aa r X i v : . [ phy s i c s . g e n - ph ] M a r Gauge Interaction as Periodicity Modulation
Donatello Dolce
Centre of Excellence for Particle Physics (CoEPP),The University of Melbourne,Parkville 3010 VIC,Australia.
Abstract
The paper is devoted to a geometrical interpretation of gauge invariance in terms of theformalism of field theory in compact space-time dimensions [1]. In this formalism, thekinematic information of an interacting elementary particle is encoded on the relativisticgeometrodynamics of the boundary of the theory through local transformations of theunderlying space-time coordinates. Therefore, gauge interaction is described as invari-ance of the theory under local deformations of the boundary, the resulting local variationsof field solution are interpreted as internal transformations, and the internal symmetriesof the gauge theory turn out to be related to corresponding local space-time symme-tries. In the case of local infinitesimal isometric transformations, Maxwell’s kinematicsand gauge invariance are inferred directly from the variational principle. Furthermorewe explicitly impose periodic conditions at the boundary of the theory as semi-classicalquantization condition in order to investigate the quantum behavior of gauge interaction.In the abelian case the result is a remarkable formal correspondence with scalar QED.
Introduction
In 1918 Weyl [2] introduced the idea of gauge invariance in field theory in an at-tempt to describe electromagnetic interaction as invariance under local transformationsof space-time coordinates. In particular he tried to extend the principle of relativityof the choice of reference frame to the choice of local units of length. For this reasonthe idea was named gauge invariance. The requirement of gauge invariance in fact ne-cessitates the introduction of a new field in the theory, named compensating field [3].It cancels all the unwanted effects of the local transformations of variables such as therelated internal transformation of the matter fields, and enable the existence of a lo-cal symmetry. Weyl noticed that such a compensating field has important analogieswith the electromagnetic potential. His proposal was very appealing because of its deepanalogies with the geometrodynamical description of General Relativity (GR). Even Email address: [email protected] (Donatello Dolce) The term “geometrodynamics” is used as a synonym to indicate a geometrical description of inter-actions [4]
Preprint submitted to Elsevier October 1, 2018 hough the compensating field associated with local Weyl transformations can be suc-cessfully used to represent gravitational interactions, further developments soon showedthat such geometrodynamical description of electromagnetism (EM) was not possible.Contrary to the experimental evidence, the compensating field associated with Weyl’sgauge interacts in the same manner with particle and antiparticles. The possibility of ageometrodynamical description of EM in terms of local transformations of the underlyingspace-time coordinates was abandoned, though the terms gauge invariance still remainfor historical reasons. In order to reproduce the correct gauge field ( i.e. compensatingfields with an imaginary unit in front with respect to the original Weyl compensatingfields) an “internal” symmetry under local phase transformations of the matter fieldswas postulated. These internal transformations of the fields originate ordinary gaugeinteraction.Another important attempt for a geometrodinamical description of gauge interactionis represented by the so-called Kaluza’s miracle. In [5] Kaluza showed that classical EMactually has a well defined geometrodynamical interpretation, but this interpretation in-volves an eXtra-Dimension (XD) — at least as a “mathematical trick”. The gauge fieldappears as entries in the space-time components on XD Kaluza’s metric and the Maxwellequations are retrieved from the fifth components of the five-dimensional (5D) Einsteinequations. This represent a de facto
5D geometrodynamical unification of electromag-netic and gravitational interaction. Also to be considered is Klein’s original proposal,which was to impose Periodic Boundary Conditions (PBCs) at the ends of a compact XD(cyclic XD with topology S ) in the attempt of a semi-classical interpretation of Quan-tum Mechanics (QM). In [6] he actually noticed that such PBCs provide an analogy withthe Bohr-Sommerfeld quantization condition — in particular he used this hypotheticalcyclic XD to interpret the quantization of the electric charge. Similarly it is importantto mention Wheeler’s program of reduction of every physical phenomenon to a purely ge-ometric aspect, as summarized by his slogan “Physics is Geometry!” [7] or the Rainich’s“already unified field theory” where the (“square”) of the electromagnetic field strengthis reinterpreted in terms of the Ricci tensor [8].In this paper we will investigate these attempts for a geometrical description of gaugeinvariance in terms of the formalism of field theory in compact space-time dimensions(compact 4D). This formalism, defined in [1] and summarized in sec.(1), see also [9, 10,11, 12, 13, 14], can be regarded as the natural realization of the de Broglie assumptionat the base of wave mechanics (wave-particle duality). In fact, by using de Broglie’swords, the formalism is based on the fundamental assumption “ of existence of a certainperiodic phenomenon of a yet to be determined character, which is to be attributed toeach and every isolated energy parcel [elementary particle] ” [15, 16]. This so-called “deBroglie periodic phenomenon” [17] or “de Broglie internal clock” [18] has been implicitlytested by 80 years of successes of QFT and indirectly observed in a recent experiment[18]. We will realize this assumption by imposing the de Broglie periodicity as a constrainto a free field. That is to say, as the solution of a bosonic action in compact 4D andPBCs. This means that the compactification lengths as well as the boundary of thetheory, are explicitly related to the de Broglie periodicity. Therefore the resulting fieldsolution will be a sum over harmonic modes similarly to a string vibrating in compactdimensions. That is, the PBCs (allowed by the variational principle) will be used asquantization condition similarly to the semi-classical quantization of a particle in a box.As noticed in [1] our description can be regarded as the full relativistic generalization of2ound waves. A sound source is a vibrating string in compact spatial dimension withina classical framework. Similarly, an elementary particle will be described as a vibratingstring in compact 4D within a relativistic framework. Indeed, as we will mention later,the theory can be regarded as a particularly simple type of string theory.In the first part of the paper we will mainly investigate classical gauge invariance, inparticular classical EM. In par.(2), we will investigate in a geometric way how the deBroglie spatial and temporal periodicities of an elementary boson, i.e. of a correspondingsingle mode of a Klein-Gordon (KG) field, varies dynamically under local transformationsof reference frame or interactions. The temporal and spatial de Broglie periodicities canbe used to describe the four-momentum of a particle, according to the de Broglie phaseharmony. They can be represented as a four-vector. Such a de Broglie four-periodicityis derived by Lorentz transformations from the invariant periodicity of the proper timeassociated with the mass of the particle. Indeed it describes a “periodic phenomenon” oftopology S . According to de Broglie, the local and retarded variations of four-momentum(or four-frequency) of a particle occurring during a given relativistic interaction schemecan be equivalently described in terms of local and retarded variations (modulations)of four-periodicity of a corresponding periodic phenomenon. In the formalism of fieldtheory in compact 4D the de Broglie four-periodicity is described by the boundary ofthe theory. Thus the kinematics of a particle in a given interaction scheme turns out tobe encoded in corresponding geometrodynamics of the boundary of the theory — in amanner of a holographic principle. Furthermore, deformations of the boundary of theaction can be obtained from corresponding transformations of the space-time variables, i.e. diffeomorphisms of the compact 4D of the theory. Hence this description of interac-tion mimics linearized gravity. In fact gravitational interaction can be described in termsof modulations of periodicity of reference clocks encoded in corresponding geometrody-namics of the underlying 4D. In [20] we have noticed that such a geometrical descriptionapplied to the Quark-Gluon-Plasma (QGP) exponential freeze-out actually provides aninteresting parallelism with phenomenological aspects of the AdS/QCD correspondence.Thus, similarly to GR, we want to describe interaction in terms of invariance of thetheory under a corresponding local transformations of variables. In ordinary field theorysimple isometric transformations of the underlying space-time dimensions have no effect.This is essentially because in ordinary field theory the BCs have a very marginal role.The KG field used for practical computations is the most generic solution of the KGequation. However, as evident in our formalism, a generic transformation of variablesimplies a corresponding deformation of the boundary of the theory and in turn, throughthe PBCs, of the de Broglie periodicity of the field solution of the theory. Indeed we havea variation of field solutions even in the case in which the transformation of variables isa simple isometry. In fact, even if the transformation is from a flat space-time to a flatspace-time, leaving the structure of the action invariant, it could involve a correspondingdeformation (rotation) of the boundary of the theory. We will show that classical gaugeinteractions can be derived by requiring invariance of the theory in compact 4D underlocal isometries and by applying the variational principle at the boundary. To figure outthe idea we can imagine to describe the trembling motion [21, 22] (we will use the germanterm zitterbewegung for the analogy of the idea to the Scrödinger’s zitterbewegung model[23]) of a charged particle interacting with an electromagnetic field in terms of localtransformations of reference frame. For this reason, in this paper we limit our study tothe approximation of local isometries; that is, particular isomorphisms where the lengths3f the four-vectors are in a first approximation preserved (contrary to the Weyl invariancewe do not consider scale transformations). The case of local scale (conformal) invariancehas been partially investigated in [20] through the dualism with XD theories.More in detail, in sec.(3) we will interpret the variation of the field solution associ-ated to a local isometry in terms of internal symmetries of the field. A local isometryinduces a minimal substitution formally equivalent to the one of classical EM. From thevariational principle it is possible to find out that this description formally reproducesthe Noether current of ordinary gauge interactions. The symmetry of the gauge transfor-mation turns out to be the symmetry of the isometry which originates it. The resultinggauge field describes the local deformation of periodicity of a matter field under a localtransformation of variables.In sec.(4), we will see the possibility to write fields with different periodicities inan action with persistent boundary by using gauge invariant terms. The requirementof gauge invariance is therefore derived from the variational principle. Gauge transfor-mations allow one to tune the periodicity of the different fields of a theory in order tominimize the action at the common boundary. For the same reason we will see that onlyparticular types of local isometries, which we will call polarized , are allowed by the vari-ational principle. These polarized local isometries reproduce Maxwell dynamics for thegauge field. Thus the geometrodynamics associated with these particular local isometriesreproduce formally classical gauge theory.In [20] we have shown that field theory in compact 4D is dual to the Kaluza-Klein(KK) field theory. Under this dualism the geometrodynamical description of gauge in-teraction proposed here can be regarded as a purely 4D formulation of Kaluza’s originalproposal.The formalism of field theory in compact 4D also provides an interesting analogywith Klein’s original proposal. In fact, in [1, 20] we have shown that the PBCs at thegeometrical boundary of the theory represent a semi-classical quantization condition.This can be regarded as the relativistic generalization of the quantization of a particlein a box. This idea is inspired to the ’t Hooft determinism [24] and the stroboscopicquantization [25] where QM is interpreted as an emerging phenomenon associated to someunderlying cyclic dynamics. In our theory the Feynman Path Integral (FPI) formulationarises in a semi-classical way as interference between classical paths with different windingnumbers associated with the underlying cyclic geometry S of the compact 4D [1, 9].Moreover the theory has implicit commutation relations. Generalizing this description,in sec.(5) we will finally find that the modulation of four-periodicity of an interactingcyclic field is formally described in local Hilbert spaces by the ordinary Scattering Matrixof QM. The space-time evolution associated to such local isometries of the compact 4Dis formally described by the ordinary FPI of scalar QED.
