge interactions. In a fewwords, the gauge field turns out to encode the modulation of periodicityassociated to local transformations of flat reference frames. The resultingclassical evolution of the modulated de Broglie “periodic phenomenon”, withall its harmonic modes, turns out to be described by the ordinary FPI scalarQED, [2].
5. AdS/CFT and Holography
In this section we review basis aspects of AdS/CFT in field theory. Inmodern physics, one of the most important issues which seems to relate clas-sical geometrodynamics with quantum dynamics is the AdS/CFT correspon-dence. Conjectured by Maldacena in 1997 [20] the original formulation statesthat, in the ’t Hooft limit g Y M N c ≫ , the IIB string theory on AdS × S is dual to the N = 4 supersymmetric SU ( N ) Yang-Mills theory, where g Y M is the Yang-Mills coupling. Roughly speaking a tree level string theory onan Anti-de Sitter (AdS) background turns out to be dual to a correspondingConformal Field Theory (CFT) Strongly Coupled (SCFT), [48].As summarized by Witten in the abstract of his paper [21]: under theAdS/CFT correspondence “ quantum phenomena [...] are encoded in classicalgeometry ” without introducing any explicit quantization condition. Such aclassical to quantum correspondence of AdS/CFT has been tested for sev-eral non-trivial aspects of modern physics, including condensed matter, andit seems to have a more general validity with respect to its original stringformulation. We will investigate the semi-classical description of relativisticQM obtained so far in terms of some basic phenomenological (4D) aspectsof AdS/CFT [48, 49, 19].For our purpose we consider the simple phenomenological formulation infield theory of the AdS/CFT correspondence given through the holographicprescription [21, 48, 49]. Similarly to (51) and (68), holography consistsof projecting the classical bulk dynamics on the boundary by imposing asconstraint to the field Φ the bulk EoMs and fixed value Φ | Σ = eφ Σ at theboundary s = Σ . In this way we are left with a lower dimensional theorywhere the only relevant d.o.f. is the so-called source or interpolating field φ Σ .The coupling of the source field with the bulk field is given by the parameter e . In this holographic prescription, a XD theory is effectively described by38he following 4D holographic action: S Holoφ Σ = S Kinφ Σ + Z d p π (cid:2) φ Σ ( − p ) φ Σ ( p )Π Holo ( p )) (cid:3) . (62)Here we have written the action in the 4-momentum space. As we will seebelow, the holographic action encodes in an effective way the information ofthe classical bulk configurations of the XD fields. By associating a conformaloperator O with source field φ Σ ( p ) to the bulk field Φ and assuming an AdSmetric, the central meaning of the AdS/CFT correspondence is summarizedby the relation [21] Z D φ SCF T e iS SCF T [ φ SCF T ]+ ie R dxφ Σ O ! e i S Holoφ Σ . (63)In other words, under the AdS/CFT correspondence any given classical configuration of a XD field in the AdS bulk has in the holographic formu-lation, a dual interpretation in terms of a 4D quantum SCFT, [48]. Thus,we have the remarkable property that the classical partition function of a4D holographic theory is related to the quantum generating functional of aSCFT [21, 48].Under this correspondence the holographic correlator Π Holo ( p ) describingthe AdS classical dynamics of the fields is dual to the quantum two-pointfunction hO ( p ) O ( − p ) i of the SCFT. The correspondence to CFT is when theholographic dimension s is non-compact (infinite compactification length),whereas if we impose branes, the conformal symmetry turns out to be broken.The holographic prescription is an useful computational technique toachieve the effective description of XD models (not necessarily in an AdSbackground) avoiding the explicit summation over the KK mass eigenmodes.The effective theory is obtained by choosing the correct source fields andcouplings to the bulk fields. Typically, [50, 51], the source field φ Σ and thecoupling e can be approximated with the fundamental KK mode ¯ φ and theboundary value of its delocalization profile: ¯Φ | Σ ∼ eφ Σ . From a computa-tional point of view, this corresponds to “integrating out” the heavy modes of The complete theoretical formulation of the conjecture involve a further functionalintegration on both sides of (63) over all the kinematic configurations of φ Σ . We will notconsider this aspect since the theory in VXD describes the quantum behavior without anyfurther quantization. Comments and Outlooks
At this point a further digression about the holography prescription isnecessary. The choice of source field and coupling described above corre-sponds to BCs effectively describing the N-BCs (or similarly of the PBCs oranti-PBCs) of the underlying XD theory. These “holographic” BCs play arole similar to the ordinary SBCs of QFT. In both cases they fix the bound-ary value of the field. In the ordinary SBCs the boundary value of the fieldis not “dynamical” (the Fourier coefficients of the generic field solution areconstants). For instance SBCs can be used to pick up only the fundamentalmode of the tower, i.e. to “eliminate” completely the higher modes of the the-ory. In the formalism of compact 4D, SBCs (instead of PBCs) can be usedto describe the non-quantum limit, eliminating all the energy eigenmodesbut the fundamental one. In fact, we have already mentioned that SBCs canbe used to select the KG mode associated to