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Torsion dynamics in multidimensional unified theories
Volume 140B, number 1,2
PHYSICS LETTERS
31 May 1984
TORSION DYNAMICS IN MULTIDIMENSIONAL UNIFIED THEORIES Claudio A. O R Z A L E S I 1 CERN, Geneva, Switzerland Received 8 July 1983
It is shown that, in quantized Einstein-Caftan gravitation in D ~ 4 dimensions, torsion propagates through virtual pair production. For D > 4, it is argued that torsion effects might determine dynamically the length scale of spontaneous compactification.
It is well k n o w n that torsion, .aT'BC , . is a d y n a m i c variable in the E i n s t e i n - C a r t a n (EC) t h e o r y [1 ]. In fact, the Cartan field e q u a t i o n ,1 tABc..
= K sAB.. c
(1)
relates the m o d i f i e d torsion,
tAf= %f
r
D
AD. '
to the canonical spin density o f matter, sAB C = 20£matter/OKAB C.
(2)
Here, K A B C = ~(TAB C + TCA B + TCBA ) is the contorsion for a metric linear c o n n e c t i o n ¢o: toAB = to [AB] = °~MAB d x M = 6°DABED '
(3)
o
°°DAB = °°DAB + K D A B '
(4)
where E A = E A M d X M are the co-vielbein one-forms 1 Also at Sezione Teorica, Dipartimento di Fisica della UniversitY, 43100 Parma, and INFN, Sezione di Milano, Milan, Italy. ,1 Conventions: Capital Latin indices run from 1 to D = 4 + K; Greek indices run from 1 to 4 and refer to ordinary spacetime; low-case Latin indices run from 5 to D and refer to those dimensions which undergo compaetification in multidimensional (D > 4) theories. The indices from the beginning of all alphabets are anholonomic and "flat", i.e., they refer to components relative to a vielbein. Indices from the middle-late parts of the alphabets refer to coordinates. The constant ~, with the length dimension D - 2 , is the D-dimensional generalization of Newton's constant K (and ~ = ~ for D = 4).
and ~ is the associated Levi-Civita c o n n e c t i o n for the metric y = ~?ABE A ® E B = TMN d X M ® d X N. In D = 4 dimensions, w h e n discussing macroscopic gravitational p h e n o m e n a , torsion is usually neglected and the Levi-Civita c o n n e c t i o n is used t h r o u g h o u t . There are good reasons for neglecting torsion in the classical D = 4 t h e o r y : w h e n m a t t e r is described by local fields, s is the density o f intrinsic angular mom e n t u m [ 1 ] and has a negligible average in the bulk m a t t e r forming galaxies and normal stars. For D i> 4, torsion effects are also neglected in most works on q u a n t u m gravity ,2 and on multidimensional unified theories ,3. In particular, the background g e o m e t r y - which determines the small field fluctuations near the q u a n t u m ground state 10) - is usually chosen by looking at solutions o f the field equations for the classical, torsion-free theory. We wish to emphasize that there is no physical justification for neglecting torsion effects in q u a n t u m gravity, especially w h e n D > 4. We will argue that, at the q u a n t u m level, torsion propagates in spite o f the algebraic character o f eq. (1) and o f the absence o f a kinetic torsion term in the EC lagrangian. O f course, already at the semi-classical level, there always occurs a " c o n v e c t i v e " torsion propagation, due to the ,2 However, some effects of a static background spin density on ferrnions have been estimated for D = 4, see ref. [2]. ,3 On multidimensional (D > 4) unified theories, see refs. [3-5] and references therein. Possible spin-torsion mechanisms of spontaneous eompactification, of the K = D - 4 "internal" dimensions, were discussed in refs. [4,5]. 39
Volume 140B, number 1,2
PHYSICS LETTERS
