ON YANG’S NONCOMMUTATIVE SPACETIME ALGEBRA,HOLOGRAPHY, AREA-QUANTIZATION AND C-SPACE RELATIVITY

Carlos CastroCenter for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta GA. 3031July, 2004

Abstract

An isomorphism between Yang’s Noncommutative space-time algebra (involving two length scales) andthe holographic area coordinates algebra of C-spaces (Clifford spaces) is constructed via a AdS5 space-time which is instrumental in explaining the origins of an extra (infrared) scale R in conjunction to the(ultraviolet) Planck scale λ characteristic of C-spaces. Yang’s space-time algebra allowed Tanaka to explainthe origins behind the discrete nature of the spectrum for the spatial coordinates and spatial momentawhich yields a minimum length-scale λ (ultraviolet cutoff) and a minimum momentum p = h/R ( maximallength R, infrared cutoff ) . The double-scaling limit of Yang’s algebra λ → 0, R →∞, in conjunction withthe large n →∞ limit, leads naturally to the area quantization condition λR = L2 = nλ2 ( in Planck areaunits ) given in terms of the discrete angular-momentum eigenvalues n. The generalized Weyl-Heisenbergalgebra in C-spaces is constructed which encodes both features from the Yang’s Noncommutative space-timealgebra and the ordinary Weyl-Heisenberg algebra. We conclude with a discussion on the implications ofNoncommutative QM and QFT’s in C-spaces.

1. Introduction

In recent years we have argued that the underlying fundamental physical principle behind string theory,not unlike the principle of equivalence and general covariance in Einstein’s general relativity, might well berelated to the existence of an invariant minimal length scale (Planck scale) attainable in nature. A scalerelativistic theory involving spacetime resolutions was developed long ago by Nottale where the Planckscale was postulated as the minimum observer independent invariant resolution [1] in Nature. Since “points”cannot be observed physically with an ultimate resolution, they are fuzzy and smeared out into fuzzy ballsof Planck radius of arbitrary dimension. For this reason one must construct a theory that includes alldimensions (and signatures) on the equal footing. Because the notion of dimension is a topological invariant,and the concept of a fixed dimension is lost due to the fuzzy nature of points, dimensions are resolution-dependent, one must also include a theory with all topologies as well. It is our belief that this may lead tothe proper formulation of string and M theory.

In [2] we applied this Extended Scale Relativity principle to the quantum mechanics of p-branes whichled to the construction of C-space (a dimension category) where all p-branes were taken to be on the samefooting; i.e. transformations in C-space reshuffled a string history for a five-brane history, a membranehistory for a string history, for example. It turned out that Clifford algebras contained the appropriatealgebro-geometric features to implement this principle of polydimensional transformations [3, 4, 5].

Clifford algebras have been a very useful tool for a description of geometry and physics [4, 5, 6, 7, 8].In [3,5] it was proposed that every physical quantity is in fact a polyvector, that is, a Clifford number or aClifford aggregate. Also, spinors are the members of left or right minimal ideals of Clifford algebra, whichmay provide the framework for a deeper understanding of sypersymmetries, i.e., the transformations relatingbosons and fermions. The Fock-Stueckelberg theory of a relativistic particle [4] can be embedded in theClifford algebra of spacetime [3]. Many important aspects of Clifford algebra are described in [3,5,6,7,8]

Using these methods the bosonic p-brane propagator, in the quenched mini superspace approximation,was constructed in [9]; the logarithmic corrections to the black hole entropy based on the geometry of Cliffordspace (in short C-space) were obtained in [14]; the action for higher derivative gravity with torsion from thegeometry of C-spaces and how the Conformal agebra of spacetime emerges from the Clifford algebra wasdescribed in [29]; the resolution of the ordering ambiguities of QFT in curved spaces was resolved by [3].

In this new physical theory the arena for physics is no longer the ordinary spacetime, but a more generalmanifold of Clifford algebra valued objects, polyvectors. Such a manifold has been called a pan-dimensional

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continuum [5] or C-space [2]. The latter describes on a unified basis the objects of various dimensionality:not only points, but also closed lines, surfaces, volumes,.., called 0-loops (points), 1-loops (closed strings)2-loops (closed membranes), 3-loops, etc.. It is a sort of a dimension category, where the role of functorialmaps is played by C-space transformations which reshuffles a p-brane history for a p′-brane history or amixture of all of them, for example.

The above geometric objects may be considered as to corresponding to the well-known physical objects,namely closed p-branes. Technically those transformations in C-space that reshuffle objects of differentdimensions are generalizations of the ordinary Lorentz transformations to C-space. In that sense, the C-space is roughly speaking a sort of generalized Penrose-Twistor space from which the ordinary spacetimeis a derived concept. In [2] we derived the minimal length uncertainty relations as well as the full blownuncertainty relations due to the contributions of all branes of every dimensionality, ranging from p = 0all the way to p = ∞. In [14] we extended this derivation to include the maximum Planck Temperaturecondition .

The contents of this work is the following. In section 2.1, 2.2. 2.3 we will review the basic features ofthe Extended Relativity Theory in C-spaces and the explicit derivation from first principles of the minimallength modfied Heisenberg uncertainty relations. This derivation is based on the effective-running Planck”constant” heff (p2) (energy dependent) that results from a breakdown of poly-dimensional covariance inC-spaces.

In section 3 we show the relationship among Yang’s 4D Noncommutative space-time algebra [ 16 ] ( interms of 8D phase space coordinates ), the holographic area coordinates algebra of the C-space associatedwith a 6D Clifford algebra, and the Euclideanized AdS5 spaces. The role of AdS5 was instrumental inexplaining the origins of an extra ( infrared ) scale R in conjunction to the ( ultraviolet ) Planck scale λcharacteristic of C-spaces. Tanaka [ 17 ] gave the physical and mathematical derivation of the discretespectra for the spatial coordinates and spatial momenta that yields a minimum length-scale λ ( ultravioletcutoff in energy ) and a minimum momentum p = h/R ( maximal length R, infrared cutoff ) . The double-scaling limit, λ → 0, R → ∞, in conjunction with the large n → ∞ limit, leads to the area-quantizationcondition λR = L2 = nλ2 in units of the Planck area, where n is the angular momentum Σ56 = (1/h)M56

eigenvalue.In section 4 we proceed with the canonical quantization in C-spaces and construct the modified Weyl-

Heisenberg algebra. We show rigorously how the Noncommutative Geometry of the C-space coordinates Xis a direct consequence of the modified Weyl-Heisenberg algebra in C-spaces in contradisctinction with theordinary phase space commutators [xi, pj ] = ihδij which imply that [xi, xj ] = 0 as a result of the Jacobiindentities. In the final section 5 we discuss some implications of Noncommutative QM and QFT’s inC-spaces.

2. The Extended Scale Relativity in Clifford -spaces

2.1 Extending Relativity from Minkowski spacetime to C-space

We embark into the extended relativity theory in C-spaces by a natural generalization of the notion ofa space- time interval in Minkwoski space to C-space:

dX2 = dΩ2 + dxµdxµ + dxµνdxµν + ... (2.1)

The Clifford valued poly-vector:

X = XMEM = Ω1 + xµγµ + xµνγµ ∧ γν + ...xµ1µ2....µDγµ1 ∧ γµ2 .... ∧ γµD. (2.2a)

denotes the position of a polyparticle in a manifold, called Clifford space or C-space. The series of terms in(2) terminates at a finite value depending on the dimension D. A Clifford algebra Cl(r, q) with r + q = Dhas 2D basis elements. For simplicity, the gammas γµ correspond to a Clifford algebra associated with a flatspacetime :

1/2γµ, γν = ηµν . (2.2b)

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but in general one could extend this formulation to curved spacetimes with metric gµν .The connection to strings and p-branes can be seen as follows. In the case of a closed string (a 1-

loop) embedded in a target flat spacetime background of D-dimensions, one represents the projections ofthe closed string (1-loop) onto the embedding spacetime coordinate-planes by the variables xµν . Thesevariables represent the respective areas enclosed by the projections of the closed string (1-loop) onto thecorresponding embedding spacetime planes. Similary, one can embed a closed membrane (a 2-loop) ontoa D-dim flat spacetime, where the projections given by the antisymmetric variables xµνρ represent thecorresponding volumes enclosed by the projections of the 2-loop along the hyperplanes of the flat targetspacetimr background.

