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On area coordinates and quantum mechanics in Yang's Noncommutative spacetime with a lower and upper scale.

We explore Yang's Noncommutative space-time algebra (involving two length scales) within the context of QM defined in Noncommutative spacetimes and the holographic area-coordinates algebra in Clifford spaces. Casimir invariant wave equations corresponding to Noncommutative coordinates and momenta in d-dimensions can be recast in terms of ordinary QM wave equations in d+2-dimensions. It is conjectured that QM over Noncommutative spacetimes (Noncommutative QM) may be described by ordinary QM in higher dimensions. Novel Moyal-Yang-Fedosov-Kontsevich star products deformations of the Noncommutative Poisson Brackets are employed to construct star product deformations of scalar field theories. Finally, generalizations of the Dirac-Konstant and Klein-Gordon-like equations relevant to the physics of D-branes and Matrix Models are presented.

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1 Introduction

Yang's noncommutative space time algebra [1] is a generalization of the Snyder algebra [2] (where now both coordinates and momenta are not commuting) that has received more attention recently, see for example [3] and references therein. In particular, Noncommutative p-brane actions, for even p + 1 = 2n-dimensional world-volumes, were written explicitly [15] in terms of the novel Moyal-Yang (Fedosov-Kontsevich) star product deformations [11, 12] of the Noncommutative Nambu Poisson Brackets (NCNPB) that are associated with the noncommuting world-volume coordinates [q.sup.A], [p.sup.A] for A = = 1, 2, 3, ... n. The latter noncommuting coordinates obey the noncommutative Yang algebra with an ultraviolet [L.sub.P] (Planck) scale and infrared (R) scale cutoff. It was shown why the novel p-brane actions in the "classical" limit [[??].sub.eff] = = [??][L.sub.P]/R [right arrow] 0 still acquire nontrivial noncommutative corrections that differ from ordinary p-brane actions. Super p-branes actions in the light-cone gauge are also amenable to Moyal-Yang star product deformations as well due to the fact that p-branes moving in flat spacetime backgrounds, in the light-cone gauge, can be recast as gauge theories of volume-preserving diffeomorphisms. The most general construction of noncommutative super p-branes actions based on non (anti) commuting superspaces and quantum group methods remains an open problem.

The purpose of this work is to explore further the consequences of Yang's Noncommutative spacetime algebra within the context of QM in Noncommutative spacetimes and the holographic area-coordinates algebra in Clifford spaces [14]. In section 2 we study the interplay among Yang's Noncommutative spacetime algebra and the former area-coordinates algebra in Clifford spaces. In section 3 we show how Casimir invariant wave equations corresponding to Noncommutative coordinates and momenta in D-dimensions, can be recast in terms of ordinary QM wave equations in D+2-dimensions. In particular, we shall present explicit solutions of the D'Alambertian operator in the bulk of AdS spaces and explain its correspondence with the Casimir invariant wave equations associated with the Yang's Noncommutative spacetime algebra at the projective boundary of the conformally compactified AdS spacetime. We conjecture that QM over Noncommutative spacetimes (Noncommutative QM) may be described by ordinary QM in higher dimensions.

In section 4 we recur to the novel Moyal-Yang (Fedosov-Kontsevich) star products [11, 12] deformations of the Noncommutative Poisson Brackets to construct Moyal-Yang star product deformations of scalar field theories. The role of star products in the construction of p-branes actions from the large N limit of SU(N) Yang-Mills can be found in [6] and in the Self-Dual Gravity/SU([infinity]) Self Dual Yang-Mills relation in [7, 8, 9, 10]. Finally, in the conclusion 5, we present the generalizations of the Dirac-Konstant equations (and their "square" Klein-Gordon type equations) that are relevant to the incorporation of fermions and the physics of D-branes and Matrix Models.

2 Noncommutative Yang's spacetime algebra in terms of area-coordinates in Clifford spaces

The main result of this section is that there is a subalgebra of the C-space operator-valued coordinates [13] which is isomorphic to the Noncommutative Yang's spacetime algebra [1, 3]. This, in conjunction to the discrete spectrum of angular momentum, leads to the discrete area quantization in multiples of Planck areas. Namely, the 4D Yang's Noncommutative space-time algebra [3] (written in terms of 8D phase-space coordinates) is isomorphic to the 15-dimensional subalgebra of the C-space operator-valued coordinates associated with the holographic areas of C-space. This connection between Yang's algebra and the 6D Clifford algebra is possible because the 8D phase-space coordinates [x.sup.[mu]], [p.sup.[mu]] (associated to a 4D spacetime) have a one-to-one correspondence to the [[??].sup.[mu]5]; [[??].sup.[mu]6] holographic area-coordinates of the C-space (corresponding to the 6D Clifford algebra). Furhermore, Tanaka [3] has shown that the Yang's algebra [1] (with 15 generators) is related to the 4D conformal algebra (15 generators) which in turn is isomorphic to a subalgebra of the 4D Clifford algebra because it is known that the 15 generators of the 4D conformal algebra SO(4, 2) can be explicitly realized in terms of the 4D Clifford algebra as shown in [13].