the fundamental mode of theperiodic field solution, see [2].In the holographic prescription, the boundary value of the field, i.e. thesource field, is “dynamical” (in the holographic action it has a kinetic term S Kinφ Σ and the Fourier coefficient of the field solution are functions, such asBessel functions in an AdS metric, containing poles). Thus, the holographicsolution is not a simple single mode ¯Φ of the tower. This implies that theholographic correlator Π Holo ( p ) and its pole encodes the effective propaga-tion and the spectrum of the higher KK modes, respectively. Thus, anotherinteresting consequence of such an holographic prescription is that the KKmodes are described in a collective way. In fact they all depend on the single4D d.o.f. φ Σ . They all fulfill the same 4D BCs similarly to the case in whichthe XD is virtual . It is important to point out that such a collective descrip-tion is similar to the one obtained in the VXD formulation where the KKmodes are virtual , i.e. they are the collective modes of the same 4D d.o.f. ,and can be interpreted as the excitations of the same quantum system.40 . XD classical geometry to 4D quantum behavior correspondence By combining together all the results obtained so far we will see that,through the assumption of PBCs, the classical geometrodynamics of the de-formed 4D boundary of (56) turns out to encode the quantum behavior ofthe corresponding interaction scheme. This aspect of the theory has beenrecently used in [2] to realize a geometrodynamical interpretation of scalarQED. In this paper we will use the VXD formalism to interpret this as acorrespondence between XD dimensional geometry and quantum behavior.We will test such a description with a very simple formalization of the Quark-Gluon-Plasma freeze-out.The analogies between XD theory and QM are well known and were firstnoted by O. Klien when, in analogy with the Bohr-Sommerfeld quantiza-tion, he introduced the idea of cyclic XD to describe the quantization of theelectric charge [4]. We may think for instance of the analogies between theresolution of the mass eigensystem and energy eigensystem in: a Kaluza-Klein (KK) field and the quantization of a “particle in a box” [6]; a XD fieldwith “brane terms” and a Schrödinger problem with Dirac delta potentials[10, 11]; an XD theory with “soft-walls” and the semi-classical quantizationof a harmonic oscillator [12] or in the Front-Light-Quantization [13, 14]. Inforthcoming papers we will extend this correspondence to interpret gaugesymmetry breaking mechanisms typical of XD or strong interacting theoriesin terms of superconductivity, [22, 42], as well as the behavior of electrons incarbon nanotubes and graphene in terms of virtual
KK modes . It is interesting to mention that, similarly to sec.(2), the correspondenceto the quantum formalism can be extended to describe ordinary KK theories Carbon nanotubes and graphene provide an exceptional arena to probe our formalismof VXD. In graphene the electron behaves as 2D massless particles. As a direction is curledup as in nanotubes, even if the drift velocity along the axial direction is set to zero (restframe), the electron can have a residual cyclic motion along the radial direction. As for theproper-time periodicity of de Broglie “internal clock” this defines the effective mass of theelectrons in nanotubes [42]. Note however that the energy spectrum in carbon nanotubescan not be directly derived from a simple KK theory with XD on a lattice of N sites, asclaimed in [58]. This would give an energy spectrum in which the momentum p n of the n -th KK particles are unrelated (non- collective description of the KK modes). To obtainthe correct energy spectrum of carbon nanotubes in which the KK modes have a collective behavior it is necessary to assume that the compact XD is virtual .
41n a Hilbert space. The KK modes form a complete set with given inner-product and they can be described by the Hilbert eigenvector | φ n i . Thus weintroduce a mass operator M whose its spectrum is the KK mass spectrum, M | φ n i ≡ M n | φ n i , the time evolution of a KK field along the XD is de-scribed by the Schrödinger equation i∂ s | φ ( s ) i = M| φ ( s ) i or by the evolutionoperator U ( s ′ ; s ) = e − i M ( s − s ′ ) , intrinsic commutation relation [ s, M ] = i , andso on. A similar notation is used in Light-Front-Quantization [13].In this section we want to combine together all the main correspondencesof field theory in compact 4D discussed so far (and in other peer-reviewpapers): i ) the correspondence between a field with persistent periodicity and afree quantum system, discussed in sec.(2) for the FPI, see [1, 2]; ii ) the dualism between a field with persistent periodicity and a field inflat XD, obtained in sec.(3) through the assumption of VXD, [1]; iii ) the geometrodynamical approach to interaction and the related modu-lation of periodicity of the field, described in sec.(4), [2]; iv ) the holographic effective description of an XD theory and its analogywith a theory with VXD, summarized in (62), see [51].The outcome of the combination of all these points will be a dualismbetween the holographic description of the classical configurations of a fieldin a deformed XD metric and the quantum behavior of the correspondinginteraction scheme.The classical evolution of a free periodic field, point i ), is described by theFPI (31) with time independent Hamiltonian H — here the results of sec.