1. Compact space-time formalism
In relativistic mechanics every isolated elementary system (free elementary particle)has associated persistent four-momentum ¯ p µ = { ¯ E/c, − ¯p } . On the other hand, the deBroglie-Planck formulation of QM prescribes that to every particle with four-momentumthere is a corresponding “periodic phenomenon” with four-angular-frequency ¯ ω µ = ¯ p µ c/ ~ , i.e. with corresponding de Broglie four-periodicity T µ = { T t , ~λ x /c } . The topology of4his so-called “de Broglie periodic phenomenon” is S . In fact it is fully characterizedby the proper time periodicity T τ [15, 16]. In a generic frame, the spatial and temporalcomponents of the de Broglie four-periodicity T µ are obtained through Lorentz transfor-mations: cT τ = cγT t − γ ~β · ~λ x . The energy and the momentum of a particle with mass ¯ M in the new reference frame is E = γ ¯ M c and ¯p = γ ~β ¯ M c , respectively. In the “deBroglie periodic phenomenon”, also known as “de Broglie internal clock”, the proper timeperiodicity is fixed by the mass of the particle, according to T τ ¯ M c = h . Therefore, in ageneric reference frame, we have de Broglie-Planck relation (de Broglie phase harmony) c ¯ p µ T µ = h . (1)Similarly to the de Broglie assumption of “periodic phenomenon”, in the compact 4Dformalism [1] we assume that every elementary bosonic particle with four-momentum ¯ p µ is described by an intrinsically periodic bosonic field with de Broglie four-periodicity T µ fixed dynamically through PBCs (1). This means that, as long as we describe free parti-cles, i.e. persistent ¯ p µ , the intrinsic four-periodicity T µ of the fields must be assumed tobe persistent. Thus we describe such a free bosonic field with persistent four-periodicity T µ as the field solution of a relativistic bosonic action in compact 4D with PBCs (PBCsare represented by the circle in the volume integral H ) S = I T µ d x L ( ∂ µ Φ( x ) , Φ( x )) . (2)Roughly speaking, in a “de Broglie periodic phenomenon” the whole physical informationis contained in a single four-period T µ , [1]. In fact, “ By a clock we understand anythingcharacterized by a phenomenon passing periodically through identical phases so that wemust assume, by the principle of sufficient reason, that all that happens in a given periodis identical with all that happens in an arbitrary period” A. Einstein [39]. Indeed, underthis assumption of intrinsic periodicity, every isolated elementary particle can be regardedas a reference clock. This aspect will play a central role since, in analogy with GR, we willdescribe interactions in terms of modulations of periodicity of these “de Broglie internalclocks”.It is important to note that the PBCs minimize the above bosonic action at theboundary. That is, every bosonic field (solution of the Euler-Lagrange equations) withfour-periodicity T µ automatically minimizes the above action [1]. This is the fundamentalreason why the theory turns out to be fully consistence with Special Relativity (SR) [1].Indeed PBCs have the same formal validity of the usual Stationary (or Synchronous)BCs (SBCs) of ordinary field theory ( i.e fixed values of the fields at the boundary). TheLorentz covariance of relativistic bosonic actions is preserved by PBCs.The theory satisfies time ordering and relativistic causality. In fact, just as Newton’slaw of inertia does not imply that every point particle moves on a straight line (persistent ¯ p µ ), our assumption of intrinsic periodicities does not mean that our field solutions havealways persistent periodicities T µ . Indeed, the four-periodicity T µ must be regardedas local and dynamical as the four-momentum ¯ p µ , according to (1). The retarded andlocal variations of four-momentum occurring during interactions imply retarded and localvariations (modulations) of the intrinsic four-periodicity of the particles. Events in time( i.e. interactions) are characterized by variations of periodic regimes of the fields. In thistheory we must interpret relativistic causality and time ordering in terms of variations5modulations) of four-periodicity rather than of variations of four-momentum (roughlyspeaking, they are two faces of the same coin).To see explicitly that the theory is Lorentz invariant we consider a global Lorentztransformation x µ → x ′ µ = Λ µν x ν (3)as a global and isometric substitution of variables in the action (2). The resulting action S = I T ′ µ d x ′ L ( ∂ ′ µ Φ ′ ( x ′ ) , Φ ′ ( x ′ )) (4)describes the same elementary system of (2) but in a new reference frame. It is importantto note that, even though we are considering an isometry and the form of the bosonicLagrangian is unchanged, the transformation of variables induces a correspondent trans-formation (rotation) of the boundary of the theory T µ → T ′ µ = Λ µν T ν . (5)Indeed we find that T µ — being an ordinary space-time interval — transforms as acontravariant four-vector.This implies that the field solution Φ ′ ( x ′ ) minimizing the transformed action (4) isin general different from the field solution Φ( x ) of the original action (2). They have thesame equations of motion but they have different BCs. This aspect will play a centralrole in the rest of the paper. Besides the naive substitution of the 4D variable x → x ′ ,to minimize the transformed action (4), the transformed field solution Φ ′ ( x ′ ) must havetransformed de Broglie four-periodicity T ′ µ . Thus, according to (1), the transformedfour-momentum associated with the transformed field Φ ′ ( x ′ ) is ¯ p µ → ¯ p ′ µ = Λ νµ ¯ p ν . (6)This actually describes the four-momentum of our elementary particle in the new ref-erence system, as expected from the Lorentz transformation (3). The de Broglie phaseharmony (1) prescribes that T µ is such that the phase of the field is invariant underfour-periodic translations. But the phase of the field is also invariant (scalar) undertransformation of variables. Thus e − i ~ x µ ¯ p µ = e − i ~ ( x µ + cT µ )¯ p µ → e − i ~ ( x ′ µ + cT ′ µ )¯ p ′ µ = e − i ~ x ′ µ ¯ p ′ µ . The intrinsic four-periodicity of the field transforms from reference frame to referenceframe in a relativistic way as in (5). This description is equivalent to the relativisticDoppler effect. By using the useful notation ¯ p µ = h/T µ c [19] the relativistic constrainon the variations of the temporal and spatial components of the de Broglie periodicitycan be expressed as T τ ≡ T µ T µ . (7)This constrain is induced by the underlying Minkowski metric c dτ = dx µ dx µ . In fact,if multiplied by the Planck constant, (7) is nothing but the geometrical description interms of four-periodicity of the relativistic constraint ¯ M c = ¯ p µ ¯ p µ , (8)6n agreement with (1). The mass ¯ M is described by the proper time periodicity T τ of the field: ¯ M = h/T τ c . This corresponds to the time for light to travel across theCompton wavelength of the elementary system λ s = cT τ . The heavier the mass. thefaster the periodicity. A massless elementary system has infinite (or frozen) proper timeperiodicity. A hypothetical bosonic particle with the mass of an electron has proper timeperiodicity of the order of ∼ − s . This is extremely fast even if compared with themodern resolution in time which is of the order of ∼ − s , or with the characteristicperiodicity of the Cesium atom which by definition is ∼ − s .The geometric constraint (7) describes the relativistic dispersion relation of the energyof our elementary system ¯ E ( ¯p ) = q ¯ p c + ¯ M c . (9)In the first part of the paper we will consider only the fundamental field solution ¯Φ( x ) of the action in compact 4D (2). This is the single mode of de Broglie four-angular-frequency ¯ ω µ associated with the PBCs at the boundary T µ . We use the normalizationof a string vibrating in compact 4D (similar to the case of “a particle in a box”), so that ¯ N depends only on the volume of the compact 4D and is invariant under isometries. Wewill denote by the bar sign the quantities related to the fundamental mode of a cyclicfield. That is, ¯Φ( x ) = ¯ N ¯ φ ( x ) = ¯ N e − i ~ ¯ p µ x µ . (10)As we will see more explicitly in sec.(5), this fundamental mode corresponds to the non-quantum (tree-level) limit of the theory. Since its dispersion relation is (9), it describesthe relativistic behavior of a corresponding classical particle with mass ¯ M . In fact, thefundamental solution ¯Φ( x ) , extended to the whole Minkowskian space-time R , formallycorresponds to a single mode of an ordinary non-quantized free Klein-Gordon (KG) field Φ KG ( x ) = ¯Φ( x ) with the same mass M KG = ¯ M and energy E KG (¯ p ) = ¯ E (¯ p ) . In otherwords, they have the same four-momentum ¯ p µ , four periodicity T µ and dispersion relation(9). Thus, in terms of the invariant mass ¯ M , the fundamental mode ¯Φ( x ) is describedby the following action in compact 4D ¯ S = 12 Z T µ d x (cid:2) ∂ µ ¯Φ ∗ ( x ) ∂ µ ¯Φ( x ) − ¯ M ¯Φ ∗ ( x ) ¯Φ( x ) (cid:3) . (11)This action is formally the KG action with boundary T µ . In this case the PBCs arereplaced by suitable SBCs in order to select the single fundamental mode with persistentfour-periodicity T µ — we have eliminated the circle from the integral symbol in order todistinguish its solution from more general field solution of the action with PBCs. As wewill see in sec.(2.1.1), the study of the variations of BCs proposed here for field theoryin compact 4D can be extended to the SBCs of ordinary field theory.When in the last part of the paper we will investigate the quantum behavior of thetheory, we will need to use the most generic field solution with intrinsic four-periodicity T µ . We will address this generic solution as cyclic field. Similarly to a vibrating string or aparticle in a box it is easy to figure out that such a generic cyclic field solution (topology S ) will be a sum of the eigenmodes associated with a quantized energy-momentumspectrum. 7 . Interaction According to the de Broglie-Planck relation (1) the local and retarded variationsfour-momentum occurring during interactions can be equivalently written as local andretarded modulations of de Broglie four-periodicity. In GR, modulations of periodicityof the reference clocks are expressed in terms of deformations of the underlying 4D.Similarly to GR, we will describe the modulation of periodicity of the cyclic fields duringinteraction in terms of deformations of the compact 4D of the theory. For these reasonswe will generically denote this description with the term “geometrodynamics”.
In relativistic mechanics a generic interaction scheme can be formalized in terms ofcorresponding variations of the four-momentum ¯ p ′ µ ( x ) with respect to the non interactingcase ¯ p µ . That is, ¯ p µ → ¯ p ′ µ ( x ) = e aµ ( x )¯ p a . (12)With this notation we mean that the persistent four-momentum ¯ p µ in any given inter-action point x = X deforms into ¯ p ′ µ ( x ) | x = X when we switch on interaction. Hence theinteraction scheme (12) is locally encoded by the tetrad e aµ ( x ) .Similarly to GR where interaction is encoded in a corresponding deformation of theunderlying 4D, we will formalize interactions in terms of local diffeomorphisms of thecompact 4D. This means that we will generalize the case of global transformation ofvariable (3) to the case of local transformations. We will therefore use field theory incurved 4D.To describe this interaction scheme in terms of cyclic fields we must apply the deBroglie-Planck condition (1) locally. It must be noted however that when interactionare concerned, the local four-periodicity of the interacting field which we will denotefrom now on by τ ′ µ ( x ) , in general does not coincide with the boundary of the theory at T µ ( x ) . That is, in general τ ′ µ ( x ) = T µ ( x ) . As we will see, T µ ( x ) transforms as a finiteand contravariant four-vector, i.e. as x ′ µ , whereas τ ′ µ ( x ) transforms as an tangent [19]and contravariant four-vector, i.e. as dx ′ µ . The former describes the recurrence period Φ ′ ( x ) = Φ ′ ( x + T ) whereas the latter describes the local or instantaneous periodicity in agiven point, similarly to the formalism of modulated signals. The local periodicity τ ′ µ ( x ) of an interacting field varies from point to point, according to the relativistic retardedpotentials. In this case the local de Broglie phase harmony is c ¯ p ′ µ ( x ) τ ′ µ ( x ) = h . (13)Therefore the local variation of four-momentum in the interaction scheme (12) corre-sponds to the contravariant local variation of the four-periodicity T µ → τ ′ µ ( x ) = T a e µa ( x ) . (14)Similarly to the variation of four-momentum induced by a given interaction scheme withrespect to the free case, the persistent four-periodicity T µ = τ µ of the free field Φ( x ) turns out to be modulated to the local four-periodicity τ ′ µ ( x ) for the interacting field Φ ′ ( x ) .The deformation of the local four-periodicity τ ′ µ ( x ) of the cyclic fields is associatedwith the corresponding stretching of the compactification four-vector T ′ µ ( x ) of the theory8hrough the PBCs. This actually induces a deformation of the original Minkowskianmetric according to the following relation η µν → g µν ( x ) = e aµ ( x ) e bν ( x ) η ab . (15)This resulting curved space-time encodes the interaction scheme (12) locally. To checkthis geometrodynamical description we consider a local transformation of variables x µ → x ′ µ ( x ) = x a Λ µa ( x ) , (16)whose tetrad matches the one used in (13) to encode our interaction scheme. That is e aµ ( X ) = (cid:18) ∂x a ∂x ′ µ (cid:19) x ′ = X . (17)For the scope of this paper we will work in the approximation e aµ ( x ′ ) ≃ e aµ ( x ) (for thesake of simplicity we neglect Christoffel symbols). Since x a Λ µa ( x ) is the primitive of thetetrad e µa ( x ) we can use the following notation (omitting prime indexes in the integrands) x a Λ µa ( x ) ≃ Z x a dx a e µa ( x ) . (18)The transformation (16) relates locally the inertial frame x ∈ S of the free cyclicfield solution Φ to the generic frame x ′ ∈ S ′ associated with the interacting cyclic fieldsolution Φ ′ . Its Jacobian is p − g ( x ) = det[ e aµ ( x )] . The Latin letters describe the freefield in an inertial frame S while the Greek letters refer to the locally accelerated frame S ′ of the interacting field Φ ′ [26]. Finally, by using (16) as a substitution of variables inthe compact 4D action (2), we find that the interaction scheme (12) is described by thefollowing action in locally deformed compact 4D S ≃ I T a Λ µa | X ( T ) d x p − g ( x ) L ( e µa ( x ) ∂ µ Φ ′ ( x ) , Φ ′ ( x )) . (19)It is important to point out that (16) induces the local deformation (or stretching) of thecompactification four-vector T ′ µ ( X ) ≃ T a Λ µa | X ( T ) ≃ Z X a + T a X a dx a e µa ( x ) . (20)This is the local deformation of the boundary associated with the modulation of localperiodicity τ ′ µ ( x ) (14), i.e. with the interaction scheme (12).As we well see later the resulting formalism will be the four-dimensional analogousof the formalism of modulated signals. Indeed our interacting system is described by thecyclic field solution Φ ′ ( x ) of the transformed action (19) in curved space-time (15) whosefour-frequency or four-momentum spectrum is modulated point by point. According tothe diffeomorphism (16) or to the phase harmony (13), if the free cyclic field Φ( x ) hasfour-momentum p µ , the transformed cyclic field Φ ′ ( x ) has corresponding modulated four-momentum ¯ p ′ µ ( x ) . The four-momentum of the field is in fact described by the derivativeoperator ∂ µ which transforms as the tangent four-vector ¯ p ′ µ ( x ) , i.e. ∂ µ → ∂ ′ µ = e aµ ( x ) ∂ a .9he explicit form of the cyclic field solution Φ ′ ( x ) will be written later for the partic-ular transformations where the normalization ¯ N is left invariant. To study the tree-levelbehavior of the system under the generic interaction scheme (12) in this first part of thepaper will be sufficient to consider only the fundamental modes ¯Φ ′ ( x ) . Similarly to (11),its interaction is described by the transformed action ¯ S ≃ Z T a Λ µa | X ( T ) d x p − g ( x ) ¯ L ( e µa ( x ) ∂ µ ¯Φ ′ ( x ) , ¯Φ ′ ( x )) . (21)This is formally the KG action in curved space-time g µν ( x ) with finite integration region T ′ µ ( X ) and suitable SBCs to select ¯Φ ′ ( x ) as solution. In the practical applications of ordinary field theory the role of the BCs is marginal.The ordinary fields used for computations are the more generic solution of the KG equa-tion, i.e. an integral over all the possible energy eigenmodes. In this paper we will seethat the formalism of fields in compact 4D has interesting properties that are not manifestin ordinary field theory. The variation of the boundary of the action describes differentfield solutions, and in turns different kinematic configurations of the same particles. Theaction (21) actually allows one to investigate the dynamical behavior of a single KG modeunder transformations of reference frame. Its variation of four-momentum ¯ p ′ µ ( x ) , i.e. itskinematics, is therefore encoded in the deformation of the boundary, in a manner of aholographic principle.Although in ordinary field theory BCs are not explicitly used in practical computa-tions, a boundary Σ µ is implicitly assumed (typically of infinite spatial lengths). SuitableSBCs at Σ µ can also be applied to ordinary KG action in order to select a particularsingle mode which locally matches a KG mode: ¯Φ ′ ( x ) | x = X ≡ ¯Φ KG ( X ) . The generaliza-tion to the ordinary field theory with generic boundary Σ µ and suitable SBCs is formallyobtained from (2) through the following formal substitution H T µ → R Σ µ . By analogy tofield theory in compact 4D, through the transformation of variable (16) it is easy to findthat — in the approximation used in this paper — this generic integration region Σ µ ( x ) transforms as (20). That is, Σ µ → Σ ′ µ ( X ) ∼ Σ a Λ µa | X (Σ) . With this we wanted to pointout that the analysis of gauge interactions that we will carry on in the next sections forthe fundamental mode, can be extended to single modes of ordinary non-quantized KGfields. We have introduced interactions for cyclic fields in terms of the local diffeomorphisms(16) from a flat manifold S to a generic manifold S ′ . Thus, under the interaction scheme(12), our system is described in terms of the geometry of the compact 4D.In order to illustrate this in a heuristic way, we will briefly discuss two examples: theweak Newtonian interaction and the QGP logarithmic freeze-out. The complete analysis Typically Σ µ has infinite size along the spatial directions, at whose boundary the field is imposed tovanish, whereas along the time dimension the boundaries are the generic initial and final time where SBCsare supposed. For a KG theory, given the mass as a parameter, the description of the deformation of thetime boundary with SBCs is sufficient to described kinematical variations of the field or interactions.
