Exotic Gauge Theories of Spin-2 Fields
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1 HESTENES' TETRAD and SPIN CONNECTIONS Frank
HESTENES’ TETRAD AND SPIN CONNECTIONS Frank Reifler and Randall Morris Lockheed Martin Corporation MS2 (137-205) 199 Borton Landing Road Moorestown, New Jersey 08057 ABSTRACT Defining a spin connection is necessary for formulating Dirac’s bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes’ tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field Ψ is mapped to a set of × K ρ SL(2,R) U(1) gauge potentials Fα and a complex scalar field . This result was previously proved for a Minkowski space-time using Fierz identities. As an application we derive several different non-Riemannian spin connections found in the literature directly from an arbitrary K linear connection acting on the tensor fields (Fα ,ρ) . We also derive spin connections for which Dirac’s bispinor equation is form invariant. Previous work has not considered form invariance of the Dirac equation as a criterion for defining a general spin connection. 1 I. INTRODUCTION Defining a spin connection to replace the partial derivatives in Dirac’s bispinor equation in a Minkowski space-time, is necessary for the formulation of Dirac’s bispinor equation in a curved space-time. All the spin connections acting on bispinors found in the literature first introduce a local orthonormal tetrad field on the space-time manifold and then require that the Dirac Lagrangian be invariant under local change of tetrad [1] – [8]. -
Modified Gravity with Vector Distortion and Cosmological Applications
universe Article Modified Gravity with Vector Distortion and Cosmological Applications Jose Beltrán Jiménez 1,∗ and Tomi S. Koivisto 2 1 Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille 13288, France 2 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden; [email protected] * Correspondence: [email protected] Academic Editors: Mariusz P. D ˛abrowski, Manuel Krämer and Vincenzo Salzano Received: 7 April 2017; Accepted: 19 May 2017; Published: 24 May 2017 Abstract: We briefly review the basics of Weyl geometry and its natural extension by a general linear ”distortion” of the metric connection by a vector field. A special class of the connections has torsion but retains the Weyl’s semi-metricity condition. We present ghost-free gravitational theories in this geometrical setup and highlight their possible cosmological applications, such as new self-tuning solutions and new bouncing solutions found in the quadratic-curvature theories. The vector distortion can mimic the cosmological effects of dark matter. Keywords: modified gravity; Weyl geometry; cosmology 1. Introduction General Relativity (GR) provides a beautiful description of the gravitational interaction within the framework of Riemannian geometry, as the curvature of the spacetime. A crucial assumption in the traditional description of GR is the identification of the connection with the Christoffel symbol of the metric, which therefore also determines the affine structure of the spacetime. However, in geometry, this structure is a priori independent, and much of its richness is lost by stipulating the connection as the Christoffel symbol of the metric. In physics, on the other hand, one usually regards connections as fundamental dynamical objects (e.g., in Yang–Mills theories), so that depriving it of independent dynamics in the case of gravity can jeopardise its gauge interpretation in the usual manner. -
Arxiv:1907.02341V1 [Gr-Qc]
Different types of torsion and their effect on the dynamics of fields Subhasish Chakrabarty1, ∗ and Amitabha Lahiri1, † 1S. N. Bose National Centre for Basic Sciences Block - JD, Sector - III, Salt Lake, Kolkata - 700106 One of the formalisms that introduces torsion conveniently in gravity is the vierbein-Einstein- Palatini (VEP) formalism. The independent variables are the vierbein (tetrads) and the components of the spin connection. The latter can be eliminated in favor of the tetrads using the equations of motion in the absence of fermions; otherwise there is an effect of torsion on the dynamics of fields. We find that the conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. We also find that for dynamically generated torsion the spin connection does not have well defined conformal properties. In particular, it affects fermions and the non-minimally coupled conformal scalar field. Keywords: Torsion, Conformal transformation, Palatini formulation, Conformal scalar, Fermion arXiv:1907.02341v1 [gr-qc] 4 Jul 2019 ∗ [email protected] † [email protected] 2 I. INTRODUCTION Conventionally, General Relativity (GR) is formulated purely from a metric point of view, in which the connection coefficients are given by the Christoffel symbols and torsion is set to zero a priori. Nevertheless, it is always interesting to consider a more general theory with non-zero torsion. The first attempt to formulate a theory of gravity that included torsion was made by Cartan [1]. -