motion of physical fermions. However, and this is one of our main points, a "quantum" propagation of torsion takes place also in the vacuum, due to virtual effects from closed fermion loops ,4. The mechanism to be discussed is similar to that of the propagation of spin waves in a crystal, and bears an even closer resemblance to the propagation of Cooper pairs in superconductors. As a matter of fact, we shall use the same functional methods which have been applied [7] to the N a m b u - J o n a Lasinio ( N - J L ) model [8]. The quantum propagation of torsion could have important effects, e.g., on the ultraviolet properties o f the theory ,s. When D = 4, from known estimates [2] o f static effects, one might expect that the effective energy threshold for virtual torsion effects might well be by many orders of magnitude lower than for pair production by gravitons *e. However, it will be seen that, when D = 4, the effective mass in the induced torsion propagator is probably of the order of the Planck mass, so that torsion effects should not really introduce a new length scale in the theory. More drastic effects can take place when D > 4, as in multidimensional unified theories. In fact, the "internal" components of torsion (i.e., those components which are orthogonal to the ordinary d = 4 spacetime) now become scalar fields in four dimensions. Through collective phenomena, analogous to those responsible for the dynamic chiral symmetry breaking in the N - J L model, such fields can drastically influence the stability of the semi-classical vacuum. Therefore, the issue of the definition of the physical vacuum should be reconsidered in this light. In particular, it becomes possible for the "internal" components of torsion to develop large vacuum expectation values, which could set dynamically the length scale o f a spontaneously compactified background geometry ,7. Since this length, L, determines (after dimensional reduction) the (bare) Yang-MiUs coupling constants in four dimensions [5], g ° M ~ V ~ / L , the pro,4 Similar effects of torsion propagation take place also in the approach of ref. [6] to induced gravity. , s In particular, torsion contributions are crucial to enforce supercovariance in supergravity ( D ) 4). *e For D = 4, the "overcritical" situation for torsion already occurs at 10 --46 times the Planck density c S / ~ G 2 = (87r/
~)~, see ref. [2]. ,7 See refs. [5,9] and eq. (28) given below. 40
31 May 1984
posed mechanism raises the hope of actually computing such coupling constants from first principles, within a multidimensional unified theory where spontaneous compactification is due [4,5] to a spin-torsion condensate ,s. Concerning our approach to the quantum dynamics of torsion, we remark that, like everybody else, we do not know if and how the quantization of geometry really makes sense. We therefore take a simplistic attitude when writing quantum path integrals: we limit ourselves to a fixed background geometry and we consider only topologically small fluctuations around the given background. In practice, this means that all the fields (including the metric and the connection) should be expanded in harmonics over the background. Where needed for convergence, gaugefixing terms and cut-offs are understood ,9 To illustrate the quantum dynamics of torsion, we consider a Dirac spinor qffX) minimally coupled to gravity in D dimensions. The first-order action is -~ -
G + '~K + "9~D'
(5)
where, with x = E 1 ^ ... A E D = Det EAMdDX the volume element,
"¢~K
=
- ~ D --
1-L f ( ld 2;¢ J " * B E .
BId DE -- K "'19.
•
B K DE ~'c
DE.
B.
•j
(7)
'
f ~ D ' --f(~D + ~(DK , ) ' '
(8)
o
£D = ---~ ~ {TA (go)' FA } xI" ,~(D K) = 1 I(
~pABC~I I
1~K'F~,
(9) (10)
with VA(go) -- EA - ~ goABC FBC" The field equations are, with GAB(W ) the generalized Einstein tensor. *8 Spin-torsion compactification can also operate together with the more traditional bosonic mechanisms [3,5 ]. In particular, for D = 1 1 supergravity [10] with bosonic spontaneous compactification [ 11 ], torsion effects could cure [12] the problem of the enormous cosmological constant induced in four dimensions. .9 Although unsatisfactory from the viewpoint of first principles, our heuristic approach might correspond to a good low-energy approximation. In any case, forD > 4, our method generalizes the "truncation to zero-modes" which is frequently advocated [3,10,11].
Volume 140B, number 1,2
PHYSICS LETTERS
31 May 1984
GAS(~) =-~ {~rA%(O~) -- [V~(~o)~rAI),I,, (11) K A B C = 1 ~ ~t[,ABCq.t '
(12)
VA VA( ~ ) ' I ' = 0.