10f the former case will be left to forthcoming papers whereas the case of the QGP hasbeen investigated in [20]. In this preliminary section we will work in the approximationin which the local four-periodicity can be identified with the compactification four-vectorof the theory τ µ ( x ) ∼ T µ ( x ) . This approximation is valid in the cases where the metricvaries sufficiently smoothly, i.e. e aµ ( x ) ∼ Λ aµ ( x ) (or logarithmically, see [20]).In the next section, by writing (12) as a minimal substitution, we will show that sucha geometric approach to interactions can be also used to describe gauge interactions. The geometric approach to interactions described above is interesting because it ac-tually mimics the usual geometrodynamical approach of linearized gravity. To showthis we consider a weak Newtonian potential V ( x ) = − GM ⊙ / | x | ≪ c . Under thisinteraction scheme the energy in a gravitational well varies with respect to the freecase as ¯ E → ¯ E ′ ∼ (cid:0) GM ⊙ / | x | c (cid:1) ¯ E , [4]. According to the geometrodynamicaldescription of interaction (14), this means that the de Broglie clocks in a gravita-tional well (the periodicity of cyclic fields) are slower with respect to the free clocks T t → τ ′ t ∼ (cid:0) − GM ⊙ / | x | c (cid:1) T t . With the schematization of interactions based on the deBroglie phase harmony (1), we have retrieved two important predictions of GR such astime dilatation and gravitational red-shift ¯ ω → ¯ ω ′ ∼ (cid:0) GM ⊙ / | x | c (cid:1) ¯ ω .Besides this we may also consider the analogous variation of spatial momentum andthe corresponding variation of spatial periodicities of cyclic fields in a gravitational well[4]. Thus, according to (15), we find that the weak Newtonian interaction turns out tobe encoded in the linearized Schwarzschild metric.Indeed, the geometrodynamical description of interaction in the formalism of compact4D actually mimics linearized gravity. In the formalism of compact 4D with PBCs, i.e. under the assumption of intrinsic periodicity, every cyclic field can be regardedas a relativistic reference clock (see again Einstein’s definition [39]), namely the “deBroglie internal clock”. The diffeomorphism (16) encodes the modulation of periodicityof this clocks occurring during interaction. This is similar to GR where gravitationalinteraction can be interpreted in terms of modulations of periodicity of reference clocksencoded in a corresponding deformed 4D background. Furthermore, it is well knownthat GR can be derived from the linearized formulation by considering self-interactions[4] — for instance by relaxing the assumption of smooth interactions. Nevertheless itis important to mention that “what is fixed at the boundary of the action principle ofGR” is not uniquely defined [28]. More considerations about this aspect has been givenin [12, 13]. SR and GR fix the differential framework of the 4D without giving anyparticular prescription about the BCs. The only requirement for the BCs is to minimizea relativistic action at the boundary. For this aspect both SBCs and PBCs have the sameformal validity and consistence with relativity. With this analysis we have provided anevidence of the consistence of our formalism of compact 4D with GR. In [20] we have applied the geometrodynamical description of interaction in compact4D to a simple Bjorken Hydrodynamical Model for QGP logarithmic freeze-out [29]. In afirst approximation the fields constituting the QGP can be supposed massless and theirfour-momentum can be supposed to decay exponentially during the freeze-out. Accordingto (12), this interaction scheme is therefore encoded by the conformal warped tetrad11 aµ ( | x | ) = δ aµ e − k | x | /c , where | x | /c = s/c = τ and k are the proper time and the gradientof the QGP freeze-out, respectively. The time periodicity T t ( s ) = e ks /k = h/E ( s ) isthe conformal parameter which describes naturally the inverse of the energy of the fieldsduring the freeze-out, according to de Broglie. The resulting variations of normalizationof the cyclic field solutions during the freeze-out reproduce formally the logarithmicrunning of the coupling constant typical of QCD [20]. In this paper we will not explorefurther the running associated with gauge interactions. This means that we will limit ourinvestigation to transformations of variables which (in a first approximation) preserve thelengths.Moreover, in [20] we have also shown that a field in deformed compact 4D is dualto a massless XD field in a corresponding 5D metric. For instance they have the sametopology S . The dualism is explicit if we assume that our cyclic world-line parameterplays the role of a cyclic XD. To address this identification we say that the theory hasa virtual XD, see also [1]. In the virtual
XD formalism, the QGP exponential freeze-outturns out to be encoded in a virtual
AdS metric and the classical configurations of cyclicfields in such a deformed background reproduces basic aspects of AdS/QCD.
3. Rotating the boundary
In order to give a mental picture of the description of gauge interaction that we wantto achieve in next sections we can imagine a charged particle (for simplicity a boson)interacting electromagnetically. Typically, the particle will be characterized by dynamics(similarly to the trembling motion of the zitterbewegung ) induced by the interaction. In-tuitively we will describe such dynamics in terms of local transformations of flat referenceframe (avoiding the use of creation and annihilation operators). The resulting modu-lations of de Broglie space-time periodicity will be used to reproduce gauge interaction.The formalism that we will adopt has analogies with that modulated signals. In analogywith an antenna, the EM field can characterized by the dynamics of the charged particlegenerating it.Indeed, in the particular approximation of local isometries, only the boundary ofthe theory is deformed without affecting the underlying flat metric. The coefficientsof these local isometries can be described in terms of vectorial fields which thereforeencode the transformation. The resulting variation of four-momentum and modulationfour-periodicity of the field solution turns out to be written formally as the minimalsubstitution and the parallel transport of ordinary electrodynamics, respectively.
