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On the limits of Poincare gauge theory
Volume 108A, number 8
PHYSICS LETTERS
22 April 1985
O N T H E L I M I T S OF P O I N C A R E G A U G E T H E O R Y * J.-F. PASCUAL-S,~NCHEZ Departamento de Ftsica Tebrica, Facultad de Ciencias, Unioersidad de Valladolid, Valladolid, Spain Received 8 August 1984; revised manuscript received 20 December 1984; accepted for publication 25 February 1985
We discuss a new general setting for the limits of the Poincar6 gauge theory of gravitation. Accordingly, in particular, the teleparallelism limit should be made in the opposite way to the usual one. A suggestion about the possible nature of a gravitational "strong" coupling constant is also proposed.
Among the theories that propose a Yang-Mills-like procedure for obtaining the gravitational interaction, it is the so-called Poincar~ gauge theory (hereafter PGT) of Hehl, Ne'eman and yon der Heyde * 1, which has focused more attention. PGT can be seen as a gauge theory of the Lorentz group plus covariance under general coordinate transformations (diffeomorphisms). It can be reformulated using the Poincar~ aff'me bundle, as it has been done in refs. [ 2 - 4 ] , for instance, at the price of introducing a new matter field of the Goldstone kind. The theory formulated in this last form can make contact with a description of gravity as a nonlinear realization of "some" group, and thus the metric can be considered as composite and hence the graviton too. This last formulation has strong analogies with the chiral breaking of QCD and, from this point of view, general relativity can be regarded as an effective phenomenological non-renormalizable theory valid at "low energies" only. Realizing this in the metric-affme (or tetrad-affine, to which PGT belongs) variational picture could be problematic and, perhaps, it is more transparent in the purely affme variational framework (see ref. [5]). However, it is not the aim of this article to make general comments about the "correct" form of the gauge theory of gravity. The main point that I would
* From work of Tesis de Doctorado. *t For the most complete review see ref. [1], and references therein. 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
like to discuss concerns new procedures to obtain limits of PGT. To begin with this topic, using the Cartan calculus of forms a lagrangian of the gravitational part can be considered in PGT of the following form
tG(h, O, ~)= LRT(h, O)+LL(h, ~) ,
(1)
where h is the tetrad, O the torsion and h the Cartan curvature. The 2-form components of torsion and Cartan curvature are 0 a = dh a +g~.a O ^ h # ,
(2)
h ag = d~ ao + go3~.t~ A ~ ' r a ,
(3)
where ^ is the exterior product of forms, ~ a , h ~ are 1-form components of the Cartan connection (a,/~ = 1,2, 3, 4) and of the tetrad field respectively and g is a dimensionless constant. In expression (1) L RT(h, O) is a rote-translational part (due to the fact that in the formula of torsion (2) a mixing of Lorentz, ~ , and translation, h, potentials appears) and LL(h, h ) is the Lorentz part o f L G. As is well known, the specific forms Of LRT and L L are not determined by the gauge principle. Assuming strictly the Yang-MiUs analogy, we take both terms of the gravitational lagrangian as quadratic in torsion and Cartan curvature respectively (and their concomitants). On the other hand, the matter lagrangian has the following dependence,
387
Volume 108A, number 8
PHYSICS LETTERS
L m ( h , cb, Dqb),
(4)
where D is the Cartan covariant derivative
6,I, = d~ +g~3 ^ ~,,
(5)
and • is a tensorial or spinorial matter field. Field equations obtained from the total lagrangian (gravity + matter) under separate variations of the tetrad and the connection (tetrad-affine principle) are respectively
D(OLG/OO")+ OLG/Oh ~ = *ta
,
D(0LG/OO at3) + 2hi# A 0LG/0® ~1 = *safJ •
(6) (7)
Here * denotes the Hodge star operator, [ ] denotes antisymmetrization, * t a a r e 3-form components of the canonical stress-energy-momentum of matter and *sa# are 3-form components of the canonical spin of matter, which are defined by • ta = -6L m/6h ~ ,
(8)
• sa~ = - 6 L m / 6 ¢ 3 ~ ,
(9)
theory with the compact group U(2), which has a direct-product structure. In PGT the first coupling constant is the conventional Newton one or in natural units (h = c = 1), the Planck length lp. The second one is the dimensionless (gauge-like) coupling constant g, which appears in formulas (2), (3), (5). If one makes a rescaling by g of the Cartan connection and an other rescaling by lp of the tetrad (i.e. ~ c~ = ~ o~ /g; hO~ = h_~/lp) in the gravitational lagrangian (1), one obtains the expression considered by Hehl et al. [1 ], which exhibits the two coupling constants (squared, since L G is quadratic in torsion and Caftan curvature) LG=/p2LRT(h,O-)+g 2LL(~,~).