(13)
Notice that, since the contorsion (12) is totally antisymmetric, one has KAB C = K[ABC ] = { TAB C, and -~K becomes
t~
t~~j
t¢3~
t~41
Fig. 1. The first four one-loop contributions to the effective action for torsion.
where N ( E A ) = exp {tr log [ - V ( ~ ) ] } and -¢{K -
1 fK.K~. -- 2~
(14)
K'
At this point, the conventional procedure is to use (12) so as to rewrite the action in second-order form:
(2) =f
(15)
qz exp [ i ( ~ + .9~j)],
(16)
where 3;t{j
--f[,7**
+
~A ® E A + g ' ~ ] x ,
(17)
and r/, r/*, 6A, cK are external sources to generate the appropriate Green's functions. For simplicity, and because we are mainly interested in torsion effects in vacuo, in what follows we shall set ~ = r/* = 6A = 0. We shall also leave out the integration over vielbeins E A , which we treat as external fields. Therefore, hereafter we shall restrict our attention to the functional Z ( E A ) = f c 3 K C D ~ @,I, e x p [ i ( ~ K + ~
+ _¢{D)],
(18) where -¢~¢)(. = f K . cK'c. With standard methods [7], by integrating over the fermions, we obtain
z(E A ) =
A) f w x exp (iS),
- i t r l o g (1 - [1/~'(6~)] U}, (21)
rABC = ¼K'F.
(22)
7~- ~ABC --
Rather than relying on eq. (15), we shall closely follow the approach which was used in ref. [7] for the N a m b u - J o n a Lasinio model [8] : we construct the second-order effective action for the contorsion KAB C. The generating functional of Green's functions reads (up to a normalization constant)
~-f@E a ~ K O ~ c - / )
st{eft = f ~ - ~ K ' K ¢ K
U= 1 ~
[(2k)-lR(Co) + £D
+ ~k~FABCq~FABC]x.
z
(20)
S = -q~Q( + st{ eff
(19)
For the effective action of contorsion, we may use the expansion - i t r l o g (1 - [1/~'(~)] U} = ~
Ql(n),
(23)
n=l
q/(n) = (i/n)tr ([//~(~)] U) n .
(24)
The first four one-loop contributions are shown in fig. 1. Notice that the fermion propagator depends on the vielbein E A , and vertices carry the matrix FABC.
The term n = 1 determines the '~¢acuum" expectation value of torsion, in the presence of the external field E A . The term n = 2 yields a kinetic term and, together with [ - ( 1 / 2 k ) K . K ] , also the mass term. The terms with n > 2 describe non-polynomial self-interactions and interactions with the vielbein E A. If a background value is used for E A, so that the "riemannian" Einstein quantum gravity is neglected, torsion propagation survives. An effective torsion propagator can be obtained, e.g., from the chain approximation shown in fig. 2. In D = 4 and with "/uv frozen to the Minkowski metric as background, K cannot acquire a vacuum expectation value without violating Poincard invariance. Hence, the occurrence of collective phenomena seems
~
°°°
Fig. 2. A typical contribution to the effective torsion propagator in the chain approximation.
41
Volume 140B, number 1,2
PHYSICS LETTERS
unlikely, and it appears reasonable to expect that the effective mass o f the torsion field should be close to the "bare" mass value, namely it should be of the order o f the Planck mass K-1/2. A reliable estimate o f the effective self-interaction strength cannot be given, because cut-offs are needed for convergence. However, it is clear dimensionally that such effective interactions should not be negligible relative to "riemannJan" gravitational interactions. More interesting collective phenomena can occur when D > 4, as in multidimensional unified theories [5] and in D = 11 supergravity [10]. To make contact with dimensional reduction procedures, as in r e f . [ 5 ] we u s e P ~ = ~ ® l , a = l ..... 4, F a = 7 5 ® o a, a = 5 ..... D, where ~ = - 1 7 5 7 ~ and {To}, {o a} generate the Dirac algebras in 4 and respectively in K dimensions. Now one obtains '/~D = _ ~ ( ~ ( 4 ) + ~(K))xlj
_ 16o[otab] ~_~,yOtoabxp_ lico[ct/~al ~TCteoaxij, (25) where
