DOCSLIB.ORG

Geometry, Topology and Physics II

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry, Topology and Physics II Lecture notes Geometry, Topology and Physics II 136.004 summer term 2009 Maximilian Kreuzer Institut f¨ur Theoretische Physik, TU–Wien +43 – 1 – 58801-13621 Wiedner Hauptstraße 8–10, A-1040 Wien [email protected] http://hep.itp.tuwien.ac.at/˜kreuzer/inc/gtp2.pdf Contents 1 Vector bundles 1 1.1 Vielbein and Lorentz connection . ........ 1 1.2 Fiberbundles .................................... ... 4 1.3 LiegroupsandLiealgebras . ...... 9 1.4 Connectionsonfiberbundles . ...... 11 2 Lie algebras and representations 14 3 Complex manifolds 23 3.1 Vector bundles on complex manifolds . ......... 23 Chapter 1 Vector bundles 1.1 Vielbein and Lorentz connection So far we described the tangent space with local coordinates (xm,vn), where the components n m v of a vector at a point with coordinates x refer to the holonomous basis ∂ m of T . As a x M consequence the transition functions for vn on the overlap of two charts is given by the Jacobi ∂ym matrix ∂xl , which is a general linear transformation. But the general linear group only has tensor representations and therefor we cannot describe spinors in this formalism. This problem a m a can be solved by introducing an orthonormal basis e = dx em for the cotangent space and m the dual basis Ea = Ea ∂m with g(Ea,Eb)= ηab for the tangent space, so that m n a n n a b gmnEa Eb = ηab, em Ea = δm, gmn = ηabem en . (1.1) We will denote Lorentz indices by letters from the beginning of the alphabet and world indices by k,l,m,...; for a Riemannian metric we have ηab = δab and in the pseudo-Riemannian case a 1 the diagonal matrix ηab has entries 1. In physics em is called vielbein, and we can use it ±m a m to write down a Dirac operator ∂/ = γ ∂mγ Ea ∂m and an action e ψ(iD/ m)ψ for spinor a b − ab fields in curved space in terms of the usual γ-matrices satisfying γR,γ = 2η . The volume a { } element can be written as e = det(em ) = g . The x-dependent γ-matrices in curved space m | a m | | | a are linear combinations γ = γ Ea of constantp representation matrices γ of the Clifford m algebra, where Ea is determined only up to a local (i.e. x-dependent) Lorentz transformation on its Lorentz index a. In order to define the covariant derivative Dψ of spinor fields we introduce the Lorentz b m b connection (or spin connection) ωa = dx ωma for objects, like spinors and tensors with Lorentz indices, that transform in some representation under local Lorentz transformations. Metricity of 1 a m a In mathematics e = dx em is called soldering form since the vielbein provides a soldering of the (principal bundle of the) cotangent bundle with an orthonormal frame bunde (see below). Geometry / M.Kreuzer — 1 — version June 2, 2009 the connection tranlates into antisymmetry in the last two indices ωab +ωba = 0, which preserves orthonormality of tangent vectors under parallel transport. If we want to use holonomous and orthogonal bases (i.e. tensors with both types of indices) at the same time we may do so by defining the total covariant derivative as D = d +Γ m∆ l + 1 ω lab, l v = η v η v . (1.2) l m 2 ab ab c ac b − bc a n m i m i with D = dx Dn, Γl = dx Γil and ωab = dx ωiab. The constant flat metric ηab is invariant under lab so that upper and lower Lorentz indices transform with the same sign. On spinors the algebra [l , l ]= η l η l η l + η l (1.3) ab cd ac bd − bc ad − ad bc bd ac 1 of Lorentz transformations is represented as labψ = 4 [γa,γb]ψ. Cartan invented a very efficient calculus by introducing differential forms and orthonormal a a bases e of contangent space: On (co)vector fields v = e va the spin-connection term of the b covariant derivative acts by matrix multiplication Dva = dva + ωa vb. Curvature and torsion can then be defined by Cartan’s structure equations (CSEq) c c b c a a a b Ra = dωa + ωa ωb , T = de + ω be , (1.4) or, in an even more compact symbolic form,2 R = D2 and T = De. The Bianchi identities dT + ω T = R e, dR + ω R R ω = 0. (1.5) ∧ ∧ ∧ − ∧ are now a trivial