ARTICLE IN PRESS Chaos, Solitons and Fractals xxx (2004) xxx–xxx www.elsevier.com/locate/chaos Local scale invariance, Cantorian space–time and unified field theory Ervin Goldfain OptiSolve Consulting, 4422 Cleveland Road, Syracuse, NY 13215, USA Accepted 5 May 2004 Abstract We develop field theoretic arguments for the unification of relativistic gravity with standard model interactions on El Naschie’s Cantorian space–time. The work proceeds by showing the equivalence between the fundamental principle of local gauge invariance and the local scale invariance of space–time and matter fields undergoing critical behavior on high-energy scales. We focus on the transition boundary between the classical and non-classical regimes, the latter being characterized by generalized scaling laws with continuously varying exponents. Both relativistic gravity and standard model interactions emerge from the underlying geometry of Cantorian space–time near this transition boundary.
2004 Elsevier Ltd. All rights reserved. 1. Introduction and motivation The theory of general relativity unifies gravitation with the geometry of the space–time continuum. It is well known that the classical electromagnetic field described by Maxwell’s equations may not be interpreted in terms of the geometrical attributes of space–time. This conceptual difficulty inspired Einstein and many others to generalize nonEuclidean geometry in order to incorporate the electromagnetic field along with the weak and strong forces within the framework of space–time [4–6,25]. The unification program has evolved along several paths, from the early models of Weyl, Kaluza and Klein to grand unified, conformal field and string-related theories [25,28]. A distinct class of theoretical efforts is built on the premise that gravitation must exhibit approximate scale invariance at large energies [4–8]. By contrast, contemporary developments due to El Naschie [10–12], Nottale [9,32] and Sidharth [16,17] rest upon the non-trivial topology of space–time at large energies or fractal attributes of trajectories in quantum physics. Motivated by these non-conventional approaches, we develop our work in the context of complexity theory. Instead of searching for a larger symmetry group capable of including both standard model fields and classical gravity in one framework, we show that all interactions are embedded in the underlying geometry of Cantorian space–time. This circumstance sets the physical basis for unification on large energy scales where the smooth space–time continuum makes the transition to the non-trivial structure of Cantorian space–time. In contrast with the theory of scale relativity [9,32], conformal field theories and other approaches relying on local scale invariance with fixed exponents, we demand invariance of physics under generalized scaling laws with continuously varying exponents. The rationale for requiring locality of scaling exponents is deeply rooted in the conceptual basis of relativity. Unlike string-related theories, no higher space–time dimensions are necessary to maintain logical integrity of the model. The organization of the paper is as follows: Section 2 outlines the concept of generalized local scaling. Sections 3 and 4 discuss the postulates and assumptions on which the paper is founded. An introduction to quantum electrodynamics E-mail address: ervingoldfain@hotmail.com (E. Goldfain). 0960-0779/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.05.020 ARTICLE IN PRESS 2 E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx (QED) as a gauge invariant theory is presented in Section 5. The equivalence of the local gauge transformation to local field scaling is formulated in Section 6. Taking as an elementary example the case of a single relativistic particle, Sections 7 and 8 apply local scale invariance to the free Dirac theory and general relativity, respectively. Summary and concluding remarks are detailed in Section 9. 