(26)
(27) The interesting point is that now Kabe can develop a non-trivial vacuum expectation value without breaking Poincar~ invariance in four dimensions [4,5], provided that (OlKabe 10) depends on X M only for M > 4. E.g., if
(Ol(~Oabc+ Kbc)lO)=O
with
(OlKabc[O):/:O, (28)
then it becomes possible to dynamically generate the kind o f spontaneous compactification advocated in ref. [5], with a background vacuum geometry (Minkowski)4 X V K, where V K is a compact space admitting a C a r t a n - S c h o u t e n [ 13 ] parallelization. The value of
31 May 1984
Let ~/0 be the lowest mass eigenvalue in the absence o f torsion: "~/0 ~ L - 1 where L is a typical length o f V K. Now, for a "realistic" multidimensional theory, one can impose the condition that )~'/0 be completely screened by the mass counterterm produced by -¼ i(OIKab c I0)o abc, leading to a fermion field with zero mass in four dimensions. This leads to a condition, which relates ~/0 ~ L - 1 to the coupling constant g, see ref. [9]. The term n = 2 in eq. (24) leads to a mass counterterm for the contorsion field Kabc; in principle, this could lead for Kabc to a physical mass much smaller than the bare mass ~ K - 1 / 2 . The terms containing Kaa b and K~;ja in eq. (24) describe composite gauge vector and tensor fields in four dimensions. Since Kaa b and Ka# a cannot acquire a non-trivial vacuum expectation value without violating (Poincar6)4-invariance , it appears unlikely that collective phenomena may be associated to these fields, and probably such fields will not acquire a relatively small physical mass. Other aspects o f torsion dynamics, especially in relation to the dynamical origin o f spontaneous compactification, are discussed in ref. [9]. I am much indebted to G. Venturi for valuable suggestions and a fruitful collaboration. I warmly thank A. D'Adda, D. Amati and T. Regge for stimulating discussions, and the CERN Theory Division for hospitality and financial support.
R eferen ces [1 ] A. Trautman, Symposia Mathem. 12 (1973) 139; F.W. Hehl, P. Von der Heyde, G.D. Kerlick and J.M. Nester, Revs. Mod. Phys. 48 (1975) 393. [2] G.D. Kerlick, Phys. Rev. D12 (1975) 3004; M. Soffel, B. M/iller and W. Greiner, Phys. Lett. 70A (1979) 167. [3] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. 966 (1921); O. Klein, Z. Phys. 37 (1925) 895;46 (1928) 188; Y.M. Cho and P.G.O. Freund, Phys. Rev. DI2 (1975) 1711; E. Cremmer and J. Seherk, Nucl. Phys. B108 (1976) 409; J.F. Luciani, Nucl. Phys. B135 (1978) 111 ; E. Witten, Nucl. Phys. B186 (1981) 412; A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316. [4] C.A. Orzalesi and M. Pauri, Phys. Lett. 107B (1981) 186. [5] C. Destri, C.A. Orzalesi and P. Rossi, Ann. Phys. 147 (1982) 321 ; Lett. Nuovo Cimento 36 (1983) 215.
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PHYSICS LETTERS
[6] D. Amati and G. Veneziano, Nucl. Phys. B204 (1982) 451. [7] T. Eguchi and H. Sugawara, Phys. Rev. D10 (1974) 4257; T. Kugo, Prog. Theor. Phys. 59 (1976) 2032; K. Kikkawa, Prog. Theor. Phys. 56 (1976) 974. [8] Y. Nambu and G. Jona Lasinio, Phys. Rev. 122 (1961) 345. [9] C.A. Orzalesi and G. Venturi, Phys. Lett. 139B (1984) 357. [10] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409; E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141.
31 May 1984
[11] P.G.O. Freund and M.A. Rubin, Phys. Lett. 94B (1980) 233; M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity, eds. S. Ferrara, J.G. Taylor and P. Van Nieuwenhuizen (World Scientific, Singapore, 1982). [12] M.J. Duff and C.A. Orzalesi, Phys. Lett. 122B (1983) 37. [13] E. Cartan and J.A. Sehouten, Proc. K. Akad. Wet. Amsterdam 29 (1926) 933. [14] A. Lichnerowicz, C.R. Acad. Sei. Paris 257 (1963) 7.