This procedure can be carried to all closed p-branes ( p-loops ) where the values of p are p =0, 1, 2, 3, ....D − 2. The p = 0 value represents the center of mass and the coordinates xµν , xµνρ.... havebeen coined in the string-brane literature [9,10] as the holographic areas, volumes, ...projections of thenested family of p-loops ( closed p-branes ) onto the embedding spacetime coordinate planes/hyperplanes.

Since the D-dimensional Planck scale is given explicitly in terms of the Newton constant : ΛD =(GN )1/(D−2), in natural units of h = c = 1, one can see that when D = ∞ the value of ΛD is thenΛ∞ = G0 = 1 ( assuming a finite value of G ). Hence in D = ∞ the Planck scale has the natural value ofunity ! . This is important if one wishes to study the convergence property of the series of terms appearingin eq-( 3 ) below in the extreme case D = ∞. For the time being, we shall focus solely on a finite value ofD to avoid any serious algebraic convergence problems.

The classification of Clifford algebras Cl(r, q) in D = r+q dimensions ( modulo 8 ) for different values ofthe spacetime signature r, q is discussed, for example, in the book of Porteous [27]. All Clifford algebras canbe understood in terms of CL(8) and the CL(k) for k less than 8 due to the modulo 8 Periodicity theorem

CL(n) = CL(8)× Cl(n− 8)

. Cl(r, q) is a matrix algebra for even n = r + q or the sum of two matrix algebras for odd n = r + q.Depending on the signature, the matrix algebras may be real, complex, or quaternionic. For furher detailswe refer to [27] .

If we take the differential dX and compute the scalar product among two polyvectors < dX†dX >scalar

we obtain the C-space extension of the particles proper time in Minkwoski space. The symbol X+ denotesthe reversion operation and involves reversing the order of all the basis γµ elements in the expansion of X. It is the analog of the transpose ( Hermitian ) conjugation. The C-space proper time associated with apolyparticle motion is then :

dΣ2 = (dΩ)2 + Λ2D2dxµdxµ + Λ2D4dxµνdxµν + .. (2.3)

Here we have explicitly introduced the Planck scale Λ since a length parameter is needed in order to tieobjects of different dimensionality together: 0-loops, 1-loops,..., p-loops. Einstein introduced the speed oflight as a universal absolute invariant in order to “unite” space with time (to match units) in the Minkwoskispace interval:

ds2 = c2dt2 − dxidxi. (2.4)

A similar unification is needed here to “unite” objects of different dimensions, such as xµ, xµν , etc... ThePlanck scale then emerges as another universal invariant in constructing an extended scale relativity theoryin C-spaces [2].

To continue along the same path, we consider the analog of Lorentz transformations in C-spaces whichtransform a poly-vector X into another poly-vector X ′ given by X ′ = RXR−1 with

R = eθAEA = exp [(θ1 + θµγµ + θµ1µ2γµ1 ∧ γµ2 .....)]. (2.5)

and

R−1 = e−θAEA = exp [−(θ1 + θνγν + θν1ν2γν1 ∧ γν2 .....)]. (2.6)

where the theta parameters in (2.5, 2.6) are the components of the Clifford-value parameter Θ = θAEA :

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θ; θµ; θµν ; .... (2.7)

they are the C-space version of the Lorentz rotations/boosts parameters.Since a Clifford algebra admits a matrix representation, one can write the norm of a poly-vectors in

terms of the trace operation as: ||X||2 = Trace X2 Hence under C-space Lorentz transformation the normsof poly-vectors behave like follows:

Trace X ′2 = Trace [RX2R−1] = Trace [RR−1X2] = Trace X2. (2.8)

These norms are invariant under C-space Lorentz transformations due to the cyclic property of the traceoperation and RR−1 = 1.

2.2 On the Minimal Planck scale, Superluminal Propagation and C-space Relativity

Long time ago L.Nottale proposed to view the Planck scale as the absolute minimum invariant (observerindependent) scale in Nature in his formulation of scale relativity [1]. We can apply this idea to C-spaces bystudying the analog of the Minkowski space-time signature (+,−,−,−) by choosing a C-space metric GMN

whose signatures relative to the scalar component Ω of the polyvectors coordinates are :

||dX||2 = dΣ2 = (dΩ)2[1− Λ2D−2 (dxµ)2

(dΩ)2− Λ2D−4 (dxµν)2

(dΩ)2− Λ2D−6 (dxµνρ)2

(dΩ)2− ..]

||dX||2 = dΣ2 = (dΩ)2[1− (Λλ1

)2D−2 − (Λλ2

)2D−4 − (Λλ3

)2D−6 − ...]. (2.9)

where the sequence of variable scales ( parameters ) λ1, λ2, λ3, .... just reflect the magnitudes of the gener-alized holographic velocities as follows:

(dxµ)2

(dΩ)2≡ (V1)2 = (

1λ1

)2D−2.

(dxµν)2

(dΩ)2≡ (V2)2 = (

1λ2

)2D−4.

(dxµνρ)2

(dΩ)2≡ (V3)2 = (

1λ3

)2D−6. (2.10).

etc.... By a simple use of the chain-rule one can relate the velocities defined w.r.t the scalar Ω variable withthose velocities defined with respect to the coordinate-clock variable t = xo. For example, V M = dXM/dΩ =(dXM/dt)/(dΩ/dt). It is clear now that if ||dX||2 ≥ 0 in ( 2.9) then the sequence of variable lengths λn

in ( 2.10) cannot be smaller than the Planck scale Λ. This is analogous to a situation with the Minkoswkiinterval:

ds2 = c2dt2[1− v2

c2]. (2.11)

ds2 ≥ 0 if, and only if, the velocity v does not exceed the speed of light. If any of the λn were smaller thanthe Planck scale the C-space interval ( 2.9 ) will become tachyonic-like dΣ2 < 0. Photons in C-space aretensionless branes/loops.

This upper holographic-velocity bound does not necessarily translate into a lower bound on the actualvalues of lengths, areas, volumes....without the introduction of quantum mechanical considerations [ 44] .One possibility is that the upper limiting speed of light and the upper bound of the momentum mpc of aPlanck-mass elementary particle (the so-called Planckton in the literature) generalizes now to an upper-bound in the p-loop holographic velocities and the p-loop holographic momenta associated with elementaryclosed p-branes whose tensions are given by powers of the Planck mass. And the latter upper bounds on the

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holographic p-loop momenta implies a lower-bound on the holographic areas, volumes,..., resulting from thestring/brane uncertainty relations that we shall derive in the next section.

Thus, Quantum Mechanics is required to implement the postulated principle of minimal lengths, areas,volumes...and which cannot be derived from the classical geometry alone. The emergence of the minimalPlanck areas occurs also in the Loop Quantum Gravity program [ 41 ] where the expecation values ofthe Area operator are given by multiples of Planck area. This area-quantization is precisely what we willshow in section 3 based on the isomorphism of Yang’s Noncommutative space-time algebra [16] and theholographic area coordinates algebra in C-space.

To finalize we will discuss briefly the possibility of superluminal propagation [44] . It is known thattachyons can induce a breakdown of causality. The simplest way to see why causality is violated whentachyons are used to exchange signals is by writing the temporal displacements δt = tB − tA between twoevents (in Minkowski space-time) in two different frames of reference:

(δt)′ = (δt)cosh(ξ) +δx

csinh(ξ) = (δt)[cosh(ξ) + (

1c

δx

δt)sinh(ξ)] =

(δt)[cosh(ξ) + (βtachyon)sinh(ξ)]. (2.12)

the boost parameter ξ is defined in terms of the velocity as βframe = vframe/c = tanh(ξ), where vframe

is is the relative velocity ( in the x-direction ) of the two reference frames and can be written in termsof the Lorentz-boost rapidity parameter ξ by using hyperbolic functions. The Lorentz dilation factor iscosh(ξ) = (1 − β2

frame)−1/2 ; whereas βtachyon = vtachyon/c is the beta parameter associated with the

tachyon velocity δx/δt . By emitting a tachyon along the negative x -direction one has βtachyon < 0 andsuch that its velocity exceeds the speed of light |βtachyon| > 1 A reversal in the sign of (δt)′ < 0 in theabove boost transformations occurs when the tachyon velocity |βtachyon| > 1 and the relative velocity of thereference frames |βframe| < 1 obey the inequality condition :

(δt)′ = (δt)[cosh(ξ)− |βtachyon|sinh(ξ)] < 0 ⇒ 1 <1

tanh(ξ)=

1βframe

< |βtachyon|. (2.13)

thereby resulting in a causality violation in the primed reference frame since the effect ( event B ) occursbefore the cause ( event A ) in the primed reference frame.

