DOCSLIB.ORG

The Strange Equation of Quantum Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Strange Equation of Quantum Gravity The strange equation of quantum gravity Carlo Rovelli Aix Marseille Universit´e, CNRS, CPT, UMR 7332, 13288 Marseille, France Universit´ede Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. (Dated: June 3, 2015) Disavowed by one its fathers, ill defined, never empirically tested, the Wheeler-DeWitt equation has nevertheless had a powerful influence on fundamental physics. A well deserved one. I. INTRODUCTION suitable for the WdW equation is far from obvious, and rivers of ink have been wasted to discuss this issue as One day in 1965, John Wheeler had a two hours stopover well. Last but obviously not least, not a single empiri- between flights at the Raleigh-Durham airport in North cal observation has so far directly supported the physical Carolina. He called Bryce DeWitt, then at the University correctness of the equation, half a century after its pub- of North Carolina in Chapel Hill, proposing to meet at lication. the airport during the wait. Bryce showed up with the Still, in spite of all these limitations, the WdW equa- Hamilton-Jacobi equation of general relativity, published tion is a milestone in the development of general relativ- by Asher Peres shortly earlier [1]: ity. It has inspired a good part of the research in quan- tum gravity for decades, and has opened new perspec- 1 δS[q] δS[q] tives for fundamental physics. Much of the original con- qabqcd qacqbd det q R[q]=0. (1) − 2 δqac δqbd − ceptual confusion raised by the equation has been clar- ified. Consensus on a theory of quantum gravity lacks, qab, with a,b =1, 2, 3 are the components of a 3d spatial but tentative theories, such as strings and loops, exist metric, R[q] its Ricci scalar, and S[q] a Hamilton-Jacobi today. Well defined versions of the equation have been functional. Bryce mumbled the idea of repeating what found, and they are utilized to produce predictions, with Schr¨odinger did for the hydrogen atom: getting a wave a hope of testing them against observation. The equa- equation by replacing the square of derivatives with (i tion is recognized as a basic tool for thinking about the times) a second derivative—a manner for undoing the quantum properties of spacetime by a large community of optical approximation. This gives [2, 3] physicists in areas ranging from early cosmology to black 1 δ δ hole physics. The equation has proven an inexhaustible qabqcd qacqbd + det q R[q] Ψ[q]=0 source of inspiration on the path towards understanding − 2 δq δq ac bd the quantum properties of space and time. (2) for a functional Ψ[q] to be interpreted as the wave func- The main reason for this is that the WdW equation has tion of the gravitational field q. Wheeler got tremen- been the icebreaker towards the construction of a quan- dously excited (he was often enthusiastic) and declared tum theory which does not presuppose a single space- on the spot that the equation for quantum gravity had time. The physics community underwent a major re- been found. This is the birth of eq. (2) ([4], pg 58). For shaping of its way of thinking when it begun to utilize a long time Wheeler called it the “Einstein-Schr¨odinger general relativity: from a conception of physics as the equation”, while DeWitt called it the “Wheeler equa- theory of what happens in space and time, to a new con- tion”, or ”that damn equation”, without hiding his un- ception, where the theory describes also of what happens arXiv:1506.00927v1 [gr-qc] 2 Jun 2015 happiness with it it. For the rest of the world it is the to space and time. This step has been difficult in the Wheeler-DeWitt, or “WdW”, equation. classical framework, where questions like whether gravi- It is a strange equation, full of nasty features. First, tational waves were physical or gauge, or concerning the it is ill-defined. The functional derivatives are distri- nature of the Schwarzschild singularity, have confused butions, which cannot be squared without yielding di- the community for decades. But it has taken longer, and vergences. Concrete calculations, indeed, tend to give in fact is still confusing the community, for the quantum meaningless results: strictly speaking, there is no equa- regime. Quantum mechanics and general relativity, taken tion. Second, the equation breaks the manifest relativis- together, imply the possibility of quantum superposition tic covariance between space and time: quite bad for the of different spacetimes. The WdW equation, which is quantum theory of general relativity. Third, there is no based on a wave function Ψ[q] over geometries, has of- time variable in the equation, a puzzling feature for an fered the first conceptual scheme for dealing with this equation with the ambition to be the dynamical equation physical possibility For this reason it is a milestone. of quantum spacetime. This has raised endless confusion, Today’s tentative quantum gravity theories take for and prompted a long lasting debate on the nature of time. granted the absence of a single predetermined smooth Furthermore, a Schr¨odinger-like equation is not suffi- spacetime at all scales. The breakthrough opening the cient to define a quantum theory: a scalar product is path to this thinking happened at the Raleigh-Durham needed to compute expectation values. A scalar product airport 1965 meeting. 