We now want to apply the geometrodynamical description of interaction (12) to thefollowing local infinitesimal transformation of variables x µ ( x ) → x ′ µ ∼ x µ − ex a Ω µa ( x ) . (22)The coefficient of the expansion is denoted by e and address as gauge coupling . In thispaper we will work in the approximation in which this transformation is a local isometries, i.e. we limit our study to the case in which the Jacobian is √− g ′ ≃ . More exactly “it is natural to attribute the origin of the zitterbewegung to the self-interaction of theelectron with its own electromagnetic field.” [21]
12n terms of the formalism (16), a local isometry is described by Λ aµ ( x ) ≃ δ aµ − e Ω aµ ( x ) , (23)whereas the tangent transformation [19] is described by the local tetrad e aµ ( x ) ≃ δ aµ − eω aµ ( x ) . (24)As we will discuss in sec.(4.4), these isometries can be regarded as belonging to some localsubgroups of the Lorentz transformations. Moreover the requirement of local isometriesimplies that ω aµ ( x ) is antisymmetric (Killing equations). For the sake of simplicity weassume that such an isometry is a unitary transformation ω aµ ( x ) ∈ U (1) . According to (18) the finite and tangent transformations are related by x a Ω µa ( x ) = Z x a dx a ω µa ( x ) . (25)To each point x = X of the inertial frame S we are associating a local Lorentzreference frame S ′ , as represented by the orthogonal tetrad e aµ ( X ) . As already noticed,the tetrad e aµ ( x ) encodes the information, point by point, of a corresponding interactionscheme. In this case the information is contained in ω µa ( x ) which in turn can be used todefine a vectorial field ¯ A µ ( x ) as ¯ A µ ( x ) ≡ ω aµ ( x )¯ p a . (26)Thus the vectorial fields ¯ A µ ( x ) can be used to encode the interaction scheme (22). Inparticular, the variation of four-momentum (12) associated with this local isometry is inthis case given by ¯ p ′ µ ( x ) ∼ ¯ p µ − e ¯ A µ ( x ) , (27)which formally is the minimal substitution of the vectorial field ¯ A µ ( x ) .Since we are assuming that the local isometry (22) is a unitary transformation ω ( x ) ∈ U (1) , we say that the vectorial field A µ ( x ) is an abelian field. In sec.(4.4) we will discussthe geometrical meaning of this assumption as well as the generalization to non-abelian isometries.Under such a local isometric change of flat manifold g µν ( x ) ≃ η µν , the resultingtransformed action (21) has the same structure of the original one, i.e. the equations ofmotion remain unchanged. Nevertheless the boundary of the transformed action turnsout to be locally rotated with respect to the original action (11). In fact, the resultingaction is ¯ S = 12 Z T ′ µ ( X ) d x (cid:2) ∂ ′ µ ¯Φ ′† ( x ) ∂ ′ µ ¯Φ ′ ( x ) − ¯ M ¯Φ ′† ( x ) ¯Φ ′ ( x ) (cid:3) . (28)This action has indeed transformed boundary at T ′ µ ( X ) with respect to the persistentboundary at T µ of the original action, according to (20). Therefore its fundamentalsolution ¯Φ ′ ( x ) in the point x = X turns out to be different from the original fundamentalsolution ¯Φ( x ) . The free solution ¯Φ( x ) has the persistent periodicity τ µ = T µ of an isolatedparticle. The transformed solution ¯Φ ′ ( x ) has modulated periodicity τ ′ µ ( X ) varying frompoint to point in order to describe interaction.13 .1.1. Global Isometry To formulate interactions in terms of gauge fields we need to express the rotations ofthe boundary in terms of internal transformations of the field.Here, as in par.(1), we consider the simple case of a global isometry Λ aµ ( x ) = e aµ ( x ) ≡ e aµ . (29)For reasons that will be clear later we address this case as pure gauge — this terminologyis not completely equivalent to ordinary QFT.In this case the tetrad is homogeneous, it does not depend on x . The resultingfour-periodicity τ ′ µ = e µa T a and four-momentum ¯ p ′ µ = e aµ ¯ p a of the transformed field ¯Φ ′ vary globally. The transformed four-periodicity (5) coincides with the compactificationfour-vector of the transformed action τ ′ µ = T ′ µ .In every interaction point x = X the fundamental solution ¯Φ ′ ( x ′ ) associated withthe transformed action (28) is related to the original solution ¯Φ( x ) by the followingtransformation ¯Φ( x ) = ¯ N e − i ~ x µ ¯ p µ → ¯Φ ′ ( x ′ ) = ¯ N e − i ~ x ′ µ ¯ p ′ µ . (30)It is easy to see that, as long as ¯Φ( x ) is a fundamental solution of the free action,under this transformation ¯Φ ′ ( x ′ ) is automatically the correct fundamental solution ofthe transformed action (28). In fact, it has transformed four-periodicity T ′ µ = e µa T a ,according to the de Broglie phase harmony.Now we expand the global transformation e aµ as in (22), so that the interaction schemeis formally the minimal substitution (27); ¯ A µ and ¯ p ′ µ are homogeneous (constant). Thephysical effect of this pure gauge is a global transformation of reference frame. Thus,under this global isometry the field transforms as ¯Φ( x ) = ¯ N e − i ~ x µ ¯ p µ → ¯Φ ′ ( x ′ ) = ¯ N e − i ~ x ′ µ (¯ p µ − e ¯ A µ ) = ¯ V ( x ′ ) ¯Φ( x ′ ) . (31)This also means that to our transformation of variables there is associated an internaltransformation of the field described by ¯ V ( x ) = e ie ~ x µ ¯ A µ . (32)We call ¯ A µ gauge connection and ¯ V ( x ) parallel-transport . Since we are assuming abeliantransformation of variables, the resulting internal transformation of the fundamentalsolution (31) generates an abelian parallel-transport ¯ V ( x ) ∈ U (1) .We also introduce the covariant derivative of the transformed field ¯Φ ′ as ∂ µ ¯Φ( x ) = ∂ µ [ ¯ V − ( x ) ¯Φ ′ ( x )] = ¯ V − ( x ) D µ ¯Φ ′ ( x ) . (33)Thus the covariant derivative associated with a global isometry is ∂ µ → D µ = ∂ µ − ie ~ ¯ A µ . (34)It is important to note that, even though the transformed fundamental solution ¯Φ ′ ( x ) has a transformed four-periodicity T ′ µ , the terms ¯ V − ( x ) ¯Φ ′ ( x ) and ¯ V − ( x ) D µ ¯Φ ′ ( x ) havethe same persistent four-periodicity T µ as the original fundamental mode ¯Φ( x ) and its14erivative ∂ µ ¯Φ( x ) , respectively. We will address this important aspect by saying that theinverse of the parallel transport , together with the covariant derivative in derivative terms, tunes the periodicity T ′ µ of the transformed fundamental mode ¯Φ ′ ( x ) to the periodicity T µ of the free fundamental mode ¯Φ( x ) . We will next generalize these considerations tolocal isometries. In the general case of local isometries, the tetrad e aµ ( x ) varies from point to point.Thus we must take into account that, at every point x = X , the transformed fundamentalmode ¯Φ ′ ( x ) must have local four-periodicity τ ′ µ ( x ) | x = X in order to be a solution of thetransformed action (28) with boundary at T ′ µ ( x ) | x = X given by (20). This also meansthat the transformed fundamental solution ¯Φ ′ ( x ) must have four-momentum ¯ p ′ µ ( x ) x = X in order to satisfy (13) in x = X . In the approximation e aµ ( x ) ∼ e aµ ( x ′ ) and consideringthat the normalization factor ¯ N is invariant under isometries, the fundamental mode ¯Φ ′ ( x ′ ) solution of the transformed action (28) can be written, according to our notation(18), as ¯Φ( x ) = ¯ N e − i ~ x · ¯ p → ¯Φ ′ ( x ′ ) = ¯ N e − i ~ R x ′ µ dx µ ¯ p ′ µ ( x ) . (35)To check this, besides the de Broglie phase harmony, the analogy with the CKM formal-ism and the modulated signals formalism, we may note that the derivative operator i ~ ∂ µ actually gives the correct transformed four-momentum ¯ p ′ µ ( x ) in the new reference frame i ~ ∂ µ ¯Φ( x ) = ¯ p µ ¯Φ( x ) → i ~ ∂ µ ¯Φ ′ ( x ′ ) = ¯ p ′ µ ( x ′ ) ¯Φ ′ ( x ′ ) . (36)According to the definition of the local tetrad e aµ ( x ) in (24), the fundamental solution ¯Φ ′ ( x ′ ) of (28) is obtained from the free fundamental solution ¯Φ( x ) by the following internal transformation of the field ¯Φ( x ) = ¯ N e − i ~ x · ¯ p → ¯Φ ′ ( x ′ ) = ¯ N e ie ~ R x ′ dx · ¯ A ( x ) e − i ~ x ′ · ¯ p = ¯ V ( x ′ ) ¯Φ( x ′ ) . (37)Hence the parallel-transport ¯ V ( x ) describing the internal transformation of funda-mental solution under this local abelian transformation of variables is formally a Wilsonline of the gauge connection ¯ V ( x ) = e ie ~ R x dx · ¯ A ( x ) . (38)This allows one to pass from a fundamental solution with persistent periodicity τ µ to afundamental solution with transformed local periodicity τ ′ µ ( x ) . In this way, as long as ¯Φ( x ) is solution of the free action, the modulated field ¯Φ ′ ( x ′ ) is automatically solution ofthe transformed action (28). The vectorial field ¯ A µ ( x ) describes the resulting modulationof periodicity under the local transformation of reference frame.The generalization to local isometry of the covariant derivative (33) of the trans-formed field ¯Φ ′ ( x ) is ∂ µ → D µ = ∂ µ − ie ~ ¯ A µ ( x ) . (39)Also in this more general case we find that the inverse of the parallel-transport , togetherwith the covariant derivative in derivative terms, tunes the locally varying periodicity τ ′ µ ( x ) of the transformed fundamental field ¯Φ ′ to the persistent periodicity τ µ = T µ ofthe free fundamental field ¯Φ . This tuning through parallel transport will be used to allowa field with locally varying four-periodicity τ ′ µ ( x ) to fulfill the variational principle in anaction with persistent boundary τ µ = T µ . 15 ( X ) T µ τ µ = T µ ¯ p µ τ ′ µ ( X )¯ p ′ µ ( X ) T ′ µ ( X )¯ J µ ( X ) ¯Φ ′ ( X )¯Φ( X ) S S ′ Figure 1: Diagrammatic description of the local isometry Λ( x ) transforming a free fundamental field ¯Φ( X ) of persistent four-momentum ¯ p µ , four-periodicity τ µ and compactification four-length T µ = τ µ ,from the inertial frame S to the reference frame S ′ . The resulting fundamental field solution ¯Φ ′ ( X ) haslocally transformed four-momentum ¯ p ′ µ ( X ) , four-periodicity τ ′ µ ( X ) and compactification four-length T ′ µ ( X ) = T µ . The conservation of stress-energy-momentum tensor involves the Noether current ¯ J µ ( X ) . The formalism of field theory in compact 4D allows one to relate, through the varia-tional principle, transformation of variables δx and internal transformations of the field δ Φ( x ) . This can be easily seen from the role of the parallel-transport (38). The internaltransformation associated to the local abelian isometry (22) is indeed δ ¯Φ( x ) = ¯Φ ′ ( x ) − ¯Φ( x ) = ie ¯ A ν ( x ) ¯Φ( x ) δx ν . (40)To understand the role of these internal transformations of the field we considerthe role of the boundary terms in the variation of the action. In the approximation oflocal isometries, the original and transformed action differ by an explicit variation of theboundary δ ¯ S = Z T ′ µ d x ′ ¯ L ′ ( ¯Φ ′ i , x ′ ) − Z T µ d x ¯ L ( ¯Φ i , x ) . (41)As represented diagrammatically in fig.(1), the stress-energy-momentum of the funda-mental mode ¯Φ is not manifestly conserved because of the local transformation of refer-ence frame. In fact, it is easy to see from (41) that the conservation of the stress-energy-momentum tensor involves the contribution of the internal transformation of the field(40); that is, of the current ¯ J µ = ie [ ¯Φ ∗ D µ ¯Φ − D µ ¯Φ ∗ ¯Φ] . (42)The interesting aspect of this analysis is that actually the current ¯ J µ has the same formas the Noether current of an abelian gauge invariant theory with internal transformation (40), as long as we identify the connection ¯ A µ ( x ) with an abelian gauge field.16 . Gauge interaction The formalism introduced so far is very useful to describe the geometrodynamicsassociated with a given interaction scheme of an elementary particle. However it doesnot explicitly take into account the conservation of four-momentum. This is because itinvolves a transformation of reference frame. As we have pointed out in the Noetheranalysis, this is related to the local variations of the boundary. From a analytic pointof view it would be easier to describe the same interaction scheme in terms of an actionwhose boundary is invariant under isometric transformation of variables. In this waythe currents related to the transformation of the boundary will be directly expressed interms of symmetries of the Lagrangian. The possibility of such a formalism is offered bythe fact that the periodicities of the fields can be tuned through parallel transport . Theresult will be an ordinary gauge invariant theory.