In the theory considered here, there are no tachyons in C-space, because physical signals in C-space areconstrained to live inside the C-space-light cone [ 44 ] . However, certain worldlines in C-space, when pro-jected onto the Minkowski subspace M4, can appear as worldlines of ordinary tachyons outside the lightconein M4. In C-space the dynamics refers to a larger space. Minkowski space is just a subspace of C-space.”Wordlines” now live in C-space that can be projected onto the Minkwoski subspace M4 . Furthermore, oneis enlarging the ordinary Lorentz group to a larger group of C-space Lorentz transformations which involvepoly-rotations and generalizations of boosts transformations. In particular, the C-space generalization of theordinary boost transformations associated with the boost rapidity parameter ξ such that tanh(ξ) = βframe

will involve now the family of C-space boost rapidity parameters θt1, θt12, θt123, ....θt123..., ... since boosts arejust ( poly ) rotations along directions involving the time coordinate. Thus, one is replacing the ordinaryboost transformations in Minkowski spacetime for the more general C-space boost transformations as we gofrom one frame of reference to another frame of reference.

Due to the linkage among the C-space coordinates (poly-dimensional covariance) when we envision anordinary boost along the x1- direction, we must not forget that it is also interconnected to the area-boosts inthe x12-direction as well, and, which in turn, is also linked to the x2 direction. Because the latter directionis transverse to the original tachyonic x1-motion, the latter x2-boosts won’t affect things and we mayconcentrate on the area-boosts along the x12 direction involving the θt12 parameter that will appear in theC-space boosts and which contribute to a crucial extra term in the transformations such that no sign-changein δt′ will occur.

More precisely, let us set all the values of the theta parameters to zero except the parameters θt1 andθt12 related to the ordinary boosts in the x1 direction and area-boosts in the x12 directions of C-space. Thisrequires, for example, that one has at least one spatial-area component, and one temporal coordinate, whichimplies that the dimensions must be at least D = 2 + 1 = 3 . Thus, we have in this case :

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X ′ = RXR−1 = eθt1γt∧γ1+θt12γt∧γ1∧γ2XMEMe−θt1γt∧γ1−θt12γt∧γ1∧γ2 ⇒ X ′N = LNMXM . (2.14)

where LNM is given by the scalar part of the Clifford geometric product LN

M =< ENREMR−1 >0 When oneconcentrates on the transformations of the time coordinate, we have now that the C-space boosts do notcoincide with ordinary boosts in the x1 direction :

t′ = LtMXM =< EtREMR−1 >0 XM 6= (Lt

t)t + (Lt1)x

1. (2.15)

because of the extra non-vanishing θ parameter θt12 .This is because the rotor R includes the extra generator θt12γt∧γ1∧γ2 which will bring extra terms into

the transformations ; i.e. it will rotate the E[12] bivector- basis , that couples to the holographic coordinatesx12, into the Et direction which is being contracted with the Et element in the definition of Lt

M . Thereare extra terms in the C-space boosts because the poly-particle dynamics is taking place in C-space and allcoordinates XM which contain the t, x1, x12 directions will contribute to the C-space boosts in D = 3, sinceone is projecting down the dynamics from C-space onto the (t, x1) plane when one studies the motion of thetachyon in M4 .

Concluding, in the case when one sets all the theta parameters to zero, except the θt1 and θt12, theX ′ = RXMEMR−1 transformations will be :

(δt)′ = LtM (θt1; θt12)(δXM ) 6= Lt

t(δt) + Lt1(δx

1). (2.16)

due to the presence of the extra term Lt12(δX

12) in the transformations. In the more general case, when thereare more non-vanishing theta parameters , the indices M of the XM coordinates must be restricted to thosedirections in in C-space which involve the t, x1, x12, x123..... directions as required by the C-space poly-particledynamics. The generalized C-space boosts involve now the ordinary tachyon velocity component of the poly-particle as well as the generalized holographic areas, volumes, hyper-volumes...velocities V M = (δXM/δt)associated with the poly-vector components of the Clifford-valued C-space velocity.

Hence, at the expense of enlarging the ordinary Lorentz boosts to the C-space Lorentz boosts, and thedegrees of freedom of a point particle into an extended poly-particle by including the holographic coordinates,in C-space one can still have ordinary point-particle tachyons without changing the sign of δt, and withoutviolating causality, due to the presence of the extra terms in the C-space boosts transformations which ensureus that the sign of δt > 0 is maintained as we go from one frame of reference to another one. Naturally,if one were to freeze all the θ parameters to zero except one θt1 one would end up with the standardLorentz boosts along the x1 -direction and a violation of causality would occur for tachyons as a result ofthe sign-change in δt′ .

What seems remarkable in this scheme of things is the nature of the signatures and the emergence oftwo times. One of the latter is the local mode, a clock, represented by t and the other mode is a “global”one represented by the volume of the space-time filling brane Ω. For more details related to this Fock-Stuckelberg-type parameter see [3]. We must emphasize that one must not confuse these global and localtime modes with the two modes of time in other branches of science [13].

Another immediate application of thistheory is that one may consider “strings” and “branes” in C-spaces as a unifying description of all branes of different dimensionality. In fact, a unified action of allp-branes was written in [ 44] . As we have already indicated, since spinors are left/right ideals elements ofa Clifford algebra, a supersymmetry is then naturally incorporated into this approach as well. In particular,one can have world volume and target space supersymmetry simultaneously [44]. We hope that the C-space“strings” and “branes” may lead us towards discovering the physical foundations of string and M-theory.For other alternatives to supersymmetry see the work by Chisholm and Baylis [33].

Related to the minimal Planck scale, an upper limit on the maximal acceleration principle in Nature wasproposed by Cainello [28]. This idea is a direct consequence of a suggestion made years earlier by Max Bornon a Dual Relativity principle operating in phase spaces [32]. There is an upper bound on the four-force(maximal string tension or tidal forces in the string case) acting on a particle as well as an upper boundin the particles velocity. One can combine the maximum speed of light with a minimum Planck scale intoa maximal proper-accleration a = c2/Λ within the framework of Finsler geometry [18]. A thorough study

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of Finsler geometry and Clifford algebras has been undertaken by Vacaru [35] . Other several new physicalimplications of the maximal acceleration principle in Nature, like neutrino oscillations, have been studied by[34]. A variable fine structure constant, with the cosmological expansion of the Universe, has been proposedby us where an exact renormalization group-like equation governing the cosmological time (scale) variationof the fine structure constant was derived explicitly from this maximal acceleration principle in Nature [37].

2.3 The Generalized String/Brane Uncertainty Relations

Below we will review how the minimal length string uncertainty relations can be obtained fromthepolyparticle dynamics C-spaces [2]. The truly C-space invariant norm of a momentum polyvector is defined(after introducing suitable powers of the Planck scale in the sum in order to match units ) :

||P ||2 = π2 + pµpµ + pµνpµν + pµνρpµνρ + .... = M2 (2.12a)

A detailed discussion of the physical properties of all the components of the polymomentum P in fourdimensions and the emergence of the physical mass m in Minkowski spacetime has been provided in thebook by Pavsic [3]. The polymomentum in D = 4 can be written as :

P = µ + pµγµ + Sµνγµ ∧ γν + πµγ5γµ + mγ5. (2.12b)

where the pseudo-scalar component mγ5 is the one which contains the physical mass in Minkwoski spacetime.This justifies using the notation m for mass.

The most salient feature of the polyparticle dynamics in C-spaces is that one can start with a constrainedaction in C-space and arrive, nevertheless, at an unconstrained Stuckelberg action in Minkowski space (asubspace of C-space). It follows that pµ is a constant of motion pµpµ = m2 but m is no longer a fixedconstant entering the action but it is now an arbitrary constant of motion. The true constraint in C-spaceis :

PAPA = µ2 + pµpµ + πµπµ −m2 − 2SµνSµν = M2. (2.12c)

This is basically the distinction between the variable m and the fixed constant M . The variable m is theconjugate to the Stuckelberg evolution parameter s that allowed Pavsic to propose a natural solution of theproblem of time in Quantum Cosmology [ 3 ] .

Nottale has given convincing arguments why the notion of dimension is resolution dependent, and at thePlanck scale, the minimum attainable distance, the dimension becomes singular, that is blows-up. Settingaside at this moment the potential algebraic convergence problems when D = ∞, if we take the dimensionat the Planck scale to be infinity, then the norm P 2 will involve an infinite number of terms. It is preciselythis infinite series expansion which will reproduce all the different forms of the Casimir invariant massesappearing in kappa-deformed Poincare algebras [11,12]. As mentioned earlier, when D = ∞, the Planckscale appearing in the series expansion of ( 2.12 ) Λ∞ = G1/0 = 1.