43
PHYSICS LETTERS
31 May 1984
TORSION DYNAMICS IN MULTIDIMENSIONAL UNIFIED THEORIES Claudio A. O R Z A L E S I 1 CERN, Geneva, Switzerland Received 8 July 1983
It is shown that, in quantized Einstein-Caftan gravitation in D ~ 4 dimensions, torsion propagates through virtual pair production. For D > 4, it is argued that torsion effects might determine dynamically the length scale of spontaneous compactification.
It is well k n o w n that torsion, .aT'BC , . is a d y n a m i c variable in the E i n s t e i n - C a r t a n (EC) t h e o r y [1 ]. In fact, the Cartan field e q u a t i o n ,1 tABc..
= K sAB.. c
(1)
relates the m o d i f i e d torsion,
tAf= %f
r
D
AD. '
to the canonical spin density o f matter, sAB C = 20£matter/OKAB C.
(2)
Here, K A B C = ~(TAB C + TCA B + TCBA ) is the contorsion for a metric linear c o n n e c t i o n ¢o: toAB = to [AB] = °~MAB d x M = 6°DABED '
(3)
o
°°DAB = °°DAB + K D A B '
(4)
where E A = E A M d X M are the co-vielbein one-forms 1 Also at Sezione Teorica, Dipartimento di Fisica della UniversitY, 43100 Parma, and INFN, Sezione di Milano, Milan, Italy. ,1 Conventions: Capital Latin indices run from 1 to D = 4 + K; Greek indices run from 1 to 4 and refer to ordinary spacetime; low-case Latin indices run from 5 to D and refer to those dimensions which undergo compaetification in multidimensional (D > 4) theories. The indices from the beginning of all alphabets are anholonomic and "flat", i.e., they refer to components relative to a vielbein. Indices from the middle-late parts of the alphabets refer to coordinates. The constant ~, with the length dimension D - 2 , is the D-dimensional generalization of Newton's constant K (and ~ = ~ for D = 4).
and ~ is the associated Levi-Civita c o n n e c t i o n for the metric y = ~?ABE A ® E B = TMN d X M ® d X N. In D = 4 dimensions, w h e n discussing macroscopic gravitational p h e n o m e n a , torsion is usually neglected and the Levi-Civita c o n n e c t i o n is used t h r o u g h o u t . There are good reasons for neglecting torsion in the classical D = 4 t h e o r y : w h e n m a t t e r is described by local fields, s is the density o f intrinsic angular mom e n t u m [ 1 ] and has a negligible average in the bulk m a t t e r forming galaxies and normal stars. For D i> 4, torsion effects are also neglected in most works on q u a n t u m gravity ,2 and on multidimensional unified theories ,3. In particular, the background g e o m e t r y - which determines the small field fluctuations near the q u a n t u m ground state 10) - is usually chosen by looking at solutions o f the field equations for the classical, torsion-free theory. We wish to emphasize that there is no physical justification for neglecting torsion effects in q u a n t u m gravity, especially w h e n D > 4. We will argue that, at the q u a n t u m level, torsion propagates in spite o f the algebraic character o f eq. (1) and o f the absence o f a kinetic torsion term in the EC lagrangian. O f course, already at the semi-classical level, there always occurs a " c o n v e c t i v e " torsion propagation, due to the ,2 However, some effects of a static background spin density on ferrnions have been estimated for D = 4, see ref. [2]. ,3 On multidimensional (D > 4) unified theories, see refs. [3-5] and references therein. Possible spin-torsion mechanisms of spontaneous eompactification, of the K = D - 4 "internal" dimensions, were discussed in refs. [4,5]. 39
Volume 140B, number 1,2
PHYSICS LETTERS
