Cartan Theory and General Relativity in Presence of Fermions »
Mémoire de Magistère en Physique
Option Physique Théorique
« ON THE EQUIVALENCE BETWEEN THE EINSTEIN- CARTAN THEORY AND GENERAL RELATIVITY IN PRESENCE OF FERMIONS »
Meriem Hadjer LAGRAA
Septembre 2010
Membres du jury:
Président : M. TAHIRI Prof. Univ. Oran
Examinateur : B. ABDESSALAM Prof. C. U. Ain Temouchent
Examinateur : S. BALASKA Prof. Univ. Oran
Examinatrice : F. IDDIR Prof. Univ. Oran
Rapporteur : M. LAGRAA Prof. Univ. Oran Contents
Acknowledgement ii
Introduction iii
1 Pure Gravity 1 1.1 General Relativity (The Einstein-Hilbert action): ...... 1 1.2 The Einstein-Cartan theory ...... 6 1.2.1 The Palatini action ...... 6 1.2.2 The Holst action ...... 12 1.2.3 The Generalized Palatini-Holst action ...... 15 1.3 Conclusion...... 19
2 Einstein-Cartan actions coupled to fermions 22 2.1 Gravity minimally coupled to fermions ...... 22 2.1.1 Torsion in presence of fermions ...... 24 2.1.2 The e¤ective theory ...... 27 2.2 The non-minimall coupling ...... 31 2.3 Conclusion...... 33
3 The Generalized Einstein-Cartan action coupled to fermions 35 3.1 The Generalized Palatini action coupled to fermions ...... 35 3.2 The Generalized Palatini-Holst action with fermions ...... 39 3.3 Conclusion...... 44
4 Conclusion 45
i Acknowledgement
First and foremost, I would like to record my gratitude to my director Pr. Mohamed LAGRAA for his supervision, advice and for giving me extraordinary experiences throughout out the work. More particularly, I would like to thank him as father for always pushing me to give my best. I am indebted to him more than he knows. Word fails to express my gratitude to all my family especially to my loving mother and to my brothers and sister for giving me their maximum supports throughout out all my life. I gratefully thank Pr. M. TAHIRI, Pr. B. ABDESSELAM, Pr. S. BALASKA and Pr. F. IDDIR for accepting to be members of the reading committee and to provide constructive comments on this thesis. My special thanks go to Pr. S. BALASKA for his hard working attitude and his incomparable generosity in dealing with the students and with all the members of the Laboratory. It is my pleasure to express my gratitude to Pr F. IDDIR who impresses me by her over‡owing energy and her sense of listening. I would also like to thank Pr. M. TAHIRI to have always pay attention to teach the scienti…c rigor to us. Thanks to Mr. Menaa who was a very pedagogue teacher and thanks to Nabila for the stimulating discussions which I had with her. I was extraordinarily fortunate to have studying the theoretical physics at the LPTO during the previous three years. I would like to thank my friends and all my colleagues for the discussions I could have with them and for sharing with me the literature, I wish a great success to them. Finally, I would thank everyone who was important to the realization of this thesis, and expressing my apology that I could not mention personally one by one.
ii Introduction
In the last few years, the addition of the Holst term to the Einstein action of gravity considered in the "tetrad-connection" framework, has been the subject of much attention in the aim of building a non-perturbative theory of the quantum gravity [2], [3] and [4]. This additional Holst term [5] depends on dimensionless real parameter called the Barbero-Immirzi parameter which does not a¤ect the classical vacuum Einstein equations which maintains the equivalence between the general relativity ("met- ric" framework) and the Einstein-Cartan theory ("tetrad-connection" framework). However, when the corresponding modi…ed action is coupled minimally to the fermionic matter the classical evolution equations are modi…ed and do depend on the Barbero-Immirzi parameter because of the non-vanishing torsion which emerges from this gravity-fermions coupling. Then, the equivalence between the general relativity and the Einstein-Cartan theory is no more established, since the Einstein-Cartan e¤ective action contains a current-current interaction term, due to the non-vanishing torsion, and where the Barbero-Immirzi parameter appears as coupling constant of this interaction. The aim of the study is to show that, from the generalized Einstein-Cartan ac- tion proposed in M. Dubois-violette and M. Lagraa [6], it is possible to obtain the standard e¤ective action of the Eintein-Cartan theory coupled minimally to fermi- onic …elds without the current-current interaction term and therefore an e¤ective action which does not depend either on the torsion or on the Barbero-Immirzi parameter [7]. This established the equivalence between the theory of general relativity and the Einstein-Cartan theory minimally coupled to fermions.
iii The plane of this thesis is presented as the following: In the …rst chapter, we will detail both of the general relativity described in the metric framework and the theory of Einstein-Cartan presented under the "tetrad-connection" form. We will see that in the Einstein-Cartan theory, the action corresponding on shell to the Einstein-Hilbert one of the general relativity, is the Palatini action added to the Holst term. Although the introduction of the Holst term has good reasons, for this study it is required to consider a more generalized action [6] obtained by adding scalar function term depending on the torsion. Where the generalized action leads to the classical vacuum Einstein equations if the added scalar function term respects some conditions.
In the second chapter, we shall consider the Palatini-Holst action coupled to the fermions which leads to a non-vanishing torsion appearing in the e¤ective action under the form of the current-current interaction with coupling constant determined by the Barbero-Immirzi parameter [8], [9] and [10]. Thus, even at the classical level the e¤ects of this parameter must be observable, leading to propose in many works various interpretations of the physical meaning of the Barbero-Immirzi parameter. Among this di¤erent approaches, we will interest to the non-minimal coupling already studied in [11] and [12], which permits to obtain an e¤ective action which does not depend on this arbitrary parameter, but still containing the current-current interaction and then di¤erent from the e¤ective action coming out from the general relativity minimally coupled to fermions.
Finally, in the last chapter the generalized action, taken with or without the Holst term, is coupled minimally to the fermionic matter [7], where we show that for particular values of the additional scalar function term we can construct the standard e¤ective action of the Einstein-Cartan theory coupled to the fermions free from the current-current interaction and from the Barbero-Immirzi parameter, which establishes the equivalence between the theory of general relativity and the theory of Einstein-Cartan coupled minimally to fermions.
Page: iv Chapter 1
Pure Gravity
In this chapter, our principal task is to expose the equivalence that exists between general relativity in vacuum and the theory of Einstein-Cartan. First, we start by deriving the Einstein equations from Einstein-Hilbert action [13] and then we pass to the tetrad formalism where we show that di¤erent actions like Palatini’s action, Palatini-Holst action [5] or the generalized one proposed in [6] lead to the same vacuum Einstein equations.
1.1 General Relativity (The Einstein-Hilbert ac- tion):
General Relativity is a theory that describes the evolution of the Riemannian space time Manifold ( ; g; ). The word evolution means that the manifold is M r M not considered as a background where the objects evaluate, but it is itself subject to evolution equations on the same footing as the usual …elds which describe matter particles or the mediators of the interaction like photons, gluons or vectorial bosons + 0 W ;W and Z . Let be a manifold equipped with a metric g = gdx dx which is non M degenerate in the sense that the components g admit an inverse g de…ned by
g g = g g = (1.1.1)
1 where the indices ; ; ::: = 0; 1; 2; 3 are tensorial indices of the four dimensional space-time manifold . In the tangent space T ( ) of the manifold , we M M M consider the covariant derivative which acts as r
v = @v + v u = @u u (1.1.2) r r where is the a¢ ne connection, v are the contravariant components of the vector v = v @ T ( ); and @ is a natural basis of T ( ). u are the covariant 2 M M components of the form u = u dx T ( ), and dx is a basis of the cotangent 2 M space T ( ) dual of the tangent space T ( ), i.e., M M
dx (@) = : (1.1.3)
A Riemannian manifold ( ; g; ) is a manifold equipped with a non M r M degenerate metric g compatible with the Levi-Civita connection
g = @ g g g = 0: (1.1.4) r
Here are the components of the Levi-Civita connection which is a torsion- free a¢ ne connection, i.e.
1 = = T = ( ) = 0 (1.1.5) ) 2
where T are the components of the torsion. By symmetrizing the indices ; and of (1.1.4), we get the expression of the Levi-civita connection in terms of the metric as
1 = g (@g + @g @g): (1.1.6) 2 The Riemannian Curvature is de…ned by
[ ; ]u = R u (1.1.7) r r where the components R of the Riemannian curvature tensor are given explic-
Page: 2 itly in terms of the Levi-civita connection by
R = @ @ + : (1.1.8)
The fully covariant form R = g R of the components of the Rie- mannian curvature tensor satisfy the following relations: (A) Symmetry
R = R (1.1.9)
(B) Antisymmetry
R = R = R = R (1.1.10) (C) Cyclicity
R + R + R = 0 (1.1.11) where we have used the metric g and its inverse to raise and to lower the tensorial indices. Note that we see From (1.1.6) and (1.1.8) that the Riemannian tensor is com- pletely determined by the metric g; therefore in the theory of general relativity the con…guration space is composed by the components of the metric and their derivatives with respect of the time "x0": In order to get an invariant action under general coordinate transformations, we present, in the following, the covariance of the di¤erent mathematical objects which describe the Riemannian manifold. 2 To get an invariant proper length ds = g(x)dx dx under the general co- ordinate transformation x x0 , the components of the metric must transform ! as @x @x g0 (x0) = g (x) (1.1.12) @x0 @x0 and the components g(x) of the inverse of the metric transform as
@x0 @x0 g0 (x0) = g (x): (1.1.13) @x @x
Page: 3 Where the coordinate transformations satisfy x0 = x0 (x)
@x0 @x @x0 @x = = : (1.1.14) @x @x0 @x @x0
The Levi-Civita connection transforms under di¤eomorphism transforma- tions inhomogeneously as
2 @x0 @x @x @x0 @ x 0 (x0) = + (1.1.15) @x @x0 @x0 @x @x0 @x0 which assures that the covariant derivative of tensors transforms as tensors.