It was discussed recently why there is an infinity of possible values of the Casimirs invariant M2 due toan infinite choice of possible bases. The parameter κ is taken to be equal to the inverse of the Planck scale.The classical Poincare algebra is retrieved when Λ = 0. The kappa-deformed Poincare algebra does not actin classical Minkwoski spacetime. It acts in a quantum-deformed spacetime. We conjecture that the naturaldeformation of Minkowski spacetime is given by C-space.

The way to generate all the different forms of the Casimirs M2 is by “projecting down” from the 2D-dim Clifford algebra to D-dim. One simply “slices” the 2D-dim mass-shell hyper-surface in C-space by aD-dimensional one. This is achieved by imposing the following constraints on the holographic componentsof the polyvector-momentum. In doing so one is explicitly breaking the poly-dimensional covariance and forthis reason one can obtain an infinity of possible choices for the Casimirs M2.

To demonstrate this, we impose the following constraints :

pµνpµν = a2(pµpµ)2 = a2p4. pµνρp

µνρ = a3(pµpµ)3 = a3p6. ...... (2.13)

What eqs-( 2.13 ) represent geometrically is the slicing of the 2D-dimensional mass-shell hypersurfacein C-space into D-dimensional regions (subspaces) parametrized solely in terms of the ordinary momentum

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coordinates pµ. This is the reason why decided to choose such constraint ( 2.13 ). There are many differentways to perform the slicing procedure in C-space depending on the choices of the coefficients an . Upondoing so the norm of the poly-momentum becomes:

P 2 =∑

n

anp2n = M2(1, a2, a3, ..., an, ...) (2.14)

Therefore, by a judicious choice of the coefficients an, and by reinserting the suitable powers of the Planckscale, which have to be there in order to combine objects of different dimensions, one can reproduce all thepossible Casimirs in the form:

M2 = m2[f(Λm/h)]2. m2 ≡ pµpµ = p2. (2.15)

To illustrate the relevance of polyvectors, we will summarize our derivation of the minimal length stringuncertainty relations [2]. Because of the existence of the extra holographic variables xµν , ... one cannotnaively impose [x, p] = ih due to the effects of the other components. The units of [xµν , pµν ] are of h2 and ofhigher powers of h for the other commutators. To achieve covariance in C-space which reshuffles objects ofdifferent dimensionality, the effective Planck constant in C-space should be given by a sum of powers of h.

This is not surprising. Classical C-space contains the Planck scale, which itself depends on h. Thisimplies that already at the classical level, C-space contains the seeds of the quantum space. At the nextlevel of quantization, we have an effective h that comprises all the powers of h induced by the commutatorsinvolving all the holographic variables. In general one must write down the commutation relations in terms ofpolyvector- valued quantities. In particular, the Planck constant will now be a Clifford number, a polyvectorwith multiple components. This will be the subject of section 5 .

The simplest way to infer the effects of the holographic coordinates of C-space on the commutationrelations is by working with the effective hthat appears in the nonlinear de Broglie dispersion relation. Themass-shell condition in C-space, after imposing the constraints among the holographic components, yieldsan effective mass M = mf(Λm/h). The generalized De Broglie relations, which are no longer linear, are [2]:

|Peffective| = |p|f(Λm/h) = heffective|k|. heffective = hf(Λm/h) =

h∑

an(Λm/h)2n. m2 = p2 = pµpµ = (hk)2. (2.16)

Using the effective heff , the well known relation based on the Schwartz inequality and the fact that|z| ≥ |Imz| we obtain:

∆xi∆pj ≥ 12| < [xi, pj ] > |. [xi, pj ] = iheffδij . (2.17)

Using the relations

< p2 >≥ (∆p)2. < p4 >≥ (∆p)4...... (2.18)

and the series expansion of the effective heff , we get for each component (we omit indices for simplicity):

∆x∆p ≥ 12h +

aΛ2

2h(∆p)2 + ............ (2.19)

This yields the minimal length string uncertainty relations:

∆x ≥ h

2∆p+

aΛ2

2h∆p..... (2.20)

By replacing lengths by times and momenta by energy one reproduces the minimal Planck time uncertaintyrelations.

The Physical interpretation of these uncertainty relations follow from the extended relativity principle.As we boost the string to higher trans-Planckian energies, part of the energy will always be invested intothe strings potential energy, increasing its length inbits of Planck scale sizes, so that the original string will

8

decompose into two, three, four....strings of Planck sizes carrying units of Planck momentum; i.e. the notionof a single particle/string loses its meaning beyond that point.

This reminds one of ordinary relativity, where boosting a massive particle to higher energy increasesitsspeed whilea part of the energy is also invested into increasing its mass. In this process the speed of lightremains the maximum attainable speed (it takes an infinite energyto recah it) and in our scheme the Planckscale is never surpassed. The effects of a minimal length can be clearly seen in Finsler geometries [18] havingboth a maximum four acceleration c2/Λ (maximum tidal forces) and a maximum speed . The Riemannianlimit is reached when the maximum four acceleration goes to infinity; i.e. The Finsler geometry “collapses”to a Riemannian one.

3. The Noncommutative Yang Spacetime algebra andArea Quantization derived from C-space Relativity

The main result of this section is that there is a subalgebra of the C-space operator-valued coordinateswhich is isomorphic to the Noncommutative Yang’s spacetime algebra [16] . This, in conjunction to thediscrete spectrum of angular momentum, leads to the discrete area-quantization in multiples of Planckareas. Namely, the 4D Yang’s Noncommutative space-time ( YNST ) algebra [16 ] ( written in terms of8D phase-space coordinates ) is isomorphic to the 15-dimensional subalgebra of the C-space operator-valuedcoordinates associated with the holographic areas of C-space. This connection between Yang’s algebra andthe 6D Clifford algebra is possible because the 8D phase-space coordinates xµ, pµ ( associated to a 4Dspacetime ) have a one-to-one correspondence to the Xµ5; Xµ6 holographic area-coordinates of the C-space(corresponding to the 6D Clifford algebra).

Furhermore, Tanaka [ 17] has shown that the Yang’s algebra [ 16 ] ( with 15 generators ) is related tothe 4D conformal algebra ( 15 generators ) which in turn is isomorphic to a subalgebra of the 4D Cliffordalgebra because it is known that the 15 generators of the 4D conformal algebra SO(4, 2) can be explicitlyrealized in terms of the 4D Clifford algebra as [ 29 ] :

Pµ = Mµ5 +Mµ6 = γµ(1 + γ5). Kµ = Mµ5 −Mµ6 = γµ(1− γ5). D = γ5. Mµν = i[γµ, γν ].. (3.1)

where the Clifford algebra generators :

1. γ0 ∧ γ1 ∧ γ2 ∧ γ3 = γ5. (3.2)

account for the extra two directions within the C-space associated with the 4D Cliffiord-algebra leavingeffectively 4 + 2 = 6 degrees of freedom that match the degrees of freedom of a 6D spacetime [29 ] . Therelevance of [ 29 ] is that it was not necessary to work directly in 6D to find a realization of the 4D conformalalgebra SO(4, 2) . It was possible to attain this by recurring solely to the 4D Clifford algebra as shown ineq-( 3.1 ) .

One can also view the 4D conformal algebra SO(4, 2) realized in terms of a 15-dim subalgebra of the6D Clifford algebra. The bivector holographic area-coordinates Xµν couple to the basis generators Γµ ∧ Γν .The bivector coordinates Xµ5 couple to the basis generators Γµ ∧ Γ5 where now the Γ5 is another generatorof the 6D Clifford algebra and must not be confused with the usual γ5 defined by eq-(3.2 ) . The bivectorcoordinates Xµ6 couple to the basis generators Γµ ∧ Γ6. The bivector coordinate X56 couples to the basisgenerator Γ5 ∧ Γ6.

In view of this fact that these bivector holographic area-coordinates in 6D couple to the bivectors basiselements Γµ ∧Γν , ... , and whose algebra is in turn isomorphic to the 4D conformal algebra SO(4, 2) via therealization in terms of the 6D angular momentum generators ( and boosts generators ) Mµν ∼ [Γµ,Γν ] ,Mµ5 ∼ [Γµ,Γ5],.... we shall define the holographic area coordinates algebra in C-space as the dual algebrato the SO(4, 2) conformal algebra ( realized in terms of the 6D angular momentum, boosts, generators interms of a 6D Clifford algebra generators as shown )

Notice that the conformal boosts Kµ and the translations Pµ in eq-( 3.1 ) do commute [Pµ, P ν ] =[Kµ,Kν ] = 0 and for this reason we shall assign the appropriate correspondence pµ ↔ Xµ6 and xµ ↔ Xµ5,up to numerical factors ( lengths ) to match dimensions, in order to attain noncommuting variables xµ, pµ .