consequence of d2 = 0. The total antisymmetrization that is implicit in these 3-forms replaces the cyclic sum over the respective indices. In order to relate the spin connection ω to the affine connection Γ we impose that parallel transport should not depend on which basis we use for tangent space. The vielbein thus has to be covariantly constant, D e a = ∂ e a Γ le a + ω a e b = ∂ e a Γ a ω a = 0, (1.6) m n m n − mn l m b n m n − mn − mn which provides a relation between Γ and the spin connection (here we use the vielbein and its inverse to convert the second index of ω and the third index of Γ to the appropriate basis). Contraction of this equation with dxmdxn leads to dea dxmdxnΓ a dxmdxnω ae b = − mn − mb n dea + ωa eb dxmdxnΓ a = 0, which shows that the definition of the torsion via the CSEq b − mn agrees with our previous definition. In order to establish the equivalence of the definitions of ω b b b the curvature we introduce the symbod (D )a = δad + ωa for the covariant derivative that only acts on Lorentz indices. Then Cartan’s definition amounts to Rω =(Dω)2 and ω b 1 i j ω ω 1 i j l l Γ j i (R )a vb = 2 dx dx (Di Dj v)a = 2 dx dx (DiDjv +ΓijDlv)a = Ea(R )i ∆j vl, (1.7) 2 Derivatives of the vector field components drop out in D2v = d2v + d(ωv) + ωdv + ω2v = (dω + ω2)v. Geometry / M.Kreuzer — 2 — version June 2, 2009 where [D ,D ] = T lD + RΓ l∆ k. In a sense, the torsion contribution to [D ,D ] is thus i j − ij l ijk l i j taken into account in Cartan’s calculus by contracting the form index of the spin connection with a differential and having the connection only act on the Lorentz indices. A formula for the spin connection as a function of vielbein and torsion can be obtained by b n b b a b using ωma = Ea ∂men Γma , gmn = em en ηab and our formula for Γ(g, T ). The same result − a a a b can also be obtained directly from the structure equation T = de +ω be if we use the formula 2ω = ω ω that allows to compute a tensor that is antisymmetric in its last [mn]r mnr mnr − rmn 3 2 indices fromP its antisymmetrization in the first two indices: ω = e ∂ e a + e ∂ e a + e ∂ e a + 1 (T + T T ) (1.8) − mnl ma [n l] la [n m] na [m l] 2 mnl lmn − nlm For fixed vielbein the components of spin connection and torsion are thus related by invertible linear equations, so that it is equivalent to use e a,ω b or e a, T a as independent of { m ma } { m mn } fields. Usually it is more convenient to work with the former set. Example: The vielbein calculus is very useful for the evaluation of the curvature tensor. As an example we derive the Schwarzschild metric for a spherically symmetric black hole. Parametrizing the time dilatation with T (r) and the length contraction in the radial direction with R(r), where 4πr2 is the surface of the sphere at fixed r: ds2 = T 2dt2 R−2dr2 r2(dθ2 + sin2 θ dϕ2). (1.9) − − A convenient choice for the vielbein is therefore e0 = Tdt, e1 = R−1 dr, e2 = rdθ, e3 = r sin θ dϕ. (1.10) de0 = T ′ drdt, de1 = 0, de2 = drdθ, de3 = sin θ dr dϕ + r cos θ dθ dϕ. (1.11) a a b a Vanishing torsion T = de + e ωb = 0 thus yields the nonvanishing connection coefficients ω 1 = T ′Rdt, ω 2 = Rdθ, ω 3 = R sin θ dϕ, ω 3 = cos θ dϕ. (1.12) 0 − 1 − 1 − 2 − and the curvatures 1 ′ ′ 2 ′ 2 3 ′ 2 R0 =(T R) dt dr, R0 = T R dt dθ, R0 = T R sin θ dt dϕ, (1.13) R 2 = R′ dr dθ, R 3 = R′ sin θ dr dϕ, R 3 = (1 R2) sin θ dθ dϕ. (1.14) 1 − 1 − 2 − The vacuum Einstein equations in the orthonormal basis are = R c = E mE nR c = 0. Rab acb a c mnb The Ricci tensor turns out to be diagonal with = R(T ′R)′/T + 2T ′R2/rT, = R(T ′R)′/T 2R′R/r, (1.15) R00 R11 − − = = T ′R2/rT R′R/r + (1 R2)/r2. (1.16) R22 R33 − − − 3 The cyclic sum gives the total antisymmetrization of such a tensor, so that both, ω[mn]r and mnr ωmnr, n r are functions of vielbein and torsion. We thus obtain ωmab = Ea Eb (ω[mn]r ω[nr]m + ω[rm]n) with ω[mn]r = a b 1 − P ηaber ∂ e Tmnr. [m n] − 2 Geometry / M.Kreuzer — 3 — version June 2, 2009 Summation of = 0 and = 0 implies that R and T are proportional and by a rescaling of R00 R11 t we can set T = cR, where c is the speed of light.