We see from (1.1.15) that the Levi-Civita connection is not a tensor, contrary to R which are the components of a true tensor transforming as
0 @x @x @x @x R0 (x0) = R : (1.1.16) @x @x0 @x0 @x0
The tensorial properties of the metric and the Riemannian curvature allow us to construct the Ricci tensor
R = g R = g R = R; (1.1.17)
where we have used (1.1.10), and the curvature R = g R which is invariant under di¤eomorphism group transformations;
@x0 @x0 @x0 @x0 R0 = R0 g0 g0 = R0 g g @x @x @x @x @x0 @x0 @x0 @x0 = R0 g g @x @x @x @x = R g g = R: (1.1.18)
The tensor R is the true mathematical object which indicates the presence of a gravitational …eld. In fact we can see from (1.1.7) that if the curvature tensor vanishes the covariant derivative commute, in this case we can show that the space-time manifold reduces to a ‡at space-time endowed with a global M Minkowski coordinate system and the metric g reduces to the Minkowski metric
Page: 4 which re‡ect the absence of the gravitational …eld.
The scalar curvature R is the simplest function which depends on the metric …eld, the derivatives of the metric …eld and the second derivatives of the metric …eld, which allows us to construct the so-called Einstein-Hilbert action
1 4 SE H = d xp gR (1.1.19) 2k ZM with k = 8G, where G is Newton’s gravitational constant in unit c = 1; g 4 denotes the determinant of the metric tensor g; and p gdx is the form-volume 4 1 de…ned by p gdx = p gdx dx dx dx where is a totally 4! ^ ^ ^ antisymmetric symbol with 0123 = 1:
One can easily check that the form volume is invariant under transformations of di¤eomorphism group,
4 1 g d x0 = g dx0 dx0 dx0 dx0 0 4! 0 ^ ^ ^ 1p @x0p@x0 @x0 @x0 = g dx dx dx dx 4! 0 @x @x @x @x ^ ^ ^ 1 p @x0 4 = g dx dx dx dx = p gdx (1.1.20) 4! 0 @x ^ ^ ^
p
@x0 where is the Jacobian of the transformation x x0. In the last line of @x ! (1.1.20) we have replaced by the relation
1 @x @x g = p g = p g 0 (1.1.21) 0 @x @x 0 p which can be easily obtained from (1.1.12). The principle of the least action gives the following equation
1 SE H = 0 = R g R = 0 (1.1.22) g ) 2 which is the Einstein …eld equation in the vacuum. In terms of the metric components, the equation (1.1.22) becomes non-linear equation whose solution gives the evolution of the metric, and therefore the char-
Page: 5 acteristics of manifold are de…ned through the Riemannian tensor components M which satisfy
g R = R = 0 (1.1.23) obtained by contracting (1.1.22) with the metric
1 g R g R = R = 0 = R = 0: (1.1.24) 2 ) It is well known that the fermionic matter is belonging to the space of spinorial representation of the Lorentz group but unfortunately the gravity presented above is a metric formalism which does not permit the coupling with fermionic matter, so we have to use a more appropriate formalism known as the "tetrad formalism".
1.2 The Einstein-Cartan theory
1.2.1 The Palatini action
As mentioned above, the principal reason that leads us to use the tetrad formalism rather than Einstein’smetric …eld g is the possibility to couple gravity to spinor …elds which are Lorentz’sgroup representations. I The co-tetrad edx is 1-form valued in Minkowski vector space, equipped with the ‡at metric IJ ; where the components de…ne a map from the tangent space T ( ) to Minkowski space M
I I v = ev : (1.2.1.1) This de…ned relation is one-to-one, hence acting with the components of the tetrad eI on a Minkowski vector give a vector belonging to the tangent space T ( ) M I u = eI u (1.2.1.2)
I where eI is the inverse of the co-tetrad e;
I I I eeI = , eeJ = J ; (1.2.1.3)
Page: 6 I; J; ::: = 0; 1; 2; 3 label a Minkowskian indices.
The ‡at metric IJ of the Minkowski space has a Lorentzian signature ( ; +; +; +): IJ The metric IJ and its inverse are used to lower and to lift the indices I; J; ::.
The metric g of the tangent space of the manifold is related to by M IJ
I J g = ee IJ : (1.2.1.4)
Unlike the usual Einstein’sformalism where one has the metric g as funda- mental …eld, in the tetrad formalism the fundamental …elds are I -The co-tetrad 1-form e(x)dx with values in Minkowski vector space. IJ -The connection 1-form ! (x)dx with values in the Lorentz Lie algebra from which we de…ne the covariant derivative operator which acts on Minkowskian D vector as I I I J v = dv + ! J v (1.2.1.5) D ^ where d is the usual exterior derivative and vI are any form de…ned on the manifold . M Under the Lorentz group, the contravariant and the covariant components of the Minkowskian vector v(x) transform respectively as
I I J 1L v0 (x) = L J (x)v (x) vI0 (x) = L I (x)vL(x) (1.2.1.6)
I where L J (x) is an element of Lorentz group which respects the following relations of orthogonality
I K JL IK I K L J (x)L L(x) = L J (x)L L(x)IK = JL (1.2.1.7)
I from which we deduce the inverse of L J (x)
1I IK L L J (x) = L K (x)LJ : (1.2.1.8)
I I I The covariant derivative J (x) = d + ! J (x) transforms under Lorentz D J transformation as
Page: 7 I I K 1L 0 J (x) = L K (x) L(x)L J (x) D D I I I K K 1L J d + !0 J (x) = L K (x)(L d + ! L(x))L J (x) I I 1K I K 1L = J d + L K (x)dL J (x) + L K (x)! L(x)L J (x) (1.2.1.9)
I hence the transformation of the connection ! J (x) with respect of the Lorentz group reads
I I K 1L I 1K !0 J (x) = L K (x)! L(x)L J (x) + L K (x)dL J (x): (1.2.1.10)
Note that Lorentz transforms do not a¤ect the coordinate systems
I I J 0u0 (x) = J (x) u (x) (1.2.1.11) D D so the Lorentz group is local gauge transformations analogous to that of the theory of Yang-Mills. In the Einstein-Cartan theory, the 2-form torsion I = 1 I dx dx is given 2 ^ by the …rst Cartan structure equation
I I I I J = e = de + ! J e : (1.2.1.12) D ^ I The components of the torsion are given in the canonical framework by
I I I N I I N = @e + ! Ne @e ! Ne (1.2.1.13) where
I 1 I J K 1 I J K = JK e e = JK e e dx dx 2 ^ 2 ^ 1 = I dx dx: (1.2.1.14) 2 ^
IJ 1 IJ The curvature 2-form = dx dx associated to the connection 2 ^
Page: 8 IJ ! is given by the second structure equation
I I J I I N J u = J u = (d! J + ! N ! J ) u (1.2.1.15) DD ^ ^ ^
IJ where the components of the curvature are in terms of the connection as
IJ IJ IJ I NJ I NJ = @! @! + ! N! ! N! : (1.2.1.17) Under the local Lorentz group, the curvature 2-form IJ transforms as
IJ I J KL 0 (x) = L K (x)L L(x) (x): (1.2.1.16)
In terms of these fundamental …elds, we introduce the Palatini’saction
1 1 I J KL SP (e; !) = IJKLe e (1.2.1.18) 2k 2 ^ ^ ZM which is an action of the …rst order type contrary to the Einstein-Hilbert one which depends on the second derivative of the metric …eld g: Here IJKL are the components of the totally antisymmetric Levi-Civita symbol with 0123 = 1 = 0123: The action (1.2.1.18) is both invariant under Lorentz group transformations and the general coordinates transformations. The equation of motion of the connection via the variational principle reads
1 KL I J I J !SP (e; !) = ! IJKL e = 0 = IJKL e = 0 (1.2.1.19) 2k ^ ^ ) ^ ZM To get (1.2.1.19) we have applied the integration by parts, ignored the surface terms and used the variation of the curvature as
KL KL K NL K NL ! = d! + ! N ! + ! N ! ^ ^ KL K NL LN K = d! + ! N ! + ! ! N ^ ^ = !KL: (1.2.1.20) D The solution of the equation of motion (1.2.1.19) is I = 0; which in terms of components reads
Page: 9 K K K = e e = 0 (1.2.1.21) D D which can be solved to have the unique torsion free connection compatible with I the tetrad e
I I I N I e = @e + ! Ne e = 0 D I I N = e + ! Ne = 0 (1.2.1.22) r
where is the Levi-Civita connection and is the covariant derivative with r respect of the di¤eomorphism (1.1.2). By contracting (1.2.1.22) with eJ we get the expression of the connection in terms of tetrad as
I I I ![e] J = (e e ) = (e e ): (1.2.1.23) J r r J This last equation will restore the number of the degree of freedom, since the IJ I total 40 degrees of freedom coming from ! and e are reduced to 16 degrees of freedom when the connection is expressed in terms of the tetrad and the co-tetrad components. In these co-tetrad’s 16 degrees of freedom, 6 degrees of freedom will be absorbed by the Lorentz symmetry while 10 degrees of freedom which left correspond to the 10 coming from the fundamental …eld g used in the Einstein metric formalism.
