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On a unified theory for interaction and matter fields
Volume 128B, number 3,4
PHYSICS LETTERS
25 August 1983
ON A UNIFIED THEORY FOR INTERACTION AND MATI'ER FIELDS N.A. BATAKIS 1
CERN, Geneva,Switzerland Received 6 May 1983 A unified theory is proposed for the gravitational and electromagnetic fields, as well as matter fields with spin and electric charge, all accommodated into the connection of a Riemann-Cartan space-time. The construction seems to admit a maximal extension, incorporating the electroweak interactions.
The relevance o f gauge theories in physics, and in particular their fundamental role in the unification o f the electromagnetic and weak interactions, has motivated analogous considerations involving general relativity, which seem to indicate that in any deeper unification scheme, gravity will participate in a fundamental manner. It is then reasonable to assume that within such a development, gravity will not lose its geometric interpretation afforded by general relativity, but, instead, a proper classical limit o f that unified theory will likewise admit an interpretation within some enlarged geometry, which would reduce to riemannian when the other interactions are switched off. Unfortunately, in spite o f enormous efforts during several decades, we do not yet have a definite scheme unifying even just the EM field with gravity. There has been, however, one interesting development involving, not another interaction, but what is considered the classical analogue o f spin. The relevant construction is known as the Einstein-Cartan theory, in which spin is given a geometric interpretation in terms o f torsion in a Riemann-Cartan (RC) spacetime, and gravity is thereby identified as the gauge theory of the Poincar~ group [ 1 ]. From both a mathematical and a physical point o f view, riemannian geometry is incomplete in comparison to RC and, o f course, it is recovered from the latter at the limit of vanishing torsion (for a discussion of these points, see, for example, refs. [ 1 - 3 ] ) . We will try to make plausible in this paper that the RC geomeI On leave from the department of Physics, University of Ioannina, Greece. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
try is indeed the "enlarged geometry" we referred to above. This would mean, o f course, that the interpretation o f torsion as classical spin is at best incomplete. On the other hand, we have recently proposed a different (and apparently incomplete as well) interpretation for torsion in terms o f the EM field [4]. We will argue that these interpretations are both special cases o f a broader unification scheme, involving the gravitational and EM fields, as well as spin and matter fields. After the exposition o f our main result, we will discuss how the construction can be generalized for the case of a non-abelian gauge field. In the following, greek indices take the values 0, 1, 2, 3, the metric isg,~t3 = diag ( - 1,1,1,1) in an orthonormal frame with In . . g = - det gat3, V refers to aageneral connecuon I', with the semi-colon reserved for r ' [see eq. (1)], and we adopt the + convention for the Riemann tensor. At the limit of a weak gravitational field, the newtonian potential is given by ½ (g00 ÷ 1), implying the standard association o f the metric tensor with the gravitational potential, while the field itself is associated with the symmetric metricpart of the connection, the usual Christoffel symbols in a holonomic frame: 0 l'~Ot
1 fly _ ( ~o~ ) _--2ge'U(gua,'r +gu"t,o-g~,u)"
(1)
One should be able to transform this field to zero at one point, in fact along any given curve (because of the principle o f equivalence), which is elegantly afforded by a corresponding property of the connection (1). On the other hand, some analogous possibility should also exist for other interactions, if they are to be incorpor165
Volume 128B, number 3,4
PHYSICS LETTERS
ated into the geometry at the level of the connection ,1. To be more precise, consider the motion o f a small charged particle in a given electric field. One can balance the electric force on the particle by a proper gravitational force, that is by a proper general coordinate transformation. Its motion will then be that of a "free" particle along some geodesic - not, of course, with respect to the connection (1), but to some other connection incorporating the EM field as well. Thus, an interaction which can be interpreted classically in terms of forces must contribute to the symmetric part of the general connection 0
P~
= Fr~
+Kr~.
(2)
A torsion T ~ = - T~..t0 does give a symmetric contribution, because it is employed in (2) in terms o f the contorsion tensor _1
K a ~ = - K ~ , - 7 (T,ro~ + T~a~, - Tam),
F
060 c~= - 16~G,
(7)
which, in general, will not be a constant. We further assume that G is given in terms of the gradient of a scalar field ~0;6 = tp,c, = ~pc~as
~o¢,~oc' = - (161rG)- I.