(2)are generalized to three spatial dimensions x . In this free case the integral R D x is trivial. In fact, the periodicity of an isolated system is persistentand the same Hilbert space, i.e. the same inner product, is defined in everypoint of its evolution. In this case, by using (34), (30) and (29) (see [1] formore detail or the introduction to the correlation function given for examplein [59]) the generating functional Z (35) can be formally written as Z = X n e − ip n · ( x f − x i ) = X n e − iM n ( s f − s i ) ↔ e i S cl ( t f ,t i ) . (64)42his summarized the correspondence i ) between a field with persistent peri-odicity,including all its harmonics, and the quantum evolution of the corre-sponding free elementary system, sec.(2).In addition to this, we have the dualism, point ii ), between a periodicfield solution with persistent periodicity and the classical configuration of afield in flat XD. This dualism, investigated in sec.(3), is manifest if we assumethat the XD is virtual , i.e. if we impose (39). It can be summarized as S cl ( t f , t i ) ↔ S Dcl ( s f , s i ) . (65)As shown in (64), this can also be seen if we evaluate the phase of the fieldin the rest frame p n · ( x f − x i ) = M n ( s f − s i ) in which the 4-momentumspectrum leads to a mass tower, see (3) and (15).Hence, the combination of point i ) and point ii ), i.e. (64-65), can beexpressed as a correspondence between classical configurations of a field inflat VXD and ordinary quantum behavior of an isolated elementary system Z = Z V x D x e i S cl ( t f ,t i ) ↔ e i S Dcl ( s f ,s i ) . (66)According to our theory the case of an infinite VXD describes the quantumbehavior of a massless system ( M n = 0 ), whereas the case of a compactVXD describes the quantum behavior of a massive system ( M n = 0 ). There-fore, similarly to AdS/CFT, a compact VXD (finite periodicity) explicitlybreaks the conformal invariance of the corresponding QFT describing freeelementary massive system.Now we consider the geometrodynamical description of interactions, point iii ), given in sec.(4), i.e modulation of periodicity, in order to generalize (66).The combination of i ), ii ), and iii ) will result in a more general correspon-dence between the classical configurations of a field in deformed-VXD andthe quantum behavior of the corresponding interaction scheme.To see this we must formalize the propagation of a periodic field with lo-cally modulated 4-periodicity T ′ µ ( x ) . As shown in [2], its resulting evolutionis described by the ordinary FPI of the corresponding interaction scheme.The derivation of the FPI (35) can in fact be generalized to the classicalevolution of a modulated de Broglie “periodic phenomenon” as follows.For a periodic field with persistent periodicity we can define a homoge-neous 4-momentum operator P µ = {H , −P i } . Similarly to (23-24), the lo-cally modulated spectrum of an interacting periodic field can be described by43 corresponding non-homogeneous 4-momentum operator P ′ µ ( x ) . As shownin detail in [2], the local operator P ′ µ ( x ) can be obtained from the homo-geneous P µ through the transformation of 4-momentum associated with theinteraction scheme (52): P µ → P ′ µ ( x ) = e aµ ( x ) P a , [2].Moreover, the space-time evolution of a locally modulated periodic fieldsolution is still Markovian (unitary), as shown in [2]. That is, the total4D evolution can be described as the product of elementary 4D evolutions,and the elementary 4D evolutions of an interacting cyclic field are formallydescribed by (30), provided that the homogeneous operators are substitutedby the corresponding local ones: H → H ′ and P i → P ′ i . Furthermore,we must consider that in case of interaction the spatial periodicity is notpersistent but it is modulated from point to point. Thus, to each localelementary 4D evolution is associated a different local Hilbert space. Infact, the inner product (20) must be locally substituted by R V ′ x ( X ) d x ′ ( X ) V ′ x ( X ) .Nevertheless, a relativistic interaction can be supposed as limited to a (finiteor infinite) region of space I . If the integration volume V ′ x ( X ) is bigger(or infinite) than the interaction region I , i.e. if we integrate over a largeor infinite number of periods N T , we can assume that the volume V ′ x ( X ) ,as well as the normalization of the fields, is overall not affected by theselocal deformations. That is, for a sufficiently large integration region wehave V ′ x ( X ) ∼ = V x . As already shown in [2], we find out that the correctmathematical tool to describe these locally modulated elementary evolutionsin which the inner product varies from point to point is again the integral R D x (which in case of interactions is not trivial).For a consistent generalization of (35) to interaction we must considerthat the action S cl ( t f , t i ) , describing a classical isolated particle, is locallytransformed under interactions. Similarly to (32) and (33) we find thatthe non-homogeneous Hamiltonian and momentum operator H ′ and P ′ de-fine the Lagrangian L ′ cl ≡ P ′ i ˙x im − H ′ , and in turn the transformed action S ′ cl ( t b , t a ) ≡ R t b t a dtL ′ cl . As already said, P ′ µ ( x ) transforms as the 4-momentum ¯ p ′ µ ( x ) of the corresponding classical particle (52). Thus, we find that thetransformed action S ′ cl ( t b , t a ) is formally the action of the corresponding in-teracting classical particle (written in terms of operators).The demonstration used to derive (35) can now be generalized to interac-tion. In fact,[2], we can plug point by point the local completeness relationsof the energy eigenmodes in between the elementary 4D evolutions. Since incase of interaction the elementary 4D evolutions are described by the local44perators H ′ and P ′ , the FPI (35) turns out to be generalized to Z = Z V x D x e i S ′ cl ( t f ,t i ) . (67)Indeed this is the ordinary FPI associated to the interaction scheme (52). Inthe l.h.s. of the correspondence (66), interaction implies the following formalsubstitution of action: S ( t f , t i ) → S ′ ( t f , t i ) .Similarly we want to generalize to interaction the r.h.s. of the correspon-dence (66). According to sec.