388
(10)
Moreover the use of the lagrangian (10) is, of course, in accordance with dimensional analysis, as has been shown in ref. [1]. Note that expressions (2), (3), (5) after the rescalings are respectively O a = d h a +~,~a
where 6 is the so-called variational derivative. The field equations (6), (7) were formulated first by Hehl and vonder Heyde (see the references cited in ref. [ 1]), using the language of tensorial components. About these field equations we would like to make several comments: (1) They are Yang-Mills like. (2) Hehl et al. [1] give the meaning of canonical stress-energy-momentum and spin of the gravitational field to the second terms in the left-hand side of the field equations. (3) The field equations have the form (6), (7) independently of the explicit expression of the gravitational part of the lagrangian. On the other hand, the gauge principle cannot determine it uniquely (at least in the usual formulation) and this is a great problem in the case of PGT, not solved even by educated guess. (4) This is the most important point for our object. Which are the coupling constants in PGT? Uncustomarily with respect to usual gauge theories involving semisimple compact groups, in PGT it is necessary to introduce two coupling constants. This is due to the semidirect product structure of the (non-compact) Poincar6 group. The same occurs in Weinberg-Salam
22 April 1985
~
~
A t f i =~)a/lp ,
= d ~ c~ + ~.~ ^ ~
= (2~lg,
= d ~ +~_ ^ ~ .
(11) (12) (13)
Now, after rescaling, the field equations (6), (7) are ~(0LG/00- ~) + lp OLG/Oh ~ = 12 *t~ ,
(14)
D_(aLG/O~ a~) + (g2/12)Zh_ [# A LG/0® _ a] = g2* Sa/3,
(15) as has been considered by Baekler [7], And finally the last comment about the field equations. (5) If one makes in the second field equation (7) the matter canonical spin zero, this does not imply, in general, that the Caftan curvature vanishes. But if *sa~ = 0, and moreover, ~at~ = 0, then the second equation (7) implies ®a = 0 and the latter through the first equation (6) leads to *ta = 0, i.e., we obtain "nothing". Hence if one considers the teleparallelism limit obtained by making ~¢~t3 = 0 in the field equations and also assumes (as in ref. [1 ]) that the teleparallelism theory is "macroscopic" if *s = 0, then one obtains the above commented inconsistency. To apparently overcome the last mentioned contradiction, Baekler [6] made manifest the coupling constants in the lagrangian and field equations in the
Volume 108A, number 8
PHYSICS LETTERS
form (14), (15). After, he considered as correct teleparallelism limit the limit g ~ 0 in the fieM equations, due to the fact that this implies that the Cartan curvature vanishes (which is the condition for obtaining a teleparallelism space-time). However, for us, this kind of limiting procedure is not adequate due to the following reasons: (i) The limit g ~ 0 should be made in the lagrangian, not in the field equations. As the so-called PGT is Yang-Millsqike, consider for example, what happens in a generic non-abelian SU(N) gauge theory with respect to limits of this kind. With the usual physicist's notation in components the total Yang-Mills lagrangian density, without showing the dimensionless coupling constant (which is also named g) is L -- -4,~~-uva~...uv.a + Lm(¢, Due) ,
(16)
where Ducb = [Ou +gAu]cb
(17)
and Fh[, a = OuA;, a - 8t, A ~ a + g f a b c A h b A ; c .