The equation of motion of the tetrad reads
1 I J KL eSP (e; !) = e IJKLe = 0 (1.2.1.24) 2k ^ ^ ZM on the canonical base (1.2.1.24) reads
1 eJ KL " = 0: (1.2.1.25) 4k IJKL
Page: 10 Inserting the relation (1.2.1.3) in the precedent equation, we obtain
1 ( eI0 eJ eK0 eL0 e e e KL ) = 4k I0 JK0 L0 I K L e e e e KL = 0 (1.2.1.26) 4k I K L
I where e is the determinant of the tetrad e. By using the contraction
= = ( + + ) (1.2.1.27) the equation (1.2.1.26) reads
e 1 (ee e KL + ee e KL ) = 0 (1.2.1.28) k I K L 2 I K L
From the relation (1.2.1.4) we obtain
I J det g = (det e)(det IJ )(det e ): (1.2.1.29) where det = 1; then the determinant e is IJ e = p g: (1.2.1.30)
The insertion of the solution (1.2.1.23) in (1.2.1.17) gives the expression of the free torsion curvature space time in terms of the Levi-Civita connection
KL K L = e e (@ @ + ) (1.2.1.31) which in terms of the Riemannian curvature is
KL K L = e e R : (1.2.1.32)
If we replace the expression (1.2.1.32) in the equation of motion (1.2.1.28) we
Page: 11 obtain
e 1 eR + eR k I 2 I e 1 = (e R e R) = 0 (1.2.1.33) k I 2 I
Finally, contracting the equation (1.2.1.33) with eI leads to
1 R gR = 0 (1.2.1.34) 2 which is the Classical Einstein …eld equation which shows the equivalence between general relativity and the Einstein-Cartan theory. Note that in general relativity one has to assume from the beginning that the space time is free torsion to preserve the Riemannian structure of the manifold , then applying the principle of the least action to the Einstein-Hilbert action M we get the Einstein equation of the gravitational …eld g. In the theory of Einstein-Cartan the torsion is not assumed to vanish, so the tetrad formalism is built on the so-called Riemannian-Cartan manifold. When we apply the principle of the least action with respect to the connection we get a vanishing torsion, and therefore, only after that, we recover the Riemannian structure of the manifold and the classical Einstein equations in the vacuum. M
1.2.2 The Holst action
We have seen in the previous section that in tetrad formalism the Palatini’s action reproduces the same equations of motion as the Einstein-Hilbert action. In order to get a polynomial Hamiltonian which is necessary to get a quantum version of the gravitational theory, in the framework of the "Loop Quantum Gravity", S. Holst has modi…ed the palatini’saction (1.2.1.18) by adding a news term [5]
1 1 I J KL 1 I J SP H (e; !) = IJKLe e + e e IJ (1.2.2.1) 2k 2 ^ ^ ^ ^ ZM
Page: 12 where is the real dimensionless parameter known as the Barbero-Immirzi para- meter [15]. As we will see, this Holst term will not a¤ect the equations of motion. The variation of the action(1.2.2.1) with respect to the connection gives
1 1 I J KL 1 IJ !SP H (e; !) = IJKLe e ! + eI eJ ! : 2k 2 ^ ^ ^ ^ ZM (1.2.2.2) Using the relation (1.2.1.20) in the equation (1.2.2.2) and the Leibnitz rule give
1 KL IJ !SP H (e; !) = ! P KL (eI eJ ) = 0 2k ^ D ^ ZM 1 IJ = P KL (eI eJ ) = 0 (1.2.2.3) ) 2k D ^
IJ 1 IJ 1 [IJ] IJ 1 I J I J where the tensor P KL = KL + with = ( ) = 2 KL [NM] 2 N M M N [IJ] NM is the antisymmetric unity in the vector space of the antisymmetric tensor.
To solve the equation of motion (1.2.2.3) one has to contract it with the inverse IJ of the tensor P KL given by
2 1KL 1 KL 1 [KL] P NM = NM (1.2.2.4) 1 + 2 2 NM to get 1 [KL] (eK eL) = 0 (1.2.2.5) 2k NM D ^ the equation (1.2.2.5) has as solution a vanishing torsion which determines the I unique spin-connection in terms of the tetrad e as in (1.2.1.23). Note that for = i we recover the action used by A. Ashtekar [16]. In this case the fundamental …elds are the co-tetrad eI and the self-dual connection which is complex leading to a theory which needs reality conditions which are di¢ cult to solve in the quantum version of gravity. J. F. Barbero adjusted this free parameter as real one to provide a real theory of gravity. However dealing with Barbero’s connection leads to more complicated Hamiltonian scalar constraint [15] and to avoid this obstacle S. Holst generalized the Palatini’s action in order to get a polynomial Hamiltonian formalism [5].
Page: 13 The equation of motion of the co-tetrad is
1 I J KL 2 J eSP H (e; !) = e IJKLe + e IJ = 0 2k ^ ^ ^ ZM 1 J KL 2 J = IJKLe + e IJ = 0 (1.2.2.6) ) 2k ^ ^ which on the space-time basis reads
1 2 eJ KL + eJ = 0: (1.2.2.7) 4k IJKL IJ Proceeding like we did it before in the Palatini’saction, the equation (1.2.2.7) reads
1 KL KL 2 J e 4e e e 2e e e + e IJ = 0 (1.2.2.8) 4k I K L I K L and by inserting the expression of the free-torsion curvature SO(1; 3) in terms of the Riemmanian curvature (1.2.1.32), the equation (1.2.2.8) reads
e K L K L (4e e e (e e R ) 2e e e (e e R ) 4k I K L I K L 2 + eJ (e e R )) = 0 (1.2.2.9) e I J leading to
e 2 4e R 2e R + e I R = 0: (1.2.2.10) 4k I I e By contracting (1.2.2.10) with e I ; one gets
e 1 1 R g R + R = 0: (1.2.2.11) k 2 2 e Using the cyclicity property (1.1.11) of the Riemmanian curvature the third term coming from Holst part vanishes and the equation of motion (1.2.2.11) reduces to the classical Einstein equations in vacuum (1.1.22).
Page: 14 One can easily check from the following identity
I I I I = d(e eI ) + e eI ^ ^ D ^ DD I IJ = d(e eI ) + eI eJ (1.2.2.12) ^ D ^ ^
1 1 I that the Holst term is equivalent to the 2k I term up to a surface term. M ^ Where the …rst term in the right hand sideR of the …rst and the second line is surface term named the Nieh-Yan topological invariant [17]. We have seen that the evolution equation of the connection gives a vanishing 1 1 I torsion which leads to a vanishing Holst term I , so the action (1.2.2.1) 2k ^ reduces to the Palatini’s action on shell and thanR the classical Einstein equation of motion remain unaltered. In the Einstein-Cartan theory, we can recover once again the classical Einstein- …eld equations from the Palatini-Holst’saction, where it is clear that the Barbero- Immirzi parameter in front of the Holst term plays no role in the classical dynamics of Gravity; however it has striking e¤ects in the non-perturbative methods of quantization of gravity, for example, in "Loop Quantum gravity", this parameter a¤ects the discrete spectra of the area and volume operators [18].