The correspondence between the holographic area coordinates [X.sup.AB] [left and right arrow] [[lambda].sup.2][[summation].sup.AB] and the angular momentum variables when A,B = 1, 2, 3, ... 6 yields an isomorphism between the holographic area coordinates algebra in Clifford spaces [14] and the noncommutative Yang's spacetime algebra in D=4. The scale [lambda] is the ultraviolet lower Planck scale. We begin by writing the exchange algebra between the position and momentum coordinates encapsulated by the commutator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

from which we can deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

hence, after using the definition N = ([lambda]/R)[[summation].sup.56], where R is the infrared upper scale, one has the exchange algebra commutator of [p.sup.[mu]] and N of the Yang's spacetime algebra given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

From the commutator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

we can deduce that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

and after using the definition N = ([lambda]/R)[[summation].sup.56] one has the exchange algebra commutator of [x.sup.[mu]] and N of the Yang's spacetime algebra

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

The other relevant holographic area-coordinates commutators in C-space are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

that yield the noncommuting coordinates algebra after having used the representation of the C-space operator holographic area-coordinates

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

where we appropriately introduced the Planck scale [lambda] as one should to match units. From the correspondence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

one can obtain nonvanishing momentum commutator

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

The signatures for Ad[S.sub.5] space are [[eta].sup.55] = +1; [[eta].sup.66] = -1 and for the Euclideanized Ad[S.sub.5] space are [[eta].sup.55] = +1 and [[eta].sup.66] = +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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.11)

Eqs-(2.1-2.11) are the defining relations of Yang's Noncommutative 4D spacetime algebra [1] involving the 8D phase-space variables. These commutators obey the Jacobi identities. There are other commutation relations like [M.sup.[mu]v], [x.sup.[rho]]], ... that we did not write down. These are just the well known rotations (boosts) of the coordinates and momenta.

When [lambda] [right arrow] 0 and R [right arrow] [infinity] one recovers the ordinary commutative spacetime algebra. The Snyder algebra [2] is recovered by setting R [right arrow] [infinity] while leaving e intact. To recover the ordinary Weyl-Heisenberg algebra is more subtle. Tanaka [3] has shown the the spectrum of the operator N = ([lambda]/R)[[summation].sup.56] is discrete given by n([lambda]/R). This is not suprising since the angular momentum generator [M.sup.56] associated with the Euclideanized Ad[S.sub.5] space is a rotation in the now compact [x.sup.5] - [x.sup.6] directions. This is not the case in Ad[S.sub.5] space since [[eta].sup.66] = -1 and this timelike direction is no longer compact. Rotations involving timelike directions are equivalent to noncompact boosts with a continuous spectrum.

In order to recover the standard Weyl-Heisenberg algebra from Yang's Noncommutative spacetime algebra, and the standard uncertainty relations [DELTA]x[DELTA]p [greater than or equal to] [??] with the ordinary [??] term, rather than the n[??] term, one needs to take the limit n [right arrow] [infinity] limit in such a way that the net combination of n [lambda]/R [right arrow] 1. This can be attained when one takes the double scaling limit of the quantities as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.12)

From eq-(2.12) one learns then that

n[[lambda].sup.2] = [lambda]R = [L.sup.2]. (2.13)

The spectrum n corresponds to the quantization of the angular momentum operator in the [x.sup.5] - [x.sup.6] direction (after embedding the 5D hyperboloid of throat size R onto 6D). Tanaka [3] has shown why there is a discrete spectra for the spatial coordinates and spatial momenta in Yang's spacetime algebra that yields a minimum length [lambda] (ultraviolet cutoff in energy) and a minimum momentum p = [??]/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-(2.12) is that the the area [L.sup.2] = [lambda]R becomes now quantized in units of the Planck area [[lambda].sup.2] as [L.sup.2] = n[[lambda].sup.2]. Thus the quantization of the area (via the double scaling limit) [L.sup.2] = [lambda]R = n[[lambda].sup.2] is a result of the discrete angular momentum spectrum in the [x.sup.5] - [x.sup.6] directions of the Yang's Noncommutative spacetime algebra when it is realized by (angular momentum) differential operators acting on the Euclideanized Ad[S.sub.5] space (two branches of a 5D hyperboloid embedded in 6D). A general interplay between quantum of areas and quantum of angular momentum, for arbitrary values of spin, in terms of the square root of the Casimir [??] ~ [[lambda].sup.2] [square root of j(j + 1)], has been obtained a while ago in Loop Quantum Gravity by using spin-networks techniques and highly technical area-operator regularization procedures [4].