(18)
Expression (16) of the lagrangian density is appropriate for the limit of "no-interaction" g ~ 0, which implies a [U(1)] N abelian gauge theory (a = 1 ..... N), noncoupled to matter, i.e., Ducb
g~0 ~ 0tt~ ,
Fi~; ~ -~ OuA b , _ a~A h"
(19) (20)
If one rescales in eqs. (17), (18) the potential in the form A h a = A h a / g , then the rescaled lagrangian density is now L = (-1/4g2)F_m'aFh'v a + Lm(~; Ouch) ,
(21)
where _D~ = [bu +A__u]~
(22)
and F h~,a = OlaA[.,a - OuAh a +fabeAhbA_~, c .
(23)
With the rescaled form (21) of the lagrangian density, if we perform the limit of "infinite coupling" g -* co, one obtains that (21) is reduced to
L = Lm(~, I)u¢b) .
22 April 1985 (24)
Thus, analogously in PGT, the limit g + 0 of ref. [6] must be taken in the lagrangian, not in the field equations, but then one obtains a "non-interacting" scheme for the Lorentz part of the theory. This is exactly the opposite of a teleparalldism limit aimed at in ref. [6]. (ii) On the other hand, if one takes the "infinite coupling limit" g -+ oo as the teleparallelism limit in the lagrangian (10), then one obtains as total lagrangian of gravity plus matter L = lp2LRT(h_, O) + Lm(~ , ~ ) .
(25)
This will be our procedure. (iii) Speculating that the "Lorentz" part of the gravitational lagrangian has some relation to a "strong gravity" interaction and that this is asymptotically free (AF) and also has infrared slavery (IS), thus, naively (without using the renormalization semigroup techniques or some kind of statistical averaging procedure of the field equations), we use the limit g -* oo as the correct "macroscopic" low energy limit. As teleparallelism theory is considered usually as macroscopic (except in refs. [7,8]), there is no contradiction. The zero coupling limit g + 0, is now the high energy (Planck length) "microscopic" limit. In this case the "Lorentz" part dominates in the gravitational lagrangian and hence in the action. Finally, we would like to comment about the possible "nature" of the gauge-like coupling constant g in the context of PGT. Is it possible to have a constant which is dimensionless and connected with gravity? Yes, it is, clue to the "universal law" J = k M 2 pointed out by Brosche in ref. [9], where J is the total angular momentum and M is the mass of a particular gravitating system (planets, stars, galaxies, ...). With this law, one obtains the following dimensionless ragio GN/kC, in which G N is the Newton constant and c is the velocity of light in vacuum, Now an other question: Can one consider this dimensionless constant, which is "macroscopically" obtained, as the coupling constant g of the "strong" gravity part of the gravitational interaction? Maybe, because there is a striking similarity between the "macroscopic" Brosche law with the "microscopic" Regga law for the high energy slope particle trajectories. 389
Volume 108A, number 8
PHYSICS LETTERS
However, for the last interpretation o f g a problem remains: In Yang-Mills theory, g measures the interaction between the potential A u and the matter field qb, which has the unit of charge. From here a new question arises: What is the meaning of the charge of • in the case of PGT? As has been considered before [1 ] this unit of charge must be the unit of spin h/2. The author wishes to thank Professor J. Mart/nez Salas for moral support and I. Martin for discussions. He also acknowledges Dr. M. Ferraris for a careful reading of this paper and the hospitality of the members of Istituto di Fisica Matematica of Turin University (Italy) where this work was first written.
390
22 April 1985
References [1] t:.W. Hehl, in: Cosmology and gravitation, NATO AS1S Vol. B58, eds. P.G. Bergmann and V. De Sabbata (Plenum, New York, 1980). [2] J. Hennig and J. Nitsch, Gen. Rel. Grav. 13 (1981) 947. [3] R. Giachetti, R. Ricci and E. Sorace, Lett. Math. Phys. 5 (1981) 85. [4] E.A. lvanov and J. Niederle, Phys. Rev. D25 (1982) 976. [5] M. Martellini, Phys. Rev. D29 (1984) 2746. [6] P. Baekler, Phys. Lett. 94B (1980) 44; 99B (1981) 329. [7] W. Kopczynski, J. Phys. A15 (1982) 493. [8] K. Hayashi and T. Shirafuji, Phys. Rev. D19 (1979) 3524. [91 P. Brosche, in: Cosmology and gravitation, NATO ASIS Vol. B58, eds. P.G. Bergmann and V. De Sabbata (Plenum, New York, 1980) pp. 375-382.