2 II. FROM EINSTEIN-HAMILTON-JACOBI TO family of solutions S[q, q0] parametrized by a 3-metric EINSTEIN-SCHRODINGER¨ 0 qab(~x). Then we can define the momenta 0 In the following I make no attempt at a full review ab 0 δS[q, q ] p0 [q, q ](~x)= 0 (7) of the vast literature where the WdW equation has been δqab(~x) discussed, utilized, and has left its mark. Instead, I focus on the main issues the equation has raised, on what we which are thus nonlocal functions of q and q0. For any 0 have learned from it, and on the possibility it has opened choice of q and p0 satisfying (4) and (5), there exists a for describing quantum spacetime. spacetime which is a solution of Einstein’s equations and I start by illustrating the path leading to the WdW for any such solution, any field q satisfying equation via Peres’ Hamilton-Jacobi formulation of gen- ab 0 ab eral relativity [1], in a bit more detail than above. This p0 [q, q ](~x)= p0 (~x). (8) is relevant not only for history, but especially because is the metric of a spacelike 3d surface imbedded in this it clarifies the meaning of the WdW equation, shedding spacetime. Since the q0 and p are related by equa- light on the confusion it raised. Some difficulties of the 0 tions (4) and (5), the last equation does not determine q WdW equation are more apparent than real. They have uniquely: the different solutions correspond to different nothing to do with quantum mechanics; they stem from spacelike surfaces in spacetime, and different coordinati- the application of the Hamilton-Jacobi language to a gen- zation of the same; in fact, all of them. Therefore the erally covariant context. solution to the Einstein-Hamilton-Jacobi system (1)-(6) Einstein wrote general relativity in terms of the provides in principle the full solution of the Einstein’s Lorentzian metric g (~x, t), where µ,ν = 0, ..., 3 are µν equations. These results follow from a simple generaliza- spacetime indices and (~x, t) are coordinates on space- tion of standard Hamilton-Jacobi theory. time. Peres’ starting point was the Arnowit-Deser- This was the starting point of Bryce DeWitt. Now re- Misner change of variables [5] call that in his milestone 1926 article [7], Schr¨odinger in- oo qab = gab, Na = gao, N = √ g . (3) troduced (what is called today) the Schr¨odinger equation − for the hydrogen atom by taking the Hamilton-Jacobi a ab where a,b =1, 2, 3. N and N = q Nb are the “Lapse” equation of an electron in a Coulomb potential: ab and “Shift” functions, and q is the inverse of qab. The interest of these variables is that the Lagrangian does ∂S(~x, t) 1 ∂S(~x, t) ∂S(~x, t) e2 + + = 0 (9) not depend on the time derivatives of Lapse and Shift. ∂t 2m ∂~x · ∂~x ~x Therefore the only true dynamical variable is the three | | metric qab(~x), the Riemannian metric of a t = constant and replacing derivatives with ( i~ times) derivative op- surface. The canonical hamiltonian vanishes, as is always erators: − the case for systems invariant under re-parametrization ∂ ~2 ∂ ∂ e2 of the lagrangian evolution parameter, and the dynamics i~ + ψ(~x, t)=0. (10) − ∂t − 2m ∂~x · ∂~x ~x is coded by the two ADM constraints | | 1 ac bd In a shortly subsequent article [8], Schr¨odinger offers a qabqcd qacqbd p p det q R[q] = 0 (4) − 2 − rationale for this procedure: the eikonal approximation of a wave equation, which defines the geometrical optic and approximation where wave packets follow definite trajec- tories, can be obtained by the opposite procedure. If ab Da p =0, (5) we interpret classical mechanics as the eikonal approx- imation to a wave mechanics, we can guess the wave on the three metric q and its conjugate momentum p. equation by this procedure. Given the immense suc- (Da is the covariant derivative of the 3-metric.) cess of Schr¨odinger’s leap, trying the same strategy for The dynamics can be written in Hamilton-Jacobi form gravity is obviously tempting. This is what led DeWitt by introducing the Hamilton-Jacobi function S[q], which and Wheeler to equation (2).