Conformally Coupled General Relativity
universe Article Conformally Coupled General Relativity Andrej Arbuzov 1,* and Boris Latosh 2 ID 1 Bogoliubov Laboratory for Theoretical Physics, JINR, Dubna 141980, Russia 2 Dubna State University, Department of Fundamental Problems of Microworld Physics, Universitetskaya str. 19, Dubna 141982, Russia; [email protected] * Correspondence: [email protected] Received: 28 December 2017; Accepted: 7 February 2018; Published: 14 February 2018 Abstract: The gravity model developed in the series of papers (Arbuzov et al. 2009; 2010), (Pervushin et al. 2012) is revisited. The model is based on the Ogievetsky theorem, which specifies the structure of the general coordinate transformation group. The theorem is implemented in the context of the Noether theorem with the use of the nonlinear representation technique. The canonical quantization is performed with the use of reparametrization-invariant time and Arnowitt– Deser–Misner foliation techniques. Basic quantum features of the models are discussed. Mistakes appearing in the previous papers are corrected. Keywords: models of quantum gravity; spacetime symmetries; higher spin symmetry 1. Introduction General relativity forms our understanding of spacetime. It is verified by the Solar System and cosmological tests [1,2]. The recent discovery of gravitational waves provided further evidence supporting the theory’s viability in the classical regime [3–6]. Despite these successes, there are reasons to believe that general relativity is unable to provide an adequate description of gravitational phenomena in the high energy regime and should be either modified or replaced by a new theory of gravity [7–11]. One of the main issues is the phenomenon of inflation. It appears that an inflationary phase of expansion is necessary for a self-consistent cosmological model [12–14]. -
Ads₄/CFT₃ and Quantum Gravity
AdS/CFT and quantum gravity Ioannis Lavdas To cite this version: Ioannis Lavdas. AdS/CFT and quantum gravity. Mathematical Physics [math-ph]. Université Paris sciences et lettres, 2019. English. NNT : 2019PSLEE041. tel-02966558 HAL Id: tel-02966558 https://tel.archives-ouvertes.fr/tel-02966558 Submitted on 14 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prepar´ ee´ a` l’Ecole´ Normale Superieure´ AdS4/CF T3 and Quantum Gravity Soutenue par Composition du jury : Ioannis Lavdas Costas BACHAS Le 03 octobre 2019 Ecole´ Normale Superieure Directeur de These Guillaume BOSSARD Ecole´ Polytechnique Membre du Jury o Ecole´ doctorale n 564 Elias KIRITSIS Universite´ Paris-Diderot et Universite´ de Rapporteur Physique en ˆIle-de-France Crete´ Michela PETRINI Sorbonne Universite´ President´ du Jury Nicholas WARNER University of Southern California Membre du Jury Specialit´ e´ Alberto ZAFFARONI Physique Theorique´ Universita´ Milano-Bicocca Rapporteur Contents Introduction 1 I 3d N = 4 Superconformal Theories and type IIB Supergravity Duals6 1 3d N = 4 Superconformal Theories7 1.1 N = 4 supersymmetric gauge theories in three dimensions..............7 1.2 Linear quivers and their Brane Realizations...................... 10 1.3 Moduli Space and Symmetries............................ -
Brst Formalism for Massive Vectorial Bosonic Fields
Dedicated to Professor Oliviu Gherman’s 80th Anniversary LAGRANGEAN sp(3) BRST FORMALISM FOR MASSIVE VECTORIAL BOSONIC FIELDS R. CONSTANTINESCU, C. IONESCU Dept. of Theoretical Physics, University of Craiova, 13 A. I. Cuza Str., Craiova, RO-200585, Romania Received August 23, 2010 In the ’90s professor Gherman promoted many PhD theses in the field of the BRST quantization, a very modern topic at that time, and contributed to the creation of a strong research group in this field at his university. This paper summarizes one of the aspects developed by this group, the extended formalism for the Lagrangian BRST quantization. The procedure will be illustrated on a generalized version of the massive bosonic field with spin 1, model known as Proca field, named from another famous Romanian physicist. We shall start from the Hamiltonian formalism and we shall end with the Lagrangian quantum master action. This way, from Hamilton to Lagrange, has a double motivation: the Proca model is not covariant and the gauge fixing procedure in the BRST Lagrangian context is simpler