Geometrodynamics of Gauge Fields on the Geometry of Yang-Mills and Gravitational Gauge Theories

Total Page：16

File Type：pdf, Size：1020Kb

E.W. Mielke Geometrodynamics of Gauge Fields On the Geometry of Yang-Mills and Gravitational Gauge Theories Series: Mathematical Physics Studies ▶ A comprehensive revision of a classic text relating geometic field theory to modern physics ▶ Offers an up-to-date overview of geometrodynamics including minimal topological models ▶ Contains new chapters on teleparalelism, Chern-Simons-induced topological 3D gravity and supergravity, quantization via topological ghosts, constrained BF model with SL (5, R) gauge group, and chiral and trace anomalies ▶ Provides a detailed exposition of frame bundles in gravity 2nd ed. 2017, XVII, 373 p. 18 illus., 8 illus. in This monograph aims to provide a unified, geometrical foundation of gauge theories color. of elementary particle physics. The underlying geometrical structure is unfolded in a coordinate-free manner via the modern mathematical notions of fibre bundles and exterior forms. Topics such as the dynamics of Yang-Mills theories, instanton solutions Printed book and topological invariants are included. By transferring these concepts to local space- time symmetries, generalizations of Einstein's theory of gravity arise in a Riemann- Hardcover Cartan space with curvature and torsion. It provides the framework in which the (broken) ISBN 978-3-319-29732-3 Poincaré gauge theory, the Rainich geometrization of the Einstein-Maxwell system, and ▶ 119,99 € | £109.99 higher-dimensional, non-abelian Kaluza-Klein theories are developed. ▶ *128,39 € (D) | 131,99 € (A) | CHF 141.50 Since the discovery of the Higgs boson, concepts of spontaneous symmetry breaking in gravity have come again into focus, and, in this revised edition, these will be exposed in geometric terms. Quantizing gravity remains an open issue: formulating it as a de Sitter type gauge theory in the spirit of Yang-Mills, some new progress in its topological form is presented. After symmetry breaking, Einstein’s standard general relativity with cosmological constant emerges as a classical background. The geometrical structure of BRST quantization with non-propagating topological ghosts is developed in some detail. The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted..

Copy Link

Recommended publications

Quantum Simplicial Geometry in the Group Field Theory Formalism
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model Aristide Baratin∗1 and Daniele Oriti†2 1Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique F-9112 Palaiseau Cedex, France 2Max-Planck-Institut für Gravitationsphysik Albert Einstein Institute Am Mühlenberg 2, 14476, Golm, Germany, EU A dual formulation of group field theories, obtained by a Fourier transform mapping functions on a group to functions on its Lie algebra, has been proposed recently. In the case of the Ooguri model for SO(4) BF theory, the variables of the dual field variables are thus so(4) bivectors, which have a direct interpretation as the discrete B variables. Here we study a modification of the model by means of a constraint operator implementing the simplicity of the bivectors, in such a way that projected fields describe metric tetrahedra. This involves a extension of the usual GFT framework, where boundary operators are labelled by projected spin network states. By construction, the Feynman amplitudes are simplicial path integrals for constrained BF theory. We show that the spin foam formulation of these amplitudes corresponds to a variant of the Barrett-Crane model for quantum gravity. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction. Introduction Group field theories (GFT) represent a second quantized framework for both spin networks and simplicial geometry [6, 7, 9], field theories on group manifolds (or Lie algebras) producing Feynman amplitudes which can be equivalently expressed as simplicial gravity path integrals [5] or spin foam models [10], in turn covariant formulations of spin networks dynamics [1]3.
[Show full text]
Symmetry and Gravity
universe Article Making a Quantum Universe: Symmetry and Gravity Houri Ziaeepour 1,2 1 Institut UTINAM, CNRS UMR 6213, Observatoire de Besançon, Université de Franche Compté, 41 bis ave. de l’Observatoire, BP 1615, 25010 Besançon, France; [email protected] or [email protected] 2 Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking GU5 6NT, UK Received: 05 September 2020; Accepted: 17 October 2020; Published: 23 October 2020 Abstract: So far, none of attempts to quantize gravity has led to a satisfactory model that not only describe gravity in the realm of a quantum world, but also its relation to elementary particles and other fundamental forces. Here, we outline the preliminary results for a model of quantum universe, in which gravity is fundamentally and by construction quantic. The model is based on three well motivated assumptions with compelling observational and theoretical evidence: quantum mechanics is valid at all scales; quantum systems are described by their symmetries; universe has inﬁnite independent degrees of freedom. The last assumption means that the Hilbert space of the Universe has SUpN Ñ 8q – area preserving Diff.pS2q symmetry, which is parameterized by two angular variables. We show that, in the absence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum ﬂuctuations break the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference—observer—and another as clock, two more continuous parameters arise, which can be interpreted as distance and time. We identify the classical spacetime with parameter space of the Hilbert space of the Universe.