(8)
The roles of 06 and ~owill be better clarified later on. To proceed with the explicit form of the connection (2) we write Koq~, : Fat ~ + Sc~ov + Xc,~ .
(4)
Up to an overall factor, Fc,t~ is uniquely determined from the requirement of linearity in 06 and f~O" plus the symmetries it inherits from the contorsion, namely F 6 ~ = - Fg~.r, F60c~ = 0, F l a i l = 0. We find
(5)
F~
so that metricity follows automatically from (3): V g6t~= 0.
ly, a classical spin field [ 1] and the EM field. The latter counts with six independent components because no field equations have been derived for it yet. In the Einstein-Caftan theory, 0 a is proportional to the velocity u 6 of the matter. Although we could make a similar identification here, we need only require that its length is proportional to the gravitational coupling G:
(3)
with T 6 ~ = Ka~tt~ - K r ~ ,
25 August 1983
We observe now that the totally antisymmetric part of the torsion, which we write as
(9)
= 3 -1/2 (f~t30v - O [ ~ f ~ l . r + O h f x l a g O l . r ) ,
(10)
and, in complete analogy:
S6ov = (e/x/J)(sc~aO~t -OtaS~l ~t +0 Xsxlc~g~l 3,), (11) T[~,r] = - 2 Xoe.r,
(6)
transforms separately, does not give a symmetric contribution to (2) and therefore does not influence (at least not directly) the geodesics. One could then argue that it is well suited for the accommodation of matter fields. X a ~ is proportional to the dual of a vector, say Xa and it therefore carries four o f the total twenty independent components of a traceless torsion. We will express the remaining 16 components in terms o f a timelike vector field 0 6 plus two antisymmetric tensor fields sa~,fa~, which will be identified with, respective* 1 This view is supported by the analogy with the gravitational field. There is an interesting relevant comment by Johnson Jr., criticizing Einstein's incorporation of the EM field at the level of the metric [5]. Our choice here in no ways upsets the standard interpretation of gauge potenrials, as the latter refers to a connection in a different principal bundle. Instead, we simply provide a linkage between the two hitherto unrelated connections (see also the discussion at the end). 166
where e is a unit of electric charge, with the current density given by
/'~ = - x/g es~a;~ .
(12)
Finally, we choose
Xr~
= 3 -1/2 (16zrG) l/2 e a ~
X~ .
(13)
The total lagrangian of the theory will be proportional to the curvature scalar: L = - x/g'(16rrG)- l R.
(14)
A direct computation involving the definitions (2), ( 7 ) - ( 1 3 ) gives, up to a total divergence,
g-ll2L=-(16zrG)-lR
+-fii f 2 - A 6 ] a * g e.1 2 s2 +2X 2, (15)
where ]-2 =fe~ f'*t~owith f~t3 = A~;~ - A~;~, s 2 = s ~ s ~ . X2 = -,, X,,X a and R is the riemannian part originating from r'c'~,. The vector 0 a has disappeared, leaving ~o, Aa,gc~t3 as the dynamical variables, plus, of course,
Volume 128B, number 3,4
PHYSICS LETTERS
those involved in X~ and sat~. Clearly, the structure of these two fields will depend on an underlying theory of matter (cf. also footnote 2). When the torsion vanishes, only the first term in (15) survives, rendering a scalar-tensor theory of gravity in riemannian spacetime• In contrast to the Brans-Dicke theory [6], the present one coincides with general relativity in the important cases of(i) a vacuum, static and spherically symmetric space-time, reproducing the Schwarzschild solution and (ii) a dust-filled, homogeneous and isotropic space-time, reproducing the Friedmann models with dust. In both instances G turns out to be a constant but, importantly, one still has a non-trivial interpretation of Mach's principles through eq. (8). These points have already been discussed in an earlier version of this scalar tensor theory, leading to the same field equations, at least in the above two cases [7]. An interesting development of our construction would be its generalization for the case o f a non-abelian Yang-Mills field [8J. To outline such a programme, it should first of all be clear that the geometric interpretation of gauge potentials as the components of a connection in a principal bundle over a curved spacetime does not automatically provide a geometric unification of the corresponding YM field with gravity. Such structures are simply generalizations of the Einstein-Maxwell system. On the other hand, for a U(I) gauge group, the present theory does provide a linkage between fibres and the base space, in terms of the connection (2)• For the case of an N-dimensional non-abelian group,fat~ and s ~ (but not Xa) will carry, as usual, an additional internal index i = 0, 1.... , (N -- 1). Since Kay.t should remain gauge invariant, 0a must also carry an internal index and (7) must be generalized to
g~O io, O~0 = - 16rrGg i/,
(16)
suggesting that 0 t is a sort o f vierbcin generating the group metricgi/("Killing form"). Consequently, eq. (10) will read
Fa# v = 3-1/2g# (fia# ojr __0 i[c,/#1 7 + oiXf/x [ag#l ,r).
