PHYSICAL REVIEW D VOLUME 35, NUMBER 4 15 FEBRUARY 1987

Geometric interpretation of a Yang-Mills theory for gravity

F. Ghaboussi and H. Dehnen Fakultat fur Physik der Universitat Konstanz, D-7750 Konstanz, Postfach 5560, West Germany

Mark Israelit University of Haifa, School of Education of the Kibbutz Movement, Oranim, Tivon 36910, Israel

(Received 26 March 1986)

The geometrical content of the classical limit of a Yang-Mills theory for gravity proposed by Dehnen and Ghaboussi is discussed in detail. It is shown that Einstein's theory can be reached without conditions which are too restrictive.

I. INTRODUCTION

Recently two of the authors (H.D. and F.G.) have pro- posed a Yang-Mills gauge theory for gravity' on the basis of Minkowski space-time. The reasons for doing this are the difficulties of usual gravity with respect to quantiza- tion and unification with the remaining physical interac- tions. In this connection we have emphasized that, in any case, quantum or microscopic physics possesses priority and follows directly from very few general first principles, whereas all macroscopic physics including Einstein's metric theory of gravity must be deduced from quantum physics in a certain classical limit.

Following the successful line of gauging of compact and unitary internal-symmetry groups for describing elec- troweak and strong interactions we have chosen, in a pre- vious paper,' as a gauge group for the Yang-Mills theory of gravity the most simple possibility beyond the U(1) phase gauge group: namely, the U(2) transformation group of the two-spinors for massless fermions. Taking into account additionally the usual principle of minimal coupling we obtained a microscopic Lorentz-invariant Yang-Mills theory which leads in its classical limit to Einstein's metric theory of gravity for massless fermionic matter, where the non-Euclidean metric is built up by the product of the expectation values of the four vector gauge potentials belonging to the group SU(2) XU(1). In this last step we have restricted ourselves for simplicity to the linearized version.

However, our theory needs further investigations. First of all, it is a fermionic theory of gravity; the interaction of gravity with the other bosonic fields may be included only within a grand unified theory of all interactions. Then it is not clear whether the product of the gauge potentials represents a stable state for the metric. We could not show until now that Einstein's exact theory can be reached with our approach. Solitonlike classical solutions and their meaning for the metric have not been investigat- ed. Furthermore, we started with massless fermions, so that mass must be introduced later dynamically by spon- taneous symmetry breaking. As mentioned in Ref. 1 the possibility exists that by this procedure also all gauge fields become massive. Finally we found that the pro- posed Yang-Mills theory possesses in its classical limit

more structures than Einstein's theory, so that strong con- straints were necessary for reducing the theory to the pure Einstein case; there were hints, that generally also torsion, etc., exists as a relic of the original Yang-Mills structure.

The content of this paper concerns only this last open question. Our aim is a detailed investigation of the geometric structures following from the microscopic theory in its macroscopic limit, restricting ourselves furthermore to the linearized form. We can show that the transition to the pure Einstein limit with pseudo- Riemannian structure is possible without conditions which are too restrictive.

11. THE MICROSCOPIC THEORY

In this section we repeat the necessary results of Ref. 1. Using the four-spinor calculus and consequently the 4 x 4 representation of the generators r" of the group SU(2) X U( 1) acting originally on the two-spinors

(a0 is the unit matrix and a ' , a 2 , a b r e the Pauli matrices) the covariant spinor derivative reads

( g is the gauge coupling constant) with the property

[ D,, yp] = 0, y',yv'= rlPv (2.3)

[yp are the generalized Dirac matrices, 7,'' = diag( - 1, 1, 1 , 1 ) is the Minkowski metric]. The gauge- covariant field strengths are defined by

[cObc are the structure constants of SU(2)] satisfying the Bianchi identities

Finally the field equations following from the minimal coupled gauge-invariant Lagrange principle take the expli- cit form2

1189 a 1987 The American Physical Society

1190 F. GHABOUSSI, H. DEHNEN, AND MARK ISRAELIT - 3 5

with the "charge" conservation law

Here k is a (second) coupling constant between the two gauge-invariant parts of the Lagrange density and comes out to be 2fiG ( G is the Newtonian gravitational constant). The coupling constant g has the dimension of a reciprocal length and remains undetermined within our theoretical approach; it can be determined only experimentally in the microscopic region.

The gauge-invariant canonical energy-momentum ten- sor belonging to (2.6) and (2.7) is given by

with the energy-momentum conservation law

In the case of a theory of gravity it is necessary that the four-force acting according to (2.8a) on the four- momentum of the matter field (* field) can be interpreted geometrically. Neglecting surface integrals over the matter field with respect to the conservation laws (2.7a) and the normalization condition J 3+3d3x = 1 one finds, from (2.8a) for the change of the four-momentum of the 3 field with time using (2.5) and (2.7) (Ref. 31,

with

K,= $ , (2.9a)

where

is the usual canonical energy-momentum tensor of the matter field. Introducing the always possible "correspon- dence condition"

where cab << 1 is valid in view of a weak-field approxi- mation, we obtain, from (2.9a), because of

where terms proportional to the spin operator a,"-[yp,yv] are neglected in view of the classical limit, the following form of the four-force within the linearized theory:

The geometrical structure of the classical limit of our theory follows directly from the interpretation of Eq. (2.12).