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 n via the double scaling limit based on Clifford algebraic methods (C-space holographic area-coordinates). This is not surprising since the norm-squared of the holographic Area operator has a correspondence with the quadratic Casimir OABOAB of the conformal algebra SO(4, 2) (SO(5, 1) in the Euclideanized Ad[S.sub.5] case). This quadratic Casimir must not be confused with the SU(2) Casimir [J.sup.2] with eigenvalues j(j +1). Hence, the correspondence given by eqs-(2.3-2.8) gives [[??].sup.2] [left and right arrow] [[lambda].sup.4][[summation].sub.AB][[summation].sup.AB].

In [5] we have shown why Ad[S.sub.4] gravity with a topological term; i. e. an Einstein-Hilbert action with a cosmological constant plus Gauss-Bonnet terms can be obtained from the vacuum state of a BF-Chern-Simons-Higgs theory without introducing by hand the zero torsion condition imposed in the McDowell-Mansouri-Chamsedine-West construction. One of the most salient features of [5] was that a geometric mean relationship was found among the cosmological constant [LAMBDA].sub.c], the Planck area [[lambda].sup.2] and the Ad[S.sub.4] throat size squared [R.sup.2] given by [([[LAMBDA].sub.c]).sup.-1] = [([lambda]).sup.2]([R.sup.2]). Upon setting the throat size to be of the order of the Hubble scale [R.sub.H] and [lambda] = [L.sub.P] (Planck scale), one recovers the observed value of the cosmological constant [L.sup.-2.sub.P] [R.sup.-2.sub.H] = [L.sup.-4.sub.P] [([L.sub.P]/[R.sub.H]).sup.2] ~ [10.sup.-120] [M.sup.4.sub.P]. A similar geometric mean relation is also obeyed by the condition [lambda]R = = [L.sup.2] (= n[[lambda].sup.2]) in the double scaling limit of Yang's algebra which suggests to identify the cosmological constant as [[LAMBDA].sub.c] = = [L.sup.-4]. This geometric mean condition remains to be investigated further. In particular, we presented the preliminary steps how to construct a Noncommutative Gravity via the Vasiliev-Moyal star products deformations of the SO(4, 2) algebra used in the study of higher conformal massless spin theories in AdS spaces by taking the inverse-throat size 1/R as a deformation parameter of the SO(4, 2) algebra. A Moyal deformation of ordinary Gravity via SU([infinity]) gauge theories was advanced in [7].

3 Noncommutative QM in Yang's spacetime from ordinary QM in higher dimensions

In order to write wave equations in non-commuting spacetimes we start with a Hamiltonian written in dimensionless variables involving the terms of the relativistic oscillator (let us say oscillations of the center of mass) and the rigid rotor/top terms (rotations about the center of mass)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

with the fundamental difference that the coordinates [x.sup.[mu]] and momenta [p.sup.[mu]] obey the non-commutative Yang's space time algebra. For this reason one cannot naively replace [p.sup.[mu]] any longer by the differential operator -i[??][differential]/[differential][x.sup.[mu]] nor write the [[summation].sup.[mu]v] generators as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The correct coordinate realization of Yang's noncommutative spacetime algebra requires, for example, embedding the 4-dim space into 6-dim and expressing the coordinates and momenta operators as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

This allows to express H in terms of the standard angular momentum operators in 6-dim. The [X.sup.A] = [X.sup.[mu]], [X.sup.5], [X.sup.6] coordinates ([mu] = 1, 2, 3, 4) and [P.sup.A] = [P.sup.[mu]], [P.sup.5], [P.sup.6] momentum variables obey the standard commutation relations of ordinary QM in 6-dim, namely - [[X.sup.A], [X.sup.B]] = 0, [[P.sup.A], [P.sup.B]] = 0, [[X.sup.A], [P.sup.B]] = i[??][[eta].sup.AB], so that the momentum admits the standard realization as [P.sup.A] = -i[??][differential]/[differential][X.sub.A].