9

Therefore, one has two possible routes to relate Yang’s algebra with Clifford algebras. One can relateYang’s algebra with the holographic area-coordinates algebra in the C-space associated to a 6D Cliffordalgebra and/or to the subalgebra of a 4D Clifford algebra via the realization of the conformal algebraSO(4, 2) in terms of the 4D Clifford algebra generators 1, γ5, γµ as shown in eq-(3.1).

Since the relation between the 4D conformal and Yang’s algebra and the implications for the AdS/CFT ,dS/CFT duality have been discussed before by Tanaka [ 17 ], in this work we shall establish the followingcorrespondence between the C-space holographic-area coordinates algebra ( associated to the 6D Cliffordalgebra ) and the Yang’s spacetime algebra via the angular momentum generators in 6D as follows :

iMµν = ihΣµν ↔ ih

λ2Xµν . (3.3)

iM56 = ihΣ56 ↔ ih

λ2X56. (3.4)

iλ2Σµ5 = iλxµ ↔ iXµ5. (3.5)

iλ2Σµ6 = iλ2 R

hpµ ↔ iXµ6. (3.6)

With Hermitian ( bivector ) operator- coordinates :

(Xµν)† = Xµν . (Xµ5)† = Xµ5. (Xµ6)† = Xµ6. (X56)† = X56. (3.7)

The algebra generators can be realized as :

Xµν = iλ2(Xµ ∂

∂Xν−Xν ∂

∂Xµ). (3.8a)

Xµ5 = iλ2(Xµ ∂

∂X5−X5 ∂

∂Xµ). (3.8b)

Xµ6 = iλ2(Xµ ∂

∂X6−X6 ∂

∂Xµ). (3.8c)

X56 = iλ2(X5 ∂

∂X6−X6 ∂

∂X5). (3.8d)

Σ56 = −i(X5 ∂

∂X6−X6 ∂

∂X5). (3.8e)

which have a one-to-one correspondence to the Yang Noncommutative space-time ( YNST ) algebra gener-ators in 4D. These generators ( angular momentum differential operators ) act on the coordinates of a 5Dhyperboloid AdS5 space defined by :

−(x1)2 + (x2)2 + (x3)2 + (x4)2 + (x5)2 − (x6)2 = R2. (3.9a)

where R is the throat size of the hyperboloid. This introduces an extra and crucial scale in addition to thePlanck scale. Notice that η55 = +1. η66 = −1. 5D de Sitter space dS5 has the topology of S4 × R1 .Whereas AdS5 space has the topology of R4×S1 and its conformal ( projective ) boundary at infinity has atopology S3 × S1 . Whereas the Euclideanized Anti de Sitter space AdS5 can be represented geometricallyas two disconnected branches ( sheets ) of a 5D hyperboloid embedded in 6D . The topology of thesetwo disconnected branches is that of a 5D disc and the metric is the Lobachevsky one of constant negativecurvature. The conformal group SO(4, 2) leaves the 4D lightcone at infinity invariant.

Thus, Euclideanized AdS5 is defined by a Wick rotation of the x6 coordinate giving :

−(x1)2 + (x2)2 + (x3)2 + (x4)2 + (x5)2 + (x6)2 = R2. (3.9b)

10

whereas de Sitter space dS5 with the topology of a pseudo-sphere S4 × R1 , and positive constant scalarcurvature is defined by :

−(x1)2 + (x2)2 + (x3)2 + (x4)2 + (x5)2 + (x6)2 = −R2. (3.9b)

( Notice that Tanaka [17 ] uses different conventions than ours in his definition of the 5D hyperboloids.He has a sign change from R2 to −R2 because he introduces i factors in iR ) .

After this discussion and upon a direct use of the correspondence in eqs-(3.3, 3.4, 3.5, 3.6 ...) yields theexchange algebra between the position and momentum coordinates :

[Xµ6, X56] = −iλ2η66Xµ5 ↔ [λ2R

hpµ, λ2Σ56] = −iλ2η66λxµ. (3.10)

from which we can deduce that :[pµ, Σ56] = −iη66 h

λRxµ. (3.11)

and after using the definition N = (λ/R)Σ56 one has the exchange algebra commutator of pµ and N of theYang’s spacetime algebra :

[pµ,N ] = −iη66 h

R2xµ. (3.12)

The other commutator is :

[Xµ5, X56] = −[Xµ5, X65] = iη55λ2Xµ6 ↔ [λxµ, λ2Σ56] = iη55λ2λ2 R

hpµ. (3.13)

from which we can deduce that :[xµ,Σ56] = iη55 λR

hpµ. (3.14)

and after using the definition N = (λ/R)Σ56 one has the exchange algebra commutator of xµ and N of theYang’s spacetime algebra :

[xµ,N ] = iη55 λ2

hpµ. (3.15)

The noncommutative algebra of the holographic area-coordinates in C-space is :

[Xµ5, Xν5] = −iη55λ2Xµν ↔ [xµ, xν ] = −iη55λ2Σµν . (3.16)

after using the representation of the C-space operator holographic area-coordinates :

iXµν → iλ2 1hMµν = iλ2Σµν iX56 → iλ2Σ56. (3.17)

where we appropriately introduced the Planck scale λ as one should to match units.From the correspondence :

pµ ↔ h

R

1λ2

Xµ6 =h

RΣµ6. (3.18)

one can obtain nonvanishing momentum commutator :

[Xµ6, Xν6] = −iη66λ2Xµν ↔ [pµ, pν ] = −iη66 h2

R2Σµν . (3.19)

The signatures for AdS5 space are η55 = +1; η66 = −1 and for the Euclideanized AdS5 space are η55 = +1and η66 = +1. Yang’s space-time algebra corresponds to the latter case.

Finally, the modified Heisenberg algebra can be read from the following C-space commutators :

[Xµ5, Xν6] = iηµνλ2X56 ↔

11

[xµ, pµ] = ihηµν λ

RΣ56 = ihηµνN . (3.20)

Eqs-(3.12, 3.15, 3.16, 3.17, 3.20 ) are the defining relations of Yang’s Noncommutative 4D spacetime algebrainvolving the 8D phase-space variables. These commutators obey the Jacobi identities. There are othercommutation relations like [Mµν , xρ], [Mµν , pρ] that we did not write down. These are just the well knownrotations ( boosts ) of the coordinates and momenta.

When λ → 0 and R → ∞ one recovers the ordinary commutative spacetime algebra. The Snyderalgebra [ 22 ] is recovered by setting R →∞ while leaving λ intact. To recover the ordinary Weyl-Heisenbergalgebra is more subtle. Tanaka [ 17 ] has shown the the spectrum of the operator N = (λ/R)Σ56 is discretegiven by n(λ/R) . This is not suprising since the angular momentum generator M56 associated with theEuclideanized AdS5 space is a rotation in the now compact x5 − x6 directions. This is not the case inAdS5 space since η66 = −1 and this timelike direction is no longer compact. Rotations involving timelikedirections are equivalent to noncompact boosts with a continuous spectrum.

In order to recover the standard Weyl-Heisenberg algebra from Yang’s Noncommutative spacetimealgebra, and the standard uncertainty relations ∆x∆p ≥ h with the ordinary h term , rather than the nhterm, one needs to take the limit n →∞ limit in such a way that the net combination of n λ

R → 1.This can be attained when one takes the double scaling limit of the quantities as follows :

λ → 0. R →∞. λR → L2.

limn→∞ nλ

R= n

λ2

λR=

nλ2

L2→ 1. (3.21)

From eq-(3.21) one learns then that :

nλ2 = λR = L2. (3.22)

The spectrum n corresponds to the quantization of the angular momentum operator in the x5−x6 direction(after embedding the 5D hyperboloid of throat size R onto 6D ) . Tanaka [ 17 ] has shown why there is adiscrete spectra for the spatial coordinates and spatial momenta in Yang’s spacetime algebra that yields aminimum length λ ( ultraviolet cutoff in energy ) and a minimum momentum p = h/R ( maximal length R, infrared cutoff ) . The energy and temporal coordinates had a continous spectrum.