Load more

Recommended publications
  • The Pursuit of Quantum Gravity
    The Pursuit of Quantum Gravity Cécile DeWitt-Morette The Pursuit of Quantum Gravity Memoirs of Bryce DeWitt from 1946 to 2004 123 Cécile DeWitt-Morette Department of Physics Center for Relativity University of Texas at Austin Austin Texas USA [email protected] ISBN 978-3-642-14269-7 e-ISBN 978-3-642-14270-3 DOI 10.1007/978-3-642-14270-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921724 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publica- tion or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protec- tive laws and regulations and therefore free for general use. Cover design:WMXDesignGmbH,Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to our daughters, Nicolette, Jan, Christiane, Abigail Preface This book is written for the curious reader. I hope it will also be a good read for the professional physicist.
    [Show full text]
  • Simulating Quantum Field Theory with a Quantum Computer
    Simulating quantum field theory with a quantum computer John Preskill Lattice 2018 28 July 2018 This talk has two parts (1) Near-term prospects for quantum computing. (2) Opportunities in quantum simulation of quantum field theory. Exascale digital computers will advance our knowledge of QCD, but some challenges will remain, especially concerning real-time evolution and properties of nuclear matter and quark-gluon plasma at nonzero temperature and chemical potential. Digital computers may never be able to address these (and other) problems; quantum computers will solve them eventually, though I’m not sure when. The physics payoff may still be far away, but today’s research can hasten the arrival of a new era in which quantum simulation fuels progress in fundamental physics. Frontiers of Physics short distance long distance complexity Higgs boson Large scale structure “More is different” Neutrino masses Cosmic microwave Many-body entanglement background Supersymmetry Phases of quantum Dark matter matter Quantum gravity Dark energy Quantum computing String theory Gravitational waves Quantum spacetime particle collision molecular chemistry entangled electrons A quantum computer can simulate efficiently any physical process that occurs in Nature. (Maybe. We don’t actually know for sure.) superconductor black hole early universe Two fundamental ideas (1) Quantum complexity Why we think quantum computing is powerful. (2) Quantum error correction Why we think quantum computing is scalable. A complete description of a typical quantum state of just 300 qubits requires more bits than the number of atoms in the visible universe. Why we think quantum computing is powerful We know examples of problems that can be solved efficiently by a quantum computer, where we believe the problems are hard for classical computers.