We want to define a new formalism to describe ¯Φ ′ ( x ) under the interaction scheme(27) such that there is an explicit conservation of four-momentum. Our strategy willbe to write a new action, which we will call tuned action, containing the same physicalinformation of the transformed action (28) but with persistent boundary at T µ . Similarlyto the transformed action, this tuned will be obtained directly from the free action (11).To build the tuned action we need to know that to a change in the periodicity ofthe fundamental field solution there is an associated internal transformation (37), i.e a parallel-transport . If we want to vary the four-periodicity of a field in an action withgiven boundary, and at the same time fulfills the variational principle at the boundary,we must use the parallel-transport to tune the four-periodicity of the field. Since the onlyterms involved in the BCs are the derivative terms (through integration by parts), thetuned action can be obtained from the free action by modifying only derivative terms.We have already noticed, for instance, that the corresponding covariant derivative (39)allows one to tune the periodicity of ¯Φ ′ ( x ) to the periodicity of ¯Φ( x ) .In the tuned action the interaction will be described in terms of symmetries of theLagrangian rather than in terms of the variations of the boundary. Hence the interactingfield ¯Φ ′ ( x ) will be described by different equations of motion with respect to the free caseand the currents (42) will be the conserved currents associated with the symmetries ofthe tuned Lagrangian.At a mathematical level this can be easily achieved through the parallel-transport ¯ V ( x ) , by explicitly writing in the free action (11) the fundamental solution ¯Φ( x ) as afunction of the transformed fundamental solution ¯Φ ′ ( x ) , i.e. by using (37). In this waywe find Z T µ d x ¯ L ( ∂ µ ¯Φ , ¯Φ) = Z T µ d x ¯ L ( ∂ µ ¯ V − ¯Φ ′ , ¯ V − ¯Φ ′ ) = Z T µ d x ¯ L ( D µ ¯Φ ′ , ¯Φ ′ ) . (43)According to the definition (39), the derivative terms ∂ µ ¯Φ( x ) of the original action mustbe replaced by the covariant derivative D µ ¯Φ ′ ( x ) in order to tune locally the periodicity τ ′ µ ( x ) to T µ . We have also used the fact that non-derivative terms, such as the massterm, contributes only to the equations of motion, but not to the BCs (the periodicity ofthe solution in such terms can be arbitrarily varied without compromising the variationalprinciple at the boundary). 17 ( X ) T µ τ µ = T µ τ ′ µ ( X ) T µ τ µγ ( X )¯Φ( X ) ¯Φ ′ ( X )¯ A µ ( X )¯ p γµ ( X )¯ p µ ¯ p µ ( X ) Figure 2: Diagrammatic description of the local isometry Λ( x ) in terms of the tuning mediated by thevectorial field A µ ( X ) with local four-momentum ¯ p γµ ( X ) and four-periodicity τ µγ ( X ) . The conservationof four-momentum is manifest from the fact that the free fundamental field ¯Φ( X ) and the interactingsystem ¯Φ ′ ( X ) and A µ ( X ) have the same compactification length T µ . Therefore the tuned Lagrangian can be directly obtained from the free Lagrangianby replacing the ordinary derivatives with covariant derivatives ¯ L tuned ( ∂ µ ¯Φ ′ , ¯Φ ′ , A µ ) = ¯ L ( D µ ¯Φ ′ , ¯Φ ′ ) . (44)In the specific interaction scheme (27) for the fundamental scalar mode of a cyclicfield with mass ¯ M , the tuned action is ¯ S = 12 Z T µ d x (cid:2) D µ ¯Φ ′∗ ( x ) D µ ¯Φ ′ ( x ) − ¯ M ¯Φ ′† ( x ) ¯Φ ′ ( x ) (cid:3) . (45)The tuned action is related to the original free action by parallel-transport . If ¯Φ( x ) isthe fundamental solution of the original free action, then the transformed fundamentalsolution ¯Φ ′ ( x ) automatically minimizes (45). On the other hand ¯Φ ′ ( x ′ ) is also the solutionof the transformed action (28). Hence the tuned action (45) contains the same physicalinformation of the transformed action (28).The vectorial field ¯ A µ ( x ) encodes the modulation of periodicity of an interactingsystem giving rise to covariant derivatives in the tuned action. If we identify the gaugeconnection ¯ A µ ( x ) with an ordinary gauge field, the tune action (45) is formally a gaugedKG action with boundary.In this way we have given a geometrical meaning to covariant derivatives, parallel-transports, and gauge connections, in terms of variations of periodicities. This picturecan be intuitively interpreted by imagining the transporting of the arms of a clock withgiven periodicity on a closed path in a curved manifold. At the end of the loop wemust either retune the arms of the clock through parallel-transport or to assume that theclock has varied its characteristic periodicity. Indeed the parallel-transport allows oneto describe the modulation of periodicity “associated with deformation of the underlyingmanifold in a background independent way”, [4].18 .2. Gauge Invariance At a mathematical level the parallel-transport ¯ V ( x ) allows one the possibility to tune periodicity of terms of the Lagrangian to the periodicity imposed by the boundary of theaction. However it is easy to note that such a parallel-transport, as well as the gaugeconnection, is not uniquely defined: the tuned action (45) has a manifest invariancewhich we call — for obvious reasons — gauge invariance .It is well known that (45) is invariant under the following transformation ¯ A µ ( x ) → ¯ A ′ µ ( x ) = ¯ A µ ( x ) − e∂ µ ¯ θ ( x ) . (46)In the parallel-transport ¯ V ( x ) , which is formally a Wilson line (38), this transformationgenerates boundary terms which can be absorbed by the following local phase transfor-mation of the fundamental scalar mode ¯Φ ′ ( x ′ ) → ¯Φ ′′ ( x ) = ¯ N e − i ~ e ¯ θ ( x ) ¯Φ ′ ( x ) . (47)The resulting local phase transformation of the field is ¯ U ( x ) = e − i ~ e ¯ θ ( x ) . (48)This local phase turns out to be of the same kind as the parallel-transport ¯ V ( x ) or ofthe local isometry generating it. Therefore ¯ U ( x ) ∈ U (1) in the case of abelian isometry.We have finally shown that a fundamental mode subject to a local abelian isometry isdescribed by an abelian gauge invariant action.The local gauge invariance U (1) is now a symmetry of the tuned Lagrangian. There-fore gauge transformations do not affect the boundary of the action. In other words wehave found that gauge invariance identifies particular class of isometries describing thesame interaction scheme. For this reason we call them gauge orbits (we only mention thatsuch a gauge orbit can be regarded as an holonomy , since it corresponds to an isometrywhose parameter ω ( x ) is a total derivative and “the boundary of a boundary is zero”).The meaning of the tuning of the field described so far can be generalized. Gaugeinvariant terms allow one to tune the periodicity of the fundamental field solution to theperiodicity imposed, through the variational principle, by the boundary of the action.This is essentially because the parallel-transport ¯ V ( x ) tunes the periodicity of the bosonicfield solutions.At this point we can repeat the variational analysis of par.(3.3). Noether’s theoremcan be applied directly to the tuned Lagrangian instead of the transformed action. Sincethe tuning formalism allows a description in which the boundary of the action doesnot vary under isometric transformation of variables, it is easy to show that the Noethercurrents (42) are directly associated with the symmetries of the tuned Lagrangian. Indeedthe Noether analysis is completely parallel to the one of ordinary gauge theory.The remaining step to prove that the interaction scheme associated with such a localabelian isometry is nothing but the usual gauge interaction, is to find that the dynamicsof the abelian gauge field A µ ( x ) is actually described by the Maxwell equations. The tuned action (45) is formally a U (1) gauge invariant action. The tuning of theinteracting field has been obtained at the expense of the introduction of a new field in19he theory. This is the gauge connection ¯ A µ ( x ) . It compensates the variation of four-periodicity of the interacting field in order to have a tuning to the persistent boundaryof the action. Therefore it is natural to interpret ¯ A µ ( x ) as a new dynamical field withgiven four-momentum, in general different from ¯ p µ or ¯ p ′ µ ( x ) , and thus with given localfour-periodicity, in general different from T µ or τ ′ µ ( x ) . This is illustrated in fig.(2).This requires one to introduce a kinetic term for ¯ A µ ( x ) . Similarly to the interactingfundamental field solution ¯Φ ′ ( x ) of the tuned action (45), this kinetic term may appearin an action with persistent integration region T µ , in order to describe explicitly theconservation of four-momentum. That is, we want to add a kinetic term for ¯ A µ ( x ) tothe tuned action (45) — a similar analysis can be done for the transformed action.From the correspondence between gauge invariance and the periodicity tuning, werequire that such a kinetic term must be gauge invariant. In fact, only in this waythe periodicity of ¯ A µ ( x ) can be tuned with the BCs imposed by the tuned action. Inparticular we have already noticed that such a tuning of the periodicity of the field inderivative terms, such as in the kinetic terms, is possible by using covariant derivatives.Through suitable covariant derivatives the four-periodicity of ¯ A µ ( x ) in the kinetic termcan be tuned to T µ .Therefore, by following the same requirement of gauge invariance as in ordinary fieldtheory, we infer that the correct kinetic terms allowed for ¯ A µ ( x ) by the variational prin-ciple is the gauge invariant term − ¯ F µν ¯ F µν , where F µν is the field strength. In fact,it is well known that the derivatives in the field strength can be replaced by arbitrarycovariant derivatives of the gauge field itself ¯ F µν ( x ) = D ′ µ ¯ A ν ( x ) − D ′ ν ¯ A µ ( x ) . This meansthat in such a kinetic term the periodicity of the gauge field is tuned to the characteristicperiodicity of the action through an appropriate covariant derivative (in general differentfrom the one of the matter field ).Finally, consistently with the variational principle, the full action describing the in-teraction scheme (22) is ¯ S Y M = Z T µ d x (cid:26) −
14 ¯ F µν ( x ) ¯ F µν ( x ) + 12 (cid:2) D µ ¯Φ ′∗ ( x ) D µ ¯Φ ′ ( x ) − ¯ M ¯Φ ′∗ ( x ) ¯Φ ′ ( x ) (cid:3)(cid:27) . (49)We have obtained nothing but the usual non-quantum (tree-level) abelian Yang-Millsaction (in a finite volume) describing a matter field solution with four-momentum ¯ p ′ µ ( x ) interacting with a gauge field ¯ A µ ( x ) and total four-momentum ¯ p µ .It must be noticed that the simultaneous minimization of the above action at theboundary for both the fields ¯Φ ′ ( x ) and ¯ A µ ( x ) , or equivalently the requirement of gaugeinvariance of the action, constrains the general form of the gauge field. In turn the so fargeneric form of the abelian transformation of variables (24) is constrained to a particularsubclass, modulo gauge orbits. In fact the fundamental field solution ¯ A µ is no more ageneric vectorial field. That is, according to the variational principle, it must be solutionof the equations of motion associated with (49), i.e. it has Maxwell dynamics. For in-stance this means that ¯ A µ ( x ) is massless and has only two transversal d.o.f. . On the otherhand, through (26), the Maxwell dynamics of gauge field ¯ A µ ( x ) corresponds to related For instance, in a given gauge ¯ A µ ( x ) , the correct covariant derivative to tune the gauge field isgiven by combining the inverse parallel-transport ¯ V − ( x ) and the gauge transformation ¯ U ( x ) where e ¯ θ ( x ) = ¯ p µ x µ . ω aµ ( x ) of the transformation of variables. Only isometries(24) satisfying this dynamics and which we will address as transversally polarized areallowed by the variational principle. The result is a geometrodynamical description ofordinary gauge interactions.The kinetic term of the gauge field has been inferred by noticing that the variationalprinciple requires gauge invariant terms. The requirement of gauge invariance is actuallythe usual way to introduce such kinetic terms in ordinary YM theory. The same argumentcan be used to introduce the kinematic terms for the gauge field in the formalism of thetransformed action (28). In this case we must assume covariant derivatives only in thefield strength such that the periodicity of the gauge field is tuned to the locally varyingperiodicity τ ′ µ ( x ) of the interacting field.“The modern viewpoint”, [30, 3], to introduce gauge interaction in ordinary field the-ory is to postulate a parallel-transport (38) — sometimes called connection — to a matterfield, i.e. to postulate internal symmetries. In this way the derivative terms generate co-variant derivatives and the Lagrangian is gauge invariant. Thus, in ordinary field theory,“ the covariant derivative and the transformation law of the connection ¯ A µ follow fromthe postulate of local phase rotation symmetry ” [30]. From the viewpoint of field theory incompact 4D the same gauge invariant Lagrangian is obtained from the geometrodynamicsallowed by the variational principle, without postulating it . In fact we have seen that the parallel-transport of a matter field arises naturally to describe the modulation of peri-odicity associated with local transformation of variables and to tune the periodicity ofthe field solution to the one imposed by the minimization of the action at the boundary.The invariance of the action under transformation of variables induces an internal trans-formation of the field solutions. This reveals an intuitive geometrodynamical nature ofgauge interactions. This important and non-trivial result is in the spirit of Weyl’s originalproposal of a geometric interpretation of gauge interactions.The formalism of compact 4D makes manifest this geometrodynamical interpretationof gauge interaction, because it explicitly relates internal transformations of the fieldsolution to variations of the BCs. The same arguments can be in principle repeated inordinary field theory by replacing the compact integration region T µ with Σ µ and thePBCs with SBCs. We conclude this section by giving a generalization of our interaction scheme to non-abelian local transformations of variables and discussing the relation to space-timesymmetries.To generalize our result to the non-abelian case we must assume that the trans-formation of variables (23) originating our interaction scheme (27) is an element of anon-abelian group H . This implies that in the equation obtained so far we must performthe following substitution from an abelian vectorial field to a non-abelian one ¯ A µ ( x ) → ¯ A iµ ( x ) τ i , where τ i are generators of H . As a consequence of the commutation relations of thegenerators the parallel-transport must be written as an path-ordered Wilson line ¯ V ( x ′ ) = e ie ~ R x ′ dx · ¯ A ( x ) → ¯ V ( x ′ ) = P[ e ie ~ R x ′ dx · ¯ A a ( x ) τ a ] . (50)21 similar redefinition must be considered for the covariant derivatives. In this way (49)turns out to describe a Yang-Mills theory with non-abelian gauge invariance H .We have described gauge interaction in the approximation where the transformationof variables (22) is a local isometry. In this approximation the lengths of the four vectorare preserved and we do not consider variations of the normalizations of the fields .Moreover the variational principle allows only a particular subclass of polarized localisometries, modulo gauge orbits . In this way the dynamics for the related vectorial fieldturns out to be the usual Maxwell’s dynamics.We can now associate the polarized isometries (23) describing our interaction schemeto corresponding polarized local Lorentz transformations. By representing the Lorentzgroup as SU L (2) ⊗ SU R (2) , the most general global isometry which we may consider is ω ∈ SU L (2) ⊗ SU R (2) . In this case eω aµ = gw aiµ τ i + g ′ y aiµ τ i where w aiµ and y aiµ , g and g ′ arethe coefficients and the coupling of SU L (2) and SU R (2) , respectively. The index i = 1 , , is associated with the generators τ i of SU (2) . The global components of this isometrydescribe pure-gauge transformations (global Lorentz transformations) whereas the localpolarized components correspond to the gauging of the corresponding subgroup H ⊂ SU L (2) ⊗ SU R (2) . For instance the abelian gauge theory describing ordinary classicalelectrodynamics can be obtained by assuming that only the corresponding polarized andabelian Lorentz transformations with generator in U em (1) ⊂ SU L (2) ⊗ SU R (2) are local.Similarly we can imagine to describe electroweak interactions by assuming that thelocal polarized isometries are only those associated with the Lorentz subgroups SU L (2) ⊗ U Y (1) ⊂ SU L (2) ⊗ SU R (2) (the gauging the electroweak group from a larger global group SU L (2) ⊗ SU R (2) is typical in technicolor models and useful to describe the custodialsymmetry of the Standard Model of electroweak interactions, see for instance [31]). Asfor the abelian case (26) we can define gauge fields associated with SU L (2) and U Y (1) as W µ ( x ) = W iµ ( x ) τ i = w aiµ ( x ) τ i ¯ p a and Y µ ( x ) = Y µ ( x ) τ = y a µ ( x ) τ ¯ p a . Indeed wehave the remarkable possibility to relate the electroweak gauge group to a local Lorentzsubgroup.A possible generalization of this geometrodynamical description of gauge interactionto fermionic fields has a natural realization on the Zitterbewegung models. Originallyproposed by Schrödinger this idea provides a semi-classical interpretation of the spinand of the magnetic momentum in terms of cyclic dynamics whose periodicity is actuallythe de Broglie periodicity of the fermions, see for instance [22]. Such a trembling motionof the fermions can be derived from the Dirac equation.Finally, it is interesting to mention that the geometrodynamical description of gaugeinteraction described so far has a deep motivation in the so called Kaluza’s miracle [5].This can be seen by considering the dualism of the theory to an XD theory [1, 20]. Underthis dualism, the metric (15) associated to the substitution of variable (22) turns out tobe a Kaluza-like XD metric. This point out interesting algebraic properties of both thecurvature tensor and the electromagnetic field tensor as already noticed by Rainich [8]and then improved by Misner and Wheeler [7].Interesting aspects of the geometrical interpretation of gauge interaction given here, Weyl’s proposal was based on conformal transformations. In future papers we will extend our analysisto transformations between two flat 4D with different norm, i.e. isomorphisms. We expect to find arunning of the coupling constant, similarly to what we have already observed for the QGP freeze-out[20]. parallel transport , holonomy and the relation with space-time symmetries, havea similar description in the “Higher Gauge Theory” of Baez, see for instance [42]. Inan appealing formalism it is in fact shown how gauge interaction can be described as“change in phase of a quantum particle”.