25 August 1983
example, 0 0 timelike and 01,2,3c~ spacelike (each with absolute square-length 167rG), the group metric is diag (l, - 1 , - I , - I ) , suggesting U(I) X SU(2) as the maximal relevant group. We conclude with a semi-qualitative discussion on how such a development could be relevant for the case of the electroweak interactions *2. Obviously, an important missing ingredient in (15), now considered in terms of (17), etc., is the Higgs field. Within our philosophy, no term should be added "by hand" to the lagrangian, so the contribution of this field to (15) could conceivably come from ×2. However, a far more economical and fruitful possibility seems to be realized by letting the Higgs complex doublet generalize the scalar field in (8). To look at this possibility more explicitly, let us start by writing (8) as ~ 0 '~ = - (16nO) -1 ,
(18)
where now ~0a = Da~0 involves the full covariant derivative with minimal coupling to the gauge fields. Then, eq. (18) not only restores scale invariance in the gravitational part o f the lagrangian but it may also induce spontaneous symmetry breaking for the U(1) X SU(2) symmetry, without the addition of any ad hoc potential [9]• Indeed, as already mentioned, G in (18) turns out to be a constant, identifiable with Newton's GN, for the case o f a Friedmann model. The corresponding classical solution ~0may be treated as a vacuum expectation value ~0 = v for the Hi~s field. Writing as usual ~o = o + ~ and rescaling ~ as R-1/2¢in terms of a new field ~0(which has now the "correct" dimensionality), we can re-write the first term in (15) as --(167rG)-l/~ = - (167rGN)-l/~ +L~.
(19)
L~0 is the lagrangian o f ¢ , which has now acquired a mass
m 2 = (In a); u (In a);U + (In a);ja;u ,
(20)
where a 2 = ~//~0 gives the ratio o f the riemannian curvature to a standard value/~0- Like v, m 2 is not a constant. It ranges from + o o (or some very high value if there is no initial singularity, due to the presence of
(17)
with a similar expression for Sao ~. The requirement of non-degeneracy f o r g i! in (27) shows that our formalism would at most allowN = 4. If the four space-time vectors 0 za are chosen mutually orthogonal, with for
*2 The restriction N = 4 indicates that no further interactions are allowed in the present form',dism, at least not on the same footing as the gravitational and electroweak ones. Presumably, the strong interactions must be considered as "hidden" within xa and saB. 167
Volume 128B, number 3,4
PHYSICS LETTERS
torsion) down to zero for an open universe. The lagrangian (15) with the first term substituted by (19) would then supply an effective lagrangian with gab and o considered as background fields. On the other hand, (15) could also be viewed at all times as a classical lagrangian, unifying gravity with the electroweak interaction and matter fields, within an enlarged general relativistic context. The author is thankful to S. Ferrara, J. Iliopoulos, Ch. Mukku, D.V. Nanopoulos, T. Regge, Th. Schiicker and other members o f the Theory Division at CERN for many useful discussions.
168
25 August 1983
References [ 11 F.W. Hehl, P. Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393, and references cited therein. [2] A. Trautman, in: Symposia Mathematica, Vol. 12 (1973) 139. [31 H.T. Nieh, Phys. Letl. 88A (1982) 388. [41 N.A. Batakis, Phys. Lett. 90A (1982) 115. [51 C.P. Johnson Jr., Phys. Rev. 87 (1953) 320. [6] C.H. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 925. [71 N.A. Batakis, Phys. Lett. 96A (1983) 331. [8] For standard results, see, for example: T. Eguchi, P.B. Gilkey and A.J. Hanson, Phys. Rep. 66 (1980) 213, and references therein. [9] A. Zee, Phys. Rev. Lett. 42 (1979) 417; see also his forthcoming article in the Proc. 20th Annum Orbis Scientiae and references cited therein.