    [Show full text]
  • 8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics
    8.962 General Relativity, Spring 2017 Massachusetts Institute of Technology Department of Physics Lectures by: Alan Guth Notes by: Andrew P. Turner May 26, 2017 1 Lecture 1 (Feb. 8, 2017) 1.1 Why general relativity? Why should we be interested in general relativity? (a) General relativity is the uniquely greatest triumph of analytic reasoning in all of science. Simultaneity is not well-defined in special relativity, and so Newton's laws of gravity become Ill-defined. Using only special relativity and the fact that Newton's theory of gravity works terrestrially, Einstein was able to produce what we now know as general relativity. (b) Understanding gravity has now become an important part of most considerations in funda- mental physics. Historically, it was easy to leave gravity out phenomenologically, because it is a factor of 1038 weaker than the other forces. If one tries to build a quantum field theory from general relativity, it fails to be renormalizable, unlike the quantum field theories for the other fundamental forces. Nowadays, gravity has become an integral part of attempts to extend the standard model. Gravity is also important in the field of cosmology, which became more prominent after the discovery of the cosmic microwave background, progress on calculations of big bang nucleosynthesis, and the introduction of inflationary cosmology. 1.2 Review of Special Relativity The basic assumption of special relativity is as follows: All laws of physics, including the statement that light travels at speed c, hold in any inertial coordinate system. Fur- thermore, any coordinate system that is moving at fixed velocity with respect to an inertial coordinate system is also inertial.
    [Show full text]
  • Manifest Covariant Hamiltonian Theory of General Relativity
    Applied Physics Research; Vol. 8, No. 2; 2016 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education Manifest Covariant Hamiltonian Theory of General Relativity Claudio Cremaschini1 & Massimo Tessarotto1;2 1 Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezrucovoˇ nam.13,´ CZ- 74601 Opava, Czech Republic 2 Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy Correspondence: Claudio Cremaschini, Institute of Physics, Faculty of Philosophy and Science, Silesian Univer- sity in Opava, Bezrucovoˇ nam.13,´ CZ-74601 Opava, Czech Republic. E-mail: [email protected] Received: January 15, 2016 Accepted: March 4, 2016 Online Published: March 16, 2016 doi:10.5539/apr.v8n2p60 URL: http://dx.doi.org/10.5539/apr.v8n2p60 Abstract The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called “DeDonder-Weyl” formalism to the treatment of classical fields in curved space-time. The theory is based on a synchronous variational principle for the Einstein equation, formulated in terms of superabundant variables. The technique permits one to determine the continuum covariant Hamiltonian structure associated with the Einstein equation. The corresponding continuum Poisson bracket representation is also determined. The theory relies on first-principles, in the sense that the conclusions are reached in the framework of a non-perturbative covariant approach, which allows one to preserve both the 4-scalar nature of Lagrangian and Hamiltonian densities as well as the gauge invariance property of the theory. Keywords: General Relativity, Einstein equation, DeDonder-Weyl formalism, Synchronous variational principle, Covariant Hamiltonian theory, Poisson bracket representation 1.
    [Show full text]
  • Synopsis of a Unified Theory for All Forces and Matter
    Synopsis of a Unified Theory for All Forces and Matter Zeng-Bing Chen National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China (Dated: December 20, 2018) Assuming the Kaluza-Klein gravity interacting with elementary matter fermions in a (9 + 1)- dimensional spacetime (M9+1), we propose an information-complete unified theory for all forces and matter. Due to entanglement-driven symmetry breaking, the SO(9, 1) symmetry of M9+1 is broken to SO(3, 1) × SO(6), where SO(3, 1) [SO(6)] is associated with gravity (gauge fields of matter fermions) in (3+1)-dimensional spacetime (M3+1). The informational completeness demands that matter fermions must appear in three families, each having 16 independent matter fermions. Meanwhile, the fermion family space is equipped with elementary SO(3) gauge fields in M9+1, giving rise to the Higgs mechanism in M3+1 through the gauge-Higgs unification. After quantum compactification of six extra dimensions, a trinity—the quantized gravity, the three-family fermions of total number 48, and their SO(6) and SO(3) gauge fields—naturally arises in an effective theory in M3+1. Possible routes of our theory to the Standard Model are briefly discussed. PACS numbers: 04.50.+h, 12.10.-g, 04.60.Pp The tendency of unifying originally distinct physical fields (together as matter), a fact called the information- subjects or phenomena has profoundly advanced modern completeness principle (ICP). The basic state-dynamics physics. Newton’s law of universal gravitation, Maxwell’s postulate [8] is that the Universe is self-created into a theory of electromagnetism, and Einstein’s relativity state |e,ω; A..., ψ...i of all physical contents (spacetime theory are among the most outstanding examples for and matter), from no spacetime and no matter, with the such a unification.