The physical interpretation of the double-scaling limit of eq-( 3.22 ) is that the the area L2 = λR becomesnow quantized in units of the Planck area λ2 as L2 = nλ2 . Thus the quantization of the area ( via thedouble scaling limit ) L2 = λR = nλ2 is a result of the discrete angular momentum spectrum in the x5− x6

directions of the Yang’s Noncommutative spacetime algebra when it is realized by ( angular momentum )differential operators acting on the Euclideanized AdS5 space ( two branches of a 5D hyperboloid embeddedin 6D ). A general interplay between quantum of areas and quantum of angular momentum, for arbitraryvalues of spin, in terms of the square root of the Casimir A ∼ λ2

√j(j + 1), has been obtained a while ago in

Loop Quantum Gravity by using spin-networks techniques and highly technical area-operator regularizationprocedures [ 41 ] . The advantage of this work is that we have arrived at similar ( not identical ) area-quantization conclusions in terms of minimal Planck areas and a discrete angular momentum spectrum nvia the double scaling limit based on Clifford algebraic methods (C-space holographic area-coordinates).

In [ 46 ] we have shown why AdS4 gravity with a topological term; i.e. an Einstein-Hilbert ac-tion with a cosmological constant plus Gauss-Bonnet terms can be obtained from the vacuum state of aBF-Chern-Simons-Higgs theory without introducing by hand the zero torsion condition imposed in theMacDowell-Mansouri-Chamsedine-West construction. One of the most salient features of [ 46 ] was thata geometric mean relationship was found among the cosmological constant Λc , the Planck area λ2 andthe AdS4 throat size squared R2 given by (Λc)−1 = (λ)2(R2). A similar geometric mean relation is alsoobeyed by the condition λR = L2(= nλ2) in the double scaling limit of Yang’s algebra which suggests toidentify the cosmological constant as Λc = L−4 . This geometric mean condition remains to be investigatedfurther. In particular, we presented the preliminary steps how to construct a Noncommutative Gravity viathe Vasiliev-Moyal star products deformations of the SO(4, 2) algebra used in the study of higher conformalmassless spin theories in AdS spaces by taking the inverse-throat size 1/R as a deformation parameter of theSO(4, 2) algebra [ 46 ] . A Moyal deformation of ordinary Gravity via SU(∞) gauge theories was advanced

12

in [ 31 ] . A new realization of holography and the geometrical intepretation of AdS2n spaces in terms ofSO(2n− 1, 2) instantons was studied in [45] .

Since the expectation valueλ2

L2< n|Σ56|n >=

nλ2

L2= 1. (3.23)

in the double-scaling limit one recovers the standard Heisenberg uncertainty relations :

∆xµ∆pµ ≥ || < [xµ, pµ] > || = h. (3.24)

and the commutators become in the double-scaling limit:

[pµ, Σ56] = −iη66 h

L2xµ. [pµ, N ] = 0. (3.25)

[xµ, Σ56] = −iη55 L2

hpµ. [xµ, N ] = 0. (3.26)

[xµ, xν ] = [pµ, pν ] = 0. [xµ, pµ] = ihηµν λ2

L2Σ56 → ihηµν1. (3.27)

Rigorously speaking, when λ → 0 the last commutator [xµ, pν ] → 0 since the generator Σ56 is well defined.It is the large n limit of the spectrum < n|Σ56|n > that reproduces the ordinary Heisenberg uncertaintyrelations.

Concluding, the ordinary Weyl-Heisenberg algebra is recovered from the double-scaling limit of theYang’s Noncommutative spacetime algebra. One ends up in the double-scaling limit with commuting space-time coordinates and commuting momentum variables and the standard commutation relations for the[xµ, pν ] = ihηµν .

4. C-space QM and the Generalized Weyl-Heisenberg Algebra

It is helpful, although not necessary, to derive the Generalized Weyl-Heisenberg Algebra in C-spaceQuantum Mechanics by recurring to matrix realizations of Clifford algebras . Namely we will be evaluatingthe commutators of Clifford-valued operators ( matrix-valued operators) in a suitable Hilbert space associatedwith the free polyparticle dynamics described in section 2 , for example. Vector coherent states have beendefined on Clifford algebras, quaternions and octonions [ 49 ] that is the starting point to construct an (overcomplete ) basis in a Hilbert space. This will be the subject of future investigations, in particular tostudy the Noncommutative Quantum oscillator in C-spaces be recurring to the polyvector coherent statesconstruction on Clifford algebras generalizing the work of [49].

Due to the noncommutative nature of the basis vectors of the Clifford algebra one has:

[EA, EB ] = FMABEM = FABMEM . EA = 1; γµ; γµ ∧ γν ; γµ ∧ γν ∧ γρ; ...... (4.1)

In order to raise and lower indices it requires the C-space metric GAB given by the scalar part of thegeometric product of < EAEB >scalar=< EAEB >0= EA ∗ EB = GAB . The geometry of curved C-spaceinvolving GMN was studied by [13] where it was shown how higher derivative gravity with torsion in ordinaryspacetime emerged naturally from the scalar curvature in C-space. As mentioned above, we will choose towork with a finite value of D to avoid algebraic convergence problems.

The quantities FMAB play a similar role as the structure constants in ordinary Lie algebras. A commutator

of two matrices is itself a matrix, which in turn, can be expanded in a suitable matrix basis due to the Cliffordalgebraic (vector space) structure inherent in C-spaces. The commutator algebra obeys the Jacobi identities,the Liebnitz rule of derivations and the antisymmetry properties.

The Clifford geometric product of two basis elements can be rewritten as :

EAEB ≡12EA, EB+

12[EA, EB ]. (4.2)

13

In general the geometric product of two multivectors EA, EB of ranks r, s, respectively, is given by anaggregate of multivectors of the form :

EAEB =< EAEB >r+s, < EAEB >r+s−2, < EAEB >r+s−4, ......

< EAEB >r+s−6 ....... < EAEB >|r−s| . (4.3)

The first term of rank r + s is the wedge product EA ∧ EB and the last term of rank |r − s| is thedot product EA • EB which is obtained by a contraction of indices and must not be confused, in general,with the scalar part of EAEB given by EA ∗EB =< EAEB >0 unless r = s. In general, the scalar productamong two equal-rank multivectors r = s cannot longer be written in terms of the anticommutator EA, EBexcept in the case when r = s = 1 : γµ, γν = 2gµν1. However, for equal-rank multivectors, the scalar part< EAEB >0= GAB1 where 1 is the unit element of the Clifford algebra and GAB is the C-space metric.

First of all we must keep track of the correct units. We will treat the P and X exactly on the samefooting. For this reason we will scale all the basis elements by judicious powers of

√h:

γµ →√

hγµ. γµ ∧ γν → hγµ ∧ γν ; ..... (4.4)

and

γµ → h−1/2γµ. γµ ∧ γν → h−1γµ ∧ γν ; ..... (4.5)

The units of XA and PB are taken to be hrA/2 and hrB/2 respectively where rA, rB are the ranks of theantisymmetric tensor components XA, PB of the polyvectors X, P respectively.

The scaling of the commutator is:

[EA, EB ] = FMABEM → h−rA/2−rB/2[EA, EB ] = (h−rA/2−rB/2FM

AB) EM . (4.6)

hence, we will absorb the powers of h appropriately in the structure constants as indicated above. Upperindices carry positive powers of

√h whereas lower indices carry negative powers. Notice that there are no

powers of√

h associated with the index M above, only w.r.t the two AB indices.Since we are scaling the basis vectors EA this means that we are choosing the quantities HC to be

dimensionless. In this fashion we automatically obtain quantities with the correct units. For example:[xµ, pν ] will contain a single power of h due to the two factors of h1/2 appearing in the Gµν as a result of thescaling of the γµ and γν . Their anticommutator yields the Gµν (after the scaling takes place). One obtainsidentical results with the other holographic components of the polyvectors. The commutator of [xµν , pρτ ]will automatically have the correct h2 power, etc......

We will begin by writing the commutator of the Clifford-valued operators X, P (matrix-valued operators):

[X, P ] = [XAEA, PBEB ] = H = HCEC . (4.7)

Notice that we have not written explicitly any i factors since these factors are re-absorbed by the HC termsin the r.h.s in addition to the suitable powers of h. Since the matrix representation of the generators EA,are hermitian and anti-hermitian in general, the commutator [X, P ] will contain both hermitian and anti-hermitian terms in the r.h.s. For this reason there is not a common i factor that factorizes in the r.h.s . Inthe ordinary Weyl-Heisenberg algebra and the Yang’s algebra when the operators A,B are both hermitian,the commutator [A,B] is anti-hermitian which justifies the presence of an explicit i factor in [x, p] = ih, forexample.