PHYSICS LETTERS
22 April 1985
O N T H E L I M I T S OF P O I N C A R E G A U G E T H E O R Y * J.-F. PASCUAL-S,~NCHEZ Departamento de Ftsica Tebrica, Facultad de Ciencias, Unioersidad de Valladolid, Valladolid, Spain Received 8 August 1984; revised manuscript received 20 December 1984; accepted for publication 25 February 1985
We discuss a new general setting for the limits of the Poincar6 gauge theory of gravitation. Accordingly, in particular, the teleparallelism limit should be made in the opposite way to the usual one. A suggestion about the possible nature of a gravitational "strong" coupling constant is also proposed.
Among the theories that propose a Yang-Mills-like procedure for obtaining the gravitational interaction, it is the so-called Poincar~ gauge theory (hereafter PGT) of Hehl, Ne'eman and yon der Heyde * 1, which has focused more attention. PGT can be seen as a gauge theory of the Lorentz group plus covariance under general coordinate transformations (diffeomorphisms). It can be reformulated using the Poincar~ aff'me bundle, as it has been done in refs. [ 2 - 4 ] , for instance, at the price of introducing a new matter field of the Goldstone kind. The theory formulated in this last form can make contact with a description of gravity as a nonlinear realization of "some" group, and thus the metric can be considered as composite and hence the graviton too. This last formulation has strong analogies with the chiral breaking of QCD and, from this point of view, general relativity can be regarded as an effective phenomenological non-renormalizable theory valid at "low energies" only. Realizing this in the metric-affme (or tetrad-affine, to which PGT belongs) variational picture could be problematic and, perhaps, it is more transparent in the purely affme variational framework (see ref. [5]). However, it is not the aim of this article to make general comments about the "correct" form of the gauge theory of gravity. The main point that I would
* From work of Tesis de Doctorado. *t For the most complete review see ref. [1], and references therein. 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
like to discuss concerns new procedures to obtain limits of PGT. To begin with this topic, using the Cartan calculus of forms a lagrangian of the gravitational part can be considered in PGT of the following form
tG(h, O, ~)= LRT(h, O)+LL(h, ~) ,
(1)
where h is the tetrad, O the torsion and h the Cartan curvature. The 2-form components of torsion and Cartan curvature are 0 a = dh a +g~.a O ^ h # ,
(2)
h ag = d~ ao + go3~.t~ A ~ ' r a ,
(3)
where ^ is the exterior product of forms, ~ a , h ~ are 1-form components of the Cartan connection (a,/~ = 1,2, 3, 4) and of the tetrad field respectively and g is a dimensionless constant. In expression (1) L RT(h, O) is a rote-translational part (due to the fact that in the formula of torsion (2) a mixing of Lorentz, ~ , and translation, h, potentials appears) and LL(h, h ) is the Lorentz part o f L G. As is well known, the specific forms Of LRT and L L are not determined by the gauge principle. Assuming strictly the Yang-MiUs analogy, we take both terms of the gravitational lagrangian as quadratic in torsion and Cartan curvature respectively (and their concomitants). On the other hand, the matter lagrangian has the following dependence,
387
Volume 108A, number 8
PHYSICS LETTERS
L m ( h , cb, Dqb),
(4)
where D is the Cartan covariant derivative
6,I, = d~ +g~3 ^ ~,,
(5)
and • is a tensorial or spinorial matter field. Field equations obtained from the total lagrangian (gravity + matter) under separate variations of the tetrad and the connection (tetrad-affine principle) are respectively
D(OLG/OO")+ OLG/Oh ~ = *ta
,
D(0LG/OO at3) + 2hi# A 0LG/0® ~1 = *safJ •
(6) (7)
Here * denotes the Hodge star operator, [ ] denotes antisymmetrization, * t a a r e 3-form components of the canonical stress-energy-momentum of matter and *sa# are 3-form components of the canonical spin of matter, which are defined by • ta = -6L m/6h ~ ,
(8)
• sa~ = - 6 L m / 6 ¢ 3 ~ ,
(9)
theory with the compact group U(2), which has a direct-product structure. In PGT the first coupling constant is the conventional Newton one or in natural units (h = c = 1), the Planck length lp. The second one is the dimensionless (gauge-like) coupling constant g, which appears in formulas (2), (3), (5). If one makes a rescaling by g of the Cartan connection and an other rescaling by lp of the tetrad (i.e. ~ c~ = ~ o~ /g; hO~ = h_~/lp) in the gravitational lagrangian (1), one obtains the expression considered by Hehl et al. [1 ], which exhibits the two coupling constants (squared, since L G is quadratic in torsion and Caftan curvature) LG=/p2LRT(h,O-)+g 2LL(~,~).