    [Show full text]
  • Prediction in Quantum Cosmology
    Prediction in Quantum Cosmology James B. Hartle Department of Physics University of California Santa Barbara, CA 93106, USA Lectures at the 1986 Carg`esesummer school publshed in Gravitation in As- trophysics ed. by J.B. Hartle and B. Carter, Plenum Press, New York (1987). In this version some references that were `to be published' in the original have been supplied, a few typos corrected, but the text is otherwise unchanged. The author's views on the quantum mechanics of cosmology have changed in important ways from those originally presented in Section 2. See, e.g. [107] J.B. Hartle, Spacetime Quantum Mechanics and the Quantum Mechan- ics of Spacetime in Gravitation and Quantizations, Proceedings of the 1992 Les Houches Summer School, ed. by B. Julia and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. LVII, North Holland, Amsterdam (1995).) The general discussion and material in Sections 3 and 4 on the classical geom- etry limit and the approximation of quantum field theory in curved spacetime may still be of use. 1 Introduction As far as we know them, the fundamental laws of physics are quantum mechanical in nature. If these laws apply to the universe as a whole, then there must be a description of the universe in quantum mechancial terms. Even our present cosmological observations require such a description in principle, although in practice these observations are so limited and crude that the approximation of classical physics is entirely adequate. In the early universe, however, the classical approximation is unlikely to be valid. There, towards the big bang singularity, at curvatures characterized by the Planck length, 3 1 (¯hG=c ) 2 , quantum fluctuations become important and eventually dominant.
    [Show full text]
  • The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity
    entropy Article The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity Massimo Tessarotto 1,2,* and Claudio Cremaschini 2 1 Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127 Trieste, Italy 2 Research Center for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezruˇcovonám.13, CZ-74601 Opava, Czech Republic; [email protected] * Correspondence: [email protected] Abstract: An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics. Keywords: quantum gravity; space-time singular solutions; quantum regularization 1. Introduction Citation: Tessarotto, M.; The discovery of the characteristic central singularity that may characterize black holes Cremaschini, C. The Quantum (BH) is due to the genius of Karl Schwarzschild who in 1916 pointed out his famous exact Regularization of Singular Black-Hole solution [1] of Albert Einstein’s namesake field equations, soon brilliantly followed by Hans Solutions in Covariant Quantum Reissner [2] and Gunnar Nordstrôm [3] who generalized it to the case of a charged BH, Gravity.
    [Show full text]
  • Dissertation
    Eccentric binaries of compact objects in strong-field gravity Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Physikalisch-Astronomischen Fakult¨at der Friedrich-Schiller-Universit¨atJena von Dipl.-Phys. Roman Gold geboren am 10.07.1981 in Bad Soden (am Taunus) Gutachter: 1. Prof. Bernd Br¨ugmann,FSU Jena 2. Prof. Luis Lehner, Perimeter Institute Waterloo 3. Prof. Deirdre Shoemaker, Georgia Tech Atlanta Tag der Disputation: 27. September 2011 Exzentrische Bin¨arsystemekompakter Objekte im starken Gravitationsfeld Zusammenfassung In dieser Arbeit untersuchen wir die Dynamik exzentrischer Bin¨arsystemekompakter Objekte und die resultierende Gravitationswellenstrahlung im nicht-linearen Regime der Allgemeinen Relativit¨atstheorie. Hierzu l¨osenwir die Einsteinschen Feldgleich- ungen numerisch in einer 3+1 Zerlegung mit der bewegten-Punktur Methode. Wir konzentrieren uns hierbei auf spezielle Orbits, die im Zusammenhang mit nicht-stabilen Kreisbahnen entstehen, und einen rein relativistischen Effekt des Zwei- k¨orperproblems der Allgemeinen Relativit¨atstheoriedarstellen. Diese werden