Therefore, concluding, the Hamiltonian H in eq-(3.1) associated with the non-commuting coordinates [x.sup.[mu]] and momenta [p.sup.[mu]] in d - 1-dimensions can be written in terms of the standard angular momentum operators in (d - 1) + 2 = d + 1-dim as H = [C.sub.2] - [N.sup.2], where [C.sub.2] agrees precisely with the quadratic Casimir operator of the SO(d - 1, 2) algebra in the spin s = 0 case,

[C.sub.2] = [[summation].sub.AB][[summation].sup.AB] = ([X.sub.A][[differential].sub.B] - [X.sub.B][[differential].sub.A])([X.sup.A][[differential].sup.B] - [X.sup.B][[differential].sup.A]). (3.4)

One remarkable feature is that [C.sub.2] also agrees with the d'Alambertian operator for the Anti de Sitter Space Ad[S.sub.d] of unit radius (throat size) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as shown by [18].

The proof requires to show that the d'Alambertian operator for the d+1-dim embedding space (expressed in terms of the [X.sup.A] coordinates) is related to the d'Alambertian operator in Ad[S.sub.d] space of unit radius expressed in terms of the [z.sup.1], [z.sup.2], ..., [z.sup.d] bulk intrinsic coordinates as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

This result is just the hyperbolic-space generalization of the standard decomposition of the Laplace operator in spherical coordinates in terms of the radial derivatives plus a term containing the square of the orbital angular momentum operator [L.sup.2]/[r.sup.2]. In the case of nontrivial spin, the Casimir [C.sub.2] = [[summation].sub.AB][[summation].sup.AB] + [S.sub.AB][S.sup.AB] has additional terms stemming from the spin operator.

The quantity [PHI]([z.sup.1], [z.sup.2], ..., [z.sup.d])[|.sub.boundary] restricted to the d - 1-dim projective boundary of the conformally compactified Ad[S.sub.d] space (of unit throat size, whose topology is [S.sup.d-2] x [S.sup.1]) is the sought-after solution to the Casimir invariant wave equation associated with the non-commutative [x.sup.[mu]] coordinates and momenta [p.sup.[mu]] of the Yang's algebra ([mu] = = 1, 2, ..., d-1). Pertaining to the boundary of the conformally compactified Ad[S.sub.d] space, there are two radii [R.sub.1], [R.sub.2] associated with [S.sup.d-2] and [S.sup.1], respectively, and which must not be confused with the two scales R, [L.sup.P] appearing in eq-(3.1). One can choose the units such that the present value of the Hubble scale (taking the Hubble scale as the infrared cutoff) is R = 1. In these units the Planck scale [L.sub.P] will be of the order of [L.sub.P] ~ [10.sup.-60]. In essence, there has been a trade-off of two scales [L.sub.P], R with the two radii [R.sub.1], [R.sub.2].

Once can parametrize the coordinates of Ad[S.sub.d] = Ad[S.sub.p+2] by writing there, according to [17], [X.sub.0] = R cosh([rho]) cos([tau]), [X.sub.p+1] = R cosh([rho]) sin([tau]), [X.sub.i] = R sinh([rho])[[OMEGA].sup.i].

The metric of Ad[S.sub.d] = Ad[S.sub.p+2] space in these coordinates is d[s.sup.2] = [R.sup.2] [-(cos[h.sup.2] [rho])d[[tau].sup.2] + d[[rho].sup.2] + (sin[h.sup.2] [rho])d[[OMEGA].sup.2]], where 0 [less than or equal to] [rho] and 0 [less than or equal to] [tau] < 2[pi] are the global coordinates. The topology of this hyperboloid is [S.sup.1] x [R.sup.p+1]. To study the causal structure of AdS it is convenient to unwrap the circle [S.sup.1] (closed-timelike coordinate [tau]) to obtain the universal covering of the hyperboloid without closed-timelike curves and take -[infinity] [less than or equal to] [tau] [less than or equal to] +[infinity]. Upon introducing the new coordinate 0 [less than or equal to] [theta] < [pi]/2 related to [rho] by tan([theta]) = sinh([rho]), the metric is

d[s.sup.2] = [R.sup.2]/[cos.sup.2] [theta] [-d[[tau].sup.2] + d[[theta].sup.2] + (sin[h.sup.2] [rho])d[[OMEGA].sup.2]]. (3.6)

It is a conformally-rescaled version of the metric of the Einstein static universe. Namely, Ad[S.sub.d] = Ad[S.sub.p+2] can be conformally mapped into one-half of the Einstein static universe, since the coordinate [theta] takes values 0 [less than or equal to] [theta] < [pi]/2 rather than 0 [less than or equal to] [theta] < [pi]. The boundary of the conformally compactified Ad[S.sub.p+2] space has the topology of [S.sup.p] x [S.sup.1] (identical to the conformal compactification of the p + 1-dim Minkowski space). Therefore, the equator at [theta] = [pi]/2 is a boundary of the space with the topology of [S.sup.p]. [[OMEGA].sub.p] is the solid angle coordinates corresponding to [S.sup.p] and [tau] is the coordinate which parametrizes [S.sup.1]. For a detailed discussion of AdS spaces and the AdS/CFT duality see [17].