(4.1), the evolution of an interacting periodicfield is also dual to the classical configurations of a field in the correspondingdeformed XD metric (58). The dualism is manifest if we assume that the XDis virtual . Therefore, in the r.h.s. of (66), the interaction can be representedby the formal substitution of the action in flat XD with the action in corre-sponding deformed XD, S Dcl ( s f , s i ) → S ′ Dcl ( s f , s i ) . Hence, combining i ), ii ),and iii ), we have found that the classical configuration of a bosonic field ina particular deformed VXD background describes the quantum behavior ofthe corresponding interaction scheme.Now we can apply the holographic prescription, point iv ), to describethe 5D theory S ′ Dcl ( s f , s i ) in an effective way. By applying the holographicprescription to a 5D theory, whose XD is interpreted as virtual , we willget an effective description of the quantum behavior of the correspondinginteracting system. That is, similarly to the boundary action in VXD (51),the holographic prescription in VXD consists in projecting the bulk dynamicsof the action (40) on the boundary. As in ordinary holography the source fieldcan be regarded as the fundamental mode of the tower. Thus the holographicformulation turns out to be an effective description of the VXD theory atenergies E eff smaller than or of the order of the fundamental energy ¯ E (thisis similar to the projection of a Euclidean partition function on the groundstate at large β [52]). That is, [48, 49, 50, 51], S Dcl ( s f , s i ) ∼ S Holo Φ | Σ = eφ Σ ( s f , s i ) + O ( E eff / ¯ M ) . (68)In an ordinary XD theory, holography provides an effective and collective de-scription of the propagation of the KK modes. However, as already noted atthe end of sec.(5), by assuming a VXD, such a collective description is alreadyexplicit, even without holography. In fact the virtual KK modes naturallydescribe the quantum excitations of the same fundamental system (string), i.e. they are not independent fields. On the other hand, the fundamental45evel of the virtual
KK tower represents the non-quantum limit of the field inVXD ( i.e. it can be locally matched with a classical KG mode). Therefore,if we explicitly assume a VXD, the only effect of the holographic prescrip-tion is an effective description of the corresponding quantum behavior. Theholographic action S Holo Φ | Σ = eφ Σ ( s f , s i ) of (62), including the kinetic term whichdescribes the “dynamics” of the source field ¯Φ | Σ ∼ eφ Σ , can be thought ofas the effective action S eff Φ | Σ of ordinary perturbation theory: S Holo Φ | Σ ∼ S eff Φ | Σ .This actually corresponds to a first order expansion in ~ and avoids the useof the functional integral in the generalization to interaction of the r.h.s ofthe correspondence (66) (the dependence on s has been eliminated).Finally we can combine points i ), ii ), iii ), and iv ), obtaining an effectivedualism between the holographic description of the classical configurations ofa field in a deformed XD background — with infinite or compact XD — andthe quantum behavior of an interacting quantum system — massless (confor-mal) or massive (non-conformal) respectively. This dualism is manifest if weassume that the XD is virtual . Therefore, the generalization of both sides ofthe correspondence (66) to interaction can be formally summarized in termsof the holographic prescription by the following mnemonic correspondence Z = Z V x D x e i S ′ ( s,s ′ ) ! e i S Holo Φ | Σ= eφ Σ ( s,s ′ ) . (69)The resulting dualism from i ), ii ), iii ), and iv ) is therefore reminiscentof the phenomenological formulation of the AdS/CFT correspondence (63).This unconventional description of AdS/CFT will be tested in the next sec-tion for the specific case of a simple model of QGP freeze-out obtaininganalogies with some basic aspects of AdS/QCD. In this paper we will limit the application of this correspondence to thestudy of warped XD geometry. However, the same approach has been used[2] to reveal the geometrodynamics scalar associated to scalar QED. Re-markably, the interpretation of this geometrodynamical description of gauge In the action S ′ on the l.h.s. of this correspondence a source term is understood. Infact, in (68), through the source field φ Σ , we are explicitly selecting a particular interactingcyclic field solution with fixed coefficient a n and fundamental mode ¯Φ . In [2] we show how to retrieve gauge interactions from particular geometrodynamicsof the compact 4D, in what can be regarded as a formulation of Kaluza’s original proposal[3] in VXD. In this case the correspondence between classical geometry and quantumbehavior, (31) will formally reproduce the ordinary FPI of scalar QED.