motion of physical fermions. However, and this is one of our main points, a "quantum" propagation of torsion takes place also in the vacuum, due to virtual effects from closed fermion loops ,4. The mechanism to be discussed is similar to that of the propagation of spin waves in a crystal, and bears an even closer resemblance to the propagation of Cooper pairs in superconductors. As a matter of fact, we shall use the same functional methods which have been applied [7] to the N a m b u - J o n a Lasinio ( N - J L ) model [8]. The quantum propagation of torsion could have important effects, e.g., on the ultraviolet properties o f the theory ,s. When D = 4, from known estimates [2] o f static effects, one might expect that the effective energy threshold for virtual torsion effects might well be by many orders of magnitude lower than for pair production by gravitons *e. However, it will be seen that, when D = 4, the effective mass in the induced torsion propagator is probably of the order of the Planck mass, so that torsion effects should not really introduce a new length scale in the theory. More drastic effects can take place when D > 4, as in multidimensional unified theories. In fact, the "internal" components of torsion (i.e., those components which are orthogonal to the ordinary d = 4 spacetime) now become scalar fields in four dimensions. Through collective phenomena, analogous to those responsible for the dynamic chiral symmetry breaking in the N - J L model, such fields can drastically influence the stability of the semi-classical vacuum. Therefore, the issue of the definition of the physical vacuum should be reconsidered in this light. In particular, it becomes possible for the "internal" components of torsion to develop large vacuum expectation values, which could set dynamically the length scale o f a spontaneously compactified background geometry ,7. Since this length, L, determines (after dimensional reduction) the (bare) Yang-MiUs coupling constants in four dimensions [5], g ° M ~ V ~ / L , the pro,4 Similar effects of torsion propagation take place also in the approach of ref. [6] to induced gravity. , s In particular, torsion contributions are crucial to enforce supercovariance in supergravity ( D ) 4). *e For D = 4, the "overcritical" situation for torsion already occurs at 10 --46 times the Planck density c S / ~ G 2 = (87r/
~)~, see ref. [2]. ,7 See refs. [5,9] and eq. (28) given below. 40
31 May 1984
posed mechanism raises the hope of actually computing such coupling constants from first principles, within a multidimensional unified theory where spontaneous compactification is due [4,5] to a spin-torsion condensate ,s. Concerning our approach to the quantum dynamics of torsion, we remark that, like everybody else, we do not know if and how the quantization of geometry really makes sense. We therefore take a simplistic attitude when writing quantum path integrals: we limit ourselves to a fixed background geometry and we consider only topologically small fluctuations around the given background. In practice, this means that all the fields (including the metric and the connection) should be expanded in harmonics over the background. Where needed for convergence, gaugefixing terms and cut-offs are understood ,9 To illustrate the quantum dynamics of torsion, we consider a Dirac spinor qffX) minimally coupled to gravity in D dimensions. The first-order action is -~ -
G + '~K + "9~D'
(5)
where, with x = E 1 ^ ... A E D = Det EAMdDX the volume element,
"¢~K
=
- ~ D --
1-L f ( ld 2;¢ J " * B E .
BId DE -- K "'19.
•
B K DE ~'c
DE.
B.
•j
(7)
'
f ~ D ' --f(~D + ~(DK , ) ' '
(8)
o
£D = ---~ ~ {TA (go)' FA } xI" ,~(D K) = 1 I(
~pABC~I I
1~K'F~,
(9) (10)
with VA(go) -- EA - ~ goABC FBC" The field equations are, with GAB(W ) the generalized Einstein tensor. *8 Spin-torsion compactification can also operate together with the more traditional bosonic mechanisms [3,5 ]. In particular, for D = 1 1 supergravity [10] with bosonic spontaneous compactification [ 11 ], torsion effects could cure [12] the problem of the enormous cosmological constant induced in four dimensions. .9 Although unsatisfactory from the viewpoint of first principles, our heuristic approach might correspond to a good low-energy approximation. In any case, forD > 4, our method generalizes the "truncation to zero-modes" which is frequently advocated [3,10,11].
Volume 140B, number 1,2
PHYSICS LETTERS
31 May 1984
GAS(~) =-~ {~rA%(O~) -- [V~(~o)~rAI),I,, (11) K A B C = 1 ~ ~t[,ABCq.t '
(12)
VA VA( ~ ) ' I ' = 0.