The d'Alambertian in Ad[S.sub.d] space (of radius R, later we shall set R = 1) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7)

where [L.sup.2] is the Laplacian operator in the p-dim sphere [S.sup.p] whose eigenvalues are l (l + p - 1).

The scalar field can be decomposed as follows [PHI]=[e.sup.[omega]R[tau]][Y.sub.l]([[OMEGA].sub.p])G([theta]); the wave equation ([D.sub.[mu]][D.sup.[mu]] - [m.sup.2])[PHI]=0 leads to the equation [[cos.sup.2] [theta] ([[omega].sup.2] + [[differential].sup.2.sub.[theta]] + p/tan [theta] [cos.sup.2] [theta]] [[differential].sub.[theta]] + + l(l+p-1)/[tan.sup.2] [theta] - [m.sup.2][R.sup.2]G([theta]) = 0, whose solution is

G([theta]) = [(sin [theta]).sup.l] [(cos [theta]).sup.[lambda][+ or -]] [sup.2][F.sub.1](a, b, c; sin [theta]). (3.8)

The hypergeometric function is defined as

[sub.2][F.sub.1](a, b, c, z) = [summation][(a).sub.k][(b).sub.k]/[(c).sub.k]k! [z.sup.n], (3.9)

where [absolute value of z] <1, [([lambda]).sub.0]=1, [([lambda]).sub.k] = [GAMMA]([lambda]+k)/[GAMMA]([lambda]) = [lambda]([lambda] + 1)([lambda] + 2) ... ([lambda] + k - 1), k = 1, 2, ..., while a = 1/2 (l + [[lambda].sub.[+ or -]] - [omega]R), b = = 1/2 (l + [[lambda].sub.[+ or -]] + [omega]R), c=l + 1/2(p+1)>0, [[lambda].sub.[+ or -]] = 1/2 (p+1) [+ or -] [+ or -] 1/2 [square root of [(p+1).sup.2] + 4[(mR).sup.2].

The analytical continuation of the hypergeometric function for [absolute value of z] [greater than or equal to] 1 is

[sub.2][F.sub.1](a, b, c, z) = = [GAMMA](c)/[GAMMA](b)[GAMMA](c - b) [[integral].sup.1.sub.0] [t.sup.b-1][(1 - t).sup.c-b-1][(1 - tz).sup.-a] dt (3.10)

with Real(c) > 0 and Real(b) > 0. The boundary value when [theta] = [pi]/2 gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.11)

Let us study the behaviour of the solution G([theta]) in the massless case: m = 0, [[lambda].sub.-] = 0, [[lambda].sub.+] = p+1.

Solutions with [[lambda].sub.+] = p+1 yield a trivial value of G([theta]) = 0 at the boundary [theta] = [pi]/2 since cos [([pi]/2).sup.p+1] = 0. Solutions with [[lambda].sub.-] = 0 lead to cos [([theta]).sup.[lambda]-] = cos [([theta]).sup.0] = 1 prior to taking the limit [theta] = [pi]/2. The expression cos [([pi]/2).sup.[lambda]-] = [0.sup.0] is ill defined. Upon using l'Hospital rule it yields 0. Thus, the limit [theta] = [pi]/2 must be taken afterwards the limit [[lambda].sub.-] = 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.12)

In this fashion the value of G([differential]) is well defined and nonzero at the boundary when [[lambda].sub.-] = 0 and leads to the value of the wavefunction at the boundary of the conformally compactified Ad[S.sub.d] (for d = p + 2 with radius R)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13a)

upon setting the radius of Ad[S.sub.d] space to unity it gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.13b)

Hence, [[PHI].sub.bound] in eq-(3.13b) is the solution to the Casimir invariant wave equation in the massless m = 0 case