A possible phenomenological application of the classical to quantum cor-respondence discussed above is represented by the simple case of the QGPexponential freeze-out as described (at a classical level) by the Bjorken Hy-drodynamic Model (BHM) [33] or by thermal QCD [34]. We therefore con-sider a collider experiment in which we imagine having a volume of quarksand gluons originally at high temperature, i.e. at high energy ¯ E UV = Λ and high spatial momentum | ¯ p UV | . We assume that the QGP is in first ap-proximation composed by massless fields, so that, (15), the energy and themomentum vary conformally during the freeze-out ¯ E UV ∼ | ¯ p UV | .During the freeze-out, the system radiates energy hadronically or elec-tromagnetically as long as the temperature of the fields is higher than thatof the surrounding environment [34]. In terms of de Broglie 4-periodicity(3), this means that the fields inside the QGP initially have small time peri-odicity T UVt and spatial periodicity ~λ UVx . These are modulated conformally during the freeze-out, i.e. T t ∼ | ~λ x | , whereas the world-line parameter canbe regarded as non-compact (infinite periodicity).According to the simple BHM model [33], during the freeze-out the energydensity ǫ ( s ) has an exponential gradient with respect to the laboratory time, i.e. proper-time, s . That is, the energy of the fields decays exponentiallyduring the freeze-out ¯ E → ¯ E ( s ) = e − ks ¯ E . Thus k describes the gradient ofthe freeze-out. It is interesting to note that in the analogy between QGP anda thermodynamic system [34], this corresponds to an exponential decay ofthe temperature T ( s ) of the fields which can also be interpreted in terms ofNewton law’s of cooling: ∂ s T ( s ) = −T ( s ) /k with gradient k . Therefore thethermodynamic system passes from a high temperature state characterizedby small time periodicity T UVt , to a state characterized by large periodici-ties T IRt ( T UVt ≪ T IRt ). The possible application of our field theory withintrinsic periodic periodicity to describe a thermodynamical system has beendiscussed in [1, 42] and will be extended in future papers. We will not givethe detail about this, but it can be easily figured out if we consider that theMinkowskian periodicity and the corresponding quantized energy spectrumof our theory play a role analogous to the “mathematical trick” of the Eu-47lidean time and the Matsubara frequencies of finite temperature field theory,respectively.In the approximation of massless fields, this means that, according to thesimple BHM model [33] prescribing in the QGP freeze-out, the 4-momentumof the fields decreases conformally and exponentially ¯ p µ → ¯ p ′ µ ( s ) ≃ e − ks ¯ p µ . (70)Indeed the interaction scheme associated with the QGP freeze-out is de-scribed by a conformal exponential of the world-line parameter s and theparameter k . In terms of our geometrodynamical description of interaction(52), the freeze-out is therefore encoded by the conformal tetrad e aµ ( s ) ≃ δ aµ e − ks . (71)Proceeding in analogy with the description of interaction given in par.(4),from (70) and (53) we find that during the freeze-out the 4-periodicity hasan exponential and conformal dilatation T µ → T ′ µ ( s ) ≃ e ks T µ . (72)According to (55), this modulation of periodicity is encoded in the substitu-tion of variables dx µ → dx ′ µ ( s ) ≃ e − ks dx µ . (73)Thus, (54), the QGP freeze-out is encoded by the warped metric ds = e − ks dx µ dx µ .As inferred in (58), by treating the world-line parameter s as a VXD,the exponential dilatation of the 4-periodicity during the QGP freeze-out ofmassless fields ( dS ≡ ) can be equivalently encoded in the virtual AdSmetric dS ≃ e − ks dx µ dx µ − ds ≡ . (74)It is interesting to note that the de Broglie-Planck relation (3) explicitlyimplies that the exponential evolution of the energy ¯ E ( s ) of the fields from s UV to s IR is parameterized, through the Planck constant, by the inverse ofthe time periodicity T t ( s ) . That is, from (70-72) we find that the character-istic time periodicity T t ( s ) of the QGP during the freeze-out is nothing otherthan the conformal parameter of AdS/CFT z ( s ) ≡ e ks k = T t ( s ) = 1¯ E ( s ) . (75)48n agreement with the AdS/CFT dictionary, we see that z naturally pa-rameterizes the inverse of the energy of the QGP fields because it actuallycorresponds to the de Broglie time periodicity of the fields.We use z UV and z IR to denote a UV energy scale Λ (Planck scale) andan IR energy scale µ (TeV scale), respectively, T UVt = 1Λ = e ks UV k , T IRt = 1 µ = e ks IR k . (76)According to our geometrodynamical description, as well as to the Bjorkenand thermal QCD description, the virtual AdS background leads to an inter-action scheme in which the 4-periodicity of the QGP particles has exponentialdilatation, i.e. the 4-momentum decays exponentially. In this approximativedescription of interaction in which the masses are neglected, the kinematicalinformation is encode on the VXD metric, in particular by the AdS curvature.The quantum evolution of the strong coupling constant can be qualitativelyobtained in a way that resembles AdS/QCD. As already said in par.(6) andsummarized by (69), from the resulting correspondence between the usualQFT and compact 4D field theory investigated in sec.(2) and sec.