1.2.3 The Generalized Palatini-Holst action
The Holst case detailed in the previous section may lead one to ask if there exist another terms depending of the torsion and reproducing the same equations of motion as the classical Einstein …eld equations in the vacuum. The most general term that we can add must be a scalar function of the torsion F (2) where 2 is scalar given by the relation I 2 I = ^
The remainder question is under which conditions on the function F (2) we obtain a vanishing torsion when we apply the principle of the least action with respect to the connection? Recently this question is studied in [6] where the Palatini’saction was generalized by adding a term such
Page: 15 1 1 I J KL 2 SG(e; !) = IJKLe e + F ( ): (1.2.3.1) 2k 2 ^ ^ ZM Remark that in the absence of the Holst term, we can consider any dimension d of the space time manifold with d 2, hence the previous action (1.2.3.1) M reads
1 IJ 2 SG(e; !) = (eI eJ ) + F ( ) (1.2.3.2) 2k ^ ^ ZM 2 1 I1 Id where F ( ) is smooth real function, = I :::I e ::: e is the form-volume, d! 1 d ^ ^ and is the dual map acting by 1 I1 Iq I1:::Iq Iq+1 Id (e ::: e ) = I :::I e ::: e : (1.2.3.3) ^ ^ (d q)! q+1 d ^ ^ The in…nitesimal variation of the action (1.2.3.1) with respect to the connection gives
1 IJ 2 !S(e; !) = (eI eJ ) ! + !F ( ) = 0 (1.2.3.4) 2k ^ ^ D ZM where we can rewrite the second term in the middle, by using ! = !;as
2 2 2 0 I I !F ( ) = F 0 ! = F 0! = F (! I + !I ) ^ ^ 0 I = 2F ! I (1.2.3.5) ^
2 0 @F ( ) where F = @2 denotes the derivative of F . While in the …rst term of the middle of (1.2.3.4) we use the Leibnitz rule, then the equation of motion (1.2.3.4) reads
1 [KL] 0 ( (eI eI )) 2F (eK L) = 0: (1.2.3.6) 2k D ^ IJ ^ h i
Page: 16 Using (1.2.3.3) on (1.2.3.6) gives
NM 1 IJKI :::I I :::I 3 d 1 K 2 d 2 KL 0 I1 I2 Id 1 I I 2F LNM e e ::: e = 0 4k (d 3)! 1 2 (d 2)! [IJ] KI1 ^ ^ ^ (1.2.3.7) acting again with the dual map (1.2.3.3) and contracting the antisymmetric Levi- Civita tensors the equation (1.2.3.7) is simpli…ed to
( 1)d P P K P K 0 P P IJ + JK IK + F I J J I = 0: (1.2.3.8) 2k I J P To solve this equation, one has to decompose the torsion IJ into 3 disjoint representations of the Lorentz group as
P P P P P IJ = IJ + ( J I ) + T IJ (1.2.3.9) I J P PQ where J is the trace component of the torsion, IJ = QIJ is completely
P PQ antisymmetric in Q; I; J and T IJ = TQIJ satis…es
T QIJ + T QJI = 0 QI TQIJ = 0
TQIJ + TJ QI + TQI J = 0 (1.2.3.10)
i.e. TQIJ has a vanishing trace and vanishing completely antisymmetric projection. By the insertion of the decomposition (1.2.3.9) into (1.2.3.8) we obtain
( 1)d 0 0 [(1 2F )QIJ + (F (d 2))( J I ) k QI QJ 0 +(1 + F )TQIJ ] = 0: (1.2.3.11)
This …nal equation leads to a vanishing torsion which gives the classical Ein- stein’s…eld equation, as long as, the image I(F 0 ) of F is disjoint of the points 1; 1 and d 2: 2 Now, in the case where we consider the Holst term, the space time must be 4
Page: 17 dimensional manifold, the action (1.2.3.1) reads
1 1 I J KL 1 I 2 SG(e; !) = IJKLe e + I + F ( ): (1.2.3.12) 2k 2 ^ ^ ^ ZM The variation of this action with respect to the connection gives
1 1 KL I J KL 2 I J !S(e; !) = !KL [ IJ e e e 2k ^ 2 ^ D ^ [IJ] ^ ZM 2F 0 KL eI J ] = 0 (1.2.3.13) [IJ] ^ which reads explicitly as
1 2 KL KL I 0 KL I PQ N M J IJ + NM + 2F PQ NM e e e = 0 4k [IJ] [IJ] ^ ^ (1.2.3.14) on which we act the operator to get 1 KL K IL L IK 0 K L L R [ R + I I + F R K k R R 1 K NML L NMK + NM R NM R ] = 0: (1.2.3.15) 2 Proceeding like we did it before, by inserting (1.2.3.9) into (1.2.3.15), we get a free-torsion if the adding function F (2) respects the following conditions (see more details about the calculation in the chapter 3)
1 (F 0 + 1)2 + = 0 (1.2.3.16) 2 6
1 1 (F 0 )(F 0 2) + = 0: (1.2.3.17) 2 2 6 When we do consider the Holst term, the generalized action gives once again the Einstein …eld equations in vacuum if the function F is such that the image
0 0 i i I(F ) of F is disjoint from the roots 1 + and 1 of (1.2.3.16) and from I the roots r1 and r2 of (1.2.3.17). In this case = 0 and the added term reduces to F (0) which plays the role of the cosmological term. F (0) = 0 corresponds to
Page: 18 a theory of gravity without a cosmological constant. Notice that if we do not take into account the function F , (i.e. F = 0) we recover the condition in the part (2.2.2) where 2 + 1 = 0. In the absence of the 6 Holst case by letting 2 we recover the condition 1; 1 and 2 for d = 4: ! 1 2
1.3 Conclusion
The main idea in this chapter is to establish the equivalence between the theory of general relativity and the theory of Einstein-Cartan in the vacuum, which both describe the evolution of space-time.
General relativity assumes from the beginning that the torsion has to be equal to zero (1.1.5) to keep the Riemannian structure of the space-time manifold. When one uses the Einstein’s metrical formalism, the Einstein-Hilbert action is the action from which we extract the classical dynamic of the theory, where the equations of motion getting from the principle of the least action with respect of the metric are the Einstein equations in vacuum (1.1.22). Assuming a zero-torsion induces a second order Einstein-Hilbert Lagrangian which giving rise to several di¢ culties, as a non-polynomial Hamiltonian if one rewrites the theory in the well-known ADM formalism [19] and [4], in the canonical quantization of gravity. Also we are unable to include fermionic matter in the theory when general relativity is described in the metric frame work. These obstacles constraint us to reformulate gravity under another way known as the tetrad formalism which considers the tetrad or the co-tetrad and the connection as two variable …elds instead of the metric g, opening the way to the canonical quantization procedure and the possibility to view this tetrad formalism of the gravity as the usual Yang- Mills theory based on the local Lorentz group and to couple gravity to elementary particles since they are Lorentz’sgroup representation.
Einstein-Cartan theory, contrary to general relativity, equips the space time manifold with a non-zero torsion which has to vanish dynamically by solving the equation of motion which comes from the variation of Palatini’saction with respect
Page: 19 to the connection, and by replacing this solution in the equations of motion of the tetrad, one can recover the vacuum Einstein’sequations (1.1.22). In Loop Quantum gravity we can not be satis…ed only by the Palatini’s ac- tion, we had to add a new term known as Holst term [5] giving more simpli…ed polynomial Hamiltonian constraints. The Holst term depends on real dimension- less parameter called the Barbero-Immirzi parameter [15], which seems to play an analogue role as of the -angle in gauge theories of Yang-Mills [20] since both of them do not a¤ect the classical equations of motion, but both of them have strik- ing e¤ects in quantum regimes . Generalizing the Palatini’saction by adding this term will give the same equation as Einstein …eld equations, as long as, the torsion vanishes dynamically which keep the equivalence between general relativity and the Einstein-Cartan theory.
Inspiring by the Holst case, the Palatini-Holst action is itself generalized in [6] 1 2 by adding smooth function depending on the torsion such 2k F ( ) which is M similar to have an in…nity of non trivial local action which leadR to the classical Ein- stein equations in vacuum if the adding function F (2) respects some conditions. So this promising new action deserves further attention like studying in details its repercussions on quantum gravity theory or what should this action give once couple to fermionic matter. In this …rst chapiter we have shown the equivalence between general relativity and Einstein-Cartan theory, where each of Palatini’saction, Palatini-Holst’saction or the Generalized action in [6] preserve this equivalence since they reproduce the same equations of motion as the Einstein’sones in vacuum when the equation of motion of the connection gives a vanishing torsion. It is clear that there are in…nitely many such functions F (with F smooth F (0) = 0) whenever 2 + 1 = 0: In particular for real, polynomials of the form 6
2p+1 F (x) = a1x + ::: + a2p+1x
2 where x = ; will satisfy (1.2.3.16) and (1.2.3.17) if the k are real such that p 8 we have: 2p a2p+1 0 and (2p + 1)a2p+1x + ::: + a1 sup(r1;r2) i i
Page: 20 or 2p a2p+1 0 and (2p + 1)a2p+1x + ::: + a1 inf(r1;r2) h h whenever r1 and r2, the roots of (1.2.3.17) are real. Then, in addition to the Barbero-Immirzi parameter which is related to the Holst term, we have showed that there exist an abundance of local actions and then an arbitrary number of not …xed parameters which do not a¤ect the classical vacuum Einstein equations. We recall that up till now, we have not increased spinor …elds to the theory of gravity yet, and it will be interesting to investigate this issue, in any case we are expecting that the equivalence between the theory of general relativity and the Einstein-Cartan theory will be still established once gravity is coupled to fermionic matter otherwise it reveals that one of these two theories is physically incorrect.
Page: 21 Chapter 2
Einstein-Cartan actions coupled to fermions
In this chapter, we shall continue with the previous analysis by coupling the gravitational …eld to the fermionic ones leading us to reconsider the question about the equivalence established between general relativity and the Einstein-Cartan theory in presence of the fermionic matter.
2.1 Gravity minimally coupled to fermions
We start with the usual Palatini-Holst action which we couple to the real minimal Dirac action [9], [8] and [10].