Load more

Recommended publications
  • 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].
    [Show full text]
  • 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].
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • Spin Connection Resonance in Gravitational General Relativity∗
    Vol. 38 (2007) ACTA PHYSICA POLONICA B No 6 SPIN CONNECTION RESONANCE IN GRAVITATIONAL GENERAL RELATIVITY∗ Myron W. Evans Alpha Institute for Advanced Study (AIAS) (Received August 16, 2006) The equations of gravitational general relativity are developed with Cartan geometry using the second Cartan structure equation and the sec- ond Bianchi identity. These two equations combined result in a second order differential equation with resonant solutions. At resonance the force due to gravity is greatly amplified. When expressed in vector notation, one of the equations obtained from the Cartan geometry reduces to the Newton inverse square law. It is shown that the latter is always valid in the off resonance condition, but at resonance, the force due to gravity is greatly amplified even in the Newtonian limit. This is a direct consequence of Cartan geometry. The latter reduces to Riemann geometry when the Cartan torsion vanishes and when the spin connection becomes equivalent to the Christoffel connection. PACS numbers: 95.30.Sf, 03.50.–z, 04.50.+h 1. Introduction It is well known that the gravitational general relativity is based on Rie- mann geometry with a Christoffel connection. This type of geometry is a special case of Cartan geometry [1] when the torsion form is zero. There- fore gravitation general relativity can be expressed in terms of this limit of Cartan geometry. In this paper this procedure is shown to produce a sec- ond order differential equation with resonant solutions [2–20]. Off resonance the mathematical form of the Newton inverse square law is obtained from a well defined approximation to the complete theory, but the relation between the gravitational potential field and the gravitational force field is shown to contain the spin connection in general.
    [Show full text]
  • Entanglement on Spin Networks Loop Quantum
    ENTANGLEMENT ON SPIN NETWORKS IN LOOP QUANTUM GRAVITY Clement Delcamp London - September 2014 Imperial College London Blackett Laboratory Department of Theoretical Physics Submitted in partial fulfillment of the requirements for the degree of Master of Science of Imperial College London I would like to thank Pr. Joao Magueijo for his supervision and advice throughout the writ- ing of this dissertation. Besides, I am grateful to him for allowing me to choose the subject I was interested in. My thanks also go to Etera Livine for the initial idea and for introducing me with Loop Quantum Gravity. I am also thankful to William Donnelly for answering the questions I had about his article on the entanglement entropy on spin networks. Contents Introduction 7 1 Review of Loop Quantum Gravity 11 1.1 Elements of general relativity . 12 1.1.1 Hamiltonian formalism . 12 1.1.2 3+1 decomposition . 12 1.1.3 ADM variables . 13 1.1.4 Connection formalism . 14 1.2 Quantization of the theory . 17 1.2.1 Outlook of the construction of the Hilbert space . 17 1.2.2 Holonomies . 17 1.2.3 Structure of the kinematical Hilbert space . 19 1.2.4 Inner product . 20 1.2.5 Construction of the basis . 21 1.2.6 Aside on the meaning of diffeomorphism invariance . 25 1.2.7 Operators on spin networks . 25 1.2.8 Area operator . 27 1.2.9 Physical interpretation of spin networks . 28 1.2.10 Chunks of space as polyhedra . 30 1.3 Explicit calculations on spin networks . 32 2 Entanglement on spin networks 37 2.1 Outlook .
    [Show full text]
  • Supergravities
    Chapter 10 Supergravities. When we constructed the spectrum of the closed spinning string, we found in the 3 1 massless sector states with spin 2 , in addition to states with spin 2 and other states i ~j NS ⊗ NS : b 1 j0; piL ⊗ b 1 j0; piR with i; j = 1; 8 yields gij(x) with spin 2 − 2 − 2 (10.1) i 3 NS ⊗ R : b− 1 j0; piL ⊗ j0; p; αiR with i; α = 1; 8 yields χi,α(x) with spin 2 2 The massless states with spin 2 correspond to a gauge field theory: Einstein's theory 3 of General Relativity, but in 9 + 1 dimensions. The massless states with spin 2 are part of another gauge theory, an extension of Einstein gravity, called supergravity. That theory was first discovered in 1976 in d = 3 + 1 dimensions, but supergravity theories also exist in higher dimensions, through (up to and including) eleven dimensions. In this chapter we discuss supergravity theories. They are important in string theory because they are the low-energy limit of string theories. We do not assume that the reader has any familiarity with supergravity. We begin with the N = 1 theory in 3 + 1 dimensions, m also called simple supergravity, which has one gravitational vielbein field eµ , and one m α fermionic partner of eµ , called the gravitino field µ with spinor index α = 1; ··· ; 4 A ¯ _ (or two 2-component gravitinos µ and µ,A_ with A; A = 1; 2) and µ = 0; 1; 2; 3. This theory has one real local supersymmetry with a real (Majorana) 4-component spinorial α A ∗ _ parameter (or two 2-component parameters and ¯A_ = (A) with A; A = 1; 2).