(13)
Notice that, since the contorsion (12) is totally antisymmetric, one has KAB C = K[ABC ] = { TAB C, and -~K becomes
t~
t~~j
t¢3~
t~41
Fig. 1. The first four one-loop contributions to the effective action for torsion.
where N ( E A ) = exp {tr log [ - V ( ~ ) ] } and -¢{K -
1 fK.K~. -- 2~
(14)
K'
At this point, the conventional procedure is to use (12) so as to rewrite the action in second-order form:
(2) =f
(15)
qz exp [ i ( ~ + .9~j)],
(16)
where 3;t{j
--f[,7**
+
~A ® E A + g ' ~ ] x ,
(17)
and r/, r/*, 6A, cK are external sources to generate the appropriate Green's functions. For simplicity, and because we are mainly interested in torsion effects in vacuo, in what follows we shall set ~ = r/* = 6A = 0. We shall also leave out the integration over vielbeins E A , which we treat as external fields. Therefore, hereafter we shall restrict our attention to the functional Z ( E A ) = f c 3 K C D ~ @,I, e x p [ i ( ~ K + ~
+ _¢{D)],
(18) where -¢~¢)(. = f K . cK'c. With standard methods [7], by integrating over the fermions, we obtain
z(E A ) =
A) f w x exp (iS),
- i t r l o g (1 - [1/~'(6~)] U}, (21)
rABC = ¼K'F.
(22)
7~- ~ABC --
Rather than relying on eq. (15), we shall closely follow the approach which was used in ref. [7] for the N a m b u - J o n a Lasinio model [8] : we construct the second-order effective action for the contorsion KAB C. The generating functional of Green's functions reads (up to a normalization constant)
~-f@E a ~ K O ~ c - / )
st{eft = f ~ - ~ K ' K ¢ K
U= 1 ~
[(2k)-lR(Co) + £D
+ ~k~FABCq~FABC]x.
z
(20)
S = -q~Q( + st{ eff
(19)
For the effective action of contorsion, we may use the expansion - i t r l o g (1 - [1/~'(~)] U} = ~
Ql(n),
(23)
n=l
q/(n) = (i/n)tr ([//~(~)] U) n .
(24)
The first four one-loop contributions are shown in fig. 1. Notice that the fermion propagator depends on the vielbein E A , and vertices carry the matrix FABC.
The term n = 1 determines the '~¢acuum" expectation value of torsion, in the presence of the external field E A . The term n = 2 yields a kinetic term and, together with [ - ( 1 / 2 k ) K . K ] , also the mass term. The terms with n > 2 describe non-polynomial self-interactions and interactions with the vielbein E A. If a background value is used for E A, so that the "riemannian" Einstein quantum gravity is neglected, torsion propagation survives. An effective torsion propagator can be obtained, e.g., from the chain approximation shown in fig. 2. In D = 4 and with "/uv frozen to the Minkowski metric as background, K cannot acquire a vacuum expectation value without violating Poincard invariance. Hence, the occurrence of collective phenomena seems
~
°°°
Fig. 2. A typical contribution to the effective torsion propagator in the chain approximation.
41
Volume 140B, number 1,2
PHYSICS LETTERS
unlikely, and it appears reasonable to expect that the effective mass o f the torsion field should be close to the "bare" mass value, namely it should be of the order o f the Planck mass K-1/2. A reliable estimate o f the effective self-interaction strength cannot be given, because cut-offs are needed for convergence. However, it is clear dimensionally that such effective interactions should not be negligible relative to "riemannJan" gravitational interactions. More interesting collective phenomena can occur when D > 4, as in multidimensional unified theories [5] and in D = 11 supergravity [10]. To make contact with dimensional reduction procedures, as in r e f . [ 5 ] we u s e P ~ = ~ ® l , a = l ..... 4, F a = 7 5 ® o a, a = 5 ..... D, where ~ = - 1 7 5 7 ~ and {To}, {o a} generate the Dirac algebras in 4 and respectively in K dimensions. Now one obtains '/~D = _ ~ ( ~ ( 4 ) + ~(K))xlj
_ 16o[otab] ~_~,yOtoabxp_ lico[ct/~al ~TCteoaxij, (25) where
(26)