5. Quantum Behavior
So far we have limited our study to the fundamental mode ¯Φ( x ) of the cyclic field Φ( x ) . This corresponds to study the theory at tree-level. In fact the fundamental modecan be matched with a corresponding single mode of a non-quantized KG field with cor-responding four-frequency. Because of this matching with a KG mode, the fundamentalmode can be in principle quantized in the usual way by imposing explicitly commutationrelations and obtaining ordinary scalar QED. However, as shown in [1] and summarizedin this section, the classical evolution of a cyclic field (with all its energy excitations) hasremarkable correspondences with the quantum evolution of ordinary second quantizedfields. This correspondence has been checked explicitly in both the canonical formulationof QM (the theory has implicit commutation relations) and the Feynman Path Integral(FPI) formulation, as well as for many other peculiar quantum phenomena and problems(including Schrodinger problems [1, 9, 32]). The correspondence with the FPI formulatedfor free systems in [1], will be generalized to the interacting system studied so far. Infact, as for the free case, an interacting cyclic field also has Markovian evolution andan Hilbert space can be defined locally. The assumption of PBCs can be regarded as asemi-classical quantization condition for relativistic fields. To investigate the quantum behavior of the theory we need to use the most genericcyclic field solutions of the action in compact 4D and PBCs. By considering the discreteFourier transform associated with the cyclic 4D of the theory, it is easy to figure outthat such a periodic field has a quantized energy-momentum spectrum. For the sake ofsimplicity we consider the simple topology S so that the spectrum is described by asingle quantum number n .From the relation ¯ E (¯ p ) = ~ ¯ ω ( ¯p ) , the intrinsic time periodicity T t (¯ p ) of cyclic field in agiven reference frame denoted by ¯p implies a quantized energy spectrum E n (¯ p ) = ~ ω n ( ¯p ) .In the free case PBCs yield ω n ( ¯p ) = n ¯ ω ( ¯p ) , which is nothing but the harmonic frequencyspectrum of a vibrating string with the characteristic time periodicity of the field . Thusthis formulation can be regarded as the full relativistic analogous to the semi-classicalquantization of a “particle” in a box. It also shares deep analogies with the Matsubaraand the Kaluza-Klein (KK) theories [33, 20]. Indeed, from this harmonic spectrum andfrom (9) we see that free cyclic fields reproduce the same quantized energy spectrum ofordinary second quantized fields (after normal ordering) ¯ E n ( ¯p ) = n q ¯ p c + ¯ M c . (51) The theory can be regarded as a particular kind of string theory. The compact world-line parameterplays the role the compact world-sheet parameter of ordinary string theory. Therefore it would be moreappropriate to speak about strings rather than fields.
23 free cyclic field Φ( x ) — here x µ = { ct, x } — with intrinsic periodicity T t (¯ p ) , solution ofthe action in compact 4D (4), is a tower of energy eigenmodes φ n ( x ) with eigenvalues (51).Furthermore, the time periodicity induces a periodicity λ x to the modulo of the spatialdimensions according to (7). Hence, similarly to the quantization of the energy spectrum,there is a quantization of the modulo of the spatial momentum | p n | = n | p | = nh/λ x .The fact that T µ is a four-vector means that there is a single periodicity inducedbetween the time dimension and the modulo of the spatial dimension. In the simplecase of topology S , this implies that the resulting quantization is to be expressed interms of the single quantum number n . It is important to note, however, that togetherwith this single fundamental periodicity S , in spherical problems (such as the Hydrogenatom, the 3D harmonic oscillator, or problems with particular bounded geometries),two other cyclic variables (or their deformations) must be considered: the sphericalangles θ ∈ [0 , π ) and ψ ∈ [0 , π ) . As well-known from ordinary QM they lead to theordinary quantization of angular momentum. In this case the cyclic field would bewritten as a sum over two additional quantum numbers, typically denoted by ( m, l ) , andthe topology of the fields would be S ⊗ S (PBCs for the transverse mode of a field,as in the front-light-quantization, can be used to calculate semi-classically importantpredictions of perturbative QED such as the anomalous-magnetic-momentum, see forinstance [27]). For simplicity we will not consider the expansion in spherical harmonics(or their deformations). We will investigate only the quantization of the four-momentumspectrum associated with S , which in the free case is harmonic p nµ = n ¯ p µ . (52)Depending whether we want to make explicit the normalization factor of the energyeigenmodes, we write the cyclic field solution by using the following notations, Φ( x ) = X n Φ n ( x ) = X n N n α n (¯ p ) φ n ( x ) = X n N n α n (¯ p ) e − i ~ p n · x . (53)As already mentioned we assume that the normalization factor N n is invariant underisometries. The coefficients of the Fourier expansion are represented by α n (¯ p ) . Thenon-quantum limit corresponds to the case where the cyclic field solutions can be ap-proximated with the fundamental mode Φ( x ) ∼ ¯Φ( x ) = ¯ N ¯ φ ( x ) , i.e. when only thefundamental level is largely populated: ¯ α (¯ p ) ∼ and α n =1 (¯ p ) ∼ , see [1, 9]. Next we summarize the correspondence of field theory in compact 4D with QFTdescribed in [1, 9]. We note that the evolution of the free cyclic field (53) along thecompact time dimension is described by the so-called “bulk” equations of motion ( ∂ t + ω n ) φ n (x , t ) = 0 . For the sake of simplicity in this section we assume a single spatialdimension x . Thus the time evolution of the energy eigenmodes φ n (x , t ) can be writtenas first order differential equations i ~ ∂ t φ n (x , t ) = E n φ n (x , t ) . (54)The cyclic field (53) is a sum of eigenmodes of an harmonic system. Actually thisharmonic system is the typical classical system which can be descried in a Hilbert space.24n fact, the energy eigenmodes of a cyclic field form a complete set with respect to theinner product h φ n ′ ( t ′ ) | φ n ( t ) i ≡ Z λ x d x λ x φ ∗ n ′ (x , t ′ ) φ n (x , t ) . (55)They energy eigenmodes can also be used to define Hilbert eigenstates h x , t | φ n i ≡ φ n (x , t ) √ λ x . (56)As obvious in the non-interacting case where the cyclic fields have persistent period-icity, the integration region λ x can be extended to an arbitrary large integer number ofperiods V x = N ′ λ x . That is, by assuming N ′ → ∞ , it can be extended to an infiniteregion.In this Hilbert space we can formally build an Hamiltonian operator defined as H | φ n i ≡ ~ ω n | φ n i . (57)Similar considerations hold for the spatial dimension and the momentum operator isdefined as P | φ n i ≡ − ~ k n | φ n i , (58)where k n = n ¯ k = nh/λ x . An Hilbert state is defined as generic superposition of energyeigenmodes | φ i ≡ X n a n | φ n i . (59)This means that cyclic fields can be repressented in an Hilbert space. With this Hilbertnotation the time evolution of a generic Hilbert state, as well as of a generic cyclic field,turns out to be described by the familiar Schrödinger equation i ~ ∂ t | φ (x , t ) i = H| φ (x , t ) i . (60)As can be easily seen in the free case, i.e. homogeneous Hamiltonian (we will gener-alize later to interactions), the finite time evolution is given by the operator U ( t ; t i ) = e − i ~ H ( t − t i ) (61)which turns out to be a Markovian (unitary) operator: U ( t f ; t i ) = N − Y m =0 U ( t i + t m +1 ; t i + t m − ǫ ) (62)where N ǫ = t ′′ − t ′ . Similarly considerations can be applied to the spatial evolution. Thefinite classical space-time evolution in the Schrödinger representation is | φ ( t, x) i = U (x , t ; 0 , | φ i = e − i ~ ( H t −P x) | φ i = X n a n e − i ~ p n · x | φ n i . (63)In order to allow an easy generalization of the following result to the interaction case,we will describe the finite space-time evolutions in terms of the infinitesimal space-time25volutions of the Markovian operator. In particular this will be useful in the interactioncase where in every point a different inner product must be considered. As a result wewill obtain the integral product R V x D x .All the elements necessary to build a FPI are already present in the theory withoutany further assumption than PBCs. In fact, we can plug the completeness relation ofthe energy eigenmodes in between the elementary time Markovian evolutions obtaining Z = Z V x N − Y m =1 d x m ! U (x f , t f ; x N − , t N − ) × . . . × U (x , t ; x , t ) U (x , t ; x ′ , t ′ ) . (64)From the notation (63), the elementary space-time evolutions of a free system can bewritten as U (x m +1 , t m +1 ; x m , t m ) = h φ | e − i ~ ( H ∆ t m −P ∆x m ) | φ i , (65)where ∆x m = x m +1 − x m and ∆ t m = t m +1 − t m . Thus, proceeding in a completelystandard way we formally obtain the ordinary Feynman Path Integral (FPI) for thetime-independent Hamiltonian (57), Z = lim N →∞ Z V x N − Y m =1 d x m ! N − Y m =0 h φ | e − i ~ ( H ∆ t m −P ∆x m ) | φ i . (66)This remarkable result has been obtained by using classical-relativistic mechanics andwithout any further assumption than intrinsic four-periodicity. As in the usual FPIformulation in phase-space we are assuming on-shell elementary space-time evolutions[34]. Though we started with homogeneous H , this derivation, being based on elementaryevolutions of a Markovian operator, can be generalized to the interacting case as we willsee in sec.(5.3.4).Proceeding in complete analogy with the ordinary derivation of the FPI in configu-ration space we note that the infinitesimal products of (65) in (66) can be genericallywritten in terms of the action of the corresponding classical particle S cl ( t f , t i ) ≡ Z t f t i dtL cl = Z t f t i dt ( P ˙x − H ) . (67)Finally the FPI in (66) can be written in the familiar form Z = Z V x D x e i ~ S cl ( t f ,t i ) . (68)This important result can be intuitively interpreted by considering that a cyclic fieldhas topology S . Indeed, for a field in such a cyclic geometry there is an infinite set ofpossible classical paths with different winding numbers linking every given initial andfinal configuration. Contrary to ordinary fields, a cyclic field can self-interfere and thisis described by the ordinary FPI [1, 9].We may also mention that, [1], by evaluating the expectation value of the observable ∂ x F (x) associated with the inner product (55) of our Hilbert space, integrating by parts,and considering that the boundary term vanishes because of the periodicity of spatialcoordinate, we find h φ f | ∂ x F (x) | φ i i = i ~ h φ f |PF (x) − F (x) P| φ i i . (69)26inally, by assuming that the observable is a spatial coordinate F (x) = x (Feynmanused a similar demonstration to show the correspondence of the FPI with canonicalQM, [34]), the above equation for generic initial and final Hilbert states | φ i i and | φ f i ,is nothing but the commutation relation of ordinary QM: [x , P ] = i ~ — or more ingeneral [ F , P ] = i ~ ∂ x F . With this result we have also checked the correspondence withcanonical QM. The commutation relations, as well as the Heisenberg uncertain relation[1], can be regarded as implicit (and not imposed ) in this theory. This is a consequenceof the intrinsically cyclic (undulatory) nature of elementary particles [1] conjectured byde Broglie in 1924 [16], implicitly tested by 80 years of successes of QFT and indirectlyobserved in a recent experiment [18]. In this section we want to generalize our geometrodynamical description of interactionto all the possible harmonic modes of the cyclic field. In this way we will find a formalcorrespondence with the ordinary FPI of interacting systems (Feynman diagrams andperturbative calculations will be discussed elsewhere ).We have already seen that the classical propagation of a free cyclic field is described bya FPI (66) written in terms of the Lagrangian of a corresponding free particle with mass ¯ M [1]. On the other hand, we have also seen that local transformations of variables induce internal transformations of field solutions can be used to describe gauge interaction.Next we will formally extend this correspondence to the quantum limit. In particularwe will see that the propagation of the cyclic bosonic field - with all its harmonics -is described by the usual FPI and Scattering Matrix of an ordinary gauge interactingbosonic fields. The Lagrangian in FPI (66) will turn out to be the usual Lagrangianof classical electrodynamics of bosonic particles. Here we assume again three spatialdimensions x .Next we will extend our semi-classical description of quantum systems to electrody-namics and we will find a formal correspondence to the corresponding QFT. The assumption of PBCs in field theory in compact 4D can be regarded as the quan-tization condition. It is easy to see that PBCs reproduce the Bohr-Sommerfeld (BS)quantization condition. Intuitively the BS condition is a periodicity condition becauseit says that the only possible orbits of the system are those with an integer number ofcycles.The correspondence with BS is immediate in the case of free fields, since it correspondsto a isochronous system T ′ µ ( x ) = T ′ µ (the analogous of to the Galilean isochronism of thependulum where