geodesic multiplication in the theory of...

Geodesic Multiplication in the Theory of Gravity Piret Kuuskt and Eugen Paa}tt

t1nstitut~ of Physics, Estonian Acad. Sci. 142 Riia Street, Tartu EE2400 '

tt . Estonia Institute of Matherr:a.tics, Tallinn Technical University,

1 Akadeemia Street, Tallinn EE0108 Estonia

Abstract

ti The. con~tructio~ of the local geodesic multiplication of space-me pomts _is desc~1~ed .. The structure constants generated by the

local geodesic mult1phcat10n are given in terms of the torsion and th curvature tensors of the space-time The tangent Ak" · 1 b e · t d · 1v1s a ge ras are l~ trio ucedd. !he ~u~ntum gravity is considered as a representation o 1e geo es1c Ak1v1s algebra.

1 Introduction

At the beginning of the XXth century, two new physical theories were pro~os~~that turned out to be extremely successful in their domain of a.pp ca ty - quantum mechanics and general relativity. However all s~tbse_;:uent attempts to merge them into a unique theory of quantum ~rav1 y ave not reached the goal. The basic concepts and mathematical apparatus of these ~wo theories turned out to be too different.

One of the ~am controversies between the general-relativistic and ~uantum-meciiarnc~ ~pproach concerns the role of space-time and its pomts. In general_ relativity, the space-time is a set of pointlike events E h event can phys1call b k · d . · ac y e rec ogrnze as an mtersection of two world lines

324

Geodesic Multiplication in the Theory of Gravity 325

of particles or light rays, and mathematically labelled by coordinates xk.

In the quantum mechanics, the basic notion is the Hilbert space of state vectors. In case of the Schrodinger representation, a state vector takes the form of a wave function ,,P( x) with arguments x that are eigenvalues of coordinate operators xi [1):

(1)

The Einstein-Podolsky-Rosen paradox demonstrates that eigenvalues (xi) cannot always and in general be identified with the coordinates of the Minkowski space-time. The role of arguments of wave functions ,,P( x) seems to be even less compatible with the concept of coordinates of the curved space-time of general relativity. In quantum mechanics, the Hilbert space is distinguished by the existence of the operation of scalar product that is an integral over the whole space, J <pt(x),,P(x)dx. In general relativity, physical processes have local character and integration over the whole space (on equal-time hypersurfaces) is an ill-defined notion.

There have been several attempts to find a mathematical framework which allows to introduce some kind of smearing out of classical events (space-time points). Geroch [2] proposed to consider as the basic notion not the "differentiable manifold M of space-time points but the commutative ring A of smooth functions on M, since the geometry of manifold

- M can be given in terms of the commutative ring A. Now let the ring A be taken noncommutative. It turns out that i~ is still possible to introduce the analogues of vector fields, one-forms and other notions of the ordinary differential geometry, although the "points" of the underlying manifold are essentially lacking. The mathematical concept of noncommutative geometry has been proposed by Connes [3, 4] and Dubois-Violette [5]. An algebraic approach to differential geometry of physical gauge fields has been developed by Landi and Marmo [6). Dubois-Violette, Kerner and Madore [7) have presented a new model of ~auge theory based on the noncommutative differential geometry of the algebra C00 (V) ® Mn( C) of smooth matrix-valued functions on a manifold V. Chamseddine, Felder and Frohlich [8) also use the nonccimmutative algebra of matrix-valued functions for describing models in particle theory and gravity. Attempts to use the smearing out of space-time points in the theory of gravity have been presented by Madore [9), Parfionov and Zapatrin [10).

326 Piret Kuusk and Eugen Paal

In the present paper we propose a novel way to reconcile the notions of general-relativistic space-time points and quantum-mechanical coordinate operators. Following the mathematical ideas of J{ikkawa [11] and Sabinin [12] we introduce a local binary operation in the set of spc;i.ce-time points, the geodesic multiplication. In general it turns out to he noncommutative and nonassociative. Our proposal is to proceed from the geodesic algebra of space-time events and to find a suitable representation of this algebra for quantizing gravity. In other words, in quantum gravity space-time points should be replaced by the points of the representation space of the geodesic algebra. The problem is ~athematically rather complicated, since the general representation theory for nonassociative algebras needed for quantum gravity is essentially still lacking.