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.14)

and (when R = 1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15)

since N = [[summation].sup.56] is the rotation generator along the [S.sup.1] component of AdS space. It acts as [differential]/[differential]T only on the [e.sup.iwRT] piece of [PHI]. Concluding: [PHI]([z.sup.1], [z.sup.2], ..., [z.sup.d])|boundary, restricted to the d - 1-dim projective boundary of the conformally compactified Ad[S.sub.d] space (of unit radius and topology [S.sup.d-2] x [S.sup.1]) given by eq-(3.12), is the sought-after solution to the wave equations (3.13, 3.14) associated with the non-commutative [x.sup.[mu]] coordinates and momenta [p.sup.[mu]] of the Yang's algebra and where the indices [micro] range over the dimensions of the boundary [micro] = 1, 2, ..., d - 1. This suggests that QM over Yang's Noncommutative Spacetimes could be well defined in terms of ordinary QM in higher dimensions! This idea deserves further investigations. For example, it was argued by [16] that the quantized Nonabelian gauge theory in d dimensions can be obtained as the infrared limit of the corresponding classical gauge theory in d + 1-dim.

4 Star products and noncommutative QM

The ordinary Moyal star-product of two functions in phase space f(x, p), g(x, p) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.1)

where C(s, t) is the binomial coefficient s!/t!(s - t)!. In the h [right arrow] 0 limit the star product f * g reduces to the ordinary pointwise product fg of functions. The Moyal product of two functions of the 2n-dim phase space coordinates ([q,sub.i], [p.sub.i]) with i = 1, 2 ... n is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.2)

The noncommutative, associative Moyal bracket is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.3)

The task now is to construct novel Moyal-Yang star products based on the noncommutative spacetime Yang's algebra. A novel star product deformations of (super) p-brane actions based on the noncommutative spacetime Yang's algebra where the deformation parameter is [[??].sub.eff] = [??][L.sub.P] / R, for nonzero values of [??], was obtained in [15] The modified (noncommutative) Poisson bracket is now given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.4)

where the entries {[q.sup.m], [q.sup.n]} [not equal to] 0, {[p.sup.m], [p.sup.n]} [not equal to] 0, and also {[p.sup.m], [q.sup.n]} = -{[q.sup.n], [p.sup.m]} can be read from the commutators described in section 2 by simply defining the deformation parameter [[??].sub.eff] [equivalent to] [??]([L.sub.P] / R). One can generalize Yang's original 4-dim algebra to noncommutative 2n-dim world-volumes and/or spacetimes by working with the 2n + 2-dim angular-momentum algebra SO(d, 2) = SO(p+1, 2) = SO(2n, 2).

The Noncommutative Poisson brackets [OMEGA]([q.sup.m], [q.sup.n]) =

= [{[q.sup.m], [q.sup.n]}.sub.NCPB], [OMEGA]([p.sup.m], [p.sup.n]) = [{[p.sup.m], [p.sup.n]}.sub.NCPB], [OMEGA]([q.sup.m], [p.sup.n]) = = -[OMEGA]([p.sup.n], [q.sup.m]) = [{[q.sup.m], [p.sup.n]}.sub.NCPB]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.5a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.5b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.5c)

defined by above expressions, where [[summation].sup.mn] is the "classical" [[??].sub.eff] = ([??][L.sub.P] / R) [right arrow] 0 limit (R [right arrow] [infinity], [L.sub.P] [right arrow] 0, R[L.sub.P] = [L.sup.2], [??] [not equal to] = 0) of the quantity [[summation.sup.mn] = 1/[??]([X.sup.m][P.sup.n] - [X.sup.n][P.sup.m]), after embedding the d-1-dimensional spacetime (boundary of Ad[S.sub.d]) into an ordinary (d - 1)+2-dimensional one. In the R [right arrow] [infinity], ... limit, the Ad[S.sub.d] space (the hyperboloid) degenerates into a flat Minkowski spacetime and the coordinates [q.sup.m], [p.sup.n], in that infrared limit, coincide with the coordinates [X.sup.m], [P.sup.n]. Concluding, in the "classical" limit (R [right arrow] [inifinity], ..., flat limit) one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.5d)

and then one recovers in that limit the ordinary definition of the angular momentum in terms of commuting coordinates q's and commuting momenta p's.

Denoting the coordinates ([q.sup.m], [p.sup.m]) by [Z.sup.m] and when the Poisson structure [[OMEGA].sup.mn] is given in terms of constant numerical coefficients, the Moyal star product is defined in terms of the deformation parameter [[??].sub.eff] = [??][L.sub.P] / R as

(F x G) (z) [equivalent to]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.6)

where the derivatives [[differential].sup.([z.sub.1]).sub.m] act only on the F([z.sub.1]) term and [[differential].sup.([z.sub.2]).sub.n] act only on the [??]([z.sub.2]) term. In our case the generalized Poisson structure [[OMEGA].sup.mn] is given in terms of variable coefficients, it is a function of the coordinates, then [differential][[OMEGA].sup.mn] [equivalent to] = 0, since the Yang's algebra is basically an angular momentum algebra, therefore the suitable Moyal-Yang star product given by Kontsevich [11] will contain the appropriate corrections [differential][[OMEGA].sup.mn] to the ordinary Moyal star product.