the orbits at different energies have the same periodicity). The PBCsapplied to the free cyclic field (53) give the following harmonic quantization condition ofthe phase of the fields I dx · p n = Z T µ dx · p n = T · p n = nh . (70) The Fourier coefficients ¯ a = α and ¯ a ∗ = α − / √− in (53) can be regarded as the annihilatingand creation operators. The anti-foundamental ( n = − ) modes can be associated to an anti-particle.Similarly, a n = α n / √ n can be regarded as Virasoro operators. Φ ′ ( x ) is a sum of eigenmodes with the same analyticalform of the fundamental mode (35), ¯Φ( x ′ ) = X n ¯ N n e − i ~ R x ′ µ dx · ¯ p ′ n ( x ) . (71)The quantized spectrum of the cyclic interacting field in x = X is given by the PBCs at T ′ µ ( X ) , i.e. Φ ′ ( X ) ≡ Φ ′ ( X + cT ( X )) , which can be written as e − ih R T ′ µ ( X ) dx · p ′ n ( x ) = e − ih H X dx · p ′ n ( x ) = e − i πn . (72)Therefore the quantization condition in the interacting case is I X dx · p ′ n ( x ) = nh , (73)according to (12) and (70). This is nothing but the generalization of the BS quantizationcondition in 4D .We now define a four-momentum operator P µ = {H /c, −P i } , and a number operator ˆ n | φ n i ≡ n | φ n i . In the Hilbert space associated with this interacting system the non-homogeneous operator P ′ µ ( x ) can be obtained from the homogeneous one similarly to(12), P µ → P ′ µ ( x ) = e aµ ( x ) P a . (74)In fact the quantization condition (73) of an interacting cyclic system can be expressedas I X dx · P ′ ( x ) = ˆ nh . (75)Therefore the quantization of interacting cyclic field through PBCs can be also re-garded as a generalization of the familiar BS quantization and described in terms ofHilbert operators. We now consider the specific case of gauge interaction. For the cyclic field solution Φ ′ ( x ) associated with the minimal substitution (27), the quantization condition (72)turns out to be e − ih H X dx · [ p n − eA n ( x )] = e − i πn . (76)As it can be seen from (70), in the case of pure gauge orbits A µ ( x ) = ∂ µ θ ( x ) , thiscondition leads to a quantization condition for the Wilson loop I X dx · eA n ( x ) = hn . (77) The Morse coefficient of the ordinary BS quantization can be retrieved by assuming a global twistfactor in the PBCs and it can be interpreted as the vev (see also Gauge-Higgs unification and Hosotanimechanism). This aspect has been shortly discussed in [1]. e A suchthat e A | φ n i ≡ eA n | φ n i . (78)In this case the assumption of PBCs reproduces formally quantization condition of aDirac string . I X dx · e A ( x ) = ˆ nh . (79)This confirms the result of the related paper [35] where the Dirac quantization con-dition (79) has been obtained in an indirect way just by assuming time periodicity foran abelian gauge field. It has been used to interpret phenomenological aspects of su-perconductivity such as the quantization of the magnetic flux, the penetration length,the Meissner effect, the Josephon effect. In this description, according to [36, 35], super-conductivity is a phenomenological consequence of the breaking of the electromagneticgauge invariance associated with S → Z . In fact, in a pure gauge, because of the PBCson the matter field Φ , the phase of the gauge transformation (47) is periodic and definedmodulo factor hn . This can be seen also from (79). Thus the “Goldstone” θ of the relatedgauge transformation can vary only by finite amounts and the electromagnetic gauge in-variance is broken — without involving a vev . Notice however that this breaking of thegauge invariance is a quantum effect since in the classical limit ~ → the quantizationcondition (79) yields a non quantized spectrum. Thus, field theory in compact 4D notonly provides a geometrodynamical description of the gauge invariance; PBCs providesalso a mechanism of gauge symmetry breaking with interesting analogy with the onestypically used in XD Higgsless and Gauge-Higgs-Unification models. As discussed in sec.(4.4), the generalization of the quantization condition (79) to thecase of a non-abelian gauge SU L (2) ⊗ U Y (1) is obtained through the following substitution e A ( x ) → g W ( x ) + g ′ Y ( x ) . According to [37], the resulting quantization condition for theneutral component I X dx · [ g W ( x ) + g ′ Y ( x )] = ˆ nh (80)could provide a realistic electroweak symmetry breaking mechanism which can be inter-preted as induced by monopole condensations. To the l.h.s. of this equation can be added a factor / by noticing that, according to the innerproduct (55), only the modulo of the field has a physical meaning. In analogy with the XD formalism itarises naturally in the orbifold notation s ∈ S / Z . This condition can also be regarded as a quantization for the electric charge. The PBCs allows thepossibility to use Noether’s theorem to directly describe the quantized variables of the theory. This ideawill be expanded in future works. In the Hosotani mechanism [40, 41], the shift of the mass eigenvalue of the KK fundamental modeis ¯ M ′ ( y ) = ¯ M − g ¯ A ( y ) where ¯ A ( y ) and g are the fifth component and the coupling of an gauge fieldwith an XD y . Under the dualism with XD studied [20] this is corresponds with the minimal substitution ¯ p ′ µ ( x ) = ¯ p µ − e ¯ A µ ( x ) . On the other hand, in complete parallelism with our 4D description, a Hilbertspace can also be introduced to describe the KK mode of an XD field. The Wilson line of the A ( y ) canbe written as an operator which takes integer number, similarly to (79). This corresponds to generalizethe Hosotani mechanism from to winding number n = 1 to all the possible n . As a results we find adeformation of the whole KK tower, mode by mode, which can be equivalently described as induced bya corresponding deformation of the XD. .3.3. Scattering Matrix The Hilbert formalism (63) used to describe the evolution of a free field Φ( x ) can beeasily extended to interacting fields Φ ′ ( x ′ ) .The transformation under local abelian (polarized) isometries reproduces ordinarygauge interaction for the fundamental mode. Similarly to (37), the corresponding trans-formation of a generic mode φ of a free cyclic field to an corresponding mode φ ′ of agauge interacting cyclic field is described by the following internal transformation φ ′ ( x ) = e ie ~ R xµ dx · A ( x ) φ ( x ) . (81)This describes the modulation of periodicity ( tuning ) of the generic mode φ ′ with respectthe free mode φ , as a function of the generic gauge mode A µ ( x ) .Thus, from the definitions of the Hilbert operator A in (78), we find that the internaltransformation of the cyclic field corresponds to pass from the Schrödinger representationto the interaction representation of perturbation theory. In fact we find | φ ′ ( x ) i = e i ~ R xµ dx · e A | φ ( x ) i = X n α n e i ~ R xµ dx · eA n | φ n ( x ) i . (82)Now we define a tuning operator, Hilbert analogous of the parallel transport , as S ( x ) = e i ~ R xµ d x · e A . (83)This is nothing but the ordinary scattering matrix of ordinary perturbation theory. Infact, it turns out to define formally the interaction term L int ( x ) of the ordinary La-grangian of classical electrodynamics e Z x µ dx µ A µ ( x ) = e Z τ µf τ µi dτ A µ ( x ) J µ ( x ) = Z x f x i d x L int ( x ) . (84)In writing this equation we have explicitly written the integration region as x µ = x µf − x µi (modulo periods) and we have defined the current J µ = dx µ /dτ .From a formal point of view, such a representation of interaction for a cyclic fieldas modulation of periodicity actually matches the ordinary interaction representationwritten in terms of the scattering matrix of ordinary QFT | φ ′ ( x ) i = S ( x ) | φ ( x ) i . (85)It is important to note that, as in ordinary QFT, this interaction representation is for-mally sufficient to describe QED in terms of the Feynman diagrams. With this formalism at hand it is finally possible to describe the evolution of an cyclicfield (including all its possible harmonics) under a given interaction scheme. We will findthat its evolution will be formally described by the ordinary FPI of the correspondingquantum interacting system. In the specific case of a cyclic field transforming underabelian polarized local isometries, the result will be the ordinary FPI of QED (for bosons).30he description of the FPI given in (66), being written in terms of elementary space-time evolutions, can be easily generalized to the non-interacting case. In case of inter-actions the four-momentum operator P ′ µ of (74) is non-homogeneous. Nevertheless thespace-time evolutions are Markovian even in case of interactions . This can be seen forinstance by considering the scattering matrix (83). In fact the time evolution in the in-teraction representation is described by the exponential operator S ( t ) = e − i ~ R t H int ( t ) dt .Similarly to the evolution of a free cyclic field (61), the contribution associated to inter-actions is Markovian S ( t + dt ) = e − i ~ H int ( t ) dt . Hence, in case of interactions the resultingevolution is Markovian: U ′ ( t + dt ; t ) = e − i ~ H ′ ( t ) dt . The generalization of the elementaryspace-time evolutions (65) of an interacting cyclic field in terms of the non-homogeneousfour-momentum operator P ′ µ ( x ) is U ′ ( x m +1 , t m +1 ; x m , t m ) = h φ | e − i ~ ( H ′ ∆ t m −P ′ i ∆ x im ) | φ i . (86)Another difference with respect to the free case is that interaction deforms pointby point the completeness relation. That is, the integration volume V x of the innerproduct (55) varies with x = X : that is, R V ′ x ( X ) d x ′ ( X ) /V ′ x ( X ) (the number of period N ′ remains fixed). The Markovian evolution of our cyclic system allows us to write thethe evolution in terms of elementary evolutions (86). This guarantees the possibility touse a different inner product at every point x = x m of (64). Moreover, in order to avoida different integration volume λ ′ x ( X ) in every integration point, the integration regionof the inner-product can be extended to a very large or infinite volume V ′ x ( X ) (large orinfinite number of periods N ′ ), much bigger than the (finite) interaction region I . In thisway the volume V ′ x ( X ) , as well as the normalization of the field, is overall not affectedby the local deformations: V ′ x ( X ) ∼ = V x . Thus the correct mathematical tool to representthis non trivial evolution is actually the integral product R V x D x .At this point, by following the same generic demonstration used in (66) (plugging lo-cally the completeness relation in the elementary Markovian evolutions), we find formallythe ordinary FPI in phase-space of an interacting particle Z = lim N →∞ Z V x N − Y m =1 d x m ! N − Y m =0 h φ | e − i ~ ( H ′ ∆ t m −P ′ i ∆x im ) | φ i . (87)Similarly to (67), it is possible to define a Lagrangian L ′ cl = P ′ i ˙x i − H ′ . Since P ′ µ in(74) transforms as the four-momentum ¯ p ′ µ of the corresponding classical particle (12), thislagrangian defines formally the action S ′ cl ( t f , t i ) ≡ R t f t i dtL ′ cl of the corresponding inter-acting classical particle — written in terms of operators. Therefore the classical evolutionof an interacting cyclic field is formally described by the ordinary FPI in configurationspace associated with the interaction scheme, Z = Z V x D x e i ~ S ′ cl ( t f ,t i ) . (88)Finally, from the analysis of the scattering matrix (84) we find that in the approxima-tion of the local isometries investigated in this paper, the resulting evolution is formallygiven by the ordinary FPI of scalar QED. In fact the resulting Lagrangian associatedwith the action S ′ in the exponential is formally the classical Lagrangian of a charged31osonic particle interacting electromagnetically: L ′ cl = L cl + L int . With this formal cor-respondence to the ordinary FPI formulation at hand we can invoke Feynman’s sayingthat “the same equations have the same solutions” [38]. Hence the assumption of intrinsicperiodicity can in principle be used for a geometrodynamical semi-classical descriptionQED.This formal correspondence must be however tested in explicit computations of QEDobservables. This will be the subject of a dedicated paper. Nevertheless we mentionthat recent studies seem to show the liability of these semi-classical computations. Wemay note that light-front quantization, a semiclassical theory which similarly to ourtheory uses the assumption of PBCs as quantization condition, shows that it is actuallypossible to reproduce semi-classically the electron anomalous magnetic momentum interm of harmonics expansion of the fields [44]. Similar results pointing in the samedirection are the computation of quantum behavior through the AdS/CFT, see discussionin sec.(2.2.2) and [20], and the calculation of Feynman diagrams in Twistor Theory [45]or other integrable theories. Here we present a digression about the physical meaning of the assumption of intrinsicperiodicity for elementary quantum systems. The motivations go beyond the de Broglieperiodic phenomenon and involve interesting aspects of modern physics, as described indetail in [1, 9, 10, 11, 12, 13, 14, 20, 35] and summarized here.We may consider the recent attempts to interpret QM as an emerging theory, suchas the ’t Hooft determinism [24] and the stroboscopic quantization [25]. According to ’tHooft [24], there is a “close relationship between the quantum harmonic oscillator anda particle moving on a circle”, both with extremely fast periodicity T t . Our field theoryin compact 4D can be intuitively derived by noticing that, as well known, the quantumharmonic oscillator is the basic element of ordinary second quantized KG fields. A cyclicfield can be intuitively derived from the ’t Hooft determinism (in the continuos limit ofthe lattice used by ’t Hooft) by considering that the characteristic periodicity T t varies ina relativistic way, as described in sec.