388
(10)
Moreover the use of the lagrangian (10) is, of course, in accordance with dimensional analysis, as has been shown in ref. [1]. Note that expressions (2), (3), (5) after the rescalings are respectively O a = d h a +~,~a
where 6 is the so-called variational derivative. The field equations (6), (7) were formulated first by Hehl and vonder Heyde (see the references cited in ref. [ 1]), using the language of tensorial components. About these field equations we would like to make several comments: (1) They are Yang-Mills like. (2) Hehl et al. [1] give the meaning of canonical stress-energy-momentum and spin of the gravitational field to the second terms in the left-hand side of the field equations. (3) The field equations have the form (6), (7) independently of the explicit expression of the gravitational part of the lagrangian. On the other hand, the gauge principle cannot determine it uniquely (at least in the usual formulation) and this is a great problem in the case of PGT, not solved even by educated guess. (4) This is the most important point for our object. Which are the coupling constants in PGT? Uncustomarily with respect to usual gauge theories involving semisimple compact groups, in PGT it is necessary to introduce two coupling constants. This is due to the semidirect product structure of the (non-compact) Poincar6 group. The same occurs in Weinberg-Salam
22 April 1985
~
~
A t f i =~)a/lp ,
= d ~ c~ + ~.~ ^ ~
= (2~lg,
= d ~ +~_ ^ ~ .
(11) (12) (13)
Now, after rescaling, the field equations (6), (7) are ~(0LG/00- ~) + lp OLG/Oh ~ = 12 *t~ ,
(14)
D_(aLG/O~ a~) + (g2/12)Zh_ [# A LG/0® _ a] = g2* Sa/3,
(15) as has been considered by Baekler [7], And finally the last comment about the field equations. (5) If one makes in the second field equation (7) the matter canonical spin zero, this does not imply, in general, that the Caftan curvature vanishes. But if *sa~ = 0, and moreover, ~at~ = 0, then the second equation (7) implies ®a = 0 and the latter through the first equation (6) leads to *ta = 0, i.e., we obtain "nothing". Hence if one considers the teleparallelism limit obtained by making ~¢~t3 = 0 in the field equations and also assumes (as in ref. [1 ]) that the teleparallelism theory is "macroscopic" if *s = 0, then one obtains the above commented inconsistency. To apparently overcome the last mentioned contradiction, Baekler [6] made manifest the coupling constants in the lagrangian and field equations in the
Volume 108A, number 8
PHYSICS LETTERS
form (14), (15). After, he considered as correct teleparallelism limit the limit g ~ 0 in the fieM equations, due to the fact that this implies that the Cartan curvature vanishes (which is the condition for obtaining a teleparallelism space-time). However, for us, this kind of limiting procedure is not adequate due to the following reasons: (i) The limit g ~ 0 should be made in the lagrangian, not in the field equations. As the so-called PGT is Yang-Millsqike, consider for example, what happens in a generic non-abelian SU(N) gauge theory with respect to limits of this kind. With the usual physicist's notation in components the total Yang-Mills lagrangian density, without showing the dimensionless coupling constant (which is also named g) is L -- -4,~~-uva~...uv.a + Lm(¢, Due) ,
(16)
where Ducb = [Ou +gAu]cb
(17)
and Fh[, a = OuA;, a - 8t, A ~ a + g f a b c A h b A ; c .