1 1 I J KL 1 I S(e; !; ) = IJKLe e + I 2k 2 ^ ^ ^ ZM i I I eI ( ): (2.1.1) 2 ^ D D ZM Note that if the Holst term is absent, the action (2.1.1) works in any dimension of the space-time manifold . M The …rst and the second term of the action (2.1.1) express the usual Palatini and Holst action which was already detailed in the …rst chapter, while the third one describes the minimall spinors coupling, with and are the Dirac spinor …elds 0 de…ned on the space-time manifold , where = y is the Dirac conjugate, M 22 I I with y is the Hermitian conjugate of the column and = e are the Dirac 1 matrices which satisfy the algebra IJ = 2 ( I J + J I ): Due to the signature of the ‡at Minkowski metric diag = ( ; +; +; +) (instead of (+; ; ; ) which is commonly used in QFT) the matrices I of Dirac satisfy the following properties:
( )2 = I ( )2 = I 0 i + I = 0 I 0: (2.1.2)
The exterior derivative acts on the Dirac spinors as D !IJ (x) = d (x) + IJ (x) (2.1.3) D 2
1 where IJ = 4 [ I ; J ] are the generators of the Lie algebra of the Lorentz group SO (1; 3) in the Dirac representation. The properties of the Dirac matrices presented above in the lines (2.1.2) imply the following expressions of the commutator and the anti-commutator
N [ ; KL] = [ ; KL]+ = iMKLN (2.1.4) M MK L ML K M 5
i IJKL 2 + with 5 = 4! I J K L = i 0 1 2 3 with ( 5) = I, 5 = 5, and 5 anti- commutes with I : From the relations (2.1.4) and
y = IJ (2.1.5) 0 IJ 0 we get !IJ (x) = d (x) IJ (x) : (2.1.6) D 2 Notice that the fermionic mass term in the action (2.1.1) is ignored because it does not change the analysis that fallows. The Dirac spinors transform under the local Lorentz transformations as
0 (x) = S (L (x)) (x) (2.1.7)
Page: 23 whereas regarding to the di¤eomorphism transformations it is scalar. S (L (x)) is matrix of the representation of the Lorentz group in the space of Dirac spinors which satis…es
+ 0 0 1 S (L (x)) = S (L (x)) : (2.1.8)
From which we get the transformations of
1 0 (x) = (x) S (L (x)) : (2.1.9)
In the same manner, from the covariance of the Dirac equations under the transformations of Lorentz, it follows
1 I I J S (L (x)) S (L (x)) = L J (x) : (2.1.10) which lead to 1 S (L (x)) 5S (L (x)) = det(L (x)) 5 (2.1.11) with det(L (x)) is the determinant of the matrix of the Lorentz transformation I L J (x) : We can see from the transformation properties of the Dirac spinors that, as the di¤erent actions presented in the …rst chapter, the action (2.1.1) is both invariant under coordinate and Lorentz transformations.
2.1.1 Torsion in presence of fermions
We will now derive the equation of motion of the connection from the action (2.1.1) and see what should result from the presence of fermionic …elds in the theory of gravity
1 1 I J KL 2 I !S(e; !; ) = IJKLe e ! + ! I 2k 2 ^ ^ ^ ZM i I I eI ( ! ! ) = 0 (2.1.1.1) 2 ^ D D ZM
Page: 24 leading to
1 KL I J 2 KL I J !S(e; !; ) = !KL (IJ e + e 2k ^ ^ [IJ] ^ ZM ik I KL + eI ; ) = 0 (2.1.1.2) 2 ^ + the …rst line of (2.1.1.2) expresses the contributions coming from the usual action of Palatini and Holst, while in the second one we have used .
!KL KL !KL KL ! = ! = D 2 D 2 and ! = ! to get the contributions coming from the fermionic matter. By inserting the second formula of (2.1.4) into the equation (2.1.1.2) and using the operator we obtain
1 KL KL 1 I k KL N M J IJ + NM J e e e = 0 (2.1.1.3) 2 [IJ] 2 [NM]J ^ ^ where J = is the axial fermionic current. J 5 J By acting once again with the operator on (2.1.1.3), it results
RKL RK IL RL IK 1 NMRK L NMRL K + I I + ( NM NM ) 2
k RKLJ = J (2.1.1.4) 2 J which shows the relation between the torsion and the fermions exhibited in term of the axial current J . J We insert the decomposition of the torsion in terms of the irreducible represen- tation of the Lorentz group (1.2.3.9) into the equation (2.1.1.4), where the totally RKL RKLP antisymmetric components is replaced by = P , to get A
Page: 25 RKLP 2 RK L 1 L RL K 1 K P + P 2 2 A A A 1 +T INM RK L + JRKL I N M NM [IJ] k RKLP = P : (2.1.1.5) 2 J
By contracting with RKLQ and with RK we obtain
2 k Q + Q = Q (2.1.1.6) A 2J
1 2L L = 0 (2.1.1.7) A respectively. Then the insertion of (2.1.1.6) and (2.1.1.7) into (2.1.1.5) gives us
1 T INM RK L + JRKL = 0 (2.1.1.8) I N M NM [IJ] S P Q 1 TS PQ which if we contract it with the tensor KL we get R K L [RT ] 1 T RKL 1 + = 0: (2.1.1.9) 2 We recall that is taken to be a real parameter, so (2.1.1.9) leads to
T RKL = 0: (2.1.1.10)
By combining (2.1.1.6) with (2.1.1.7) we obtain the irreducible components of the torsion in terms of the axial current
2k 2 k I = I I = I : (2.1.1.11) A 4(1 + 2)J 4(1 + 2)J Contrary to the vacuum where a vanishing torsion results from the equation of motion of the connection, in the presence of fermionic …elds the torsion is no
Page: 26 longer equal to zero, which seems that fermionic matter is acting as source of the torsion [8]. One must notice that the solution (2.1.1.11) violates the parity, in fact in the second one the trace component I of the torsion which has to be a proper vector, is proportional to an axial-vector.
I I I J 0 = L J (2.1.1.12) !
I I I J 0 = det (L) L J (2.1.1.13) J ! J J I with det (L) is the determinant of the matrix L J belonging to the SO(1; 3) group: For the transformation (2.1.1.13) we have used (2.1.11).
IJK IJK I J K LNM 0 = L LL N L M ! IJK I J K R 1P LNMS 0 = L LL N L M L SL R P A IJK IJKR IJKR 1P 0 = 0 = det () L R P (2.1.1.14) AR A hence the components P transforms as axial-vector A
1P 0 = det (L) L R P : (2.1.1.15) AR A We can check out easily that this violating-parity is emerging from the Holst term because in the case where the solutions (2.1.1.11) read ! 1 k I = I I = 0: (2.1.1.16) A 2 J
2.1.2 The e¤ective theory
In order to determine the contributions coming from the di¤erent irreducible components (1.2.3.9) of the torsion to the action (2.1.1), we have to solve the Cartan structure equation (1.2.1.12) by splitting the connection !IJ into
!IJ = ! [e]IJ + CIJ (2.1.2.1)
e
Page: 27 where ![e]IJ is the uniquely de…ned torsion-free SO(1; 3) spin connection compat- ible with the tetrad I I I J e e = de + ![e] J e = 0 (2.1.2.2) D ^ and CIJ = CIJ dx = CIJ eK is the contorsion 1-form. eK e To get a metric connection, IJ = 0; the contorsion must satisfy CIJ +CJI = D 0: It is explicitly given in terms of torsion by
1 CIJK = (JIK + KIJ IJK ) (2.1.2.3) 2 from which we deduce
I I I J I I J I J e = de + ! J e = de + ![e] J e + C J e D ^ ^ ^ I I J I K J = e + C J e = C JK e e D ^ ^ 1 I J K I e = JK e e = : (2.1.2.4) 2e ^ In terms of the irreducible decomposition (1.2.3.9) of the torsion, the contorsion (2.1.2.3)
IJK 1 IJK L IK J JK I KIJ C = L + 2 2 + 2T : (2.1.2.5) 2 A We act by the square of the exterior covariant derivative on the vectorial D representation V I
I I J I I N J V = J V = (d! J + ! N ! J )V DD ^ I I I I N N J = d![e] J + dC J + ![e] N + C N ![e] J + C J V ^ I I I N J = J [e] + C J + C N C J V (2.1.2.6) e D e ^ e the equation (2.1.2.6)e is thee relation between the curvature IJ associated with the connection !IJ ; the curvature
IJ IJ I NJ [e] = d![e] + ![e] N ![e] : (2.1.2.7) ^ associated with the torsione free soe(1; 3) spine connectione !~ [e]IJ , which is expressed
Page: 28 in terms of the Riemannian curvature (1.2.1.32), and the contortion as
I I I I N J = J [e] + C J + C N C J (2.1.2.8) D ^ In this manner, the actione (2.1.1) composede by the torsion-free part and the contorsion CIJ reads
1 1 I J KL i I I S(e; !;C;~ ) = IJKLe e ~ eI ( ~ ~ ) 2k 2 ^ ^ 2 ^ D D ZM ZM 1 1 I J KN NL 1 I + IJKLe e C C + I 2 2 ^ ^ ^ ^ ZM i I KL eI CKL ; (2.1.2.9) 4 ^ + ZM where the term depending on CIJ in (2.1.2.8) is not taken into account because D it will vanishes when we integrate by part and using eI = 0: D The …rst line of the actione (2.1.2.9) expresses the part which is free from the torsion, and then depends uniquely on the tetrad ande the Dirac spinors, whereas the second and the third lines are the action parts giving the non-propagating torsion due to the presence of fermions. By using the formula eI eJ eK eL = IJKL and replacing the expres- ^ ^ ^ sions of (1.2.3.9) and (2.1.2.5), the action (2.1.2.9) reads explicitly in terms of the irreducible components of the torsion as
1 1 I J KL i I I S(e; !;C;~ ) = IJKLe e ~ eI ( D~ ~ ) 2k 2 ^ ^ 2 ^ D ZM ZM 1 1 I IJKL N + (24 I T IJ TNKL) 2k 4 A ZM 1 1 I I IJK + ( 24 I + 6 I 4T TJKI ) 2k 4 A A ZM 3! I + I : (2.1.2.10) 8 A J ZM In the …nal step, we will replace the set of equations (2.1.1.11) and (2.1.1.10) which are the equations of motions of the di¤erent irreducible components of the
Page: 29 torsion (1.2.3.9) when the usual Palatini-Holst Lagrangian is coupled to the mini- mal Dirac Lagrangian for getting the e¤ective action of the theory
1 1 I J KL i I I Seff (e; !; ) = IJKLe e ~ eI ( ~ ~ ) 2k 2 ^ ^ 2 ^ D D ZM 2 ZM 3k I e I : (2.1.2.11) 16 (1 + 2) J J ZM In the e¤ective action (2.1.2.11) results a current-current interaction term with the parameter as constant of coupling due to the non-vanishing torsion. This emerging term is not predicted by the general relativity when it is mini- mally coupled to the fermions by consequence the equivalence established before in the vacuum case is no more institute once fermionic matter is present in the theory. More over according to (2.1.2.11), the Barbero-Immirzi parameter has physically e¤ects even at the classical level then in principle the Barbero-Immirzi parameter has to be observable. This leads to bring up the question about its physical origin which is still debated in many works where di¤erent interpretations are proposed. In the references [21] and [22] a possible topological interpretation of the para- meter is suggested where the interesting analogy between and the -angle of QCD is exploited [20]. In this manner takes place in front of a topological term played by the Nieh-Yan topological invariant [17] rather than the Holst term, and then the fermionic matter can be introduced in the theory without implying the Barbero-Immirzi parameter on shell. In others attempts [23] and [24], the parameter is no more considered as multiplicative constant in front of the Holst term but promoted to be a scalar …eld interacting with gravity, where later in [25] the parity properties of the Barbero- Immirzi …eld was investigated and shown that the BI …eld must be a pseudoscalar rather than scalar in order to preserve the parity properties of the torsion compo- nents under the Lorentz transformations. Recently in [26] it is shown that the cosmological constant is linked to the Barbero-Immirzi parameter in the extension of the Plebanski theory.