(27) The interesting point is that now Kabe can develop a non-trivial vacuum expectation value without breaking Poincar~ invariance in four dimensions [4,5], provided that (OlKabe 10) depends on X M only for M > 4. E.g., if
(Ol(~Oabc+ Kbc)lO)=O
with
(OlKabc[O):/:O, (28)
then it becomes possible to dynamically generate the kind o f spontaneous compactification advocated in ref. [5], with a background vacuum geometry (Minkowski)4 X V K, where V K is a compact space admitting a C a r t a n - S c h o u t e n [ 13 ] parallelization. The value of
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Let ~/0 be the lowest mass eigenvalue in the absence o f torsion: "~/0 ~ L - 1 where L is a typical length o f V K. Now, for a "realistic" multidimensional theory, one can impose the condition that )~'/0 be completely screened by the mass counterterm produced by -¼ i(OIKab c I0)o abc, leading to a fermion field with zero mass in four dimensions. This leads to a condition, which relates ~/0 ~ L - 1 to the coupling constant g, see ref. [9]. The term n = 2 in eq. (24) leads to a mass counterterm for the contorsion field Kabc; in principle, this could lead for Kabc to a physical mass much smaller than the bare mass ~ K - 1 / 2 . The terms containing Kaa b and K~;ja in eq. (24) describe composite gauge vector and tensor fields in four dimensions. Since Kaa b and Ka# a cannot acquire a non-trivial vacuum expectation value without violating (Poincar6)4-invariance , it appears unlikely that collective phenomena may be associated to these fields, and probably such fields will not acquire a relatively small physical mass. Other aspects o f torsion dynamics, especially in relation to the dynamical origin o f spontaneous compactification, are discussed in ref. [9]. I am much indebted to G. Venturi for valuable suggestions and a fruitful collaboration. I warmly thank A. D'Adda, D. Amati and T. Regge for stimulating discussions, and the CERN Theory Division for hospitality and financial support.
R eferen ces [1 ] A. Trautman, Symposia Mathem. 12 (1973) 139; F.W. Hehl, P. Von der Heyde, G.D. Kerlick and J.M. Nester, Revs. Mod. Phys. 48 (1975) 393. [2] G.D. Kerlick, Phys. Rev. D12 (1975) 3004; M. Soffel, B. M/iller and W. Greiner, Phys. Lett. 70A (1979) 167. [3] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. 966 (1921); O. Klein, Z. Phys. 37 (1925) 895;46 (1928) 188; Y.M. Cho and P.G.O. Freund, Phys. Rev. DI2 (1975) 1711; E. Cremmer and J. Seherk, Nucl. Phys. B108 (1976) 409; J.F. Luciani, Nucl. Phys. B135 (1978) 111 ; E. Witten, Nucl. Phys. B186 (1981) 412; A. Salam and J. Strathdee, Ann. Phys. 141 (1982) 316. [4] C.A. Orzalesi and M. Pauri, Phys. Lett. 107B (1981) 186. [5] C. Destri, C.A. Orzalesi and P. Rossi, Ann. Phys. 147 (1982) 321 ; Lett. Nuovo Cimento 36 (1983) 215.
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PHYSICS LETTERS
[6] D. Amati and G. Veneziano, Nucl. Phys. B204 (1982) 451. [7] T. Eguchi and H. Sugawara, Phys. Rev. D10 (1974) 4257; T. Kugo, Prog. Theor. Phys. 59 (1976) 2032; K. Kikkawa, Prog. Theor. Phys. 56 (1976) 974. [8] Y. Nambu and G. Jona Lasinio, Phys. Rev. 122 (1961) 345. [9] C.A. Orzalesi and G. Venturi, Phys. Lett. 139B (1984) 357. [10] E. Cremmer, B. Julia and J. Scherk, Phys. Lett. 76B (1978) 409; E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141.
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[11] P.G.O. Freund and M.A. Rubin, Phys. Lett. 94B (1980) 233; M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity, eds. S. Ferrara, J.G. Taylor and P. Van Nieuwenhuizen (World Scientific, Singapore, 1982). [12] M.J. Duff and C.A. Orzalesi, Phys. Lett. 122B (1983) 37. [13] E. Cartan and J.A. Sehouten, Proc. K. Akad. Wet. Amsterdam 29 (1926) 933. [14] A. Lichnerowicz, C.R. Acad. Sei. Paris 257 (1963) 7.
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