    [Show full text]
  • Introduction to Supergravity
    Introduction to Supergravity Lectures presented by Henning Samtleben at the 13th Saalburg School on Fundamentals and New Methods in Theoretical Physics, September 3 { 14, 2007 typed by Martin Ammon and Cornelius Schmidt-Colinet ||| 1 Contents 1 Introduction 3 2 N = 1 supergravity in D = 4 dimensions 4 2.1 General aspects . 4 2.2 Gauging a global symmetry . 5 2.3 The vielbein formalism . 6 2.4 The Palatini action . 9 2.5 The supersymmetric action . 9 2.6 Results . 14 3 Extended supergravity in D = 4 dimensions 16 3.1 Matter couplings in N = 1 supergravity . 16 3.2 Extended supergravity in D = 4 dimensions . 17 4 Extended supergravity in higher Dimensions 18 4.1 Spinors in higher dimensions . 18 4.2 Eleven-dimensional supergravity . 20 4.3 Kaluza-Klein supergravity . 22 4.4 N = 8 supergravity in D = 4 dimensions . 26 A Variation of the Palatini action 27 2 1 Introduction There are several reasons to consider the combination of supersymmetry and gravitation. The foremost is that if supersymmetry turns out to be realized at all in nature, then it must eventually appear in the context of gravity. As is characteristic for supersymmetry, its presence is likely to improve the quantum behavior of the theory, particularly interesting in the context of gravity, a notoriously non-renormalizable theory. Indeed, in supergravity divergences are typically delayed to higher loop orders, and to date it is still not ruled out that the maximally supersymmetric extension of four-dimensional Einstein gravity might eventually be a finite theory of quantum gravity | only recently very tempting indications in this direction have been unvealed.
    [Show full text]
  • 15 Gravity As a Gauge Theory
    15 Gravity as a gauge theory Aim of this short chapter is develop the formalism necessary to describe the interaction of fermions with gravity. Moreover, we want to stress the similarity of gravity as gauge theory with the group GL(4) to “usual” Yang-Mills theories. Finally, we want to understand how gravity selects among the many mathematically possible connections on a Riemannian manifold. 15.1 Vielbein formalism and the spin connection For fields transforming as tensor under Lorentz transformation, the effects of gravity are accounted for by the replacements {∂µ, ηµν } →{∇µ, gµν } in the matter Lagrangian Lm and the resulting physical laws. Imposing the two requirements ∇ρgµν = 0 (“metric connection”) α α and Γ βγ = Γ γβ (“torsionless connection”) selects uniquely the Levi-Civita connection. In the following, we want to understand if these conditions are a consequence of Einstein gravity or necessary additional constraints. The substitution rule {∂µ, ηµν }→{∇µ, gµν } cannot be applied to the case of spinor repre- sentations of the Lorentz group. Instead, we apply the equivalence principle as physical guide line to obtain the physical laws including gravity: More precisely, we use that we can find at any point P a local inertial frame for which the physical laws become those known from Minkowski space. We demonstrate this first for the case of a scalar field φ. The usual Lagrange density without gravity, 1 L = ∂ φ∂µφ − V (φ) , (15.1) 2 µ is still valid on a general manifold M({xµ}), if we use in each point P locally free-falling coordinates, ξa(P ).