Page: 30 2.2 The non-minimall coupling
This subsection is devoted to the study of the gravity-fermions system when the non-minimal coupling term is added to the usual action (2.1.1) showing that the e¤ects of the Barbero-Immirzi parameter does not a¤ect the equation of motions coming from the e¤ective action since is resulting as constant beside a total derivative term. the action proposed in the references [11] and [12].
1 1 I J KL 1 I S(e; !; ) = IJKLe e + I 2k 2 ^ ^ ^ ZM i I i i eI 1 + 1 + (2.2.1) 2 ^ 5 D D 5 ZM where is a real constant otherwise the non-minimall fermionic part in the action (2.2.1) is no more Hermitian and then we need of reality conditions. inserting the expressions (2.1.2.8) and (2.1.2.1) in the action (2.2.1) leads to
1 1 I J KL K NL 1 I S(e; !; ) = IJKLe e + C N C + I 2k 2 ^ ^ ^ ^ ZM i I I CKL I KL eI [ ~ ~ e + ; 2 ^ D D 2 + ZM i i C I ~ + ~ I KL I ; KL ] (2.2.2) 5 D D 5 2 5 and by replacing the formulas (2.1.4), the di¤erent components of the torsion (1.2.3.9) and the contorsion (2.1.2.5) in (2.2.2) we obtain
1 1 I J KL i I I S(e; !; ) = IJKLe e eI ~ ~ 2k 2 ^ ^ 2 ^ D D ZM ZM 1 I I eI ~e + ~ 2 ^ 5 D D 5 ZM 1 1 I IJKL N + (24 I T IJ TNKL) 2k 4 A ZM 1 1 I I IJK + ( 24 I + 6 I 4T TJKI ) 2k 4 A A ZM 3! I 3 I I + I : (2.2.3) 8 A J 2 J ZM ZM
Page: 31 Now, applying the principle of the least action with respect to the di¤erent components I ; I and T IJK coming from the contribution of the torsion and the A contorsion, we obtain the following set of equations
4 AS(e; !; ) = I + 2 I + k I = 0 (2.2.4) A J
2 k S(e; !; ) = I 4I + I = 0 (2.2.5) A J
1 KL [KL] T S(e; !; ) = IJ 4 TNKL = 0 (2.2.6) IJ the equation (2.2.6) leads to TRNM = 0 once it is contracted with the tensor 1 IJ [NM] NM + 4IJ since is real. While combining between the equations of motion (2.2.4) and (2.2.5) leads to
k ( 2 + ) k ( 2) I = I I = I (2.2.7) A 2 (1 + 2) J 4 (1 + 2) J these equations coincides with those resulting from the gravity minimally coupled to fermions (2.1.1.11) by letting : The solutions (2.2.7) depend both on ! 1 the Barbero-Immirzi parameter and the constant ; but if we state = ; (2.2.7) reduce to k I = I I = 0 (2.2.8) A 2 J remark that even in presence of the Holst term, there is no more violating parity in (2.2.8) contrary to (2.1.1.11). Finally, we insert the set of equations (2.2.8) into (2.2.3) in order to get the e¤ective action when gravity is coupled to fermions by the non-minimall coupling as
1 1 I J KL i I I Seff (e; !; ) = IJKLe e eI ~ ~ 2k 2 ^ ^ 2 ^ D D ZM ZM 3k I 1 ~ I e I e eI (2.2.9) 16 J J 2 ^ D 5 ZM ZM which depends on the free Barbero-Immirzi parameter , however the equations of motion deriving from the e¤ective action (2.2.9) are not a¤ected by this parameter
Page: 32 because appears only in front of the last term of the e¤ective action (2.2.9) which, if we integrate by part and take into account the solutions in the lines (2.2.8), T IJK = 0 and eI = 0, gives D
1 e I 1 I I eI ~ = d( eI ) + ~ ( eI ) 2 ^ D 5 2 J D J ZM ZM 1 1 J K L I 1 K L J = d IJKLe e e = d JKLe e e 2 3! ^ ^ J 3 k ^ ^ ZM ZM 2 J = d J e (2.2.10) 3k ^ ZM which is the well-known Nieh-Yan topological invariant [17], thus the non-minimall coupling procedure allows to discard the e¤ects of the Barbero-Immirzi parameter, nevertheless the current-current interactions is still present at the on-shell level due to the non-propagating emerging torsion and then the resulting e¤ective action is di¤erent from the e¤ective action of the theory of general relativity coupled minimally to fermionic matter.
2.3 Conclusion
The aim of this chapter was to show the results inducing by the presence of the fermionic matter in the theory of gravity especially concerning the question about the equivalence established between the theory of general relativity and the Einstein-Cartan theory.
We have already seen that in the vacuum the Einstein-Cartan theory is equiv- alent to the general relativity, where we have seen that the …rst order Palatini- Holst action [5] in the tetrad framework is equivalent on-shell to the second order Einstein-Hilbert action in the Einstein’smetric framework, but it is not the same once the fermions are included in gravity. Coupling the usual Palatini-Holst action to the minimal Dirac action induces a non vanishing torsion. The variation of this action with respect to the connection gives arise to a current-current interaction term with the Barbero-Immirzi parameter as coupling constant. This is totally di¤erent from the case of the theory of general relativity coupled minimally to fermions where the e¤ective theory is free from such interaction or from the para-
Page: 33 meter . Therefore according to the e¤ective action (2.1.2.10) the current-current interaction and the Barbero-Immirzi parameter have to be observable but it is not the case, the physical e¤ects of such quantities have never been detectable yet, which opened the way to more investigate and try to explain these results where in many attempts, it is claimed that the parameter has to be scalar …eld or pseudoscalar …eld which interacts with the gravitational …eld, or is constant which is related to the cosmological one. The other approach is to stress the gravity-fermions system described by the usual Palatini-Holst-Dirac action by the non-minimall coupling. Indeed the e¤ec- tive theory coming out depends on the Barbero-Immirzi parameter, but as long as the parameter appears only in front of a total derivative played by the Nieh-Yan term, the evolution equations will not be a¤ected by this parameter. However, it remains that this e¤ective action depends on the axial current-current coupling hence the equations of motion of the fermions are no more linear and makes the predictions of the usual Einstein-Cartan theory di¤erent from those of the theory of general relativity. Needless to say that the experimental successes of the theory of general relativity and the standard model are in favor of the minimal approach where the framework of the standard model is the Yang-Mills theory based on the minimal coupling at the fundamental level. This contradicting result leads us to reconsider the action at the start in purpose to discard both the Barbero-Immirzi parameter and the current-current interaction from the e¤ective action.
