Energy-Momentum Tensor in Quantum Field Theory
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Minkowski Space-Time: a Glorious Non-Entity
DRAFT: written for Petkov (ed.), The Ontology of Spacetime (in preparation) Minkowski space-time: a glorious non-entity Harvey R Brown∗ and Oliver Pooley† 16 March, 2004 Abstract It is argued that Minkowski space-time cannot serve as the deep struc- ture within a “constructive” version of the special theory of relativity, contrary to widespread opinion in the philosophical community. This paper is dedicated to the memory of Jeeva Anandan. Contents 1 Einsteinandthespace-timeexplanationofinertia 1 2 Thenatureofabsolutespace-time 3 3 The principle vs. constructive theory distinction 4 4 The explanation of length contraction 8 5 Minkowskispace-time: thecartorthehorse? 12 1 Einstein and the space-time explanation of inertia It was a source of satisfaction for Einstein that in developing the general theory of relativity (GR) he was able to eradicate what he saw as an embarrassing defect of his earlier special theory (SR): violation of the action-reaction principle. Leibniz held that a defining attribute of substances was their both acting and being acted upon. It would appear that Einstein shared this view. He wrote in 1924 that each physical object “influences and in general is influenced in turn by others.”1 It is “contrary to the mode of scientific thinking”, he wrote earlier in 1922, “to conceive of a thing. which acts itself, but which cannot be acted upon.”2 But according to Einstein the space-time continuum, in both arXiv:physics/0403088v1 [physics.hist-ph] 17 Mar 2004 Newtonian mechanics and special relativity, is such a thing. In these theories ∗Faculty of Philosophy, University of Oxford, 10 Merton Street, Oxford OX1 4JJ, U.K.; [email protected] †Oriel College, Oxford OX1 4EW, U.K.; [email protected] 1Einstein (1924, 15). -
The Dirac-Schwinger Covariance Condition in Classical Field Theory
Pram~na, Vol. 9, No. 2, August 1977, pp. 103-109, t~) printed in India. The Dirac-Schwinger covariance condition in classical field theory K BABU JOSEPH and M SABIR Department of Physics, Cochin University, Cochin 682 022 MS received 29 January 1977; revised 9 April 1977 Abstract. A straightforward derivation of the Dirac-Schwinger covariance condition is given within the framework of classical field theory. The crucial role of the energy continuity equation in the derivation is pointed out. The origin of higher order deri- vatives of delta function is traced to the presence of higher order derivatives of canoni- cal coordinates and momenta in the energy density functional. Keywords. Lorentz covariance; Poincar6 group; field theory; generalized mechanics. 1. Introduction It has been stated by Dirac (1962) and by Schwinger (1962, 1963a, b, c, 1964) that a sufficient condition for the relativistic covariance of a quantum field theory is the energy density commutator condition [T°°(x), r°°(x')] = (r°k(x) + r°k(x ') c3k3 (x--x') + (bk(x) + bk(x ') c9k8 (x--x') -+- (cktm(x) cktm(x ') OkOtO,3 (X--X') q- ... (1) where bk=O z fl~k and fl~--fl~k = 0mY,,kt and the energy density is that obtained by the Belinfante prescription (Belinfante 1939). For a class of theories that Schwinger calls ' local' the Dirac-Schwinger (DS) condition is satisfied in the simplest form with bk=e~,~r,, = ... = 0. Spin s (s<~l) theories belong to this class. For canonical theories in which simple canonical commutation relations hold Brown (1967) has given two new proofs of the DS condition. -
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. -
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. -
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. -
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. -
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 -
Interpreting Supersymmetry
Interpreting Supersymmetry David John Baker Department of Philosophy, University of Michigan [email protected] October 7, 2018 Abstract Supersymmetry in quantum physics is a mathematically simple phenomenon that raises deep foundational questions. To motivate these questions, I present a toy model, the supersymmetric harmonic oscillator, and its superspace representation, which adds extra anticommuting dimensions to spacetime. I then explain and comment on three foundational questions about this superspace formalism: whether superspace is a sub- stance, whether it should count as spatiotemporal, and whether it is a necessary pos- tulate if one wants to use the theory to unify bosons and fermions. 1 Introduction Supersymmetry{the hypothesis that the laws of physics exhibit a symmetry that transforms bosons into fermions and vice versa{is a long-standing staple of many popular (but uncon- firmed) theories in particle physics. This includes several attempts to extend the standard model as well as many research programs in quantum gravity, such as the failed supergravity program and the still-ascendant string theory program. Its popularity aside, supersymmetry (SUSY for short) is also a foundationally interesting hypothesis on face. The fundamental equivalence it posits between bosons and fermions is prima facie puzzling, given the very different physical behavior of these two types of particle. And supersymmetry is most naturally represented in a formalism (called superspace) that modifies ordinary spacetime by adding Grassmann-valued anticommuting coordinates. It 1 isn't obvious how literally we should interpret these extra \spatial" dimensions.1 So super- symmetry presents us with at least two highly novel interpretive puzzles. Only two philosophers of science have taken up these questions thus far. -