    [Show full text]
  • Exotic Gauge Theories of Spin-2 Fields
    Imperial College of Science, Technology and Medicine Department of Physics Exotic Gauge Theories of Spin-2 Fields Nicholas Andrew Ondo Supervised by Dr. Andrew J. Tolley Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in in Physics of Imperial College London and the Diploma of Imperial College, October 2017 Abstract There has been a resurgence of interest in theories of massive spin-2 fields, owing to the recent discovery of ghost-free self-interactions. In addition to reviewing the historical and recent progress in this subject, I discuss my contributions, including the derivation of the complete decoupling limit of dRGT ghost-free massive gravity, proving no-go theorems on ghost-free interactions for charged spin-2 fields, updating the method of Dimensional Deconstruction for fermions to obtain massive supersym- metric gauge theories, and my progress towards supergravity theories with non-zero graviton mass. i ii Statement of Originality All of original research presented in this thesis are works that were done either by myself or alongside contributors. My original contributions to these works represent a substantial contribution. Any non-original works are appropriately cited to the best of the author's knowledge. iii iv Publications (*)\ The Complete Decoupling Limit of Ghost-free Massive Gravity" by Nicholas A. Ondo, Andrew J. Tolley. Published in JHEP 1311 (2013) 059. [ARXIV:1307.4769] (*)\ Interactions of Charged Spin-2 Fields" by Claudia de Rham, Andrew Matas, Nicholas A. Ondo, and Andrew J. Tolley. Published in Class.Quant.Grav. 32 (2015) no.17, 175008. [ARXIV:1410.5422] (*)\ Deconstructing Supergravity, I: Massive Supermultiplets" by Nicholas A.
    [Show full text]
  • General Relativity with Torsion
    TOKAI-HEP/TH-04-09 General Relativity with Torsion Tomoki Watanabe∗ and Mitsuo J. Hayashi† Department of Physics, Tokai University, 1117, Kitakaname, Hiratsuka, 259-1292, Japan ∗[email protected]; †[email protected] Abstract General Relativity with nonvanishing torsion has been investigated in the first order formalism of Poincar´egauge field theory. In the presence of torsion, either side of the Einstein equation has the nonvanishing covariant divergence. This fact turned out to be self-consistent in the framework under consideration. By using Noether’s procedure with the definition of the Lie derivative where the general coordinate transformation and the local Lorentz rotation are combined, the revised covariant divergence of the energy momentum is consistently obtained. Subsequently we have definitely derived the spin correction to the energy momentum tensor for the Dirac field and the Rarita-Schwinger field in the Einstein equation. We conjecture that the accelerated expansion of the universe possibly arises due to the spin correction in our framework. Keywords: torsion, Dirac fields, Rarita-Schwinger fields, the accelerating universe. 1 Introduction In the well-known formulation of General Relativity, gravity is appropriately described by Riemann geometry which is torsion-free. The metric of the space-time represents the gravitational field. The connections are given by the Christoffel symbol compatible with the metric structure. However in particle physics, the spinor fields are the essential ingredients in the description of natural law. Therefore, in order to bring spinors into the curved space-time, the vierbein (tetrad) had been introduced to formulate General arXiv:gr-qc/0409029v1 7 Sep 2004 Relativity by the local gauge transformation invariance, such as the local translation or the local Poincar´egroup[1]-[3].
    [Show full text]
  • “RELATIVISTIC FERMION SYSTEMS” Contents Preface 2 1. Introduction 3
    LECTURE NOTES ON \RELATIVISTIC FERMION SYSTEMS" FELIX FINSTER AND JAN-HENDRIK TREUDE Contents Preface 2 1. Introduction 3 1.1. The Schr¨odingerEquation 3 1.2. The Dirac Equation 4 1.3. A First Glance at Causal Fermion Systems 9 2. Fourier Methods 12 2.1. The Green's Functions 12 2.2. The Causal Fundamental Solution and Time Evolution 14 2.3. The Lippmann-Schwinger Equation 16 3. The Cauchy Problem, Linear Symmetric Hyperbolic Systems 17 3.1. Finite Propagation Speed and Uniqueness of Solutions 18 3.2. Global Existence of Smooth Solutions 22 3.3. Existence of Causal Green's Functions 26 4. The Fermionic Projector 28 4.1. The External Field Problem 28 4.2. The Dirac Sea and the Fermionic Projector of the Vacuum 29 4.3. Perturbative Description 32 5. A Non-Perturbative Construction of the Fermionic Projector 33 5.1. The Mass Oscillation Property 34 5.2. A Self-Adjoint Extension of S2 35 5.3. The Operator k and its Extended Domain 37 5.4. The Fermionic Projector as an Operator-Valued Measure 38 5.5. Proof of the Mass Oscillation Property 39 6. Spinors in Curved Space-Time 42 6.1. Curved Space-Time and Lorentzian Manifolds 42 6.2. The Dirac Equation in Curved Space-Time 43 7. Causal Fermion Systems and a Lorentzian Quantum Geometry 50 7.1. Constructing a Causal Fermion System from the Fermionic Projector 50 7.2. Geometric Structures on a Causal Fermion System 51 7.3. The Correspondence to Lorentzian Spin Geometry 57 8. Outlook 59 8.1.
    [Show full text]