2. Review of generalized local scaling Many collective systems in statistical physics share the property of global scale invariance, i.e., their behavior remains unchanged under rescaling of all independent observables and control parameters [2,3]. Let the behavior of a system having N independent observables and control parameters ðx1 ; x2 ; . . . ; xN Þ be described by the generic scalar function f ðx1 ; x2 ; . . . ; xN Þ. Global scale invariance implies that f is quasi-homogeneous and satisfies the functional equation f ðsm1 x1 ; sm2 x2 ; . . . ; smN xN Þ ¼ sf ðx1 ; x2 ; . . . ; xN Þ ð1Þ in which s is an arbitrary scale parameter and m1 ; m2 ; . . . ; mN are constant exponents. By analogy with (1), a generalized local scale transformation is defined by f ðsm1 ðx1 ;::;xN Þ x1 ; sm2 ðx1 ;::;xN Þ x2 ; . . . ; smN ðx1 ;::;xN Þ xN Þ ¼ sf ðx1 ; x2 ; . . . ; xN Þ ð2Þ where all exponents depend on ðx1 ; x2 ; . . . ; xN Þ and their choice is constrained by a group property that requires [1] mi ðx1 ; x2 ; . . . ; xN Þ ¼ mi ðsm1 x1 ; sm2 x2 ; . . . ; smN xN Þ for i ¼ 1; 2; . . . ; N. It can be shown that, for an integer number m (0 6 m < N ), the scaling exponents satisfy
 ln jxj j ln jx1 j ln jx2 j ln jxm j mj ¼ mN ; ;...; ln jxN j ln jxN j ln jxN j ln jxN j where j ¼ 1; 2; . . . ; m. The set of functions

 ln jx1 j ln jxm j ln jx1 j ln jxm j mmþ1 ; . . . ; mN ;...; ;...; ln jxN j ln jxN j ln jxN j ln jxN j ð3Þ ð4Þ ð5Þ can be chosen freely, while the rest are related through (4) [1]. 3. Postulates Based on Renormalization Group arguments regarding critical behavior in the high-energy regime [22,29,30] and the relativistic principle of locality, we now postulate the following: P1. Space–time and fields acting in the high-energy regime obey generalized scaling laws with continuously varying exponents having the form x ¼ x0 smx ðx;oxÞ ðoxÞ ¼ ðoxÞ0 smox ðx;oxÞ u ¼ u0 smu ðu;ouÞ ð6Þ ou ¼ ðouÞ0 smou ðu;ouÞ Here x ¼ ðx0 ; x1 ; x2 ; x3 Þ is the four-dimensional space–time and u ¼ uðxÞ is a generic scalar, vector, spinor or tensor field that can be expressed as a base function times a scaling function. In the above o is the four-dimensional gradient o ¼ ðo0 ; o1 ; o2 ; o3 Þ and s 2 ½0; 1 represents the resolution scale, that is, the minimum resolvable separation of either space–time or field observables normalized to the characteristic unit of measurement. We study the high-energy behavior of field theory in a region of interest centered about the Cohen–Kaplan threshold ( 100 TeV). This threshold sets the upper limit of validity for any quantum field theoretical description of nature [19,20,27]. Let lc represent the Cohen–Kaplan threshold and let ½x and ½w stand for the units of space–time and field. Then, in this high-energy regime, ARTICLE IN PRESS E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx ½x ¼ lc 1 ½u ¼ lDc 3 ð7Þ where the numerical value of exponent D depends on the nature of the field [21]. If the minimum resolvable energy separation is Dl ¼ jl lc j, the resolution scale is defined by s¼ Dl lc ð8Þ (8) is the analogue of the reduced control parameter, a key concept in the theory of critical phenomena [2,3]. An important point regarding (8) is now in order. Compliance with relativity demands that s be locally defined since, under the most general circumstances, Dl ¼ DlðxÞ. However, since both Dl and lc are expected to scale in the same fashion with x to a first-order approximation, it is reasonable to consider s a global parameter in the critical region l ! lc . Scaling relations (6) reveal the close connection between the approach to criticality in statistical physics and selfsimilarity of fractal structures. Assuming that all exponents entering (6) are finite, both space–time coordinates and fields diverge or vanish as s ! 