PHYSICS LETTERS
25 August 1983
ON A UNIFIED THEORY FOR INTERACTION AND MATI'ER FIELDS N.A. BATAKIS 1
CERN, Geneva,Switzerland Received 6 May 1983 A unified theory is proposed for the gravitational and electromagnetic fields, as well as matter fields with spin and electric charge, all accommodated into the connection of a Riemann-Cartan space-time. The construction seems to admit a maximal extension, incorporating the electroweak interactions.
The relevance o f gauge theories in physics, and in particular their fundamental role in the unification o f the electromagnetic and weak interactions, has motivated analogous considerations involving general relativity, which seem to indicate that in any deeper unification scheme, gravity will participate in a fundamental manner. It is then reasonable to assume that within such a development, gravity will not lose its geometric interpretation afforded by general relativity, but, instead, a proper classical limit o f that unified theory will likewise admit an interpretation within some enlarged geometry, which would reduce to riemannian when the other interactions are switched off. Unfortunately, in spite o f enormous efforts during several decades, we do not yet have a definite scheme unifying even just the EM field with gravity. There has been, however, one interesting development involving, not another interaction, but what is considered the classical analogue o f spin. The relevant construction is known as the Einstein-Cartan theory, in which spin is given a geometric interpretation in terms o f torsion in a Riemann-Cartan (RC) spacetime, and gravity is thereby identified as the gauge theory of the Poincar~ group [ 1 ]. From both a mathematical and a physical point o f view, riemannian geometry is incomplete in comparison to RC and, o f course, it is recovered from the latter at the limit of vanishing torsion (for a discussion of these points, see, for example, refs. [ 1 - 3 ] ) . We will try to make plausible in this paper that the RC geomeI On leave from the department of Physics, University of Ioannina, Greece. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
try is indeed the "enlarged geometry" we referred to above. This would mean, o f course, that the interpretation o f torsion as classical spin is at best incomplete. On the other hand, we have recently proposed a different (and apparently incomplete as well) interpretation for torsion in terms o f the EM field [4]. We will argue that these interpretations are both special cases o f a broader unification scheme, involving the gravitational and EM fields, as well as spin and matter fields. After the exposition o f our main result, we will discuss how the construction can be generalized for the case of a non-abelian gauge field. In the following, greek indices take the values 0, 1, 2, 3, the metric isg,~t3 = diag ( - 1,1,1,1) in an orthonormal frame with In . . g = - det gat3, V refers to aageneral connecuon I', with the semi-colon reserved for r ' [see eq. (1)], and we adopt the + convention for the Riemann tensor. At the limit of a weak gravitational field, the newtonian potential is given by ½ (g00 ÷ 1), implying the standard association o f the metric tensor with the gravitational potential, while the field itself is associated with the symmetric metricpart of the connection, the usual Christoffel symbols in a holonomic frame: 0 l'~Ot
1 fly _ ( ~o~ ) _--2ge'U(gua,'r +gu"t,o-g~,u)"
(1)
One should be able to transform this field to zero at one point, in fact along any given curve (because of the principle o f equivalence), which is elegantly afforded by a corresponding property of the connection (1). On the other hand, some analogous possibility should also exist for other interactions, if they are to be incorpor165
Volume 128B, number 3,4
PHYSICS LETTERS
ated into the geometry at the level of the connection ,1. To be more precise, consider the motion o f a small charged particle in a given electric field. One can balance the electric force on the particle by a proper gravitational force, that is by a proper general coordinate transformation. Its motion will then be that of a "free" particle along some geodesic - not, of course, with respect to the connection (1), but to some other connection incorporating the EM field as well. Thus, an interaction which can be interpreted classically in terms of forces must contribute to the symmetric part of the general connection 0
P~
= Fr~
+Kr~.
(2)
A torsion T ~ = - T~..t0 does give a symmetric contribution, because it is employed in (2) in terms o f the contorsion tensor _1
K a ~ = - K ~ , - 7 (T,ro~ + T~a~, - Tam),
F
060 c~= - 16~G,
(7)
which, in general, will not be a constant. We further assume that G is given in terms of the gradient of a scalar field ~0;6 = tp,c, = ~pc~as
~o¢,~oc' = - (161rG)- I.
(8)
The roles of 06 and ~owill be better clarified later on. To proceed with the explicit form of the connection (2) we write Koq~, : Fat ~ + Sc~ov + Xc,~ .