bes- timmt durch schnelle, quasi-zirkul¨areUml¨aufebei kleinen Abst¨anden,gefolgt von langsamen radialen Bewegung auf quasi-elliptischen Trajektorien. Auf Grund der besonderen Gestalt dieser Bahnen werden sie als "Zoom-Whirl-Orbits" bezeichnet. Wir analysieren, wie sich diese besondere Dynamik in der emittierten Gravitation- swellenstrahlung ¨außert,und inwieweit sich aus Beobachtungen dieser Strahlung auf die Eigenschaften der Orbits schließen l¨asst. Im ersten Teil betrachten wir Bin¨arsystemeSchwarzer L¨ocher. Wir f¨uhreneine umfassende Parameterstudie durch, indem wir die Anfangsexzentrizit¨atvariieren, und die entstehende Gravitationswellenform berechnen und charakterisieren. Hier befassen wir uns insbesondere mit Aspekten, die nur mit Hilfe nicht-st¨orungstheoret- ischer Methoden analysiert werden k¨onnen,und f¨urdie astrophysikalische Relevanz dieser Orbits entscheidend sind.
    [Show full text]
  • Download Report 2010-12
    RESEARCH REPORt 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Cover: Aurora borealis paintings by William Crowder, National Geographic (1947). The International Geophysical Year (1957–8) transformed research on the aurora, one of nature’s most elusive and intensely beautiful phenomena. Aurorae became the center of interest for the big science of powerful rockets, complex satellites and large group efforts to understand the magnetic and charged particle environment of the earth. The auroral visoplot displayed here provided guidance for recording observations in a standardized form, translating the sublime aesthetics of pictorial depictions of aurorae into the mechanical aesthetics of numbers and symbols. Most of the portait photographs were taken by Skúli Sigurdsson RESEARCH REPORT 2010—2012 MAX-PLANCK-INSTITUT FÜR WISSENSCHAFTSGESCHICHTE Max Planck Institute for the History of Science Introduction The Max Planck Institute for the History of Science (MPIWG) is made up of three Departments, each administered by a Director, and several Independent Research Groups, each led for five years by an outstanding junior scholar. Since its foundation in 1994 the MPIWG has investigated fundamental questions of the history of knowl- edge from the Neolithic to the present. The focus has been on the history of the natu- ral sciences, but recent projects have also integrated the history of technology and the history of the human sciences into a more panoramic view of the history of knowl- edge. Of central interest is the emergence of basic categories of scientific thinking and practice as well as their transformation over time: examples include experiment, ob- servation, normalcy, space, evidence, biodiversity or force.
    [Show full text]
  • General Relativity
    GENERALRELATIVITY t h i m o p r e i s1 17th April 2020 1 [email protected] CONTENTS 1 differential geomtry 3 1.1 Differentiable manifolds 3 1.2 The tangent space 4 1.2.1 Tangent vectors 6 1.3 Dual vectors and tensor 6 1.3.1 Tensor densities 8 1.4 Metric 11 1.5 Connections and covariant derivatives 13 1.5.1 Note on exponential map/Riemannian normal coordinates - TO DO 18 1.6 Geodesics 20 1.6.1 Equivalent deriavtion of the Geodesic Equation - Weinberg 22 1.6.2 Character of geodesic motion is sustained and proper time is extremal on geodesics 24 1.6.3 Another remark on geodesic equation using the principle of general covariance 26 1.6.4 On the parametrization of the path 27 1.7 An equivalent consideration of parallel transport, geodesics 29 1.7.1 Formal solution to the parallel transport equa- tion 31 1.8 Curvature 33 1.8.1 Torsion and metric connection 34 1.8.2 How to get from the connection coefficients to the connection-the metric connection 34 1.8.3 Conceptional flow of how to add structure on our mathematical constructs 36 1.8.4 The curvature 37 1.8.5 Independent components of the Riemann tensor and intuition for curvature 39 1.8.6 The Ricci tensor 41 1.8.7 The Einstein tensor 43 1.9 The Lie derivative 43 1.9.1 Pull-back and Push-forward 43 1.9.2 Connection between coordinate transformations and diffeomorphism 47 1.9.3 The Lie derivative 48 1.10 Symmetric Spaces 50 1.10.1 Killing vectors 50 1.10.2 Maximally Symmetric Spaces and their Unique- ness 54 iii iv co n t e n t s 1.10.3 Maximally symmetric spaces and their construc- tion