0, that is, near the transition boundary separating the classical regime ðl < lc Þ from its non-classical counterpart ðl > lc Þ. At this point it is instructive to introduce the following two observations regarding the first postulate: (a) By the definition stated in (6), the gradients transform covariantly with the original observables (that is, they transform in the same manner). (b) A linear relationship exists between the coordinate exponent and the Hausdorff dimension dH ðx; oxÞ. Based on the Hausdorff dimension definition [31,32], we expect mx ðx; oxÞ ¼ dH ðx; oxÞ 1 ð9Þ The transfinite topology of Cantorian space–time [10,12] implies that dH ðxÞ and exponent mx ðxÞ are dimensions belonging to sets of infinite support. As indicated earlier, the important distinction here is that all exponents are locally defined as continuous functions of space–time coordinates. We now advance the second postulate in connection with the topic of the previous section. P2. Let L L½ui ðxÞ; oui ðxÞ ð10Þ denote the Lagrangian of a prototype free-field theory depending on N fields and their first order derivatives. The differential action corresponding to (10) maintains full covariance to the local scaling transformation defined by (2), that is dS½smi;u ðxÞ ui ðxÞ; smi;ou ðxÞ oui ðxÞ; smd4 x ðxÞ d4 x ¼ dS½ui ðxÞ; oui ðxÞ; d4 x ð11aÞ dS½ui ðxÞ; oui ðxÞ; d4 x ¼ L½ui ðxÞ; oui ðxÞd4 x ð11bÞ 1 dS½smi;u ðxÞ ui ðxÞ; smi;ou ðxÞ oui ðxÞ; smd4 x ðxÞ d4 x ¼ sL½ui ðxÞ; oui ðxÞ d4 x s ð11cÞ where for i ¼ 1; 2; . . . ; N . It is seen that (11a) translates the fundamental ansatz of relativistic quantum field theory according to which differential action represents a Lorentz scalar [18,21]. 4. Assumptions and conventions In addition to postulates P1 and P2, we introduce the following set of assumptions and conventions: (1) For the sake of simplicity it is assumed that all observables and their exponents, along with scalar functions dependent on them, maintain their analytic properties in the high-energy regime. ARTICLE IN PRESS 4 E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx (2) For the sake of simplicity and following (1), space–time coordinates are treated as continuous random variables. As a result, observables and their exponents, along with scalar functions dependent on them, are also continuous random entities [22]. Functional relationships are therefore understood as being exclusively defined in a statistical sense. (3) Matter fields are interpreted as classical Dirac particles due to the phenomenon of decoherence induced by the continuous exposure to random fluctuations [23,24]. (4) The analysis is limited to free massless particles because the high-energy regime of field theory warrants the approximation of such ultra-relativistic objects. This viewpoint is consistent with the philosophy of conformal field theory where all particle-like excitations are necessarily massless as a result of scale invariance [33]. (5) Space–time and fields become scale independent in the classical regime as all exponents vanish for l < lc . (6) Einstein summation convention and natural units are assumed throughout the remainder of the paper ð h ¼ c ¼ 1Þ. (7) Space–time and fields are suitably normalized as dimensionless observables. 5. Local gauge transformation and quantum electrodynamics All field theories that make up the standard model of particle physics are based on the fundamental principle of local gauge invariance. This principle asserts that the Lagrangian of the theory and equations of motion derived from it are invariant to internal transformation of fields that are continuously dependent on space–time. The direct consequence of local gauge invariance is that all matter particles (i.e., leptons and quarks) interact via gauge bosons that mediate the interaction between these particles. Free matter particles are fermions whose space–time evolution is governed by the Dirac equation. The typical prototype of a field theory that obeys local gauge invariance is quantum electrodynamics (QED). For convenience, we outline