Finally we note that according to (2.10) the three SU(2)

gauge potentials are spacelike and the U(1) gauge poten- tial is timelike choosing as a group-space metric the Min- kowski matrix vab This choice is essential for reaching the pseudo-Riemannian structure of the metric in the fol- lowing.

111. THE GEOMETRIC INTERPRETATION

Let us now turn to a geometric interpretation of the theory. In the linearized approximation the space-time is characterized by a metric tensor

with I h,, << 1 and by a connection I-"p,. In our case the fundamental variables are the Yang-Mills gauge po- tentials op, and their counterparts:

Regarding the gauge potentials as tetrads we can construct the metric tensor as done in Ref. 1 by

ab gPv=wpaw:= ?7 mpaovb

and introduce gp" as

g""gvu = iY, . Using the normalization condition (2.10) one finds the re- lations

and

We note, as shown in Ref. 1, that an (infinitesimal) gauge transformation of wpa implies, according to (3.1) and (3.3), an (infinitesimal) coordinate transformation of h,,.

In the geometric framework given by the metric g,, and by the connection I-',, we describe the behavior of matter by means of an energy-momentum tensor Tpv(&) which satisfies the dynamical equation:

V;T""= Tpv .+ T ~ ~ , T ~ " + T v u v T ~ u = O . (3.8)

This equation may be rewritten as

T p v ,= - (Tpvp+6vpTuBo)~vP

Considering an insular distribution of matter and apply- ing Gauss's theorem one obtains from (3.9) the four-force acting on the matter system:

On the other hand, this four-force expressed in terms of the gauge field is, according to (2.12),

Comparing (3.10) to (3.1 1) we get (in the linearized ap- proximation)

GEOMETRIC INTERPRETATION OF A YANG-MILLS THEORY . .

rpvS+ 6 v ~ r u S u = - ? 7 p ~ ~ p B a wv a . (3.12) rApv= - 71*~a PV o pa

On contracting one obtains - - ? 7 A p ( ~ ; v-@z l P ) ~ P a + ~ c ~ ~ ~ w ~ o ~ ~ w ~ . (3.14)

2 r u v u = - g B p ~ p S a ~ : = ~ (3.13) For the purpose of the following discussion let us

and hence the connection is given with the use of (2.4) by rewrite (3.14) as

where

is the linearized Christoffel symbol and gPv is given by (3.3). In order to treat the non-Christoffel terms in (3.15) one has to turn to a generalized representation of affine connections. One can introduce the nonmetricity tensor

the torsion tensor

and the contortion tensor

By means of (3.16) and (3.18) the connection r",, can be expressed (cf. Ref. 4) as

From (3.15) one obtains immediately nonmetricity and contortion in the linearized theory:

QipV= - ( @ p a @ ~ v + ~ , ~ ~ f f ~ ) (3.20)

and

Substituting (3.201 and (3.21) into (3.19) and replacing in (3.19) g A u by we get of course (3.15). Consequently, in the classical limit of our Yang-Mills theory there exist, in general, a connection containing the Christoffel sym- bol, contorsion, and nonmetricity.

The Christoffel symbol {,h,] represents the phenomena of classical gravitation. It may be interesting to recall that nonmetricity may be treated (under certain assump- tions) as a classical geometric representation of elec- tromagnetism (cf. Refs. 5-7), whereas contortion is re- garded as a classical spinlike concept of matter (cf. Refs. 8 and 9). The linearized theory discussed here provides an adequate geometric description of the above-mentioned physical phenomena in terms of a Yang-Mills gauge field.

IV. THE EINSTEIN LIMIT

The Einstein general theory of relativity has been re- markably successful in describing macroscopic gravita- tional phenomena. It has also provided a conceptual framework for discussing large-scale phenomena. In view of the outstanding, dominating role of Einstein's theory, agreement with general relativity has to be considered as a crucial test for any new gravitational theory. In this sec- tion we show that the linearized Yang-Mills gauge theory proposed in Ref. 1 results in the general-relativity theory (in the first approximation). One could try to achieve this purpose by setting to zero the nonmetricity in (3.20) and the contorsion in (3.21) (Ref. 10). It turns out, however, that these two requirements impose too strong restrictions on the degrees of freedom of the gauge fields.