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. -
5 Lorentz Covariance of the Dirac Equation
5 Lorentz Covariance of the Dirac Equation We will set ~ = c = 1 from now on. In E&M, we write down Maxwell's equations in a given inertial frame, x; t, with the electric and magnetic fields E; B. Maxwell's equations are covariant with respecct to Lorentz transformations, i.e., in a new Lorentz frame, x0; t0, the equations have the same form, but the fields E0(x0; t0); B0(x0; t0) are different. Similarly, Dirac equation is Lorentz covariant, but the wavefunction will change when we make a Lorentz transformation. Consider a frame F with an observer O and coordinates xµ. O describes a particle by the wavefunction (xµ) which obeys @ iγµ m (xµ): (97) @xµ − In another inertial frame F 0 with an observer O0 and coordinates x0ν given by 0ν ν µ x = Λ µx ; (98) O0 describes the same particle by 0(x0ν ) and 0(x0ν ) satisfies @ iγν m 0(x0ν ): (99) @x0ν − Lorentz covariance of the Dirac equation means that the γ matrices are the same in both frames. What is the transformation matrix S which takes to 0 under the Lorentz trans- formation? 0(Λx) = S (x): (100) Applying S to Eq. (97), @ iSγµS−1 S (xµ) mS (xµ) = 0 @xµ − @ iSγµS−1 0(x0ν) m 0(x0ν) = 0: (101) ) @xµ − Using @ @ @x0ν @ = = Λν ; (102) @xµ @x0ν @xµ µ @x0ν we obtain @ iSγµS−1Λν 0(x0ν ) m 0(x0ν ) = 0: (103) µ @x0ν − Comparing it with Eq. (99), we need µ −1 ν ν µ −1 −1 µ ν Sγ S Λ µ = γ or equivalently Sγ S = Λ ν γ : (104) 16 We will write down the form of the S matrix without proof. -
Quantum Theory of Gravity. II. the Manifestly Covariant Theory* BRVCE S
I'HVSI CAI REVIEW VOLUME 162, NUMBER 5 2& OCTOBER 1967 Quantum Theory of Gravity. II. The Manifestly Covariant Theory* BRVCE S. DEWITT Institute for Advanced Study, Princeton, New Jersey Cwd Department of Physics, University of North Carolina, Chape/ Hill, North Caroiinat (Received 25 July 1966; revised manuscript received 9 January 1967) Contrary to the situation which holds for the canonical theory described in the first paper of this series, there exists at present no tractable pure operator language on which to base a manifestly covariant quantum theory of gravity. One must construct the theory by analogy with conventional 5-matrix theory, using the c-number language of Feynman amplitudes when nothing else is available. The present paper undertakes this construction. It begins at an elementary level with a treatment of the propagation of small disturbances on a classical background. The classical background plays a fundamental role throughout, both as a technical instrument for probing the vacuum (i.e., analyzing virtual processes) and as an arbitrary fiducial point for the quantum fluctuations. The problem of the quantized light cone is discussed in a preliminary way, and the formal structure of the invariance group is displayed. A condensed notation is adopted which permits the Yang-Mills field to be studied simultaneously with the gravitational field. Generally covariant Green s functions are introduced through the imposition of covariant supplementary conditions on small dis- turbances. The transition from the classical to the quantum theory is made via the Poisson bracket of Peierls. Commutation relations for the asymptotic fields are obtained and used to define the incoming and outgoing states. -
General Relativity 2020–2021 1 Overview
N I V E R U S E I T H Y T PHYS11010: General Relativity 2020–2021 O H F G E R D John Peacock I N B U Room C20, Royal Observatory; [email protected] http://www.roe.ac.uk/japwww/teaching/gr.html Textbooks These notes are intended to be self-contained, but there are many excellent textbooks on the subject. The following are especially recommended for background reading: Hobson, Efstathiou & Lasenby (Cambridge): General Relativity: An introduction for Physi- • cists. This is fairly close in level and approach to this course. Ohanian & Ruffini (Cambridge): Gravitation and Spacetime (3rd edition). A similar level • to Hobson et al. with some interesting insights on the electromagnetic analogy. Cheng (Oxford): Relativity, Gravitation and Cosmology: A Basic Introduction. Not that • ‘basic’, but another good match to this course. D’Inverno (Oxford): Introducing Einstein’s Relativity. A more mathematical approach, • without being intimidating. Weinberg (Wiley): Gravitation and Cosmology. A classic advanced textbook with some • unique insights. Downplays the geometrical aspect of GR. Misner, Thorne & Wheeler (Princeton): Gravitation. The classic antiparticle to Weinberg: • heavily geometrical and full of deep insights. Rather overwhelming until you have a reason- able grasp of the material. It may also be useful to consult background reading on some mathematical aspects, especially tensors and the variational principle. Two good references for mathematical methods are: Riley, Hobson and Bence (Cambridge; RHB): Mathematical Methods for Physics and Engi- • neering Arfken (Academic Press): Mathematical Methods for Physicists • 1 Overview General Relativity (GR) has an unfortunate reputation as a difficult subject, going back to the early days when the media liked to claim that only three people in the world understood Einstein’s theory.