[Show full text]
Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings.
[Show full text]
Geometrodynamics of Gauge Fields on the Geometry of Yang-Mills and Gravitational Gauge Theories
springer.com Physics : Classical and Quantum Gravitation, Relativity Theory Mielke, Eckehard W. Geometrodynamics of Gauge Fields On the Geometry of Yang-Mills and Gravitational Gauge Theories A comprehensive revision of a classic text relating geometic field theory to modern physics Offers an up-to-date overview of geometrodynamics including minimal topological models Contains new chapters on teleparalelism, Chern-Simons-induced topological 3D gravity and supergravity, quantization via topological ghosts, constrained BF model with SL (5, R) gauge group, and chiral and trace anomalies Provides a detailed exposition of frame bundles in gravity Springer This monograph aims to provide a unified, geometrical foundation of gauge theories of 2nd ed. 2017, XVII, 373 p. elementary particle physics. The underlying geometrical structure is unfolded in a coordinate- 2nd 18 illus., 8 illus. in color. free manner via the modern mathematical notions of fibre bundles and exterior forms. Topics edition such as the dynamics of Yang-Mills theories, instanton solutions and topological invariants are included. By transferring these concepts to local space-time symmetries, generalizations of Einstein's theory of gravity arise in a Riemann-Cartan space with curvature and torsion. It Printed book provides the framework in which the (broken) Poincarégauge theory, the Rainich geometrization Hardcover of the Einstein-Maxwell system, and higher-dimensional, non-abelian Kaluza-Klein theories are Printed book developed. Since the discovery of the Higgs boson, concepts of spontaneous symmetry breaking in gravity have come again into focus, and, in this revised edition, these will be Hardcover exposed in geometric terms. Quantizing gravity remains an open issue: formulating it as a de ISBN 978-3-319-29732-3 Sitter type gauge theory in the spirit of Yang-Mills, some new progress in its topological form is £ 109,99 | CHF 141,50 | 119,99 € | presented.
[Show full text]
Dimensional Topological Quantum Field Theory from a Tight-Binding Model of Interacting Spinless Fermions
This is a repository copy of (3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/99993/ Version: Accepted Version Article: Cirio, M, Palumbo, G and Pachos, JK (2014) (3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions. Physical Review B, 90 (8). 085114. ISSN 2469-9950 https://doi.org/10.1103/PhysRevB.90.085114 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ (3+1)-dimensional topological quantum ﬁeld theory from a tight-binding model of interacting spinless fermions Mauro Cirio,1 Giandomenico Palumbo,2 and Jiannis K. Pachos2 1Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, North Ryde, NSW 2109, Australia 2School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom (Dated: July 15, 2014) Currently, there is much interest in discovering analytically tractable (3 + 1)-dimensional models that describe interacting fermions with emerging topological properties.
[Show full text]
Construction and Examples of Higher Gauge Theories∗
Construction and examples of higher gauge theories∗ Tijana Radenkovićy Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia Marko Vojinovićz Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia Abstract We provide several examples of higher gauge theories, constructed as gener- alizations of a BF model to 2BF and 3BF models with constraints. Using the framework of higher category theory, we introduce appropriate 2-groups and 3- groups, and construct the actions for the corresponding constrained 2BF and 3BF theories. In this way, we can construct actions which describe the correct dynamics of Yang-Mills, Klein-Gordon, Dirac, Weyl, and Majorana fields coupled to Einstein-Cartan gravity. Each action is naturally split into a topological sector and a sector with simplicity constraints. The properties of the higher gauge group structure opens up a possibility of a nontrivial unification of all fields. 1. Introduction The quantization of the gravitational field is one of the fundamental open problems in modern physics. There are various approaches to this prob- lem, some of which have developed into vast research frameworks. One of such frameworks is the Loop Quantum Gravity approach, which aims to establish a nonperturbative quantization of gravity, both canonically and covariantly [1, 2, 3]. The covariant approach is slightly more general, and ∗ This work was supported by the project ON171031 of the Ministry of Education, Sci- ence and Technological Development (MPNTR) of the Republic of Serbia, and partially by the bilateral scientific cooperation between Austria and Serbia through the project \Causality in Quantum Mechanics and Quantum Gravity - 2018-2019", no.
