Structure of Constrained Systems in Lagrangian Formalism and Degree of Freedom Count
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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. -
Reduced Phase Space Quantization and Dirac Observables
Reduced Phase Space Quantization and Dirac Observables T. Thiemann∗ MPI f. Gravitationsphysik, Albert-Einstein-Institut, Am M¨uhlenberg 1, 14476 Potsdam, Germany and Perimeter Institute f. Theoretical Physics, Waterloo, ON N2L 2Y5, Canada Abstract In her recent work, Dittrich generalized Rovelli’s idea of partial observables to construct Dirac observables for constrained systems to the general case of an arbitrary first class constraint algebra with structure functions rather than structure constants. Here we use this framework and propose how to implement explicitly a reduced phase space quantization of a given system, at least in principle, without the need to compute the gauge equivalence classes. The degree of practicality of this programme depends on the choice of the partial observables involved. The (multi-fingered) time evolution was shown to correspond to an automorphism on the set of Dirac observables so generated and interesting representations of the latter will be those for which a suitable preferred subgroup is realized unitarily. We sketch how such a programme might look like for General Relativity. arXiv:gr-qc/0411031v1 6 Nov 2004 We also observe that the ideas by Dittrich can be used in order to generate constraints equivalent to those of the Hamiltonian constraints for General Relativity such that they are spatially diffeomorphism invariant. This has the important consequence that one can now quantize the new Hamiltonian constraints on the partially reduced Hilbert space of spatially diffeomorphism invariant states, just as for the recently proposed Master constraint programme. 1 Introduction It is often stated that there are no Dirac observables known for General Relativity, except for the ten Poincar´echarges at spatial infinity in situations with asymptotically flat boundary conditions. -
Theory of Systems with Constraints
Theory of systems with Constraints Javier De Cruz P´erez Facultat de F´ısica, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain.∗ Abstract: We discuss systems subject to constraints. We review Dirac's classical formalism of dealing with such problems and motivate the definition of objects such as singular and nonsingular Lagrangian, first and second class constraints, and the Dirac bracket. We show how systems with first class constraints can be considered to be systems with gauge freedom. We conclude by studing a classical example of systems with constraints, electromagnetic field I. INTRODUCTION II. GENERAL CONSIDERATIONS Consider a system with a finite number, k , of degrees of freedom is described by a Lagrangian L = L(qs; q_s) that don't depend on t. The action of the system is [3] In the standard case, to obtain the equations of motion, Z we follow the usual steps. If we use the Lagrangian for- S = dtL(qs; q_s) (1) malism, first of all we find the Lagrangian of the system, calculate the Euler-Lagrange equations and we obtain the If we apply the least action principle action δS = 0, we accelerations as a function of positions and velocities. All obtain the Euler-Lagrange equations this is possible if and only if the Lagrangian of the sys- tems is non-singular, if not so the system is non-standard @L d @L − = 0 s = 1; :::; k (2) and we can't apply the standard formalism. Then we say @qs dt @q_s that the Lagrangian is singular and constraints on the ini- tial data occur. -
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. -
Arxiv:Math-Ph/9812022 V2 7 Aug 2000
Local quantum constraints Hendrik Grundling Fernando Lledo´∗ Department of Mathematics, Max–Planck–Institut f¨ur Gravitationsphysik, University of New South Wales, Albert–Einstein–Institut, Sydney, NSW 2052, Australia. Am M¨uhlenberg 1, [email protected] D–14476 Golm, Germany. [email protected] AMS classification: 81T05, 81T10, 46L60, 46N50 Abstract We analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space–time regions. In particular we find “weak” Haag–Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag–Kastler axioms. Gupta– Bleuler electromagnetism is developed in detail as an example of a theory which satisfies the “weak” Haag–Kastler axioms but not the usual ones. This analysis is done by pure C*– algebraic means without employing any indefinite metric representations, and we obtain the same physical algebra and positive energy representation for it than by the usual means. The price for avoiding the indefinite metric, is the use of nonregular representations and complex valued test functions. We also exhibit the precise connection with the usual indefinite metric representation. We conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the generators of the translation group, we also find a “weak” version, and show that the Gupta–Bleuler example satisfies it. arXiv:math-ph/9812022 v2 7 Aug 2000 1 Introduction In many quantum field theories there are constraints consisting of local expressions of the quantum fields, generally written as a selection condition for the physical subspace (p). -
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 -
Lagrangian Constraint Analysis of First-Order Classical Field Theories with an Application to Gravity
PHYSICAL REVIEW D 102, 065015 (2020) Lagrangian constraint analysis of first-order classical field theories with an application to gravity † ‡ Verónica Errasti Díez ,1,* Markus Maier ,1,2, Julio A. M´endez-Zavaleta ,1, and Mojtaba Taslimi Tehrani 1,§ 1Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany 2Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 Munich, Germany (Received 19 August 2020; accepted 26 August 2020; published 21 September 2020) We present a method that is optimized to explicitly obtain all the constraints and thereby count the propagating degrees of freedom in (almost all) manifestly first-order classical field theories. Our proposal uses as its only inputs a Lagrangian density and the identification of the a priori independent field variables it depends on. This coordinate-dependent, purely Lagrangian approach is complementary to and in perfect agreement with the related vast literature. Besides, generally overlooked technical challenges and problems derived from an incomplete analysis are addressed in detail. The theoretical framework is minutely illustrated in the Maxwell, Proca and Palatini theories for all finite d ≥ 2 spacetime dimensions. Our novel analysis of Palatini gravity constitutes a noteworthy set of results on its own. In particular, its computational simplicity is visible, as compared to previous Hamiltonian studies. We argue for the potential value of both the method and the given examples in the context of generalized Proca and their coupling to gravity. The possibilities of the method are not exhausted by this concrete proposal. DOI: 10.1103/PhysRevD.102.065015 I. INTRODUCTION In this work, we focus on singular classical field theories It is hard to overemphasize the importance of field theory that are manifestly first order and analyze them employing in high-energy physics. -
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. -
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. -
On Gauge Fixing and Quantization of Constrained Hamiltonian Systems
BEFE IC/89/118 INTERIMATIOIMAL CENTRE FOR THEORETICAL PHYSICS ON GAUGE FIXING AND QUANTIZATION OF CONSTRAINED HAMILTONIAN SYSTEMS Omer F. Dayi INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1989 MIRAMARE-TRIESTE IC/89/118 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Quantization of a classical Hamiltonian system can be achieved by the canonical quantization method [1]. If we ignore the ordering problems, it consists of replacing the classical Poisson brackets by quantum commuta- tors when classically all the states on the phase space are accessible. This is no longer correct in the presence of constraints. An approach due to Dirac ON GAUGE FIXING AND QUANTIZATION [2] is widely used for quantizing the constrained Hamiltonian systems |3|, OF CONSTRAINED HAMILTONIAN SYSTEMS' [4]. In spite of its wide use there is confusion in the application of it in the physics literature. Obtaining the Hamiltonian which Bhould be used in gauge theories is not well-understood. This derives from the misunder- standing of the gauge fixing procedure. One of the aims of this work is to OmerF.Dayi" clarify this issue. The constraints, which are present when there are some irrelevant phase International Centre for Theoretical Physics, Trieste, Italy. space variables, can be classified as first and second class. Second class constraints are eliminated by altering some of the original Poisson bracket relations which is equivalent to imply the vanishing of the constraints as strong equations by using the Dirac procedure [2] or another method |Sj. ABSTRACT Vanishing of second class constraints strongly leads to a reduction in the number of the phase apace variables.