Eddington-Borm-Infeld Action and the Dark Side of General Relativity where Kuv(C)is the(traced)curvature associated to the connection C No metric is needed for this construction what we have in mind is the existence of a connection Cva that existed before the Universe, as a Riemannian manifold, was created. This has to be interpreted with care because without guv there are no causality relations. At this point we have departed from the treatment suggested in [9]. In that ref- nce the field which acquires an spontaneous non-zero expectation value was the metric itself We shall now turn on the metric and consider the whole self consistent system. The metric guv generates its own connection, the Christoffel symbol r(g). Thus, our theory will contain two independent connections. One, called C, is generated spontaneously. The other, called r(g) is driven by the external field gu The action describing the coupled system is notably simple. The pair g, I are of course described by the standard Einstein-Hilbert action either in first or second order form. On the other hand the field C is described by Eddingtons action Now, to make things more interesting we shall couple both connections through the metric. Again, there are no two many couplings one can write. An attractive possibility is the Einstein-Hilbert-Eddington-Borm-Infeld action [2] (R-2A)+V/ a/- This action has several interesting properties. First, note that as g-0 we re- over Eddingtons action for the field C. (We shall not need this limit in what follows. )Second, (10) is a tensor Born-Infeld theory analogous to the scalar det(guy+duoduo)and vector vdet(guv+ Fuv)Born-Infeld theories. Observe that the equations of motion for the whole action are of second order. This is differ- ent from the gravitational Bl action written in [7 where extra terms had to be added in order to eliminate the ghost. Finally, the action(10) contains dynamical proper ties which makes it an attractive candidate for dark matter and dark energy [2]. Both appear in a unified way, just like the Chapligyn formulation [11]. It is also curious to observe that the Chapligyn gas can be derived from a scalar Born-Infeld theory. More specifically, for Friedman models, it follows that the Eddington field behaves like matter for early times and as dark energy for late times: its equation of state w=p/p evolve from w=0 near the big bang to w=-1 for late times One can also analyze the dynamics of objects moving around spherically symmetric sources. The Eddington field in this case yields asymptotically flat rotation curve and thus again provides a candidate for dark matter The reader may wonder what does guy =0 have to do with dark matter and dark energy. Could one had predicted this(suggested)relationship? Our initial motivation to look at guv=0 came from the following analogy. A particle at rest has an energy mc2. The most direct manifestation of this energy is through gravity, and in fact mc2 is a source of curvature. From a Newtonian point of view, the energy of a particle at an energy-density associated to it? In the standard choice for zero point of energy flat space has zero energy. The very definition of energy in general relativity require
Eddington–Born–Infeld Action and the Dark Side of General Relativity 31 where Kμν(C) is the (traced) curvature associated to the connectionC. No metric is needed for this construction. What we have in mind is the existence of a connection Cμ να that existed before the Universe, as a Riemannian manifold, was created. This has to be interpreted with care because without gμν there are no causality relations. At this point we have departed from the treatment suggested in [9]. In that reference the field which acquires an spontaneous non-zero expectation value was the metric itself. We shall now turn on the metric and consider the whole self consistent system. The metric gμν generates its own connection, the Christoffel symbol Γ (g). Thus, our theory will contain two independent connections. One, called C, is generated spontaneously. The other, called Γ (g) is driven by the external field gμν. The action describing the coupled system is notably simple. The pair g,Γ are of course described by the standard Einstein–Hilbert action either in first or second order form. On the other hand the field C is described by Eddington’s action. Now, to make things more interesting we shall couple both connections through the metric. Again, there are no two many couplings one can write. An attractive possibility is the Einstein–Hilbert–Eddington–Born–Infeld action [2], I = √g(R−2Λ) + 2 αl2 |l2Kμν −gμν|. (10) This action has several interesting