We shall also impose the hermitian condition on the polyvector components (XM )† = XM , etc...however, for each value of the index M , the operator-valued matrices XMEM can be either hermitian or anti-hermitian depending on the hermitian, anti-hermitian, properties of the matrix representation (EM )† = ±EM

of the Clifford algebra. For example, if we take the signature (+,−,−,−) one has :

(γ0)† = γ0. (γi)† = −γi. (γ5)† = −γ5. (4.8)

14

(γ0)2 = 1. (γi)2 = (γ5)2 = −1. (4.9)

After this detour, a direct evaluation of the commutator of two polyvectors operators in terms of theClifford-valued Planck constant H = HCEC is :

[X, P ] = [XAEA, PBEB ] = H = HCEC =

[XA, PB ]EAEB + PBXA[EA, EB ] =

[XA, PB ]EAEB + PBXAFCABEC (4.10))

Notice that the EAEB terms in the r.h.s of (4.10 ) can be written as the sum of an anti-commutatorand commutator as :

EAEB =12EA, EB+

12[EA, EB ] =

12KC

ABEC +12FC

ABEC ≡ ΩCABEC (4.11)

This is important in what follows. Continuing with :

[XA, PB ]EAEB + PBXAFCABEC = HCEC ⇒

[XA, PB ] EAEB = (−PBXAFCAB + HC)EC . (4.12)

and after renaming the indices A,B in the r.h.s of eq-( 4.12 ) for M,N yields due to the antisymmetryproperty of FC

MN = −FCNM :

[XA, PB ]EAEB = (PNXMFCNM + HC)EC . (4.13)

The next step is to take the scalar product of both sides of ( 4.13 ) by (EAEB)−1 = (EB)−1(EA)−1 =EBEA in order to eliminate the EAEB terms from the l.h.s of ( 4.13 ). The C-space metric

GMN ≡ EM ∗ EN =< EMEN >0 . (4.14)

permits us to raise and lower indices. A flat C-space metric, for example, has for components :

Gµν = ηµν . G[µ1µ2] [ν1ν2] = ηµ1ν1ηµ2ν2 − ηµ1ν2ηµ2ν1 , ..... (4.15a)

It is convenient to order the indices in an increasing sequence:

µ1 < µ2 < µ3...... < µn. ν1 < ν2 < ν3..... < νn. (4.15b)

Thus, upon eliminating the EAEB terms from the l.h.s of ( 4.13 ) gives:

[XA, PB ] = (PNXMFCNM + HC) < ECEBEA >0= (PNXMFC

NM + HC) ΩBAC . (4.16)

The scalar part of the triple geometric product is defined as :

ΩBAC ≡ < ECEBEA >0 = EC ∗ (EBEA). (4.17)

Eq-(4.16) is the fundamental result of this section.If XA, PB are hermitian then one must have the condition :

[XA, PB ]† = −[XA, PB ] ⇒ [(HC + PNXMFCNM )ΩBA

C ]† = −(HC + PNXMFCNM )ΩBA

C . (4.18)

which is required because rather than introducing explicit i factors in the r.h.s , we may absorb them inthe definitions of HC , the structure FC

MN constants and ΩBAC . The condition ( 4.18) is the analog of the

vector-calculus result L = r ∧ p = −p ∧ rThe Weyl-Heisenberg algebra in C-space can also be written compactly in the ”spin” plus ”orbital”

angular momentum form:

15

[XA, PB ] = HAB + JAB . (4.19)

with the standard Planck constant-like terms of the form :

HAB = HCΩBAC . (4.20)

Notice the mixed symmetry of the expressions HAB ,ΩBAC , a symmetric plus antiymmetric piece in A,B.

The elements HAB involving the components of the polyvector-valued Planck constant resemble the familiarquaternionic and octonionic expansions of a quaternion and octonion in terms of their components. Thisindicates that QM in C-spaces may be intrinsically linked with Quaternionic and Octonionic QM [23]. Hadone had commuting basis elements, like in ordinary space-time, the scalar component for H = HCEC = H01survives and one would have had the standard Weyl-Heisenberg algebra [XA, PB ] = H0GAB . Since thepowers of h are absorbed by the metric GAB this implies that H0 = i. For example, [xµ, pν ] = ihηµν . Theextra term in eq-(4.16) is the analog of the orbital-angular momentum in C-spaces given by:

JAB = PNXMFCNM < ECEBEA >0 = PNXMFC

NMΩBAC . (4.21)

An immediate consequence of the C-space Weyl-Heisenberg algebra is that it induces automatically a Non-commutative Geometric structure in the XA coordinates. To satisfy the Jacobi identities among the triplesXA, PB , XC and XA, XB , XC it is fairly clear now that the coordinates cannot commute due to the explicitX, P terms in the modified Weyl-Heisenberg algebra. Hence, the Jacobi identities require in general that :

[XA, XB ] = ΣAB . [XC ,ΣAB ] 6= 0. (4.22)

Notice that we should have a non-vanishing commutator [XC ,ΣAB ] 6= 0 otherwise we will be introducingvery strong constraints on ΣAB that could not otherwise be satisfied unless one had trivial solutions madeout of numerical constants. ΣAB is a tensor-like operator-valued object in C-space that does not destroyC-space Lorentz invariance and which is implicitly defined by the Jacobi identities. Suitable powers of thePlanck scale are absorbed in the defining relations for ΣAB in order to match units. Since the Planck scaleis a C-space invariant one will maintain C-space Lorentz invariance.

One should have as well as a result of the Jacobi indentities :

[PA, PB ] = ΘAB 6= 0. [XA, XB ] = ΣAB 6= 0. (4.23)

Notice that a flat C-space does not mean it has a trivial geometry. A flat C-space has nontrivial torsion. Thenonvanishing of momentum commutators are compatible with a curved C-space. We have shown that theanalog of the scalar curvature in C-spaces can be decomposed as sums of powers of the Riemann curvaturescalar ( and other contractions of the curvature tensors ) including torsion terms. Thus, Relativity in C-spaces involves a higher derivative gravity theory with torsion [ 13 ] and a vanishing cosmological constantin C-spaces does not amount to a vanishing cosmological constant in ordinary spacetimes ! [ 44 ] .

The other commutators are :

[XC ,ΣAB ] 6= 0. [PC ,ΘAB ] 6= 0. (4.24)

and :

[XC ,ΘAB ] 6= 0. [PC ,ΣAB ] 6= 0. (4.25)

The above equations must be consistent with the Jacobi identities. To find a realization of the C-spaceWeyl-Heisenberg algebra, in terms of generators given by differential operators in C-space, which is consistentwith the Jacobi identities is far more difficult than it was in the case of the Yang’s spacetime algebra in theprevious section.

The open problem remains (if possible) to find explicit nontrivial solutions ( realizations) of ΣAB ,ΘAB

that define the full fledged C-space Quantum Algebra. By nontrivial one means where ΣAB ,ΘAB are notcomprised of ordinary c-numbers like it is assumed in the Noncommutative Geometry literature. At the

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moment, this is beyond the scope of the present work. Based on the nature of the C-space variables, weshould expect that the quantities ΣAB ,ΘAB could represent a generalized area element in the Phase-spacesassociated with C-spaces, not unlike the Yang’s spacetime algebra which contains the angular momentumgenerators that are proportional to phase-space areas . We leave this difficult problem for future work.The most salient feature is that the C-space Quantum algebra is a generalization of both the ordinaryWeyl-Heisenberg algebra and Yang’s spacetime algebras. It contains features from both algebras.

5. Concluding Remarks on Noncommutative QM and QFT in C-spaces

We have explained in the introduction and in section 2 why the Planck scale is a true invariant ofC-space that is required in order to combine p-loops ( closed p-branes ) of different dimensions. C-spaceRelativity contains two fundamental constants, the speed of light and the Planck scale. The authors in [11,1221] have interpreted the Planck scale as a deformation parameter in the kappa-deformed Poincare algebrasl = 1/κ = λ, where the ordinary four-dim Lorentz invariance is broken explicitly and only the rotationalsymmetry is preserved. This is not the case in C-space Relativity where the C-space Lorentz invariance isan essential feature of the theory. In addition, the Weyl-Heisenberg algebra and the XA coordinate algebrasin C-space are indeed compatible with the C-space Lorentz invariance without introducing extra dimensionslike it occurs in the Snyder and Yang’s Noncommutative spacetime algebras.