below a brief discussion on how QED is constructed as a gauge theory. Additional details may be found in [18,21]. The Dirac Lagrangian for a free-electron field wðxÞ is given by a   L½wðxÞ; wðxÞ; owðxÞ ¼ wðxÞðic oa mÞwðxÞ ð12Þ  the positron field. We demand that (12) stays where ca are Dirac matrices, a ¼ 0; 1; 2; 3 is the Lorentz index and wðxÞ invariant under the local gauge transformation w0 ðxÞ ¼ e 0 ieðxÞ wðxÞ ð13Þ ieðxÞ  w ðxÞ ¼ e wðxÞ Invariance of (12) under (13) is equivalent to the invariance of the corresponding differential action since (13) is an internal field transformation independent of space–time. The derivative term in (12) transforms as  w0 ðxÞoa w0 ðxÞ ¼ wðxÞo a wðxÞ  iwðxÞo a eðxÞwðxÞ ð14Þ It is seen that the presence of the second term makes the Lagrangian dependent on oa eðxÞ and prohibits its invariance under (13). The canonical way to eliminate this dependence is to add a new vector field Aa ðxÞ, called a gauge field, and require the local transformation law for this new field to cancel the oa eðxÞ dependence in (12). The gauge field can be introduced into the theory by generalizing the ordinary differential operator oa to Da , in which Da wðxÞ ¼ ½oa þ ieAa ðxÞwðxÞ ð15Þ Here e is a free coupling parameter that can be identified with the electric charge and Aa ðxÞ with the photon field. Da wðxÞ is named the covariant derivative of the Dirac field because it transforms in the same manner as wðxÞ under the local gauge transformation, that is ½Da wðxÞ0 ¼ e ieðxÞ Da wðxÞ ð16Þ Invariance of the theory describing electrons interacting with photons under the local gauge transformation (13) makes QED an Abelian U(1) symmetry. Standard model accommodates richer systems of non-Abelian gauge theories corresponding to SU(2), SU(2) · U(1) and SU(3) · SU(2) · U(1) symmetries. These local symmetries give rise to the electroweak and QCD models that account for the existence of weak vector bosons (W, Z) and the gluon octet [18,21,25]. ARTICLE IN PRESS E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx 5 6. Equivalence of the local gauge transformation to a local field scaling The standard representation of the Dirac field as a doublet of left and right components is [18,25]   wR ðxÞ wðxÞ ¼ wL ðxÞ ð17Þ A local internal transformation wðxÞ ! w0 ðxÞ is equivalent to a U(1) rotation in the w––space and is known as a local gauge transformation. The generic form of this operation is " #    cos eðxÞ sin eðxÞ wR ðxÞ w0R ðxÞ 0 w ðxÞ ¼ ¼ ð18Þ sin eðxÞ cos eðxÞ wL ðxÞ w0L ðxÞ in which eðxÞ is an arbitrary angle of rotation dependent on the space–time point under consideration. If the rotation is infinitesimal, then " #    1 eðxÞ wR ðxÞ w0R ðxÞ ð19Þ ¼ eðxÞ 1 wL ðxÞ w0L ðxÞ In a similar fashion, starting from (6) wðxÞ ¼ w0 ðxÞsmw ðxÞ ð20Þ and using the representation (17) of the Dirac field, it can be seen that the scaling function must be a 2 · 2 matrix. It follows that the same local gauge transformation wðxÞ ! w0 ðxÞ can be formulated as a local scale transformation according to " #    m1w ðxÞ m2w ðxÞ wR ðxÞ w0R ðxÞ k k ð21Þ ¼ m3w ðxÞ w0L ðxÞ k km4w ðxÞ wL ðxÞ upon a redefinition of the resolution scale as s ! ks. Considering again the infinitesimal U(1) rotation and equating the respective entries of the two transformation matrices we obtain km1w ðxÞ ¼ km4w ðxÞ ¼ 1 km2w ðxÞ ¼ eðxÞ km3w ðxÞ ¼ eðxÞ ð22Þ proving the formal equivalence of the two operations. Demanding invariance of Lagrangian (12) under a local gauge transformation amounts to demanding invariance of (12) under a local transformation of the scaling function. Stated differently, (12) must not depend on the arbitrary choice of either rotation angle eðxÞ or scaling function kmw ðxÞ . 