following this way, as far as the ghost spectrum. Key words: extended BRST symmetry, Proca model, gauge fixing procedure. PACS: 11.10.Ef 1. INTRODUCTION The BRST quantization method is one of the most powerful tools for describing the quantization of constrained dynamical systems and, in particular, the gauge field theories. The BRST symmetry is expressed either as a differential operator s, or in a canonical form, by the antibracket ( ; ) in the Lagrangean (Batalin-Vilkovisky) case [1] and by the extended Poisson bracket [ ; ] in the Hamiltonian (Batalin-Fradkin- Vilkovisky) formulation [2]: s∗ = [∗;Ω] = (∗;S): (1) The BRST charge Ω and the BRST generator S are both defined in extended spaces generated by the real and by the ghost-type variables. -
Cosmologies of Extended Massive Gravity
Cosmologies of extended massive gravity Kurt Hinterbichler, James Stokes, and Mark Trodden Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Dated: June 15, 2021) We study the background cosmology of two extensions of dRGT massive gravity. The first is variable mass massive gravity, where the fixed graviton mass of dRGT is replaced by the expectation value of a scalar field. We ask whether self-inflation can be driven by the self-accelerated branch of this theory, and we find that, while such solutions can exist for a short period, they cannot be sustained for a cosmologically useful time. Furthermore, we demonstrate that there generally exist future curvature singularities of the “big brake” form in cosmological solutions to these theories. The second extension is the covariant coupling of galileons to massive gravity. We find that, as in pure dRGT gravity, flat FRW solutions do not exist. Open FRW solutions do exist – they consist of a branch of self-accelerating solutions that are identical to those of dRGT, and a new second branch of solutions which do not appear in dRGT. INTRODUCTION AND OUTLINE be sustained for a cosmologically relevant length of time. In addition, we show that non-inflationary cosmological An interacting theory of a massive graviton, free of solutions to this theory may exhibit future curvature sin- the Boulware-Deser mode [1], has recently been discov- gularities of the “big brake” type. ered [2, 3] (the dRGT theory, see [4] for a review), al- In the second half of this letter (which can be read lowing for the possibility of addressing questions of in- independently from the first), we consider the covariant terest in cosmology. -
Arxiv:1511.03101V2
PI/UAN-2015-589FT Generalized Proca action for an Abelian vector field Erwan Allys∗ and Patrick Peter† Institut d’Astrophysique de Paris, UMR 7095, 98 bis bd Arago, 75014 Paris, France UPMC Université Paris 6 et CNRS, Sorbonne Universités Yeinzon Rodríguez‡ Centro de Investigaciones en Ciencias Básicas y Aplicadas, Universidad Antonio Nariño, Cra 3 Este # 47A-15, Bogotá D.C. 110231, Colombia Escuela de Física, Universidad Industrial de Santander, Ciudad Universitaria, Bucaramanga 680002, Colombia and Simons Associate at The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34151, Trieste, Italy (Dated: January 27, 2016) We revisit the most general theory for a massive vector field with derivative self-interactions, ex- tending previous works on the subject to account for terms having trivial total derivative interactions for the longitudinal mode. In the flat spacetime (Minkowski) case, we obtain all the possible terms containing products of up to five first-order derivatives of the vector field, and provide a conjecture about higher-order terms. Rendering the metric dynamical, we covariantize the results and add all possible terms implying curvature. I. INTRODUCTION Modifying general relativity in order to account for otherwise unresolved issues like, e.g., the cosmological constant or even dark matter, has become common practice lately. One way of doing so, following the scalar-tensor proposal of the 1930s, is the so-called galileon method, which provides a means to write down the most general theory allowing for Ostrogradski-instability-free [1, 2] second-order equations of motion [3–7], which happens to be equivalent [8], in the single-field case1 and in four dimensions, to that proposed much earlier by Horndeski [9]. -
3. Introducing Riemannian Geometry
3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4). -
A General Theory of Linear Cosmological Perturbations: Stability Conditions, the Quasistatic Limit and Dynamics