The paper is organized as follows. In Sec. II, the construction of the local geodesic multiplication of space-time points is described. In Sec. III, the structure constants generated by the local geodesic multiplication are given in terms of the torsion and the curvatrne of the space-time. In Sec.IV, the tangent Aki vis algebras are introduced. In Sec. V, the quantum gravity is considered as a representation of the geodesic Akivis algebra.

2 Local Geodesic Multiplication

At first let us introduce some basic algebraic notions. A quasigroup [13, 14] is a. set G of points with a binary operation (multiplication) which has the following property: in the equation gh = k, the knowledge of any two elements specifies the third one uniquely. A quasigroup with a unit element e is called a loop [13, 14].

For each element g of the loop there exist the left and the right inverse l t -1 -1 d fi d b -1 -1 e emen s gL ,gR e ne y gL g = e, ggR = e.

As in case of groups one can define the left (L) and the right (R) translations by ·

(2)

From the definition of the quasigroup it follows that these translations are bijections. Translations L9 and R9 generate a group that is called the Albert group of the loop G (13, 14). Nonassocia.tivity of G can now be measured by the deviation from unity of the following elements of the


Albert group of G

) R R-1 R-1 M(g, h) ~ R9Lf:1 R;1

Lh· L(g,h)::L;~L9Lh, R(g,h = 9h 9 h' .

f M with an affine connection, For a fixed point e of the space- ime TM f Mate Consider

t t X from the tangent space e o . choose a tangen vec or h tl . t e with the tangent vector a local path t -+ g( t; X) in M throug ie pom

X at e . ) dgi(O; X) _ Xi (3) g'(O;X = e, dt - ·

t . 11 known that this path is a unique local geodesic path through e

I is we . "al f on holds· · in the direction of X iff the following different! equa i .

. . k h dgi d2g' i dg' dg = 0 (4) M di = dt2 + r ik dt dt ,

· th affi connection coefficients. where fjk denote. e n~ ·-ex X := g(l;X) ate is known

The exponential mappmg X -+ g ·-:- Pe . borhood of the origin of (15) to be a local diffeomo~phism of a~)t~b~b:~~!od of e E M. Note that TeM onto the correspondin~ (n~rm en:~; ent vectors of the space-time this property allows us to ldent~y th th ~ every event from the normal

as .infi.nithesimdal feven::_ :eg~:ra::ds:ra t:e exponential. ma~ping by the neighbor oo o e c corresponding tangent vector from ~eM. tructed in such a neighborhood

The local geodesic loop at e can e cons . well defined local . d t" al mappings are

Me of e wh~re all reqmr~ . exp:::h~r local geodesic arc h( s; Y) through di:ffeomorph1sms. C~oos~ m Me TM T rforro a parallel transport of the point e in the direction. y E e .t ~ pe the linear Cauchy problem X E TeM along this geodesic, we mus so ve

6X'i - dX'i ri. dhk x'i = O; X'(O) = x. (5) -- = -d- + 3k ds 6s s

t f X E T M we obtain at h := expe Y Performing the parallel tra~s(p)or ; M No~ draw the local geodesic arc the tangent vector X' := X 1 E h · k . 't X' on it This point

. h d" t" n X' and mar pom exph . through h m t e irec 10

' . h and it will be denoted as is cal.led the geodesic product of pomts g and ' d 11 12] gh. Explicitly, the geodesic multiplication formula rea s [ '

e -1) (6) gh = Rhg::: (exph orh o expe g,

328 Piret Kuusk and Eugen PaaI

where r~ : TeM _,. Tli.M denotes the parallel transport mapping of tangent I

vectors from TeM int6 ThM along the unique local geodesic arc joining the points e and h: rJ:(X) = X'. fu respect of multiplication (6) only right 1

translation can be seen explicitly

(7)

The neighborhood Me of e with the multiplication rule (6) turns out to be a local differentiable loop [11, 12, 16] denoted henceforth by Me as well. The unit element of Me is e, and local geodesic paths through the unit element e are one-parameter subgroups of Me. From defining formula (6) it follows that the local geodesic multiplication is power associative:

for all m,nE N. (8)

Note that crucial part of the construction lies in the Cauchy problems (3)-( 4) and (5), in the existence and uniqueness of their solutions, and also in the local diffeomorphism property of the exponential mapping [15].