properties. First, note that as g → 0 we recover Eddington’s action for the field C. (We shall not need this limit in what follows.) Second, (10) is a tensor Born–Infeld theory analogous to the scalar det(gμν +∂μφ∂νφ) and vector det(gμν +Fμν) Born–Infeld theories. Observe that the equations of motion for the whole action are of second order. This is different from the gravitational BI action written in [7] where extra terms had to be added in order to eliminate the ghost. Finally, the action (10) contains dynamical properties which makes it an attractive candidate for dark matter and dark energy [2]. Both appear in a unified way, just like the Chapligyn formulation [11]. It is also curious to observe that the Chapligyn gas can be derived from a scalar Born–Infeld theory. More specifically, for Friedman models, it follows that the Eddington field C behaves like matter for early times and as dark energy for late times: its equation of state w = p/ρ evolve from w = 0 near the big bang to w = −1 for late times. One can also analyze the dynamics of objects moving around spherically symmetric sources. The Eddington field in this case yields asymptotically flat rotation curves and thus again provides a candidate for dark matter. The reader may wonder what does gμν = 0 have to do with dark matter and dark energy. Could one had predicted this (suggested) relationship? Our initial motivation to look at gμν = 0 came from the following analogy. A particle at rest has an energy mc2. The most direct manifestation of this energy is through gravity, and in fact mc2 is a source of curvature. From a Newtonian point of view, the energy of a particle at rest is zero. Can we ask the same question in general relativity? Could flat space have an energy-density associated to it? In the standard choice for zero point of energy, flat space has zero energy. The very definition of energy in general relativity requires
M. Bardos knowledge of boundary conditions and certainly guv =0 falls outside all examples However at the level of energy density, namely, the Einstein tensor Guv(g),one may wonder about its value at guv=0. Interestingly this tensor depends only on g"dg and thus the limit can be defined in a way that Guv(O)becomes finite. By a mixture f Bianchi identities and some reasonable assumptions its value can be computed ind yields a contribution to Einstein equations similar to those expected from dark matter 3]. The ideas presented in this contribution are highly speculative and need for- malization. The analogy with spin systems is the most challenging and difficult problem. Other applications like fluctuations and the CMB spectra are presently under analysis [4]. The action(10)has several interesting formal properties under duality transformations. This theory can be written as a bigravity theory, which have been under great scrutiny in the past and also recently [1]. a detailed analysis will be reported in [5]. Rotation curves for several galaxies has been analyzed in [12] with interesting results Acknowledgments Some of the ideas presented here have been developed in collaboration with A Gomberoff and D. Rodriguez. I would like to thank them for their permission to include here me unpublished material. I would also like to thank s. Theisen for many useful conversations and for bringing to my attention[8]. P. Ferreira and C. Skordis have been crucial to keep this idea References N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Ann. Phys. 305, 96(2003)[arXiv: hep-th 2. M. Barados, Class. Quant. Grav. 24, 5911(2007)[arXiv: hep-th/07011691 3. M. Banados, Phys. Rev. D77, 123534(2008)arXiv: 0801.4103 [hep-th]. 4. M. Barados, P. Ferreira and C. Skordis, ar Xiv: 0811 1272[astro-ph](unpublished 5. M. Banados, A Gomberoff and D. Rodrigues(unpublished 6. N. Boulanger, S. Cnockaert and M. Henneaux, JHEP 0306, 060(2003)[arXiv: hep-th 0306023 7.S. Deser and G. w. Gibbons, Class. Quant. Grav. 15, L35(1998)arXiv: hep-th/9803049 8. E.S. Fradkin and A. A. Tseytlin, Ann. Phys. 162, 31(1985 9. S B. Giddings, Phys. Lett. B 268, 17(1991) 10. G. T. Horowitz, Class. Quant. Grav. 8, 587(1991). 11. A. Y.Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, 265(2001) [arXiv: grqc0103004 egger and N Rojas(unpublished) 13. E. Witten, Nucl. Phys. B 311, 46(1988)