[Show full text]
Aspects of Loop Quantum Gravity
Aspects of loop quantum gravity Alexander Nagen 23 September 2020 Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 1 Contents 1 Introduction 4 2 Classical theory 12 2.1 The ADM / initial-value formulation of GR . 12 2.2 Hamiltonian GR . 14 2.3 Ashtekar variables . 18 2.4 Reality conditions . 22 3 Quantisation 23 3.1 Holonomies . 23 3.2 The connection representation . 25 3.3 The loop representation . 25 3.4 Constraints and Hilbert spaces in canonical quantisation . 27 3.4.1 The kinematical Hilbert space . 27 3.4.2 Imposing the Gauss constraint . 29 3.4.3 Imposing the diffeomorphism constraint . 29 3.4.4 Imposing the Hamiltonian constraint . 31 3.4.5 The master constraint . 32 4 Aspects of canonical loop quantum gravity 35 4.1 Properties of spin networks . 35 4.2 The area operator . 36 4.3 The volume operator . 43 2 4.4 Geometry in loop quantum gravity . 46 5 Spin foams 48 5.1 The nature and origin of spin foams . 48 5.2 Spin foam models . 49 5.3 The BF model . 50 5.4 The Barrett-Crane model . 53 5.5 The EPRL model . 57 5.6 The spin foam - GFT correspondence . 59 6 Applications to black holes 61 6.1 Black hole entropy . 61 6.2 Hawking radiation . 65 7 Current topics 69 7.1 Fractal horizons . 69 7.2 Quantum-corrected black hole . 70 7.3 A model for Hawking radiation . 73 7.4 Effective spin-foam models .
[Show full text]
Gravitation Law and Source Model in the Anisotropic Geometrodynamics
Gravitation law and source model in the anisotropic geometrodynamics Sergey Siparov State University of Civil Aviation, 38 Pilotov str., St-Petersburg, 196210; Research Institute for Hyper Complex Systems in Geometry and Physics, 3 bg 1 Zavodskoy pr., Fryazino, Moscow region, 141190; Russian Federation Abstract. The GRT modification taking into account the dependence of metric on the velocities of the sources is built. It is shown that this dependence follows from the equivalence principle and from the inseparability of the field equations and geodesics equations. As it is known, the latter are the conditions of the field equations solvability, and their form coincides with Newtonian one only in the lowest approximation. The obtained modification provides the explanation for the flat character of the rotation curves of spiral galaxies, for Tully-Fisher law, for some specific features of globular clusters behavior and for the essential excess of the observable gravitational lens effect over the predicted one. Neither dark matter nor arbitrary change of dynamics equations appeared to be needed. Important cosmological consequences are obtained. Keywords: modified theory of gravity, rotation curves, Tully-Fisher law, anisotropic metric PACS: 04.20.Cv, 04.50.Kd, 95.30.Sf, 98.20.Gm, 98.52.Nr, 98.80.Es 1. INTRODUCTION The flat character of the rotation curves (RC) of spiral galaxies is the most pronounced problem of the known GRT problems on the galactic scale. This is a simple, not small and statistically verified effect which doesn’t find explanation not only in the GRT but in Newton gravity as well. It is to the fact that the orbital velocities of stars in the spiral galaxies don’t tend to zero with the distance from the center but approach constant values of the order 105m/s for all galaxies.
[Show full text]
Do Global Virtual Axionic Gravitons Exist?
NeuroQuantology | March 2014 | Volume 12 | Issue 1 | Page 35-39 35 Glinka LA., Do global virtual axionic gravitons exist? Do Global Virtual Axionic Gravitons Exist? Lukasz Andrzej Glinka ABSTRACT Recently, the author has proposed the global one-dimensionality conjecture which, throughout making use of the method of the primary canonical quantization of General Relativity according to the Dirac-Arnowitt-Deser-Misner Hamiltonian formulation, formally avoids the functional differential nature which is the basic obstacle of quantum geometrodynamics, a model of quantum gravity based on the Wheeler-DeWitt equation and resutling from quantization of the Einstein field equations. The emergent quantum gravity model is straightforwardly integrable in terms of the Riemann-Lebesgue integral, has an unambiguous physical interpretation within quantum theory, and introduces a consistent concept of global graviton. In this article, the axionic nature of the global gravitons, that is the hypothetical particles responsible for gravitation in the global one-dimensional quantum gravity model, is shortly described and, making use of fundamental methods of quantum field theory, the virtual states of such gravitons are generated. Key Words: global one-dimensionality conjecture; quantum gravity; quantum geometrodynamics; graviton; axion DOI Number: 10.14704/nq.2014.12.1.716 NeuroQuantology 2014; 1: 35-39 1. Introduction1 results in replacement of the functional Recently the Wheeler-DeWitt equation was evolution parameter which is the metric of considered in the framework of the global one- space by a variable which is the metric dimensionality conjecture (Glinka, 2012; 2010). determinant. The ramifications of this change Recall that this conjecture formally reduces the are highly non-trivial.