Translations L9 and R9. of the geodesic loop Me generate a subgroup in the group of diffeomorphisms, the Albert group of Me. Representations of the Albert group could be considered as representations of the geodesic loop. However, the group of diffeomorphisms has a rather complicated structure that is not ·well understood up to now [18]. The theory of representations of nonassociative algebraic systems has been elaborated only for some simple cases as Lie algebras, Jordan algebras, Mal'tsev algebras and Lie triple systems (see ref. [19] for a recent review). The Lie algebras of the Albert groups of analytic Moufang loops are calculated in refs. [19, 20].

We can repeat the above construction and attach a local geodesic loop to all reasonable (nonsingular) points of the space-time. Patching conditions for local geodesic loops attached to different points of a manifold with an affine connection have been described by Sabinin [12], for a recent review see ref. [21].


3 Structure of the Local Geodesic Loop

all aki the aeodesic multiplication need not be commutative Gener Y spe ng, 0

• f · h k M th t . t" Th re mav exist such a tnple o points g' ' E a. and a.ssocia ive. e 'J

gh i= hg, (gh)k i= g(hk). (9)

. . M where e' = o for all i. The deviati-Let us choose lo~al coor~a;;;: co~mutativity and associativity can be

. on of the geodesic loop e i d Ai defined by measured (161 by the structure constants c,m an lmn

( (hg)£1(gh)y = cfmg'hm + ... '

(lg(hk)lil[(gh)k]) i = Aimnglhmkn + ... '

(10)

(11)

d t the higher order terms. cal. where o s mean h t mmutativity and nonassociativity of the lo

It turns out t a nonco d h t re of . . 1 related to the torsion an t e curva u

geodesic loops are mtimatealy di t be the Riemannian normal coor-the spate-time. Let the loc coor na. es .. cllilates determined by _the coordinate conditions

(12)

. . · f om the unit element e are Then the equations of geodesic lines emerging r

simply . - i X' E T. M (13) g'(t)=Xt, e •

. . · f m some other point h E The equations of'geodesic lines em~rgmg. ro . . . :finitesimal M can be solved in any approximation usmg expansions m im i

ev~nts. Denote the torsion and the cur~ature tens~rs as Sim and R1mn

respectively.lOur sign conventions are as m ref. [17].

. i Sim= -r(lm1'

The direct computations (16] show that

Cfm = 2Sfm(e), (14)

1/2 21 3aKa3 63f>

330 Piret Ku usk and Eugen Paa]

(15) where V n is the covariant di:ff erentiation operator.

In case of a R.iemannian space-time without torsion, the only structure constants are those of the associator, Afmn = Rfmn ( e). The first non vanishing term of the associator can be read out from eqs. (11), (15) but for finding the the first nonvanishing term of the commutator the higher order terms in eq. (10) are needed. From the direct computations [21] we get

((hg)£1(gh)y = Rfmn(e)(g1hmgn - h1gmhn) + ... , {16)

(r!i(hk)]£1[(gh)kJY = Rfmn(e)g1hmkn +.. .. {17)

We see that all geodesic loops of the Minkowski space-time are Abelian groups. In this particular case, the geodesic multiplication coincides with the common vector addition rule [22]. The Abelian property manifests algebraically the fact that affine spaces are globally torsionless and fiat.

4 Akivis Algebras

Let us introduce the tangent algebras of loops. Geometrically, the tangent algebra Ae of Me coincides with the tangent space TeM of Me at e. The product [X, YJ of X, Y E Ae is defined in Ae by means of the structure constants Cfm of the geodesic loop (10):

{18)

We can equip Ae with a ternary operation as well (16, 23, 24). For a triple X, Y, Z E Ae, define their triple product (X, Y, Z) E Ae by using the other structure constants Afmn of a geodesic loop (11):

(19) /

The tangent algebra Ae is thus a binary-ternary algebra, and it need not be a Lie algebra. In other words, there may be a triple X, Y, Z E Ae, such that the Jacobi identity fails in Ae

J(X, Y, Z) := [[X, Y], ZJ + [[Y, Z], XJ + [[Z,X], YJ -::fi 0. (20)

~

I Geodesic Multiplication in the Theory of Gravity

Instead, for all X, Y, Z E Ae, we have a more general identity [16]

J(X, Y, Z) = (X, Y, Z) + (Y, Z,X) + (Z, X.Y)

- (X, Z,Y)- (Z,Y,X)- (Y,X, Z),

•'

331

(20)

called the Akivis identity. The binary-ternary algebra Ae is called the Akivis algebra..