(18)
Expression (16) of the lagrangian density is appropriate for the limit of "no-interaction" g ~ 0, which implies a [U(1)] N abelian gauge theory (a = 1 ..... N), noncoupled to matter, i.e., Ducb
g~0 ~ 0tt~ ,
Fi~; ~ -~ OuA b , _ a~A h"
(19) (20)
If one rescales in eqs. (17), (18) the potential in the form A h a = A h a / g , then the rescaled lagrangian density is now L = (-1/4g2)F_m'aFh'v a + Lm(~; Ouch) ,
(21)
where _D~ = [bu +A__u]~
(22)
and F h~,a = OlaA[.,a - OuAh a +fabeAhbA_~, c .
(23)
With the rescaled form (21) of the lagrangian density, if we perform the limit of "infinite coupling" g -* co, one obtains that (21) is reduced to
L = Lm(~, I)u¢b) .
22 April 1985 (24)
Thus, analogously in PGT, the limit g + 0 of ref. [6] must be taken in the lagrangian, not in the field equations, but then one obtains a "non-interacting" scheme for the Lorentz part of the theory. This is exactly the opposite of a teleparalldism limit aimed at in ref. [6]. (ii) On the other hand, if one takes the "infinite coupling limit" g -+ oo as the teleparallelism limit in the lagrangian (10), then one obtains as total lagrangian of gravity plus matter L = lp2LRT(h_, O) + Lm(~ , ~ ) .
(25)
This will be our procedure. (iii) Speculating that the "Lorentz" part of the gravitational lagrangian has some relation to a "strong gravity" interaction and that this is asymptotically free (AF) and also has infrared slavery (IS), thus, naively (without using the renormalization semigroup techniques or some kind of statistical averaging procedure of the field equations), we use the limit g -* oo as the correct "macroscopic" low energy limit. As teleparallelism theory is considered usually as macroscopic (except in refs. [7,8]), there is no contradiction. The zero coupling limit g + 0, is now the high energy (Planck length) "microscopic" limit. In this case the "Lorentz" part dominates in the gravitational lagrangian and hence in the action. Finally, we would like to comment about the possible "nature" of the gauge-like coupling constant g in the context of PGT. Is it possible to have a constant which is dimensionless and connected with gravity? Yes, it is, clue to the "universal law" J = k M 2 pointed out by Brosche in ref. [9], where J is the total angular momentum and M is the mass of a particular gravitating system (planets, stars, galaxies, ...). With this law, one obtains the following dimensionless ragio GN/kC, in which G N is the Newton constant and c is the velocity of light in vacuum, Now an other question: Can one consider this dimensionless constant, which is "macroscopically" obtained, as the coupling constant g of the "strong" gravity part of the gravitational interaction? Maybe, because there is a striking similarity between the "macroscopic" Brosche law with the "microscopic" Regga law for the high energy slope particle trajectories. 389
Volume 108A, number 8
PHYSICS LETTERS
However, for the last interpretation o f g a problem remains: In Yang-Mills theory, g measures the interaction between the potential A u and the matter field qb, which has the unit of charge. From here a new question arises: What is the meaning of the charge of • in the case of PGT? As has been considered before [1 ] this unit of charge must be the unit of spin h/2. The author wishes to thank Professor J. Mart/nez Salas for moral support and I. Martin for discussions. He also acknowledges Dr. M. Ferraris for a careful reading of this paper and the hospitality of the members of Istituto di Fisica Matematica of Turin University (Italy) where this work was first written.
390
22 April 1985
References [1] t:.W. Hehl, in: Cosmology and gravitation, NATO AS1S Vol. B58, eds. P.G. Bergmann and V. De Sabbata (Plenum, New York, 1980). [2] J. Hennig and J. Nitsch, Gen. Rel. Grav. 13 (1981) 947. [3] R. Giachetti, R. Ricci and E. Sorace, Lett. Math. Phys. 5 (1981) 85. [4] E.A. lvanov and J. Niederle, Phys. Rev. D25 (1982) 976. [5] M. Martellini, Phys. Rev. D29 (1984) 2746. [6] P. Baekler, Phys. Lett. 94B (1980) 44; 99B (1981) 329. [7] W. Kopczynski, J. Phys. A15 (1982) 493. [8] K. Hayashi and T. Shirafuji, Phys. Rev. D19 (1979) 3524. [91 P. Brosche, in: Cosmology and gravitation, NATO ASIS Vol. B58, eds. P.G. Bergmann and V. De Sabbata (Plenum, New York, 1980) pp. 375-382.