In order of set uv a not-too-restrictive vrocedure of achieving general relativity one can use the approach pro- posed by ~ o s e n " and developed in Ref. 12. This ap- proach is based on a modification of the law of parallel displacement in spaces with nonmetricity. Let us apply this approach to our problem. Suppose that a vector quantity is given by its components Vp and VP = g P ~ vA in the space characterized by gpv [cf. (3.3)] and by rAPV which is defined in (3.19)-(3.21) and (3.15a). If the vec- tor V is carried from a point with coordinates x v to a neighboring point with coordinates x '+dx by parallel displacement, the change of the contravariant component will be

Then for the change of the covariant component one can write

Substituting (4.1) and (3.19) into (4.2) one obtains

with

From (4.1) and (4.3) one finds the change in length of the vector:

1192 F. GHABOUSSI, H. D E H N E N, A N D M A R K ISRAELIT 3 5 -

a result that is of course not surprising in view of the non- metricity of the space.

Alternatively one can start with the covariant com- ponent of the vector writing the law of parallel displace- ment instead of (4.1) as

d 2 V p = r f , v V p d ~ v . 14.51

Subsequently one finds

with T*f,, given in (4.3a1, and for the change in the length one obtains

d 2 ( VgVu)= V,d2 VU+ VUd2 V,

= v ~ v ~ Q , ~ ~ x . (4.7)

Thus one has two possibilities of introducing a parallel displacement: the first is ruled by (4.1) and the second by (4.5). These two kinds of displacement lead to opposite signs of the change in the length of a given vector V. The existence of these two kinds of displacement suggests the definition of a third kind of parallel displacement:

with

For the covariant component it follows

so that, with respect to (4.81,

Nonmetricity with respect to FA,, vanishes automatical- ly, i.e.,

- r QA, '~ = - Vhgpv=O , (4.1 1)

whereas the contortion belonging to ? ",, is glven by - C ~ , , = C ~ , ~ + ~ g p u ( ~ p v , - Q,,~ ) . 14.12)

It is worth noting that the affine properties given by the connection (4.8a) are still affected by the three essential quantities: cApV, and Q*,,, although nonmetricity is canceled out. Substituting (3.20) and (3.21) into (4.8a)

we obtain the connection without nonmetricity for the linearized Yang-Mills gauge theory:

where the general Christoffel symbol has to be replaced by its linearized counterpart (3.15a). For reasons mentioned above we adopt the standpoint that the geometry evoked by the gauge field is described by the metric tensor (3.3) and by the connection (4.13). In general the connection 14.13) has 64 components.

Now, let us consider the torsionless case. We have to impose 24 conditions, namely,

- T h [ p v l = ~ . 14.14)

This causes contortion to vanish,

and hence the connection 14.13) results in 40 Chr~stoffel symbols only. Thus, the geometric description is identical with that of the Einstein general theory of relativity. The field equations for the pure Einstein limit are discussed in Ref. 1 in detail.

V. F I N A L R E M A R K S

It was shown in this paper that the linearized Yang- Mills gauge theory results in the geometry of general rela- tivity under plausible assumptions and by means of a logi- cally based procedure. Of course, the approach ~lsed in this work is a special one. Thus, one can think about oth- er interpretations of the classical limit of our Yang-Mills theory of gravity. An interesting possibility is to set non- metricity (3.20) equal to its Weylian counterpart:

QApV= - ( W ~ , C ~ ; v + ~ ~ , , m ~ , p ) = - 2 g PV . (5.1)

By this choice the Weyl connection vector LO^, would be defined by the Yang-Mills gauge potentials. However the discussion of the theory in this direction lies beyond the scope of this paper.

ACKNOWLEDGMENT

M.I. is very grateful to the University of Konstanz for its hospitality.

'H. Dehnen and F. Ghaboussi, Phys. Rev. D 33, 2205 (1986). 21n our content 3 is given by 8=+btj, where the matrix j is a

Lorentz scalar defined by 1gyP)+=fyp and [D,, j]=O [ j and yp are functions of x", cf. (2.311; one proves easily that a spe- cial representation of j is g = yr'.

3A reliable foundation for vanishing of the second term on the left-hand side of 14.4) in Ref. 1 can be given only after time integration of this equation. This corresponds to the fact that

from a quantum-mechanical standpoint only the difference between the initial and final momentum has a classical mean- ing.

4J. A. Schouten, Kicci-Culculus (Springer, Berl~n, 19541, p. 132. 5H. Weyl, Ann. Phys. 59, 101 (1919). hP. A. M. Dirac, Proc. R . Soc. London A333,403 ( 1 973). 'M. Israelit and N. Rosen, Found. Phys. 13, 1023 11983). 8D. W. Sciama, Rev. Mod. Phys. 36, 463 (19641: F . W. Hehl,

3 5 - GEOMETRIC INTERPRETATION O F A YANG-MILLS THEORY

Gen. Relativ. Gravit. 4, 333 (1973). ishes. 9M. Israelit, Found. Phys. (to be published). l lN . Rosen, Found. Phys. 12, 213 (1982). I0In Ref. 1 we have used the weaker requirement that only the 12M. Israelit, Found. Phys. (to be published).

symmetric part of contorsion and nonmetricity in (3.19) van-