(1). This cyclic behavior of a “particle on a circle”of the ’t Hooft determinism has motivated the “stroboscopic quantization” [25]. In thiscase we explicitly find the idea of dimensions compactified in a torus and, even moreinteresting, the fact that the “ticks” resulting from ergodic dynamics yield an effectivedescription of the arrow of time. Similarly, in our theory every instant in time can becharacterized by a different combination of the phases of all the de Broglie clocks con-stituting an isolated system of elementary particles. This is similar to a calendar or astopwatch which allows us to fix events in time in term of combinations of the phasesof periods that we call years, months, days, hours, minutes, and so on. If the elemen-tary cycles constituting our systems of periodic phenomena have irrational periods, thetotal evolution will result in an ergodic evolution; if we also allow interactions betweenelementary periodic phenomena, i.e. exchange energy or equivalently variation of peri-odicity, the resulting evolution of the non elementary system will be chaotic. Indeed, theassumption of intrinsic periodicity, realized in terms of field theory in compact 4D withPBCs, has important motivations in the interpretation of the notion of time in physics[1, 13]. In particular, as Galileo taught us with the experiment of the pendulum in thePisa dome, or as explicitly stated in the Einstein [39] definition of relativistic clock, oraccording to the operative definition of a second through the Cs-133 atom, time can be32nly defined by counting the number of periods of a phenomenon supposed to be periodicin order to guarantee the constancy of the unit of time. Thus every free elementary parti-cle can be regarded as a reference clock , the so-called “de Broglie internal clock”. As wehave seen, the modulation of periodicity of these internal clocks can be used to describeinteractions similarly to GR. We also mention the geometric quantization [46] which isan attempt to reproduce quantum behavior by introducing two grassmannian partnersof the physical time. This could be interesting for a possible semi-classical descriptionof spinning particle typical of the zitterbewegung models [43] (this brings elements ofsupersymmetry in the theory and it could be investigated in a dedicated paper).The time periodicity of the de Broglie internal clock is bounded by the inverse ofthe mass, T τ ≤ T τ = h/ ¯ M c , i.e. by the Compton wavelength divided by the seed oflight T τ = λ s /c . In this way it is easy to see that these de Broglie intrinsic clocks ofelementary particles are typically extremely fast (except neutrinos). The heavier theparticle, the faster the periodicity. As already mentioned, a light particle such as theelectron has a periodicity faster than − s which means that for every “tick” of theCs-133 atomic clock ( T Cs ∼ − s ) an electron does a number of cycles of the orderof the age of the universe expressed in years. Even with the modern time resolutionit is not yet possible to resolve such small time scales, though the internal clock of theelectron has been indirectly observed in a recent interference experiment [17]. Thus theobservation of such a fast de Broglie internal clock is similar to the observation of a “clockunder a stroboscopic light”, [25]. That is, at every observation the particle appears tobe in an aleatory phase of its cyclic evolution and, similarly to a dice rolling too fast (deBroglie deterministic dice) with respect to our resolution in time, the outcomes can beonly described in a statistical way [12]. Similarly to the deterministic models mentionedabove, the results of the preview section show that the statistics associated to these cyclicbehavior have formal correspondences to ordinary QM. They also suggest that the directexperimental exploration of microscopical time scales (smaller that − s in the case ofthe electron) is of primary interest in understanding the inner nature of the elementarysystems .Another motivation is given by the variational analysis of the BCs discussed in [1] andmentioned in sec.(1). Roughly speaking we may say that relativity fixes the differentialstructure of the continuous space-time of a theory without giving particular prescriptionsabout BCs, [1, 28]. The important requirement for the BCs is that they must fulfill thevariational principle. As already discussed in this paper, the role of the BCs is marginalin ordinary QFT. On the other hand, BCs have played an important role since the earliestdays of QM, according to de Broglie, Bohr, Sommerfeld, et.al.. We have also noticedthat the non-quantum limit of a massive cyclic field corresponds to the limit whereonly the fundamental mode ¯Φ( x ) is exited, see [1] and [9] for more detail. This also isthe non-relativistic limit of a massive particle ( | ¯ p | ≪ ¯ M c ). Thus, the classical particledescription, corresponding to the limit of large mass ( ¯ M → ∞ ) or equivalently the spatialmomentum to zero ( ¯ p → ), the cyclic field reduces to ¯Φ( x ) ∼ exp[ − i ¯ Mc ~ t + i ¯ M ~ x t ] , see[1]. Neglecting the de Broglie rest clock represented by the first term, the modulo squareof a massive cyclic field is a distribution centered along the path of the corresponding In this way the local character of the relativistic time turns out to be enforced. We can say that LHC is exploring indirectly time scale of the order of − s corresponding toenergy scale of the order of the T eV . λ s ,as can be easily shown by performing an explicit plot [9]. Thus in the classical limit( λ s → ) this distribution reproduces the ordinary non-relativistic limit of the FPI, i.e. a Dirac delta distribution. In this limit the spatial compactification lengths tend toinfinity ( ¯ p → ) whereas the compactification length along the time dimensions tends tozero ( ¯ M → ∞ ). Indeed a classical particle turns out to be actually described by a pointlike distribution in R . Similar arguments can be used to interpret other interestingaspects of the wave-particle duality of QM. The assumption of periodic phenomenonenforces the wave-particle dualism, giving rise to implicit commutations relations andHeisenberg uncertain relations, [1]. A massless field, i.e. a field leaving in the light-cone( ds = 0 ), is always relativistic. Since its Compton wavelength is infinite we say thatthe rest de Broglie clock of a massless field, such as the EM field, is frozen . Thus theenergy spectrum of a massless cyclic field can be approximated to a continuous in the IRregion where the compactification length tend to infinity ( T t → ∞ ) and the PBCs can beneglected. In the UV limit however the PBCs are important because the compactificationlength are very small ( T t → ). Therefore, the quantized nature of the energy spectrumbecomes manifest avoiding the UV catastrophe of the black-body radiation. Indeed athigh frequencies the field theory has an effective corpuscular description, [1].A further conceptual motivation to use BCs as quantization condition is that we havethe remarkable property that QM emerges without involving any (local) hidden variablein the theory. Since the hypothesis of existence of local hidden variable is not realized, theBells’s theorem can not be applied to our theory. The assumption of intrinsic periodicityintroduces an element of non locality which, however, can be regarded as consistent withSR since the periodicity varies in a relativistic way. Thus the theory can in principleviolate the Bells’s inequality (if we try to adapt the Bell’s theorem to our case, i.e. toevaluate the expectation values of an observable in the Hilbert space described above,we find again a formal parallelism with ordinary QM). For this reason we speak about— mathematically — deterministic theories [1].We have seen that the assumption of intrinsic periodicity of elementary systems pro-vides a semi-classical description of scalar QED. The study of the limit where quantumcorrections become relevant for gravitational interaction is beyond the scope of this pa-per. In particular quantum gravity represents another important subject where to testthe consistence and the validity of the theory. Some more detail about this point is givenin [1, 13].
6. Conclusions
Field theory in compact 4D represents a natural realization of the “periodic phe-nomenon” associated to every elementary particle, as conjectured by de Broglie in 1924[15, 16], at the base of the wave-particle duality, implicitly tested by 80 years of QFT andindirectly observed in a recent experiment [18] (Schrödinger used a similar assumptionin his zitterbewegung model of the electron).In this formalism the kinematical information of an interaction scheme is encoded inthe relativistic geometrodynamics of the boundary of the theory — in a sort of holo-graphic description. The resulting description has shown remarkable relationships withthe following fundamental approaches to interactions:34 the approach typical of classical-relativistic mechanics in which interaction is de-scribed in terms of retarded and local variations of four-momentum; • the approach typically used in QM to describe systems in generic potentials in termswaves and BCs. The dynamical boundary of the theory reproduces the retardedand local modulation of de Broglie four-periodicity and thus the local and retardedvariation of four-momentum of an interacting quantum system; • in GR, gravitational interaction can be described as local modulations of period-icity of reference clocks encoded in corresponding deformations of the underlyingspace-time coordinates. Similarly, in our description, local modulations of the four-periodicity are described as deformations of the compact 4D. • the typical approach to interaction of gauge theory is obtained because the varia-tions of field solution associated with the variations of the boundary of the theorydefines an internal transformations which, in the approximation of the local isome-tries described in the paper, formally matches classical electrodynamics.Remarkably, we have found that gauge interaction can be derived from the invarianceof the theory under local transformations of variables as gravitational interaction can bederived by requiring invariance under diffeomorphisms. Gauge symmetries are relatedto space-time symmetries. This cab be regarded as in the spirit of Weyl’s, Kaluza’s andWheeler’s and original proposal of a geometrodynamical description of gauge invariance.On the other hand the assumption of PBCs (or similar BCs such as anti-PBCs, N-BCs, D-BCs) provides a semi-classical quantization condition for fields. In fact the theorycan be regarded as the full relativistic generalization of the quantization of a “particlein a box”. This geometric quantization method, without introducing hidden-variables, reproduces formal correspondences between well established quantization methods : • the correspondence to canonical formulation of QM, arises from the fact that acyclic field is naturally described by an Hilbert space, that it evolves accordingto the Schrödinger equation and its cyclic variables implicitly satisfy commutationrelations and Heisenberg uncertain relations; • the Feynman Path Integral is obtained as interference of the classical paths withdifferent winding numbers associated with the intrinsically cyclic geometry of fieldsin compact 4D [1, 9]; • The Bohr-Sommerfeld quantization condition is nothing but a periodicity condition.It simply states that the allowed orbits are those with an integer number of cycles(close orbits) and can be described in terms of PBCs; • the Dirac quantization condition is obtained as a result of the periodicity inducedby the matter cyclic field on pure gauge transformations. • the Scattering Matrix naturally describes the modulations of periodicity betweena free cyclic field (Schrödinger representation) and an interacting cyclic field (in-teraction representêation). 35emarkably, field theory in cyclic 4D, without any further assumption than intrinsicperiodicity, provides the possibility of a geometrodynamical and semi-classical descriptionof scalar QED .To the above list of well established quantization methods it should be added that, asshown in [20] through the dualism of cyclic fields to XD massless fields, the theory yieldsinteresting analogies with the classical XD geometry to quantum behavior correspon-dence typical of AdS/CFT. The dualism to XD theory will be used in future papers toinvestigate the geometrodynamical description of gauge invariance in terms of Kaluza’soriginal proposal. Acknowledgements
I would like to thank M. Neubert, M. Reuter, E. Manrique and T. Gherghetta fortheir interest in new ideas; N. Liu for her help in writing; and the reviewers for theirvaluable work. This paper is part of the project “
Compact Time and Determinism ” andit has been presented at
QTS7
Aug 2011, Prague, and
FFP12
Nov 2011, Udine.
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