32 M. Ba˜nados knowledge of boundary conditions and certainly gμν = 0 falls outside all examples. However at the level of energy density, namely, the Einstein tensor Gμν(g), one may wonder about its value at gμν = 0. Interestingly this tensor depends only on g−1∂g and thus the limit can be defined in a way that Gμν(0) becomes finite. By a mixture of Bianchi identities and some reasonable assumptions its value can be computed and yields a contribution to Einstein equations similar to those expected from dark matter [3]. The ideas presented in this contribution are highly speculative and need formalization. The analogy with spin systems is the most challenging and difficult problem. Other applications like fluctuations and the CMB spectra are presently under analysis [4]. The action (10) has several interesting formal properties under duality transformations. This theory can be written as a bigravity theory, which have been under great scrutiny in the past and also recently [1]. A detailed analysis will be reported in [5]. Rotation curves for several galaxies has been analyzed in [12], with interesting results. Acknowledgments Some of the ideas presented here have been developed in collaboration with A. Gomberoff and D. Rodriguez. I would like to thank them for their permission to include here some unpublished material. I would also like to thank S. Theisen for many useful conversations and for bringing to my attention [8]. P. Ferreira and C. Skordis have been crucial to keep this idea alive. References 1. N. Arkani-Hamed, H. Georgi and M. D. Schwartz, Ann. Phys. 305, 96 (2003) [arXiv:hep-th/ 0210184]. 2. M. Ba˜nados, Class. Quant. Grav. 24, 5911 (2007) [arXiv:hep-th/0701169]. 3. M. Ba˜nados, Phys. Rev. D77, 123534 (2008) arXiv:0801.4103 [hep-th]. 4. M. Ba˜nados, P. Ferreira and C. Skordis, arXiv:0811.1272[astro-ph] (unpublished). 5. M. Ba˜nados, A. Gomberoff and D. Rodrigues (unpublished). 6. N. Boulanger, S. Cnockaert and M. Henneaux, JHEP 0306, 060 (2003) [arXiv:hep-th/ 0306023]. 7. S. Deser and G. W. Gibbons, Class. Quant. Grav. 15, L35 (1998) [arXiv:hep-th/9803049]. 8. E. S. Fradkin and A. A. Tseytlin, Ann. Phys. 162, 31 (1985). 9. S. B. Giddings, Phys. Lett. B 268, 17 (1991). 10. G. T. Horowitz, Class. Quant. Grav. 8, 587 (1991). 11. A. Y. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. B 511, 265 (2001) [arXiv: gr-qc/0103004]. 12. A. Reisenegger and N. Rojas (unpublished). 13. E. Witten, Nucl. Phys. B 311, 46 (1988)
Light-Cone Field Theory, Maximal Supersymmetric Theories and e 7(7) in Light-Cone Superspace Lars brink Abstract In this lecture I describe the light-cone formulation of quantum field the ories especially the maximally supersymmetric ones. This is a formalism in which we keep only the physical degrees of freedom for both bosons and fermions. I show how N=4 Yang-Mills Theory and N=8 supergravity come out very naturally and that they look very much alike. I finally show how to implement the E7( symmetry for the supergravity theory. The new feature in this formulation is that all fields of the supermultiplet including the graviton transform under e- 1 Introduction When we study supersymmetric theories, the maximally supersymmetric ones (r=8,d=4)supergravity [1, 2]and(r=4, d=4) Yang-Mills Theory [3, 4] or their 1 1-dimensional or ten-dimensional versions always show up. The =I supergravity in 1l dimensions [5], is the largest supersymmetric local field theory with maximum helicity two(on reduction to d= 4). This theory has gained re- newed prominence since its recognition as the infrared limit of M-Theory [6, 7] its actual structure remains a mystery. We must therefore glean all we can fron.lds, Although M-Theory casts well-defined shadows on lower-dimensional manifold y=l supergravity theory or its dimensionally reduced versions before tackling M-Theory. =l supergravity is ultraviolet divergent in d= 1l but this divergence is presumably tamed by M-Theory and the hope is that an understanding of this di- vergent structure, will give us a window into the workings of M-Theory. Similarly the w=l Yang-Mills Theory in ten dimensions [3, 4], which is the low-energy limit of the open string theory in ten dimensions has been shown to play an important Brink Chalmers Uniy of Technology, Goteborg, Sweden e-mail: lars brink@chalmers. se M. Henneaux, J Zanelli(eds ) Quantum Mechanics of Fundamental Systems: The Quest 33 for Beauty and Simplicity. DOI 10.1007/978-0-387-87499-9-5, C Springer Science+ Business Media LLC 2009