(4)
Up to an overall factor, Fc,t~ is uniquely determined from the requirement of linearity in 06 and f~O" plus the symmetries it inherits from the contorsion, namely F 6 ~ = - Fg~.r, F60c~ = 0, F l a i l = 0. We find
(5)
F~
so that metricity follows automatically from (3): V g6t~= 0.
ly, a classical spin field [ 1] and the EM field. The latter counts with six independent components because no field equations have been derived for it yet. In the Einstein-Caftan theory, 0 a is proportional to the velocity u 6 of the matter. Although we could make a similar identification here, we need only require that its length is proportional to the gravitational coupling G:
(3)
with T 6 ~ = Ka~tt~ - K r ~ ,
25 August 1983
We observe now that the totally antisymmetric part of the torsion, which we write as
(9)
= 3 -1/2 (f~t30v - O [ ~ f ~ l . r + O h f x l a g O l . r ) ,
(10)
and, in complete analogy:
S6ov = (e/x/J)(sc~aO~t -OtaS~l ~t +0 Xsxlc~g~l 3,), (11) T[~,r] = - 2 Xoe.r,
(6)
transforms separately, does not give a symmetric contribution to (2) and therefore does not influence (at least not directly) the geodesics. One could then argue that it is well suited for the accommodation of matter fields. X a ~ is proportional to the dual of a vector, say Xa and it therefore carries four o f the total twenty independent components of a traceless torsion. We will express the remaining 16 components in terms o f a timelike vector field 0 6 plus two antisymmetric tensor fields sa~,fa~, which will be identified with, respective* 1 This view is supported by the analogy with the gravitational field. There is an interesting relevant comment by Johnson Jr., criticizing Einstein's incorporation of the EM field at the level of the metric [5]. Our choice here in no ways upsets the standard interpretation of gauge potenrials, as the latter refers to a connection in a different principal bundle. Instead, we simply provide a linkage between the two hitherto unrelated connections (see also the discussion at the end). 166
where e is a unit of electric charge, with the current density given by
/'~ = - x/g es~a;~ .
(12)
Finally, we choose
Xr~
= 3 -1/2 (16zrG) l/2 e a ~
X~ .
(13)
The total lagrangian of the theory will be proportional to the curvature scalar: L = - x/g'(16rrG)- l R.
(14)
A direct computation involving the definitions (2), ( 7 ) - ( 1 3 ) gives, up to a total divergence,
g-ll2L=-(16zrG)-lR
+-fii f 2 - A 6 ] a * g e.1 2 s2 +2X 2, (15)
where ]-2 =fe~ f'*t~owith f~t3 = A~;~ - A~;~, s 2 = s ~ s ~ . X2 = -,, X,,X a and R is the riemannian part originating from r'c'~,. The vector 0 a has disappeared, leaving ~o, Aa,gc~t3 as the dynamical variables, plus, of course,
Volume 128B, number 3,4
PHYSICS LETTERS
those involved in X~ and sat~. Clearly, the structure of these two fields will depend on an underlying theory of matter (cf. also footnote 2). When the torsion vanishes, only the first term in (15) survives, rendering a scalar-tensor theory of gravity in riemannian spacetime• In contrast to the Brans-Dicke theory [6], the present one coincides with general relativity in the important cases of(i) a vacuum, static and spherically symmetric space-time, reproducing the Schwarzschild solution and (ii) a dust-filled, homogeneous and isotropic space-time, reproducing the Friedmann models with dust. In both instances G turns out to be a constant but, importantly, one still has a non-trivial interpretation of Mach's principles through eq. (8). These points have already been discussed in an earlier version of this scalar tensor theory, leading to the same field equations, at least in the above two cases [7]. An interesting development of our construction would be its generalization for the case o f a non-abelian Yang-Mills field [8J. To outline such a programme, it should first of all be clear that the geometric interpretation of gauge potentials as the components of a connection in a principal bundle over a curved spacetime does not automatically provide a geometric unification of the corresponding YM field with gravity. Such structures are simply generalizations of the Einstein-Maxwell system. On the other hand, for a U(I) gauge group, the present theory does provide a linkage between fibres and the base space, in terms of the connection (2)• For the case of an N-dimensional non-abelian group,fat~ and s ~ (but not Xa) will carry, as usual, an additional internal index i = 0, 1.... , (N -- 1). Since Kay.t should remain gauge invariant, 0a must also carry an internal index and (7) must be generalized to
g~O io, O~0 = - 16rrGg i/,
(16)
suggesting that 0 t is a sort o f vierbcin generating the group metricgi/("Killing form"). Consequently, eq. (10) will read
Fa# v = 3-1/2g# (fia# ojr __0 i[c,/#1 7 + oiXf/x [ag#l ,r).