Denoting by [[differential].sub.m] = [differential]/[differential][z.sup.m] = ([differential]/[differential][q.sup.m]; [differential]/[differential][p.sup.m]) the Moyal-Yang-Kontsevich star product, let us say, of the Hamiltonian H(q, p) with the density distribution in phase space [rho] (q, p) (not necessarily positive definite), H(q, p) x [rho] (q, p) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.7)

where the explicit components of [[OMEGA].sup.mn] are given by eqs-(4.5a-4.5d). The Kontsevich star product is associative up to second order [11] (f * g) * h = f * (g x h) + O([[??].sup.3.sub.eff].

The most general expression of the Kontsevich star product in Poisson manifold is quite elaborate and shall not be given here. Star products in curved phase spaces have been constructed by Fedosov [12]. Despite these technical subtlelties it did not affect the final expressions for the "classical" Noncommutative p-brane actions as shown in [15] when one takes the [[??].sub.eff] [right arrow] 0 "classical" limit. In that limit there are still nontrivial noncommutative corrections to the ordinary p-brane actions.

In the Weyl-Wigner-Gronewold-Moyal quantization scheme in phase spaces one writes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (4.8)

where the Wigner density function in phase space associated with the Hilbert space state |[PSI] > is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.9)

plus their higher dimensional generalizations. It remains to be studied if this Weyl-Wigner-Gronewold-Moyal quantization scheme is appropriate to study QM over Noncommutative Yang's spacetimes when we use the above Moyal-Yang-Kontsevich star products. A recent study of the Yang's Non-commutative algebra and discrete Hilbert (Buniy-Hsu-Zee) spaces was undertaken by Tanaka [3].

Let us write down the Moyal-Yang-Konstevich star deformations of the field theory Lagrangian corresponding to the scalar field [PSI]= [PSI]([X.sup.AB]) which depends on the holographic-area coordinates [X.sup.AB] [13]. The reason one should not try to construct the star product of [PSI]([x.sup.m) x [PSI]([x.sup.n]) based on the Moyal-Yang-Kontsevich product, is because the latter star product given by eq-(4.7) will introduce explicit momentum terms in the r.h.s of [PSI]([x.sup.m]) x [PSI]([x.sup.m]), stemming from the expression [[summation].sup.mn] = [x.sup.m][p.sup.n] - [x.sup.n][p.sup.m] of eq-(4.5d), and thus it invalidates writing [phi] = [phi](x) in the first place. If the [[summation].sup.mn] were numerical constants, like [[THETA].sup.mn], then one could write the [PHI]([x.sup.m]) * [PHI]([x.sup.m]) in a straightforward fashion as it is done in the literature.

The reason behind choosing [PHI] = [PHI]([X.sup.AB]) is more clear after one invokes the area-coordinates and angular momentum correspondence discussed in detail in section 2. It allows to properly define the star products. A typical Lagrangian is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.10)

and leads to the equations of motion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.11)

when the multi-symplectic [[OMEGA].sup.ABCD] form is coordinate-independent, the star product is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.12)

where [[summation].sup.ABCD] is derived from the structure constants of the holographic area-coordinate algebra in C-spaces [14] as: [[X.sup.AB],[X.sup.CD] = [[summation].sup.ABCD] [equivalent to] i[L.sup.2.sub.P] ([n.sup.AD] [X.sup.BC] - [n].sup.AC] [X.sup.BD] + + [n.sup.BC] [X.sup.AD] - [n.sup.BD] [X.sup.AC]. There are nontrivial derivative terms acting on [[summation].sup.ABCD] in the definition of the star product ([PHI] x [PHI])([Z.sup.MN]) as we have seen in the definition of the Kontsevich star product H(x, p) x [rho] (x, p) in eq-(4.7). The expansion parameter in the star product is the Planck scale squared [lambda] = [L.sup.2.sub.P]. The star product has the same functional form as (4-7) with the only difference that now we are taking derivatives w.r.t the area-coordinates [X.sup.AB] instead of derivatives w.r.t the variables x, p, hence to order O([L.sup.4.sub.P]), the star product is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.13)

Notice that the powers of i[L.sup.2.sub.P] are encoded in the definition of [[summation].sup.ABCD]. The star product is noncommutative but is also nonassociative at the order O([L.sup.6.sub.P]) and beyond. The Jacobi identities would be anomalous at that order and beyond. The derivatives acting on [[summation].sup.ABCD] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4.14)

where [[delta].sup.AB.sub.CD] = [[delta].sup.AC][[delta].sup.BD] - [[delta].sup.AD][[delta].sup.BC] and the higher derivatives like [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be zero.