Light-Cone Field Theory, Maximal Supersymmetric Theories and E7(7) in Light-Cone Superspace Lars Brink Abstract In this lecture I describe the light-cone formulation of quantum field theories especially the maximally supersymmetric ones. This is a formalism in which we keep only the physical degrees of freedom for both bosons and fermions. I show how N = 4 Yang–Mills Theory and N = 8 supergravity come out very naturally and that they look very much alike. I finally show how to implement the E7(7) symmetry for the supergravity theory. The new feature in this formulation is that all fields of the supermultiplet including the graviton transform under E7(7). 1 Introduction When we study supersymmetric theories, the maximally supersymmetric ones, (N = 8, d = 4) supergravity [1, 2] and (N = 4, d = 4) Yang–Mills Theory [3, 4] or their 11-dimensional or ten-dimensional versions always show up. The N = 1 supergravity in 11 dimensions [5], is the largest supersymmetric local field theory with maximum helicity two (on reduction to d = 4). This theory has gained renewed prominence since its recognition as the infrared limit of M-Theory [6, 7]. Although M-Theory casts well-defined shadows on lower-dimensional manifolds, its actual structure remains a mystery. We must therefore glean all we can from the N = 1 supergravity theory or its dimensionally reduced versions before tackling M-Theory. N = 1 supergravity is ultraviolet divergent in d = 11 but this divergence is presumably tamed by M-Theory and the hope is that an understanding of this divergent structure, will give us a window into the workings of M-Theory. Similarly the N = 1 Yang–Mills Theory in ten dimensions [3, 4], which is the low-energy limit of the open string theory in ten dimensions has been shown to play an important L. Brink Chalmers University of Technology, G¨oteborg, Sweden e-mail: lars.brink@chalmers.se M. Henneaux, J. Zanelli (eds.), Quantum Mechanics of Fundamental Systems: The Quest 33 for Beauty and Simplicity, DOI 10.1007/978-0-387-87499-9 5, c Springer Science+Business Media LLC 2009
L. Brink role in the AdS/CFT duality [8]. The four-dimensional version w =4 Yang-Mills Theory is also very special since it is free of ultraviolet divergences [9, 10] In the normal covariant treatments of these two theories they look quite different, one being a reparametrization invariant gravity theory, while the other is a Yang- Mills gauge theory. However, in the light-cone formulations, the so-called LC2 formulations where all auxiliary degrees of freedom have been eliminated, [11, 12] the two superfields describing the field content of the two theories are particularly simple and and very much alike. Indeed these superfields can be regarded as mas ter fields for a series of theories. Since they are natural partners in string theory this imilarity must be quite important and much of my research in recent years has been to use this similarity and to try to use it to learn more about these theories and the underlying string theory. They are, of course, very well studied during a long time ut they have consistently shown themselves to be more interesting than what meets Writing the two theories in the LC2 formulation has a price. We loose a lot of nformation from the geometry and the only guideline will be the non-linearly real ized superPoincare algebra. However, it is important to view these very importan heories from different angles and for certain question this formulation is the most In this talk I will start from the beginning to build up light-cone field theories and then go over to the supersymmetric ones. I will start the analysis in four dimensions of space-time and then them to higher dimensions keeping the specific form of the superfields. Finally I will show how the E7() symmetry is implemented 2 Light-Frame formulation of Field theories In his famous paper of 1949 Dirac [13] argued that for a relativistically invariant theory any direction within the light-cone can be the evolution parameter, the"time In particular we can use one of the light-cone directions. For this discussion we will usex+=1(+x)as the time. The coordinates and the derivatives that we will use will then :(x+ix2);d=一=(O1-i02) );d=一(+i2)
34 L. Brink role in the AdS/CFT duality [8]. The four-dimensional version N = 4 Yang–Mills Theory is also very special since it is free of ultraviolet divergences [9, 10]. In the normal covariant treatments of these two theories they look quite different, one being a reparametrization invariant gravity theory, while the other is a Yang– Mills gauge theory. However, in the light-cone formulations, the so-called LC2 formulations where all auxiliary degrees of freedom have been eliminated, [11,12], the two superfields describing the field content of the two theories are particularly simple and and very much alike. Indeed these superfields can be regarded as master fields for a series of theories. Since they are natural partners in string theory this similarity must be quite important and much of my research in recent years has been to use this similarity and to try to use it to learn more about these theories and the underlying