Page: 34 Chapter 3
The Generalized Einstein-Cartan action coupled to fermions
The main goal of the last chapter is to construct the standard e¤ective Einstein- Cartan action coupled to fermions free from the usual current-current interaction and from the Barbero-Immirzi parameter : This is establishing the equivalence between the general relativity and the theory of Einstein-Cartan coupled minimally to the fermionic matter
3.1 The Generalized Palatini action coupled to fermions
we start by coupling the generalized Einstein-Cartan action presented in the …rst chapter with the fermions [7] without taking in consideration the Holst term where the space-time is no more restricted to four dimension whatever the dimen- sion d of the manifold is d 2: M The action (1.2.3.2) coupled to the standard Dirac action reads
1 IJ 2 S(e; !; ) = (eI eJ ) + F 2k ^ ^ ZM i I I eI ((3.1.1)) 2 ^ D D ZM 35 where is the usual dual map (1.2.3.3). We shall now proceed by decomposing of the connection !IJ in the SO(1; 3) spin connection free from the torsion !IJ and the contorsion part CIJ as in (2.1.2.1). After that the insertion of (2.1.2.1) and (2.1.2.6) in the action (3.1.1) gives e
1 IJ i I I S(e; !;C; ) = (eI eJ ) eI 2k ^ ^ 2 ^ D D ZM ZM 1 IN J 2 e + (eI eJ ) eC CN + F e e 2k ^ ^ ^ ZM i KL I eI C ; KL (3.1.2) 4 ^ + ZM where the …rst line of (3.1.2) corresponds only to the torsion-free parts of the gen- eral relativity coupled minimally to fermions whereas the lines which left depend on the scalar torsion function F (2), the contorsion CIJ and the fermionic current after using the second formula of (2.1.4), which reads
1 IJ i I I S(e; !;C; ) = (eI eJ ) eI 2k ^ ^ 2 ^ D D ZM ZM 1 I NJ INJ 2 e + C NI C Je C CNJI + F e e 2k ZM 1 d IKL + ( 1) CKLI (3.1.3) 4 J ZM where IKL = i I ; JK is the tensorial current which is also completely J + antisymmetric in I; J; K: Now we will replace the irreducible components coming from the decomposition
Page: 36 of the torsion (1.2.3.9) and the contorsion (2.1.2.5) in the action (3.1.3) which reads
1 IJ i I I S(e; !;C; ) = (eI eJ ) eI 2k ^ ^ 2 ^ D D ZM ZM 1 1 IJK I IJK e + F eIJK + 2 (d 1) I +e T eTIJK 2k 2 ZM 1 1 N IJK IJK + 4 (d 1) (d 2) N IJK 4T TJKI 2k 4 ZM 1 d IKL ( 1) IKL (3.1.4) 8 J ZM where for obtaining of the scalar function term F (2) in terms of the irreducible decomposition of the torsion we have used this expression
2 I 1 IKL = I = IKL ^ 2 1 IKL L IKL = IKL + 2 (d 1) L + T TIKL : (3.1.5) 2 The variational principle will give the evolution equations of the di¤erent irre- ducible components of the torsion
S(e; !;C; ) = 0 = 2 (d 1) (F 0 + 2 d) I = 0 (3.1.6) I )
e S(e; !;C; ) = 0 = (F 0 + 1) TIJK = 0 (3.1.7) T IJK ) d e ( 1) k S(e; !;C; ) = 0 = IJK = IJK : (3.1.8) IJK ) 4 F 1 J 0 2 From the above equations,e we can see that the di¤erent components I ;TIJK and IJK of the torsion are de…ned if the function F is such that the image Im F 0 of F 0 is disjoint of the points 1; d 2; and 1 leading to 2 ( 1)d k I = 0 TIJK = 0 IJK = IJK : (3.1.9) 4 F 1 J 0 2 The insertion of the values (3.1.9) of the torsion components into the action
Page: 37 (3.1.4) gives
1 IJ i I I S(e; !;C; ) = (eI eJ ) eI 2k ^ ^ 2 ^ D D ZM ZM 1 1 IJK IJK 1 IJK 1 IJK + e F IJK eIJK F 0 IJK +e eIJK 2k 2 2 4 ZM (3.1.10) where we have replaced in the last line of the action (3.1.4) the tensorial current by its expression in terms of the totally antisymmetric component IJK ;
IJK d 4 1 IJK = ( 1) F 0 : (3.1.11) J k 2 If we try to get an action without the usual current-current coupling and then an equivalence between the theory of general relativity and the theory of Einstein- Cartan, we have to cancel the second line of the action (3.1.10) which consists to solve the following equation
x F (x) 2xF 0 (x) + = 0 (3.1.12) 2 1 IJK where x = 2 IJK : x The smooth solution F (x) = 2 is not acceptable because it contradicts the conditions F 0 = 1 required for having a well-de…ned relation between the torsion 6 2 and the fermionic current given in the set of equations (3.1.9). In the case where we accept real non-polynomial solutions, the general solution of the equation (3.1.12) is x F (x) = + cpx (3.1.13) 2 where c is any real scalar function which does not depends on the torsion. The 1 IJK solution (3.1.13) is de…ned when x = IJK 0: 2 i In this case, F 0 1 = c ; leads to a non-singular value of the com- 2 1 IJK 6 2p 2 IJK pletely antisymmetric irreducible component of the torsion
( 1)d 2c IJK = IJK (3.1.14) 1 NML J k 2 NML q Page: 38 IJK IJKL which reduces, if we replace by = L and by the explicit form of the IJK I JK A IJKL IJKL axial-current = i ; + = i i 5 L = 5 L = IJKL J L for d = 4 to J 2c L = L (3.1.15) K kp 3 K A J A A K K de…ned if the vector is time-like, it means that K 0: A A A h Then, with this non-polynomial choice of the function F; the action (3.1.1) reads
2 1 IJ 2 S(e; !; ) = (eI eJ ) + + cp 2k ^ ^ 2 ZM i I I eI (3.1.16) 2 ^ D D ZM leading to the e¤ective action
1 IJ i I I Seff (e; !; ) = (eI eJ ) eI (3.1.17) 2k ^ ^ 2 ^ D D ZM ZM which ise the standard second-order tetrade form of the Einstein-Cartane e theory min- imally coupled to fermionic matter without the current-current coupling term and which is free from the torsion and from the arbitrary function c:
3.2 The Generalized Palatini-Holst action with fermions
In this subsection, we will consider the generalized Palatini-Holst action (1.2.3.12), which works in four-dimensional space-time manifold only, coupled minimally M to the fermionic matter which reads
1 1 I J KL 1 I 2 S(e; !; ) = IJKLe e + I + F 2k 2 ^ ^ ^ ZM i I I eI : (3.2.1) 2 ^ D D ZM
Page: 39 The in…nitesimal variation of the action (3.2.1) with respect of the connection S(e;!; ) = 0 gives the following equation !KL
1 KL KL 1 I 1 KL I PQ ( + + F 0 4 IJ [IJ] 2 NM 4 [IJ] PQ NM k KL N M J J )e e e = 0: (3.2.2) 4 [NM]J ^ ^ By acting with the operator on the equation (3.2.2), we obtain
1 RKL RK IL RL IK 1 KRL LRK + I I + F 0 2 2 1 K RLNM L RKNM k RKLJ NM NM = J (3.2.3) 2 4 J where J = is the axial fermionic current. J 5 J The insertion of the di¤erent irreducible components of the torsion (1.2.3.9) in the precedent equation (3.2.3) gives
1 RKLP 2 (1 2F 0) P + P 2 A 1 RK L 1 L RL K 1 K (2 F 0) (2 F 0) 2 A A 1 INM R K L 1 JR KL k RKLP + T (1 + F 0) + NM = P (3.2.4) 2 I N M [IJ] 4 J where the totally antisymmetric component RKL of the torsion is replaced by RKL RKLP = P : A Contracting with the Levi-Civita tensor RKLJ and with metric RK will lead respectively to the following equations
2 k (1 2F 0) J + J = J (3.2.5) A 2J L 1 L (2 F 0) = 0: (3.2.6) A Inserting the equations (3.2.5) and (3.2.6) into (3.2.4), we obtain
Page: 40 INM R K L 1 JR KL T (1 + F 0) + = 0: (3.2.7) I N M NM [IJ] By combining (3.2.5) with (3.2.6) we get the irreducible components of the torsion in terms of the axial fermionic current as
k 1 L = L (3.2.8) 4 F 1 (F 2) + 1 J 0 2 0 2 k (F 0 2) L = L (3.2.9) A 4 F 1 (F 2) + 1 J 0 2 0 2 1 1 de…ned when F 0 (F 0 2) + 2 = 0 from which follows that the function F 2 6 is such that the image Im (F 0) of F 0 is disjoint from the roots r1 and r2 of the 1 1 equation F 0 2 (F 0 2) + 2 = 0 . S P Q 1 TS PQ By contracting (3.2.7) with (1 + F 0) KL we get R K L [RT ]
SPQ 2 1 T (1 + F 0) + = 0 (3.2.10) 2 leading to T SPQ = 0: (3.2.11)
Remark that in the irreducible components of the torsion are well-de…ned if the function F respects the conditions
2 1 (1 + F 0) + = 0 (3.2.12) 2 6
1 1 F 0 (F 0 2) + = 0 (3.2.13) 2 2 6 which are the same conditions as (1.2.3.16) and (1.2.3.17) obtained from the equa- tion (1.2.3.15). The action (3.2.1) given explicitly in terms of the SO(1; 3) torsion-free spin
Page: 41 connection !IJ [e]and the contorsion CIJ reads
1 1 I J KL i I I S(e; !;C; )e = IJKLe e eI 2k 2 ^ ^ 2 ^ D D ZM ZM 1 1 I J K NL 1 I 2 e + IJKLe e e C N C + eI + Fe 2k 2 ^ ^ ^ ^ ZM 1 I J K + e e e CIJ K : (3.2.14) 4 ^ ^ ^ J ZM By inserting (1.2.3.9) and (2.1.2.5) into the action (3.2.14), we obtain
1 1 I J KL i I I S(e; !;C; ) = IJKLe e eI 2k 2 ^ ^ 2 ^ D D ZM ZM 1 1 I I IKL e + F 6 Ie+ 6 I + T TIKL e e 2k 2 A A ZM 1 1 I IJKL N + 24 I T IJ TNKL 2k 4 A ZM 1 1 I I IJK + 24 I + 6 I 4T TJKI 2k 4 A A ZM 3! I + I (3.2.15) 8 A J ZM where the …rst line in (3.2.15) expresses the free-torsion part of the gravity mini- mally coupled to fermions. 1 For regular value F 0 = 2; 2 = 0; the equations of motion (3.2.8), (3.2.9) and 6 (3.2.11) reduce to