5 On the generalized Dirac-Konstant equation in Clifford spaces

To conclude this work we will discuss the wave equations relevant to fermions. The "square" of the Dirac-Konstant equation ([[gamma].sup.[[mu]v]][[summation].sub.[mu]v]) [PSI] = [lambda][PSI] yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.2)

where we omitted numerical factors. The generalized Dirac equation in Clifford spaces is given by [13]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.3)

where [sigma], [x.sup.[mu]], [x.sup.[mu]v], ... are the generalized coordinates associated with the Clifford polyvector in C-space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.4)

after the length scale expansion parameter is set to unity. The generalized Dirac-Konstant equations in Clifford-spaces are obtained after introducing the generalized angular momentum operators [14]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.5)

by writing

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.6)

and where we sum over all polyvector-valued indices (antisymmetric tensors of arbitrary rank). Upon squaring eq-(5.4), one obtains the Clifford space extensions of the D0-brane field equations found in [3] which are of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5.6)

where A,B = 1, 2, ..., 6. It is warranted to study all these equations in future work and their relation to the physics of D-branes and Matrix Models [3]. Yang's Noncommutative algebra should be extended to superspaces, meaning nonanti-commuting Grassmanian coordinates and noncommuting bosonic coordinates.

Acknowledgments

We are indebted to C. Handy and M. Bowers for encouragement and support.

References

[1.] Yang C. N. Phys. Rev., 1947, v. 72, 874; Proc. of the Intern. Conference on Elementary Particles, 1965, Kyoto, 322-323.

[2.] Snyder S. Phys. Rev., 1947, v. 71, 38; ibid., 1947, v. 71, 78.

[3.] Tanaka S. Nuovo Cimento, 1999, v. 114B, 49; arXiv: hep-th/ 0511023; hep-th/0406166; hep-th/0002001; hep-th/0303105.

[4.] Ashtekar A., Rovelli C., and Smolin L. Phys. Rev. Lett., 1992, v. 69, 237.

[5.] Castro C. Mod. Phys. Lett., 2002, v. A17, No. 32, 2095.

[6.] Castro C. arXiv: hep-th/9908115; Ansoldi S., Castro C., and Spallucci E. Phys. Lett., 2001, v. B504, 174; Class. Quan. Gravity, 2001, v. 18, L17-L23; Class. Quan. Gravity, 2001, v. 18, 2865; Ansoldi S., Castro C., Guendelman E., and Spallucci E. Class. Quant. Gravity, 2002, v. 19, L135.

[7.] Castro C. General Relativity and Gravitation, 2004, v. 36, No. 12, 2605.

[8.] Castro C. Jour. Math. Phys., 1993, v. 34, 681; Phys. Lett., 1992, v. B288, 291.

[9.] Park Q. H. Int. Jour. Mod. Phys., 1992, v. A7, 1415; Garcia-Compean H., Plebanski J., and Przanowski M. arXiv: hep-th/ 9702046.

[10.] Hoppe J. Quantum theory of a relativistic surface. Ph.D Thesis, MIT, 1982.

[11.] Kontsevich M. arXiv: q-alg/9709040.

[12.] Fedosov B. J. Diff. Geom., 1994, v. 40, No. 2, 213.

[13.] Castro C. and Pavsic M. Progress in Physics, 2005, v. 1, 31-64.

[14.] Castro C. Foundations of Physics, 2005, v. 35, No. 6, 971.

[15.] Castro C. Phys. Lett., 2005, v. B626, 209; Foundations of Physics, 2000, v. 8, 1301.

[16.] Biro T., Mueller B., and Matinyan S. arXiv: hep-th/0301131.

[17.] Aharony O., Gubser S., Maldacena J., Ooguri H., and Oz Y. Phys. Rep., 2000, v. 323, 183.

[18.] de Witt B. and Herger I. arXiv: hep-th/9908005.

Carlos Castro

Center for Theoretical Studies of Physical Systems, Clark Atlanta University,

Atlanta, Georgia, USA

* Email: czarlosromanov@yahoo.com; castro@ctsps.cau.edu
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