In case of a space-time M with non.vanishing torsion it follows that for all X, Y, Z E Ae we have

[X, Y]i = 2SfmX1Ym,

(X,Y,z)i = (Rlmn -VnSlm)X1Ymzn.

{21)

(22)

We see that the tangent algebra. of a. R.iemannian space-time with vanishing torsion is commutative but nonassociative.

5 Quantum Gravity as a Representation of the Geodesic Akivis Algebra

Quantization of a physical system is essentially a. heuristic procedure with the aim of finding a mathematical framework that allows .us to describe the results of experiments carried out in the quantum realm. In ca.se of gravitation, up to now no experiments have detected gravitons, quanta of the gravitational field. So the only guiding principle can be the analogy with the already existing quantum theories.

The quantization procedure that seems to be best understood is the canonical quantization given by Dirac [1]. Let us consider a classical system with the phase space (pi, qk) and classical observables F(p, q). The Poisson brackets

(23)

determine the Poisson algebra of observables. The canonical quantizatiOn amounts to finding a representation of classical observables as operators acting on the Hilbert space of state vectors. The algebra of Poisson brackets (23) is replaced by the commutator algebra of operators Pi, Qk

{24)


Canonical quantization scheme has been applied to the Einstein-Hilbert lagrangian of the gravitational field by Arnowitt, Deser and Misner [25], Ashtekar [26] and others, for pedagogical reviews see refs. [17, 27}. The resulting theories are ambiguous, mathematically complicated and physically not easily understandable.

We propose to consider as the guiding idea of canonical quantization finding a suitable representation of the algebra that characterizes the classical system. The algebra that naturally belongs to the curved spacetime (gravitational field) is the geodesic Akivis algebra. So the quantization of the gravitational field amounts to finding a suitable representation of the geodesic Akivis algebra. As an analogue of the usual canonical quantum conditions (24) we have proposed [21, 22) the following geodesic quantum conditions:

[Qx,Qy) == QxQy -QyQx = qQ[x,Y], (25)

(Qx, Qy, Qz) == QxQy · Qz - Qx · QyQz = q2Qcx,Y,Z)- (26)

Here, the quantization parameter (geodesic quantum deformation parameter) is denoted as q, and Qx (quantum event) belong to a representation space of the geodesic Akivis algebra X ~ Qx.

Recall that X E TeM denotes a tangent vector, or equivalently, a spacetime point from the infinitesimal neighborhood of the unit element e. In quantum gravity we have replaced the space-time points by the elements Qx of the representatfon space of the geodesic Akivis algebra. Unfortunately, the representation theory for nonassociative non-Lie algebras is essentially still lacking and we cannot indicate the precise mathematical nature of Q-s.

The canonical quantum conditions (24) are ~terpreted as describing impossibility to measure the generalized coordinate qk and the corresponding generalized momentum Pk simultaneously. If we preserve this interpretation also for geodesic quantum conditions ( 25) restrictions on the measurability of infinitesimal points (vectors) X of the space-time arise due to the torsion. The physical interpretation of the second set (26) of quantum conditions is unclear yet. They are reasonable only in the neighborhood~ of space-time p~ints 'vhich are nonsingular, i.e. the components of the lliema.nn tensor Rimn(e) have finite values. Note that the Akivis algebra (18), (19) and the quantum conditjons (25), (26) are not the only ones that can be introduced on the basis of the geodesic loop.


The geodesic quantization has one more novel feature - quantum conditions depend on the dynamics of the gravitational field. Indeed, the geodesic multiplication (6) is determined by geodesic lines. In the framework of the conventional general relativity they can be found only upon solving the Einstein equations for metric tensor and finding connection coefficients. Note that in case of a Riemannian space-time with vanishing torsion geodesic lines allow to determine all components of the curvature tensor via geodesic deviation equation [17). This means that quantum conditions contain all dynamics. As a result geodesic quantization cannot be divided into quantum conditions and dynamical evolution equations as quantum mechanics.

Acknowledgments

This work was supported by the Estonian Science Foundation under Grants No. 359 and No. 348. One of the authors (P.K.) acknowledges stimulating discussions and warm hospitality during the Friedmann Seminar.

References

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