k L = L L = 0 T IJK = 0 (3.2.16) 4 J A which make the action (3.2.15) more simple. Indeed, from the solution F 0 = 2 ; the 2 I t F -term of the action (3.2.1) writes F ( ) = I + C where the constant ^ is proportional to the cosmological constant. For gravity without a cosmological constant, Ct = 0; the action (3.2.1) reads
1 1 I J KL 1 I I S(e; !; ) = IJKLe e + I + 2 I 2k 2 ^ ^ ^ ^ ZM i I I eI : (3.2.17) 2 ^ D D ZM Page: 42 In spite of the fact that the nonvanishing torsion is given by the relation (3.2.16) which violates the parity transformation, it does not appear in the action (3.2.18). L IJK 2 I In fact by inserting = 0;T = 0 and F ( ) = 6 I into (3.2.17), which A consists in making this insertion into (3.2.15), we see that the Holst term which contains and the last one which describes the coupling of the torsion with the current vanish while the second and the fourth lines cancel each other out, and then the action (3.2.17) reduces to the e¤ective action
1 1 I J KL i I I Seff (e; !; ) = IJKLe e eI 2k 2 ^ ^ 2 ^ D D ZM ZM (3.1.18) e e e which ise the standard second tetrad form of the Einstein-Cartan theory minimally coupled to fermionic matter without the current-current term and which does not depend both on the torsion and on the Barbero-Immirzi parameter : Note that although the e¤ective action (3.1.18) does not depend on the Barbero- Immirzi parameter ; the Holst term is necessary to get a well-de…ned relation between the trace component of the torsion and the fermionic current (3.2.16). Remark that in the absence of the fermionic matter which reduces the action (3.2.15) to
1 1 I J KL S(e; !;C; ) = IJKLe e [e] 2k 2 ^ ^ ZM 1 1 I I IKL e + F 6 Ie+ 6 I + T TIKL 2k 2 A A ZM 1 1 I IJKL N + 24 I T IJ TNKL 2k 4 A ZM 1 1 I I IJK + 24 I + 6 I 4T TJKI (3.1.19) 2k 4 A A ZM 2 I where if we state the scalar function F ( ) = 2 I ; the second, the third ^ and the fourth lines are cancelled which reduces the action (3.1.19) to
1 1 I J KL Seff (e; !; ) = IJKLe e [e] (3.1.20) 2k 2 ^ ^ ZM which is the second-ordere tetrad form of the Einstein-Cartane theory in spit of the
Page: 43 2 I fact that stating the scalar function F ( ) = 2 I contradicts the condition @F (2) ^ F 0 = 2 = 2 in (1.2.3.11) for having a well-de…ned vanishing trace component @ 6 I : So we can get the second-order Einstein-Cartan action (3.1.18) with unde- termined trace torsion hence the space-time is not four-dimensional Riemannian manifold but a four-dimensional Riemannian-Cartan manifold.
3.3 Conclusion
We have shown in this …nal chapter that we can construct the e¤ective action of the theory of Einstein-Cartan coupled minimally to fermions free from the usual current-current coupling term, and by consequence an e¤ective action which does not depend both on the torsion and on the Barbero-Immirzi parameter : In the …rst case the usual action of Palatini, considered in any dimension d 2 of the space-time manifold , is generalized by adding a non-polynomial term M 1 2 p 2 in the torsion under the form of 2k 2 + c where c is any function M which does not depend on the torsion.R From this generalized Palatini action we have shown that we can construct the e¤ective action of the theory of Einstein- Cartan coupled minimally to fermions which does not depends on the emerging torsion which induces the current-current coupling term, thus the torsion becomes an inert quantity which does not a¤ect the evolution equations. This result shows the equivalence between the theory of general relativity and the theory of Einstein- Cartan coupled to the fermionic matter by the minimal coupling. In the second case we add to the Palatini-Holst action coupled minimally to 1 2 fermions a term of the form 2k F ( ) where for particular value of this added 1 I M scalar function term 2k 2 I , it is shown that we can obtain the standard M R^ e¤ective action of the Einstein-CartanR theory coupled to the fermionic matter with- out the usual current-current interaction and therefore an e¤ective action which is free from the arbitrary Barbero-Immirzi parameter and the torsion, hence the equivalence established between the theory of general relativity and the theory of Einstein-Cartan is maintained even in presence of fermionic …elds.
Page: 44 Chapter 4
Conclusion
We have shown in this dissertation the possibility to obtain an equivalence between the general relativity and the Einstein-Cartan theory in the presence of fermions. The "tetrad-connection" framework of the Einstein-Cartan theory o¤ers the possibility to recast gravity as gauge theory analogous to Yang-Mills theory, which is more familiar to manipulate than the metric formulation of the theory of general relativity in attempt to ward a quantum theory of gravity. In the vacuum, the Einstein-Cartan theory and the general relativity predict the same physics, in the sense that the resulting evolution equations of these two theories are the classical vacuum Einstein equations. Hence, there is an equivalence between the theory of the general relativity and the Einstein-Cartan theory. In presence of fermionic matter …elds which couple to the connection, it is no more the case. The usual tetrad form of the gravitational action induces a non- vanishing torsion which appears under the form of current-current interaction, and which depends either on the arbitrary Barbero-Immirzi parameter coming from the Holst term. Considering a more general action, where in additional to the Holst-modi…ed 1 I Einstein-Cartan theory we have a torsion term of the form of 2k 2 I we M ^ obtain an Einstein-Cartan e¤ective action which does not containsR the current- current coupling term and the Barbero-Immirzi parameter. Therefore the general- ized Einstein-Cartan action is equivalent to the theory of general relativity mini-
45 mally coupled to fermions. Note that this equivalence is not based on a completely vanishing torsion; its irreducible trace component does not vanish. This leads to a space-time manifold with a nonvanishing torsion depending on the Barbero- Immirzi parameter which does not induce physical e¤ects because of its absence in the e¤ective action. 1 I The other interesting result, from the addition of a term such 2k 2 M ^ I to the usual Einstein-Cartan action without fermions, suggests that we can R construct a theory of the Einstein gravity of Riemannian-Cartan manifold : M Where, althought the torsion is arbitrary, it does not a¤ect the evolution of the gravitational …eld described by the Einstein equations. We have also shown that if we add to the Einstein-Cartan action, without considering the Holst term, a non-polynomial term in the torsion of the form 1 2 p 2 2k 2 + c we obtain an e¤ective action equivalent to the theory of the M generalR relativity minimally coupled to the fermionic matter. This equivalence is not based on a vanishing torsion also, its irreducible completely antisymmetric component does not vanish. In spit of the fact that is related to the fermionic current, it is an inert quantity which does not a¤ect the evolution equations. We conclude by, these promising actions deserve further attentions which it might lead, even in the absence of fermionic …elds, to di¤erent algebras resulting from the Hamiltonian approach since the canonical variables are sensitive to the additional terms, as was the case for the Holst term.
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Page: 49
« On The Equivalence Between The Einstein- Cartan Theory and General Relativity in Presence of Fermions »
Meriem Hadjer LAGRAA
Keywords : Einstein-Cartan theory, Torsion, Fermions
Résumé / Abstract From the freedom exhibited by the generalized Palatini’s proposed in Dubois-Violette and Lagraa (2010 Lett. Math. Phys. 91 83-91), we show that we can obtain the standard effective Einstein-Cartan action coupled to the fermions without the usual current-current interaction and therefore an effective action which does not depend either on the torsion or on the Barbero-Immirzi parameter. This establishes the equivalence between the Einstein-Cartan theory and the theory of general relativity minimally coupled to the fermionic matter.