    [Show full text]
  • Realism About the Wave Function
    Realism about the Wave Function Eddy Keming Chen* Forthcoming in Philosophy Compass Penultimate version of June 12, 2019 Abstract A century after the discovery of quantum mechanics, the meaning of quan- tum mechanics still remains elusive. This is largely due to the puzzling nature of the wave function, the central object in quantum mechanics. If we are real- ists about quantum mechanics, how should we understand the wave function? What does it represent? What is its physical meaning? Answering these ques- tions would improve our understanding of what it means to be a realist about quantum mechanics. In this survey article, I review and compare several realist interpretations of the wave function. They fall into three categories: ontological interpretations, nomological interpretations, and the sui generis interpretation. For simplicity, I will focus on non-relativistic quantum mechanics. Keywords: quantum mechanics, wave function, quantum state of the universe, sci- entific realism, measurement problem, configuration space realism, Hilbert space realism, multi-field, spacetime state realism, laws of nature Contents 1 Introduction 2 2 Background 3 2.1 The Wave Function . .3 2.2 The Quantum Measurement Problem . .6 3 Ontological Interpretations 8 3.1 A Field on a High-Dimensional Space . .8 3.2 A Multi-field on Physical Space . 10 3.3 Properties of Physical Systems . 11 3.4 A Vector in Hilbert Space . 12 *Department of Philosophy MC 0119, University of California, San Diego, 9500 Gilman Dr, La Jolla, CA 92093-0119. Website: www.eddykemingchen.net. Email: [email protected] 1 4 Nomological Interpretations 13 4.1 Strong Nomological Interpretations . 13 4.2 Weak Nomological Interpretations .
    [Show full text]
  • Gravitation Without Curved Space-Time
    Gravitation Without Curved Space-time Kris Krogh Neuroscience Research Institute University of California, Santa Barbara, CA 93106, USA email: k [email protected] July 2, 2006 Abstract A quantum-mechanical theory of gravitation is presented, where the motion of particles is based on the optics of de Broglie waves. Here the large-scale geometry of the universe is inherently flat, and its age is not constrained to < 13 Gyr. While this theory agrees with the standard experimental tests of Einstein’s general relativity, it predicts a differ- ent second-order deflection of light, and measurement of the Lense- Thirring effect in the upcoming NASA experiment Gravity Probe B. arXiv:astro-ph/9910325v23 4 Jul 2006 1 1 Introduction Modern physics has two different representations of gravitation: in addition to the geometric one of Einstein’s general relativity, there is also the quantum-mechanical description. According to general relativity’s weak equivalence principle, the mo- tion of a test particle in a gravitational field is independent of its mass. However, in quantum mechanics, the motion depends intimately on particle mass. The math- ematical structures of the two representations “seem utterly incompatible,” in the words of Francis Everitt. Weinberg [1] suggests that the prevailing geometric model of gravitation “has driven a wedge between general relativity and the theory of elementary particles.” He also points out that this approach is unnecessary: Einstein and his successors have regarded the effects of a gravitational field as producing a change in the geometry of space and time. At one time it was even hoped that the rest of physics could be brought into a geometric formulation, but this hope has met with disappointment, and the geometric interpretation of the theory of gravitation has dwindled to a mere analogy, which lingers in our language in terms like “metric,” “affine connection,” and “curvature,” but is not otherwise very useful.
    [Show full text]