We have argued in [ 37 ] why the kappa-deformed Poincare algebras could be obtained directly from an8D Phase space via a Moyal star product deformation procedure taking the Planck scale as the deformationparameter and by choosing an appropriate basis Q(x, p, κ) , P (x, p, κ) in the 8D phase-space It is warrantedto explore the relationship among all these algebras on a unified footing. Two-parameter quantum Hopfalgebraic deformations of ordinary algebras have been studied in the past in the context of Quantum Groups[ 20 ] and more recently by the authors [ 38, 42 ]. Unfortunately the latter authors were not aware of the oldwork by Yang [ 16 ] on Noncommutative spacetime algebras which involved two different length scales andof Tanaka’s work [ 17 ] about the connection of Yang’s algebra to de Sitter and Anti de Sitter spaces and thephysical explanation of the orgins of a discrete spectrum for the spatial coordinates and spatial momentathat yields a minimum length λ ( ultaviolet cutoff in energy ) and a minimum momentum p = h/R (maximal length R , infrared cutoff ) . The importance of Yang’s algebra and the Lie-algebraic stability inthe construction of physical theories within the context of new length scales was addressed by [ 51, 52] .

Quantum group deformations of the Poincare and Conformal group has been developed by Castellani[20] and used in his construction of a bicovariant q-Gravity theory . One may interpret the q-deformationparameter in terms of a minimal Planck scale and an upper impassible scale R by setting q = exp(λ/R) sothat when λ goes to zero, or when R goes to infinity, the deformation parameter collapses to the classicalundeformed values q = 1. Hence, the classical gravitational theory is recovered in the short and large distancelimits of Castellani’s q-Gravity theory. This sort of Ultraviolet/Infrared entanglement duality has received alot of attention in the past years within the framework of Noncommutative QFT defined on Noncommutativespcetimes and in M-theory [ 19, 43],

An upper limiting scale in cosmology was long ago advocated by Nottale [1] in his proposal for theresolution of the cosmological constant problem. It is unknown at the present if Nottale’s Fractal spacetimeconstruction belongs to the class of Noncommutative Geometries studied by Connes. The importance ofnonlinear dynamics, chaos and fractals in Particle physics and Cosmology has been raised by Nottale andothers. Smith [ 50] has derived the values of all the coupling constants and masses of the Standard Modelbased on Clifford algebraic methods associated with hyper-diamond discrete lattices ( a Feyman Chess-Boardmodel ) generalizing the celebrated Wyler’s mathematical expression for the fine structure constant. Mostrecently, Beck [ 39 ] gave convincing numerical results to support why Chaotic Strings dynamics determinesthe values of all the Standard Model parameters.

It is fairly clear why C-space QM differs from the ordinary QM in many aspects. To start with itinvolves a Noncommutative QM due to noncommutative geometric structure of the XM , PM operatorsin a C-phase-space. Despite the fact that Clifford algebras could be interpreted already as the quantumextensions of Grassmann algebras we believe that a main task in the near future will be to construct QFT’sin C-spaces based on Quantum Clifford Algebras, like the Braided Hopf Quantum Cliiford algebras [15]. Ithas been argued by Ablamowicz and Fauser [15] that these Hopf algebraic structures will replace groupsand group representations as the leading paradigm in forthcoming times and why the Grassmann-Cayley

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bracket algebras and other algebraic structures are all covered by the Hopf algebraic framework. Recently,a Dirac-Kahler fermion action based on a new Clifford product with a noncommutative differential form ona lattice was introduced in [ 40 ] .

A Moyal-like star product construction in C-spaces deserves further study as well. C-space involves thephysics of all p-loops ( closed p-branes ), thus it is warranted to use methods of multisymplectic geometry(mechanics) due to the presence of antisymmetric tensors of arbitrary rank. Nambu-Poisson QM seems tobe the most appropriate one to study C-space QM. In particular the use of the Zariski and Fedosov starproduct deformations versus the Moyal one [24] will be welcome. A relativistic variant of the Moyal-Wignerfunction was proposed by [47].

To finalize this work we discuss how one could take contractions of the C-space Weyl-Heisenberg algebrato obtain an effective matrix-valued Planck ” constant”, or for that matter, how to generate a single effectivePlanck ” constant ” heff (p2) by looking for a matrix representation of HCEC → Hµν = heffηµν1, afterperforming a dimensional reduction from the 16-dim C-space down to 4-dim. The most natural procedureis to look at the norm-squared :

HABHAB + JABJAB + HABJAB + JABHAB . (5.1)

C-space polydimensional covariance can be broken by imposing similar type of constraints like we had insection 2 , relating the holographic norms of polyvectors to the powers of ordinary vector norms:

JABJAB =∑

n

an(J/h)2n. J2 = JµνJµν . Jµν = pµxν − pνxµ. (5.2)

and in this way the norm-squared (4.26 ) reduces to:

HABHAB + HABJAB + JABHAB +∑

n

an(J/h)2n (5.3)

Eq-(4.28) is more general than the ones discussed in section 2.3 . In particular, the last terms of (4.28)do contain the required terms

∑p2n which defined the effective h in section 2.3 . This can be seen after

relating the angular momentum J to the mass-shell condition m2 = p2 of the center of mass coordinates. Inthe ordinary mechanics of a rigid-top there are two Casimirs, the angular momentum J2 and the Energy. Inthe case of a symmetric top, the energy is E = J2/2I where I is the moment of inertia of the symmetrictop. The importance of the de Sitter top and the Regge relations have been recently analyzed by [48 ] . Thecovariant matrix model of N D0-branes discussed by Tanaka also explores these Casimirs relationships of arigid-top in de Sitter ( anti de Sitter ) spaces

If one assumes a similar relationship among Energy and angular momentum in the relativistic case, onewill have the familiar Regge-type of relation asociated with the string specrum: J ∼ (α)′ m2, where m2 = p2

( on-shell) and (α)′ is the inverse string tension, of the order of λ2 ( Planck area ) . In this case, one will beable to explain the results of section 2.3 , involving an effective heff (p2), in terms of the JABJAB terms ofeq- (4.28), after breaking the C-space polydimensional covariance through the constraints in eq-(4.27) andafter using the desired Regge relation J ∼ α′m2 = α′p2. Namely, a power series expansion in Jn is recast asa power series expansion in p2n that defined the heff (p2) in our derivation of the generalized string-p-braneuncertainty relations in section 2.3 from the dispersion relations in C-spaces.

To conclude this last section : Quantization in C-spaces contains a very rich Noncommutative structurefrom which many old results can be derived after breaking the C-space Lorentz covariance/invariance (pandimensional covariance ) . There is no need to introduce ad-hoc nontrivial commutation relations forthe spacetime coordinates. These are induced from the mere quantization process; i.e. from the C-spaceWeyl-Heisenberg algebra as a result of the Jacobi identies. No extra dimensions are required to introducea length scale. C-space Relativity already has a natural invariant minimum Planck scale by definition.The Weyl-Heisenberg algebra in C-spaces is naturally modified due to the noncommutativity of the Cliffordalgebra basis elements.

In section 3 we presented the relationship among Yang’s 4D Noncommutative space-time algebra [ 16 ] (in terms of 8D phase space coordinates ), the holographic area coordinates algebra of the C-space associatedwith a 6D Clifford algebra, and the Euclideanized AdS5 spaces. The role of AdS5 was instrumental in

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explaining the origins of an extra ( infrared ) scale R in conjunction to the ( ultraviolet ) Planck scale λcharacteristic of C-spaces. Tanaka [ 17 ] gave the physical and mathematical derivation of the discrete spectrafor the spatial coordinates and spatial momenta that yields a minimum length-scale λ ( ultraviolet cutoffin energy ) and a minimum momentum p = h/R ( maximal length R, infrared cutoff ) . The double-scalinglimit, in conjunction with the large n → ∞ limit, led to the area-quantization condition λR = L2 = nλ2

in units of the Planck area, where n is the angular momentum Σ56 = (1/h)M56 eigenvalue. These resultsare compatiible with the minimal Planck length scale principle advocated many years ago in Nottale’s ScaleRelativity Theory [ 1 ] and in Born’s Dual Relativity Principle in Phase spaces [ 32 ] which led to a Maximal-acceleration Relativity in Phase spaces [ 37 ] and in Rindler geometries [18 ] . In essence, when both QMand Relativity are extended to C-spaces both QM and Relativity theories are modified accordingly whichmaybe what is required in order to formulate a consistent Quantum Theory of Gravity.

Acknowledgements

We are indebted T. Smith and Matej Pavsic for many conversations which led to this work and toM.Bowers for invaluable assistance.

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