(3), we infact expect to find out that the exponential dilatation of the 4-periodicitieswritten in terms of the conformal parameter z describes the quantum behav-ior of the QGP at different energy scales.This correspondence indeed resembles AdS/QCD. The topic is too wideto be treated in detail here. However, our strategy to describe the quantumbehavior arising from the classical geometrodynamics associated to (72) willbe to follow the line of some of the founding papers on the AdS/QCD cor-respondence [19, 18, 17, 53]. In this way we will find out an unconventionalinterpretation of basic aspects of the correspondence similarly to the inter-pretation already given for the conformal parameter z and the curvature k .According to our theory, the quantum behavior of massless fields ( M n = 0 )is described by an infinite warped VXD (infinite proper-time periodicity ofthe fields). This means that the conformal parameter z , i.e. the time peri-odicity of the fields is allowed to take any value. Since in this case we arenot introducing any scale, the energy of the fields in VXD is allowed to varyfrom zero to infinity.As shown in sec.(6), a non-compact XD geometry describes the quantum49ehavior of conformal fields . Here we consider the case of a 5D gauge the-ory with bulk coupling g and an infinite VXD. In this way all the modes areflat, i.e. independent on the VXD A ( flat ) µ ( x, z ) = A ( flat ) µ ( x ) . At this point,similarly to (14), the quantum behavior associated to the classical configura-tions of the fields in this virtual AdS metric can be extracted by integratingout the VXD (i.e. the conformal parameter describing the periodicity of theQGP), so that, for every (massless) mode of the tower we get S AdS ≃ − Z /µ / Λ dzkz Z d x g F ( flat ) µν ( x, z ) F ( flat ) µν ( x, z )= − log µ Λ kg Z d xF ( flat ) µν ( x ) F ( flat ) µν ( x ) . (77)Hence, [19, 18], the effective 4D coupling behaves logarithmically with respectto the infrared scale g ≃ g k log µ Λ . (78)This reproduces the quantum behavior with the energy of the strong couplingconstant, a.k.a. the asymptotic freedom , as long as we suppose [17, 18, 53] k ∼ N c g π . (79)Thus, according to our description, the curvature encodes the strength ofthe interaction, i.e. the gradient of the QGP freeze-out. In agreement withthe AdS/CFT dictionary, the case of infinite VXD actually corresponds to aconformally invariant quantum theory.The analogy with AdS/CFT phenomenology can also be extended to thecase of compact VXD. In an Orbifold description, the PBCs can be substi-tuted with N and N BCs at the IR and UV energy scales, respectively. Since the mass has a geometrical meaning in the de Broglie “periodic phenomenon”,it can be shown that the terminology “conformal invariance” has an extended validity forcyclic fields. We will give more detail about this in future publications Since in the Hilbert space (20) of our theory of topology S only the modulo of thefield has physical meaning, we can adopt the formalism of XD field theory in an Orbifold S / Z and describe the PBCs in terms of N-N BCs (or D-D BCs depending if we want toconsider or not the purely translational mode n = 0 ). By assuming explicit branes the QGP freeze-out is limited between an initial and finaltemperature (the branes could be interpreted as phase transition points). virtual
KK modes of this VXD theoryare the quantum excitations of the same fundamental conformal system, infact they can be regarded as describing the hadrons as we will see below.However, according to our theory, a compact VXD with PBCs generates amass spectrum, which is non-trivial in the case of warped VXD . This meansthat the conformal invariance is broken in the corresponding quantum the-ory, in agreement with the AdS/CFT dictionary. The quantum behavior canbe described in an effective way by applying the holographic prescription tothis compact VXD geometry. In fact, as already said, holography implicitlyprovides an effective description of the propagation of the virtual KK modes.That is, the KK modes are already described in an effective collective wayin the holographic propagator. Thus, we introduce an IR source field ¯ A µ ( q ) [19] with coupling e , whereas we keep NBCs at the UV scale. As argued inpar.(6) the assumption of a source field at the IR brane corresponds to an ef-fective description of NBCs [17], i.e. of the underlying theory with compactVXD and N-N BCs. Hence, by assuming Euclidean momentum ( q → iq )and expanding in the limit of large 4-momentum with respect to the IR scale( Λ ≫ | q | ≫ µ ), we find out that the leading behavior of the holographiccorrelator of the action (68) is Π Holo ( q ) ∼ − q kg log q Λ . (80)By identifying the source field ¯ A µ with the vectorial field of a chiral theory, Π Holo ( q ) approximately matches the vector-vector two-point function of or-dinary QCD. Analogously to (78) this gives an estimation of the logarithmicrunning of the gauge coupling of QCD, i.e. asymptotic freedom, e eff ( q ) ≃ e − N c π log q Λ . (81)This quantum behavior has been obtained without imposing any explicitquantization except BCs.We must note that, even though we have generated massive modes byimposing a compact VXD, we are still adopting a conformal metric in which The massless mode of this model behaves as in the conformal case above [19]. ThisZero Mode Approximation can be obtained by substituting the NBCs with appropriateSBCs in order to eliminate all the higher massive modes. t ∼ dx . But according to (12) and (58), this conformal behavior is justi-fied only in the massless approximation. That is, the AdS metric is not fullyconsistent with a massive dispersion relation since it describes a conformaldeformation of the space-time dimensions. This reveals the limit of this for-mulation of the QGP and at the same time it allows