string theory. They are, of course, very well studied during a long time but they have consistently shown themselves to be more interesting than what meets the eye. Writing the two theories in the LC2 formulation has a price. We loose a lot of information from the geometry and the only guideline will be the non-linearly realized superPoincar´e algebra. However, it is important to view these very important theories from different angles and for certain question this formulation is the most adequate one. In this talk I will start from the beginning to build up light-cone field theories and then go over to the supersymmetric ones. I will start the analysis in four dimensions of space–time and then ‘oxidize’ them to higher dimensions keeping the specific form of the superfields. Finally I will show how the E7(7) symmetry is implemented in this formulation. 2 Light-Frame Formulation of Field Theories In his famous paper of 1949 Dirac [13] argued that for a relativistically invariant theory any direction within the light-cone can be the evolution parameter, the “time”. In particular we can use one of the light-cone directions. For this discussion we will use x+ = √ 1 2 (x0 +x3) as the time. The coordinates and the derivatives that we will use will then be x± = 1 √2 (x0 ±x3 ); ∂± = 1 √ 2 (−∂0 ±∂3); (1) x = 1 √2 (x1 +ix2); ¯ ∂ = 1 √ 2 (∂1 −i∂2); (2) x¯ = 1 √2 (x1 −ix2); ∂ = 1 √ 2 (∂1 +i∂2), (3) so that ∂+ x− = ∂− x+ = −1; ¯ ∂ x = ∂ x¯ = +1. (4)
Light-Cone Field Theory The derivatives are of course. related to the momenta through the usual formula P=-id and we use the light-cone decomposition also for p. We will only consider massless theories so we solve the condition p-=0. We then find en The generator p- is really the Hamiltonian conjugated to the light cone time x+ we see that the translation generators of the Poincare algebra are written with just three operators. We will use Dirac's vocabulary that generators that involve the time"are called dynamical (or Hamiltonians) and the others kinematical. Usin light-cone notation and the complex one from above for the transverse directions, the most general form of the generators of the full Poincare algebra at xt=0 is ther given by the four momenta i5,(6 the kinematical transverse space rotation the other kinematical generators +=ixo+,+=i+ (8) as well as the dynamical boosts d+ir d+ia. There is one degree of freedom in the algebra, namely the parameter A which the helicity. At this stage it is arbitrary and checking the corresponding spin one finds, of course, that it is a Hence the algebra covers all possible free field theo- ries. We can let the generators act on a complex field o(x) with helicity 2, with complex conjugate having the opposite helicity. This is the"first-quantized "version We can also consider the fields as operators having the commutation relation a+d(x),p(x)=-6(x-x), (12) where hence the momentum field conjugate to o is a+o
Light-Cone Field Theory 35 The derivatives are, of course, related to the momenta through the usual formula pμ = −i∂μ and we use the light-cone decomposition also for pμ . We will only consider massless theories so we solve the condition p2 = 0. We then find p− = pp¯ p+ . (5) The generator p− is really the Hamiltonian conjugated to the light cone time x+ and we see that the translation generators of the Poincare algebra are written with just three operators. We will use Dirac’s vocabulary that generators that involve the “time” are called dynamical (or Hamiltonians) and the others kinematical. Using light-cone notation and the complex one from above for the transverse directions, the most general form of the generators of the full Poincar´e algebra at x+ = 0 is then given by the four momenta p− = −i ∂ ¯ ∂ ∂+ , p+ = −i∂+, p = −i∂, p¯ = −i ¯ ∂, (6) the kinematical transverse space rotation j = j 12 = x ¯ ∂ −x¯∂ +λ, (7) the other kinematical generators j + = ix∂+, ¯j + = ix¯∂+, (8) and j +− = ix− ∂+, (9) as well as the dynamical boosts j − = ix ∂ ¯ ∂ ∂+ − ix− ∂ +iλ ∂ ∂+ , (10) ¯j − = ix¯ ∂ ¯ ∂ ∂+ − ix− ¯ ∂ +iλ ¯ ∂ ∂+ . (11) There is one degree of freedom in the algebra, namely the parameter λ which is the helicity. At this stage it is arbitrary and checking the corresponding spin one finds, of course, that it is |λ|. Hence the algebra covers all possible free field theories. We can let the generators act on a complex field φ(x) with helicity λ, with its complex conjugate having the opposite helicity. This is the “first-quantized” version. We can also consider the fields as operators having the commutation relation. [∂+φ¯(x),φ(x )] = − i 2 δ(x−x ), (12) where hence the momentum field conjugate to φ is ∂+φ¯