7. Local scale invariance of the free Dirac theory Following (2) and postulate P2, the principle of local scale invariance applied to (12) is explicitly defined by the ansatz   kmow ðxÞ oa wðxÞ ¼ kL½wðxÞ; oa wðxÞ L½kmw ðxÞ wðxÞ; ð23Þ  mw ðxÞ ¼ mw ½wðxÞ; oa wðxÞ  mow ðxÞ ¼ mow ½wðxÞ; oa wðxÞ ð24Þ where Scaling the Lagrangian by k maintains the full covariance of the equation of motion since [18,21] o dðkLÞ dðowÞ dðkLÞ ¼0 dw Setting m ¼ 0 in (12), from (12) and (23) we find ð25Þ ARTICLE IN PRESS 6 E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx mw ðxÞ þ mow ðxÞ ¼ 1 ð26Þ Taking into consideration (4) and using the shorthand notations uw ðxÞ ¼ ln jwðxÞj  uw ðxÞ ¼ ln jwðxÞj ð27Þ uow ðxÞ ¼ ln jowðxÞj yields   uw ðxÞ uw ðxÞ uw ðxÞ mow ; uow ðxÞ uow ðxÞ uow ðxÞ   uw ðxÞ uw ðxÞ uw ðxÞ mow ; mw ðxÞ ¼ uow ðxÞ uow ðxÞ uow ðxÞ mw ðxÞ ¼ ð28Þ Here mow is selected to represent the freely chosen exponent whereas mw , mw are related via (26) and (28). On the basis of (23), (26) and (28) it is concluded that invariance of Lagrangian (12) under a local scale transformation is accomplished without introducing the electromagnetic interaction. Maxwell field has been effectively removed and replaced by geometrical constraints (26) and (28) imposed on the field exponents (24). The same line of arguments applies to richer local symmetries involving non-abelian gauge fields mentioned at the end of Section 5. For the sake of simplicity and concision, these models are not discussed here. Next section applies the concept of local scale invariance to general relativity. 8. Local scale invariance in general relativity It is known that a deep connection exists between the concept of gauge invariance and the geometry of curved space– time continuum in general relativity [18,21]. Demanding invariance of the relativistic action in relation to arbitrary frames of reference leads to the introduction of gravitational field and the passage from Euclidean to the Riemannian metric of space–time. It was shown in the last section that the requirement of local scale invariance applied to the Dirac theory of free electrons offsets the need to introduce electromagnetic interactions between electrons, as it is done in the standard model. In this sense, local scale invariance effectively generates a gauge-free theory. The goal of this section is to prove that a similar argument holds for general relativity, whereby local scaling of space–time coordinates results in local cancellation of the gravitational field. 8.1. General relativity as a gauge invariant field theory In the absence of gravitation, special relativity is invariant under global Lorentz coordinate transformations that leave the action and the Minkowski metric tensor invariant. Let xr be the four-dimensional coordinates of a point (r ¼ 0; 1; 2; 3) referred to an inertial frame specified in some arbitrary manner. Let xq represent the coordinates of the same point relative to another inertial frame. In standard notation, the global Lorentz transformation between the two frames is described by the following set of relations [26] xq ¼ Kqr xr gqr ¼ Kaq Kbr gab gqr ¼ diagð1; 1; 1; 1Þ ð29Þ ds2 ¼ gqr dxq dxr ¼ gab dxa dxb where a; b; q; r ¼ ð0; 1; 2; 3Þ are Lorentz indices, Kqr is the 4 · 4 Lorentz transformation matrix, gqr is the Minkowski metric tensor and ds2 is the square of the proper time differential. Scalar and vector fields transform according to  ðxÞ ¼ uðxÞ u q V ðxÞ ¼ Kqr V r ðxÞ ð30Þ The standard procedure for introducing gravitational interaction is to convert these global transformations into local and arbitrary coordinate transformations according to ARTICLE IN PRESS E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx xq ¼ xq ðxÞ dxq ¼ Xqr ðxÞ dxr 7 ð31Þ The metric tensor is a coordinate dependent field that generalizes the Minkowski metric tensor such as to keep the proper time differential invariant a b gqr ðxÞ ¼ Xq Xr gab ðxÞ ds2 ¼ gqr ðxÞ dxq dxr ¼ gab ðxÞ dxa dxb ð32Þ in which q Xr ¼ ½Xrq  1 ð33Þ In the presence of gravitation, scalar and vector fields obey local transformation laws defined by  ðxÞ ¼ uðxÞ u ð34Þ q V ðxÞ ¼ Xqr ðxÞV r ðxÞ As with the case of