us to speculate about thefact that deformations of the AdS metric, for instance through the assump-tion of a dilaton in (58) or “soft-walls”, provide more realistic predictions ofthe hadronic mass spectrum . In future studies we will try to interpretthe meaning of these deformations and the related Regge-like behavior of thehadronic mass spectrum in terms of Kaluza’s matrix and winding numbersaround compact 4D. As further evidence of the relevance of semi-classicalmethods, based on PBCs, in AdS/QCD phenomenology, we mention thatthe “soft-wall” was originally justified [12] in terms the semi-classical Bohr-Sommerfeld quantization— i.e. a periodicity condition—associated with aharmonic potentials. Similar results are obtained in Light-Front quantiza-tion in which PBCs at the ends of the Light-Front coordinate are imposedas semi-classical quantization condition, [13].The semi-classical quantization by PBCs of field theory in compact 4Dand its (unconventional) interpretation in terms of AdS/CFT presented inthis paper must be tested for other known aspects of the AdS/QCD. Never-theless the remarkable correspondences between 5D geometry and 4D quan-tum behavior pointed out in this paper can provide an intuitive tool to de-velop more realistic phenomenological AdS/QCD models. In future paperswe will extend this analysis by using the analogy ( i.e. mode expansion associ-ated to BCs) with the Light-Front quantization and “soft-walls” (see below).Indeed we have a description of the AdS/CFT which, on the one hand,mimics the dilatation of periodicity of reference clocks and red-shift typical ofa gravitational interaction, as described in par.(4.2) and on the other hand,it can be interpreted in terms of wave-particle duality, through the “periodicphenomenon”. In a bottom-up approach to AdS/QCD, one can implement 5D chiral models with asoft-wall in the IR or with a dilaton which give realistic predictions for the masses and forthe decay constants of light vector and axial-vector mesons, for their form factors, for thevector meson dominance, for the Weinberg sum rules, etc, in sufficient agreement with theexperimental data [18, 19, 17, 12, 56]. .2. Comments and Outlooks Finally we introduce another intuitive element to interpret the good prop-erties and the consistency of field theory in compact 4D and fundamentaltopology S . By imposing the constraint of intrinsic periodicity, the result-ing field solutions Φ( x ) of the theory can be regarded as a relativistic stringvibrating in compact 4D. The harmonics resulting from its space-time vi-brations, together with the corresponding energy-momentum spectrum, canbe regarded as quantum excitations according to the results of sec.(2). Re-markably this picture turns out to be the full relativistic generalization ofsound theory. A sound signal is a packet of classical waves whose harmonicspectrum is determined by the vibration on compact spatial dimensions of asource, for instance a one-dimensional string. Loosely speaking, in our for-malization of the de Broglie “periodic phenomenon” every matter particle canbe regarded as a relativistic string ( i.e. governed by relativistic wave equa-tions) vibrating in compact 4D (in particular, with PBCs). These generate(unconstrained) relativistic waves playing the role of the (massless) media-tors of the interaction and thus defining the underlying reference space-timefor a relational description of the sources ( e.g. the atomic orbitals can beregarded as the harmonics of space-time vibrations of topology S × S ). Itis also interesting to point out that, from a historical point of view, the for-malism of QM has its origin in the formalism of sound theory developed byRayleigh at the end of the 19th century.The idea to describe nature in terms of the mathematics of vibratingstrings has indeed solid historical origins (which can be postdated to Pythago-ras). This idea is also at the origin of modern String Theory (ST) whosemathematical beauty has attracted the interest of several generations ofphysicists. It is interesting to mention that, our formalism being based on theassumption of a compact world-line parameter, we have a fundamental anal-ogy with ST. In fact, ST is based on the fundamental assumption that oneof the two world-sheet parameters is compact (with PBCs, or D-BCs and N-BCs, in the case of closed strings or open strings, respectively). In our theory,the compact world-line parameter, whose length is determined by the mass,plays a role similar to the compact world-sheet parameter of ST, so that thebosonic action in compact 4D defined in (6) can be regarded as the prototypeof a simple, purely 4D, ST. In this way it is possible to figure out that ourtheory inherits some of the interesting mathematical properties of ST, with-out the problematic requirement of the actual existence of unobserved XDsin the theory. A exhaustive description of the analogy with string theory53s beyond the scope of this paper. However, in [2] we have already men-tioned that the energy levels of a “periodic phenomenon” can be described interms Virasoro algebra (with implicit commutation relations). Furthermore,we may note that the dualism to XD theories allows us to associate to ourvibrating strings a Veneziano amplitude whose poles, for instance calculatedthrough the holographic prescription, depend on the VXD geometry, [55, 54].In the holographic prescription of a warped VXD , assuming dilatons or soft-walls in order to reproduce Regge poles, we find that the mass spectrum andthe propagation of virtual KK modes approximately matches the