[Show full text]
Loop and Spin Foam Quantum Gravity: a Brief Guide for Beginners
AEI-2006-004 hep-th/0601129 January 18th, 2006 Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners Hermann Nicolai and Kasper Peeters Max-Planck-Institut für Gravitationsphysik Albert-Einstein-Institut Am Mühlenberg 1 14476 Golm, GERMANY hermann.nicolai, [email protected] Abstract: We review aspects of loop quantum gravity and spin foam models at an introductory level, with special attention to questions frequently asked by non-specialists. arXiv:hep-th/0601129 v2 16 Feb 2006 Contributed article to “An assessment of current paradigms in the physics of fundamental interactions”, ed. I. Stamatescu, Springer Verlag. 1. Quantum Einstein gravity ics approach [7], addresses several aspects of the problem that are currently outside the main focus The assumption that Einstein’s classical theory of of string theory, in particular the question of back- gravity can be quantised non-perturbatively is at the ground independence and the quantisation of geom- root of a wide variety of approaches to quantum etry. Whereas there is a rather direct link between gravity. The assumption constitutes the basis of sev- (perturbative) string theory and classical space-time eral discrete methods [1], such as dynamical tri- concepts, and string theory can therefore rely on fa- angulations and Regge calculus, but it also implic- miliar notions and concepts, such as the notion of a itly underlies the older Euclidean path integral ap- particle and the S-matrix, the task is harder for LQG, proach [2, 3] and the somewhat more indirect ar- as it must face up right away to the question of what guments which suggest that there may exist a non- an observable quantity is in the absence of a proper trivial fixed point of the renormalisation group [4–6].
[Show full text]
Plasma Modes in Surrounding Media of Black Holes and Vacuum Structure - Quantum Processes with Considerations of Spacetime Torque and Coriolis Forces
COLLECTIVE COHERENT OSCILLATION PLASMA MODES IN SURROUNDING MEDIA OF BLACK HOLES AND VACUUM STRUCTURE - QUANTUM PROCESSES WITH CONSIDERATIONS OF SPACETIME TORQUE AND CORIOLIS FORCES N. Haramein¶ and E.A. Rauscher§ ¶The Resonance Project Foundation, [email protected] §Tecnic Research Laboratory, 3500 S. Tomahawk Rd., Bldg. 188, Apache Junction, AZ 85219 USA Abstract. The main forces driving black holes, neutron stars, pulsars, quasars, and supernovae dynamics have certain commonality to the mechanisms of less tumultuous systems such as galaxies, stellar and planetary dynamics. They involve gravity, electromagnetic, and single and collective particle processes. We examine the collective coherent structures of plasma and their interactions with the vacuum. In this paper we present a balance equation and, in particular, the balance between extremely collapsing gravitational systems and their surrounding energetic plasma media. Of particular interest is the dynamics of the plasma media, the structure of the vacuum, and the coupling of electromagnetic and gravitational forces with the inclusion of torque and Coriolis phenomena as described by the Haramein-Rauscher solution to Einstein’s field equations. The exotic nature of complex black holes involves not only the black hole itself but the surrounding plasma media. The main forces involved are intense gravitational collapsing forces, powerful electromagnetic fields, charge, and spin angular momentum. We find soliton or magneto-acoustic plasma solutions to the relativistic Vlasov equations solved in the vicinity of black hole ergospheres. Collective phonon or plasmon states of plasma fields are given. We utilize the Hamiltonian formalism to describe the collective states of matter and the dynamic processes within plasma allowing us to deduce a possible polarized vacuum structure and a unified physics.
[Show full text]
Arxiv:1108.1178V2 [Gr-Qc]
Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model Aristide Baratin∗1 and Daniele Oriti†2 1Centre de Physique Théorique, CNRS UMR 7644, Ecole Polytechnique F-9112 Palaiseau Cedex, France 2Max-Planck-Institut für Gravitationsphysik Albert Einstein Institute Am Mühlenberg 2, 14476, Golm, Germany, EU A dual formulation of group field theories, obtained by a Fourier transform mapping functions on a group to functions on its Lie algebra, has been proposed recently. In the case of the Ooguri model for SO(4) BF theory, the variables of the dual field variables are thus so(4) bivectors, which have a direct interpretation as the discrete B variables. Here we study a modification of the model by means of a constraint operator implementing the simplicity of the bivectors, in such a way that projected fields describe metric tetrahedra. This involves a extension of the usual GFT framework, where boundary operators are labelled by projected spin network states. By construction, the Feynman amplitudes are simplicial path integrals for constrained BF theory. We show that the spin foam formulation of these amplitudes corresponds to a variant of the Barrett-Crane model for quantum gravity. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction. Introduction Group field theories (GFT) represent a second quantized framework for both spin networks and simplicial geometry [6, 7, 9], field theories on group manifolds (or Lie algebras) producing Feynman amplitudes which can be equivalently expressed as simplicial gravity path integrals [5] or spin foam models [10], in turn covariant formulations of spin networks dynamics [1]3.
[Show full text]