the free Dirac theory discussed in Sections 6 and 7, it is apparent from (31) that a local coordinate transformation is equivalent to a local coordinate scaling upon performing the matrix substitution Xqr ðxÞ ! kmq;r ðxÞ ð35Þ 8.2. Invariance of the action and proper time under local scale transformations Consider now the simple case of a single classical particle in motion. The differential action (11a) for a free massless particle embedded in Euclidean space–time is given by dS ¼ Lðv; p0 Þ dt ð36Þ where the Lagrange function is [13–15] Lðv; p0 Þ ¼ p0 g va vb ¼ 2 ab p0 dxa dxb gab 2 ds ds ð37Þ vÞ is the four-velocity vector, s is the proper time of the particle and p0 Here gab is the Minkowski metric tensor, va ¼ ð1;~ its energy. The Lagrange function (37) vanishes identically since massless particles travel at the speed of light, which implies va va ¼ 0. The differential action (36) and the proper time differential ds stay invariant under the local and arbitrary coordinate change (31) from the first frame of reference (unbarred) to the second one (barred) if p0 dxa dxb p0 dxq dxr gab dt ¼ gqr dt 2 ds ds ds ds 2 ð38Þ Here p0 and dt are the energy and time differential measured relative to the barred frame. Proceeding by analogy with Sections 6 and 7, one can invoke the equivalence of a local coordinate change to a local coordinate scale transformation according to (35). As a result, we demand invariance of (37) and ds to the local scaling of coordinates 1 dxq ¼ kmq;a ðxÞ dxa dxr ¼ kmr;b ðxÞ dxb ð39Þ 1 For simplicity we assume here that exponents mq;a ðxÞ and mr;b ðxÞ are independent of ox. Note also that grouping of index pairs ðq; aÞ and ðr; bÞ is chosen to reflect the transition from the barred frame of reference to the unbarred one. ARTICLE IN PRESS 8 E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx The energy of a relativistic particle is dependent on the coordinate frame chosen to represent its motion and thus p0 ¼ p0 ðxÞ. Near criticality (as l ! lc ), all observables are quasi-homogeneous functions of the resolution scale and p0 ðxÞ transforms as 2 p0 ðxÞ ¼ p0 ðxÞkmp0 ðxÞ ð40Þ where mp0 ðxÞ represents the corresponding energy scaling exponent. From (39), the time differential is expected to change as dt ¼ km0;a ðxÞ dxa ð41Þ To streamline the derivation and cast results in a simpler form, we choose to work in the barred frame having the dominant connection along the temporal component, that is jkm0;a ðxÞ j  jkm0;0 ðxÞ j ð42Þ for a ¼ 1; 2; 3. Under this assumption (41) becomes dt ¼ km0;0 ðxÞ dt ð43Þ Following (2) and postulate P2, the requirement of local scale invariance is Lðkmq;a ðxÞ va kmr;b ðxÞ vb ; kmp0 ðxÞ p0 Þkm0;0 ðxÞ dt ¼ kLðva vb ; p0 Þ dt 0 ð44Þ for the action differential and ds2 ¼ gqr dxq dxr ¼ gab dxa dxb 0 ð45Þ for the square of the proper time differential. Substitution of (39), (40) and (43) into (44) and (45) yields mq;a ðxÞ þ mr;b ðxÞ þ mp0 ðxÞ þ m0;0 ðxÞ ¼ 1 ð46aÞ mq;a ðxÞ þ mr;b ðxÞ ¼ 1 ð46bÞ from which we obtain, by comparison mp0 ðxÞ þ m0;0 ðxÞ ¼ 0 According to (4), the above exponents satisfy additional constraints, namely
 wp0 ðxÞ wp0 ðxÞ wq;a ðxÞ wr;b ðxÞ mp0 ðxÞ ¼ ; m0;0 ; wt ðxÞ wt ðxÞ wt ðxÞ wt ðxÞ
 wp0 ðxÞ wq;a ðxÞ wr;b ðxÞ wq;a ðxÞ ; mq;a ðxÞ ¼ m0;0 ; wt ðxÞ wt ðxÞ wt ðxÞ wt ðxÞ
 wp0 ðxÞ wq;a ðxÞ wr;b ðxÞ wr;b ðxÞ ; mr;b ðxÞ ¼ m0;0 ; wt ðxÞ wt ðxÞ wt ðxÞ wt ðxÞ ð47Þ ð48Þ Here we have employed the shorthand notations wp0 ðxÞ ¼ lnðp0 ðxÞÞ wt ðxÞ ¼ ln jtðxÞj wq;a ðxÞ ¼ ln jdxq ðxÞj wr;b ðxÞ ¼ ln jdxr ðxÞj ð49Þ and assumed that m0;0 is freely chosen. On the basis of (44), (45), (46), (47) and (48) it is concluded that invariance of the differential action (36) and proper time under a local and arbitrary coordinate transformation is accomplished without introducing the gravitational interaction. In 2 This scaling behavior follows from the relationship between linear momentum and wavelength of a free particle in the de Broglie