hadronicmass spectrum and quantum two-point function of QCD, respectively, [17].Thus in our description the hadrons can be actually interpreted as quantumexcitations encoded in the harmonic modes of a vibrating string describingQCD. This interpretation of QCD is at the very origin of string theory, but itis also well known that, [57], because of confinement, hadrons have a naturalformalization in terms of field theory in compact 4D and BCs.Our purely 4D reinterpretation of AdS/QCD can also be seen by consider-ing the analogies with Light-Front quantization. For instance, in the isotropyQGP freeze-out should also involve the expansion in spherical harmonics as-sociated to the periodicities in ( θ, ϕ ) . Thus, the Light-Front coordinates,especially in the case of massless fields, represent a natural parameterizationof the system, so that our semi-classical quantization is achieved by impos-ing PBCs to the Front-Light coordinates. Indeed, PBCs for the Front-Lightcoordinates is at the base of Light-Front quantization [13]. In this way, wecan intuitively interpret the analogies between the Front-Light quantizationand AdS/QCD, [13, 14]. As noticed recently, Light-Front-quantization hasalso been used to reproduce semi-classical results of perturbative QED [14] interms of expansion of the harmonic modes allowed by the BCs. In agreementwith the motivations of sec.(5) and with the results of sec.(2), this can be re-garded as an indirect confirmation of the other relevant results of field theoryin compact 4D, that is to say the semi-classical description of scalar QEDobtained in [2]. Another relevant application of field theory in compact 4Dwould be a purely 4D reinterpretation of phenomenological XD models suchas Randall-Sundrum, in terms of the formalism of VXD. All these heuristicconsiderations provide a general picture of the fundamental physical motiva-tions for our formulation of field theory in compact 4D, and will be used infuture research for other phenomenological applications and predictions.54 onclusions According to Witten, in the AdS/CFT correspondence, quantum behavior[...] are encoded in classical geometry , [21]. In this paper we have proposedan unconventional interpretation of fundamental aspects of this “classical toquantum” correspondence of AdS/CFT in terms of the undulatory mechan-ics and relativistic geometrodynamics, [22]. In recent papers, [1, 2], we haveformulated a possible realization of the so-called de Broglie “periodic phe-nomenon”, at the base the wave-particle duality and whose character is “yetto be determined” [28]. This formulation is based on the assumption of PBCsas semi-classical quantization conditions at the geometrodynamical boundaryof a field theory defined in compact 4D. The resulting semi-classical solutionof the theory has remarkable formal correspondences with the fundamentalaspects of ordinary QFT. It can also be regarded as a “vibrating string”,and similarly to a “particle in a box”, its harmonics describes the quantumexcitation of the elementary system.Field theory in compact 4D has an exact dualism to XD field theory[1]. In fact it can be formally retrieved by identifying the cyclic XD of a5D massless field with a world-line parameter. That is, the cyclic XD ofour purely 4D theory and its compactification length turns out to encodethe so-called de Broglie “internal clock” and the Compton wavelength of acorresponding particle, respectively. We address this identification by sayingthat field theory in compact 4D has a virtual
XD. In this dual description theKK modes are virtual in the sense that they encode quantum excitations.This idea is in the spirit of Kaluza’s and Klein’s original proposals [3, 4]:Kaluza introduced the XD as a “mathematical trick” rather than as a “real”XD whereas Klein used PBCs for a semi-classical interpretation of QM.According to undulatory mechanics, the variation of 4-momentum oc-curring during interaction can be equivalently described as modulations of4-periodicity. In the formalism of field theory in compact 4D, through theassumption of PBCs, the modulation of 4-periodicity associated to a giveninteraction scheme can be described in terms of geometrodynamics of theboundary (similarly to the holographic principle), or equivalently in termsof deformations on a corresponding metric in virtual
XD. Moreover this ge-ometrodynamical description of interaction mimics linearized gravity. In fact,the spacial and temporal modulations of reference lengths and clocks are en-coded in local deformations of the space-time metric, respectively. Remark-ably, this description typical of GR can also be extended to describe gauge55nteractions in terms of local variations of reference frames, as proven in [2].Similarly to AdS/CFT, by combining the correspondence of field theoryin compact 4D with ordinary QFT, its dualism to XD field theory, and thegeometrodynamical description of interaction, we have inferred that he clas-sical configuration of a field in a deformed VXD reproduces the quantumbehavior of a corresponding interaction scheme. The application of this ap-proach to the exponential freeze-out of the QGP actually yields fundamentalanalogies with basic aspects AdS/QCD.
Acknowledgements
I would like to thank in chronological order V. Ahrens, M. Neubert, M.Reuter and T. Gherghetta for fruitful and fair discussions, and for theirinterest in new ideas in physics. This paper is part of the project “CompactTime and Determinism”. Its main results have been partially published on-line in [22].
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