Prepared for submission to JCAP A general theory of linear cosmological perturbations: stability conditions, the quasistatic limit and dynamics. Macarena Lagos,a Emilio Bellini,b Johannes Noller,c Pedro G. Ferreira,b Tessa Bakerb aKavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637, USA bAstrophysics, University of Oxford, DWB, Keble road, Oxford OX1 3RH, UK cInstitute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. We analyse cosmological perturbations around a homogeneous and isotropic background for scalar-tensor, vector-tensor and bimetric theories of gravity. Building on previous results, we propose a unified view of the effective parameters of all these theories. Based on this structure, we explore the viable space of parameters for each family of models by imposing the absence of ghosts and gradient instabilities. We then focus on the quasistatic regime and confirm that all these theories can be approximated by the phenomenological two-parameter model described by an effective Newton’s constant and the gravitational slip. Within the quasistatic regime we pinpoint signatures which can distinguish between the broad classes of models (scalar-tensor, vector-tensor or bimetric). Finally, we present the equations of motion for our unified approach in such a way that they can be implemented in Einstein-Boltzmann -
Weyl's Spin Connection
THE SPIN CONNECTION IN WEYL SPACE c William O. Straub, PhD Pasadena, California “The use of general connections means asking for trouble.” —Abraham Pais In addition to his seminal 1929 exposition on quantum mechanical gauge invariance1, Hermann Weyl demonstrated how the concept of a spinor (essentially a flat-space two-component quantity with non-tensor- like transformation properties) could be carried over to the curved space of general relativity. Prior to Weyl’s paper, spinors were recognized primarily as mathematical objects that transformed in the space of SU (2), but in 1928 Dirac showed that spinors were fundamental to the quantum mechanical description of spin—1/2 particles (electrons). However, the spacetime stage that Dirac’s spinors operated in was still Lorentzian. Because spinors are neither scalars nor vectors, at that time it was unclear how spinors behaved in curved spaces. Weyl’s paper provided a means for this description using tetrads (vierbeins) as the necessary link between Lorentzian space and curved Riemannian space. Weyl’selucidation of spinor behavior in curved space and his development of the so-called spin connection a ab ! band the associated spin vector ! = !ab was noteworthy, but his primary purpose was to demonstrate the profound connection between quantum mechanical gauge invariance and the electromagnetic field. Weyl’s 1929 paper served to complete his earlier (1918) theory2 in which Weyl attempted to derive electrodynamics from the geometrical structure of a generalized Riemannian manifold via a scale-invariant transformation of the metric tensor. This attempt failed, but the manifold he discovered (known as Weyl space), is still a subject of interest in theoretical physics. -
Milgrom's Law and Lambda's Shadow: How Massive Gravity
Journal of the Korean Astronomical Society preprint - no DOI assigned 00: 1 ∼ 4, 2015 June pISSN: 1225-4614 · eISSN: 2288-890X The Korean Astronomical Society (2015) http://jkas.kas.org MILGROM'S LAW AND Λ'S SHADOW: HOW MASSIVE GRAVITY CONNECTS GALACTIC AND COSMIC DYNAMICS Sascha Trippe Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea; [email protected] Received March 11, 2015; accepted June 2, 2015 Abstract: Massive gravity provides a natural solution for the dark energy problem of cosmology and is also a candidate for resolving the dark matter problem. I demonstrate that, assuming reasonable scaling relations, massive gravity can provide for Milgrom’s law of gravity (or “modified Newtonian dynamics”) which is known to remove the need for particle dark matter from galactic dynamics. Milgrom’s law comes with a characteristic acceleration, Milgrom’s constant, which is observationally constrained to a0 1.1 10−10 ms−2. In the derivation presented here, this constant arises naturally from the cosmologically≈ × required mass of gravitons like a0 c√Λ cH0√3ΩΛ, with Λ, H0, and ΩΛ being the cosmological constant, the Hubble constant, and the∝ third cosmological∝ parameter, respectively. My derivation suggests that massive gravity could be the mechanism behind both, dark matter and dark energy. Key words: gravitation — cosmology — dark matter — dark energy 1. INTRODUCTION On smaller scales, the dynamics of galaxies is in ex- cellent agreement with a modification of Newtonian dy- Modern standard cosmology suffers from two criti- namics (the MOND paradigm) in the limit of small ac- cal issues: the dark matter problem and the dark celeration (or gravitational field strength) g, expressed energy problem.