25 August 1983
example, 0 0 timelike and 01,2,3c~ spacelike (each with absolute square-length 167rG), the group metric is diag (l, - 1 , - I , - I ) , suggesting U(I) X SU(2) as the maximal relevant group. We conclude with a semi-qualitative discussion on how such a development could be relevant for the case of the electroweak interactions *2. Obviously, an important missing ingredient in (15), now considered in terms of (17), etc., is the Higgs field. Within our philosophy, no term should be added "by hand" to the lagrangian, so the contribution of this field to (15) could conceivably come from ×2. However, a far more economical and fruitful possibility seems to be realized by letting the Higgs complex doublet generalize the scalar field in (8). To look at this possibility more explicitly, let us start by writing (8) as ~ 0 '~ = - (16nO) -1 ,
(18)
where now ~0a = Da~0 involves the full covariant derivative with minimal coupling to the gauge fields. Then, eq. (18) not only restores scale invariance in the gravitational part o f the lagrangian but it may also induce spontaneous symmetry breaking for the U(1) X SU(2) symmetry, without the addition of any ad hoc potential [9]• Indeed, as already mentioned, G in (18) turns out to be a constant, identifiable with Newton's GN, for the case o f a Friedmann model. The corresponding classical solution ~0may be treated as a vacuum expectation value ~0 = v for the Hi~s field. Writing as usual ~o = o + ~ and rescaling ~ as R-1/2¢in terms of a new field ~0(which has now the "correct" dimensionality), we can re-write the first term in (15) as --(167rG)-l/~ = - (167rGN)-l/~ +L~.
(19)
L~0 is the lagrangian o f ¢ , which has now acquired a mass
m 2 = (In a); u (In a);U + (In a);ja;u ,
(20)
where a 2 = ~//~0 gives the ratio o f the riemannian curvature to a standard value/~0- Like v, m 2 is not a constant. It ranges from + o o (or some very high value if there is no initial singularity, due to the presence of
(17)
with a similar expression for Sao ~. The requirement of non-degeneracy f o r g i! in (27) shows that our formalism would at most allowN = 4. If the four space-time vectors 0 za are chosen mutually orthogonal, with for
*2 The restriction N = 4 indicates that no further interactions are allowed in the present form',dism, at least not on the same footing as the gravitational and electroweak ones. Presumably, the strong interactions must be considered as "hidden" within xa and saB. 167
Volume 128B, number 3,4
PHYSICS LETTERS
torsion) down to zero for an open universe. The lagrangian (15) with the first term substituted by (19) would then supply an effective lagrangian with gab and o considered as background fields. On the other hand, (15) could also be viewed at all times as a classical lagrangian, unifying gravity with the electroweak interaction and matter fields, within an enlarged general relativistic context. The author is thankful to S. Ferrara, J. Iliopoulos, Ch. Mukku, D.V. Nanopoulos, T. Regge, Th. Schiicker and other members o f the Theory Division at CERN for many useful discussions.
168
25 August 1983
References [ 11 F.W. Hehl, P. Heyde, G.D. Kerlick and J.M. Nester, Rev. Mod. Phys. 48 (1976) 393, and references cited therein. [2] A. Trautman, in: Symposia Mathematica, Vol. 12 (1973) 139. [31 H.T. Nieh, Phys. Letl. 88A (1982) 388. [41 N.A. Batakis, Phys. Lett. 90A (1982) 115. [51 C.P. Johnson Jr., Phys. Rev. 87 (1953) 320. [6] C.H. Brans and R.H. Dicke, Phys. Rev. 124 (1961) 925. [71 N.A. Batakis, Phys. Lett. 96A (1983) 331. [8] For standard results, see, for example: T. Eguchi, P.B. Gilkey and A.J. Hanson, Phys. Rep. 66 (1980) 213, and references therein. [9] A. Zee, Phys. Rev. Lett. 42 (1979) 417; see also his forthcoming article in the Proc. 20th Annum Orbis Scientiae and references cited therein.