picture. ARTICLE IN PRESS E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx 9 this manner the gravitational field is effectively transformed away by using a local scale transformation that leaves the proper time invariant and the metric Euclidean. The Riemannian metric is replaced by geometrical constraints (46), (47) and (48). We close this section with the following three observations: (a) The statistical nature of exponents outlined in Section 4 allows a straightforward interpretation of (26) and (46b): for each random value of the first term in the sum there is a corresponding random value for the second term that identically satisfies these constraints. (b) (46b) suggests the non-commutative nature of on Cantorian space–time. This fact is related to the asymmetry of the Lorentz index since, in general dxq 6¼ dxr dxa 6¼ dxb and mq;a ðxÞ 6¼ mr;b ðxÞ (c) (47) may be naturally linked to the opposite scaling behavior of time and energy as conjugate observables. 9. Summary and concluding remarks We have developed field theoretic arguments supporting the unification of relativistic gravity with standard model interactions on Cantorian space–time. The approach is built on the equivalence between local gauge invariance and the local scale invariance of space–time and matter fields near the transition boundary l ! lc , where lc is the Cohen– Kaplan threshold. It is found that the non-commutative nature of Cantorian space–time follows from the local scale invariance of the proper time differential. Relativistic gravity and gauge fields are embedded in the underlying dimensionality of Cantorian space–time for l > lc and branch out as distinct interactions for l < lc . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] Sittler L, Hinrichsen HJ. Phys A: Math Gen 2002;35:10531–8. Binney JJ et al. The theory of critical phenomena. Oxford: Clarendon Press; 1992. Grosse H. Models in statistical physics and quantum field theory. Berlin Heidelberg: Springer-Verlag; 1988. Bock RD. arXiv:gr-qc/0312024. Wesson P. Gravity, particles and astrophysics. Dordrecht: Reidel; 1980. Hehl W et al. Foundations of Physics 1989;19:1075. Bjorken JD. Phys Rev 1967;163:1767. Bjorken JD. Phys Rev 1969;179:1547. Nottale L. Chaos, Solitons & Fractals 2001;12:1577–83, and included references. El Naschie MS. Chaos, Solitons & Fractals 2003;17:797–807. El Naschie MS. Chaos, Solitons & Fractals 2000;11(7):1149–62. El Naschie MS. Chaos, Solitons & Fractals 2000;11(9):1459–69. Brizard A.J, et al. arXiv:hep-ph/9911462. Brink L et al. Phys Lett B 1976;64:435. Deser S, Zumino B. Phys Lett B 1976;65:369. Sidharth BG. Chaos, Solitons & Fractals 2001;12:1449–57. Sidharth BG. Chaos, Solitons & Fractals 2001;12:2143–7. Cheng TP, Li LF. Gauge theory of elementary particle physics. Oxford: Clarendon Press; 1984. Cohen AG et al. Phys Rev Lett 1999;82:4971. Carmona JM, Cortes JL. Phys Rev D 2001;65:025006. Ryder LH. Quantum field theory. Cambridge University Press; 1989. Goldfain E. Chaos, Solitons & Fractals 2004;19:1023–30. Elze HT. arXiv: quant-ph/9/10063; Zurek WH. Phys Today 1991;44(10):36. [24] Zeh HD. Phys Lett A 1993;172:189. [25] Kaku M. Quantum field theory. Oxford University Press; 1993. ARTICLE IN PRESS 10 [26] [27] [28] [29] [30] E. Goldfain / Chaos, Solitons and Fractals xxx (2004) xxx–xxx Landau LD, Lifschitz EM. The classical theory of fields. Butterworth–Heineman; 1997. Goldfain E. Chaos, Solitons & Fractals 2004;22:513–20. Cooper NG, West GB. Particle physics: a Los Alamos primer. Cambridge University Press; 1989. Hochberg D, Peres Mercader J. J Phys Lett A 2002;296:272. Creswick RJ, Farach HA, Poole Jr CP. Introduction to renormalization group methods in physics. New York: John Wiley and Sons; 1992. [31] Peitgen HO, Jurgens H, Saupe D. Chaos and fractals: new frontiers in science. New York: Springer-Verlag; 1992. [32] Nottale L. Int J Mod Phys A 1989;4(19):5047–117. [33] Gaberdiel MR. arXiv:hep-th/9910156.