A First Class Constraint Generates Not a Gauge Transformation, but a Bad Physical Change: the Case of Electromagnetism∗
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Immirzi Parameter Without Immirzi Ambiguity: Conformal Loop Quantization of Scalar-Tensor Gravity
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Aberdeen University Research Archive PHYSICAL REVIEW D 96, 084011 (2017) Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity † Olivier J. Veraguth* and Charles H.-T. Wang Department of Physics, University of Aberdeen, King’s College, Aberdeen AB24 3UE, United Kingdom (Received 25 May 2017; published 5 October 2017) Conformal loop quantum gravity provides an approach to loop quantization through an underlying conformal structure i.e. conformally equivalent class of metrics. The property that general relativity itself has no conformal invariance is reinstated with a constrained scalar field setting the physical scale. Conformally equivalent metrics have recently been shown to be amenable to loop quantization including matter coupling. It has been suggested that conformal geometry may provide an extended symmetry to allow a reformulated Immirzi parameter necessary for loop quantization to behave like an arbitrary group parameter that requires no further fixing as its present standard form does. Here, we find that this can be naturally realized via conformal frame transformations in scalar-tensor gravity. Such a theory generally incorporates a dynamical scalar gravitational field and reduces to general relativity when the scalar field becomes a pure gauge. In particular, we introduce a conformal Einstein frame in which loop quantization is implemented. We then discuss how different Immirzi parameters under this description may be related by conformal frame transformations and yet share the same quantization having, for example, the same area gaps, modulated by the scalar gravitational field. DOI: 10.1103/PhysRevD.96.084011 I. -
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. -
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). -
An Invitation to the New Variables with Possible Applications
An Invitation to the New Variables with Possible Applications Norbert Bodendorfer and Andreas Thurn (work by NB, T. Thiemann, AT [arXiv:1106.1103]) FAU Erlangen-N¨urnberg ILQGS, 4 October 2011 N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 1 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the New Variables ILQGS, 4 October 2011 2 Plan of the talk 1 Why Higher Dimensional Loop Quantum (Super-)Gravity? 2 Review: Hamiltonian Formulations of General Relativity ADM Formulation Extended ADM I Ashtekar-Barbero Formulation Extended ADM II 3 The New Variables Hamiltonian Viewpoint Comparison with Ashtekar-Barbero Formulation Lagrangian Viewpoint Quantisation, Generalisations 4 Possible Applications of the New Variables Solutions to the Simplicity Constraint Canonical = Covariant Formulation? Supersymmetry Constraint Black Hole Entropy Cosmology AdS / CFT Correspondence 5 Conclusion N. Bodendorfer, A. Thurn (FAU Erlangen) An Invitation to the -
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. -
Asymptotically Flat Boundary Conditions for the U(1)3 Model for Euclidean Quantum Gravity
universe Article Asymptotically Flat Boundary Conditions for the U(1)3 Model for Euclidean Quantum Gravity Sepideh Bakhoda 1,2,* , Hossein Shojaie 1 and Thomas Thiemann 2,* 1 Department of Physics, Shahid Beheshti University, Tehran 1983969411, Iran; [email protected] 2 Institute for Quantum Gravity, FAU Erlangen—Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany * Correspondence: [email protected] (S.B.); [email protected] (T.T.) 3 Abstract: A generally covariant U(1) gauge theory describing the GN ! 0 limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we the study boundary conditions and asymptotic symmetries of the U(1)3 model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that the non-Abelian part of the SU(2) Gauss constraint, which is absent in the U(1)3 model, plays a crucial role in obtaining boost and rotation generators. Keywords: asymptotically flat boundary conditions; classical and quantum gravity; U(1)3 model; asymptotic charges Citation: Bakhoda, S.; Shojaie, H.; 1. Introduction Thiemann, T. Asymptotically Flat In the framework of the Ashtekar variables in terms of which General Relativity (GR) Boundary Conditions for the U(1)3 is formulated as a SU(2) gauge theory [1–3], attempts to find an operator corresponding to Model for Euclidean Quantum the Hamiltonian constraint of the Lorentzian vacuum canonical GR led to the result [4] that Gravity. -
Cosmological Plebanski Theory
General Relativity and Gravitation (2011) DOI 10.1007/s10714-009-0783-0 RESEARCHARTICLE Karim Noui · Alejandro Perez · Kevin Vandersloot Cosmological Plebanski theory Received: 17 September 2008 / Accepted: 2 March 2009 c Springer Science+Business Media, LLC 2009 Abstract We consider the cosmological symmetry reduction of the Plebanski action as a toy-model to explore, in this simple framework, some issues related to loop quantum gravity and spin-foam models. We make the classical analysis of the model and perform both path integral and canonical quantizations. As for the full theory, the reduced model admits two disjoint types of classical solutions: topological and gravitational ones. The quantization mixes these two solutions, which prevents the model from being equivalent to standard quantum cosmology. Furthermore, the topological solution dominates at the classical limit. We also study the effect of an Immirzi parameter in the model. Keywords Loop quantum gravity, Spin-foam models, Plebanski action 1 Introduction Among the issues which remain to be understood in Loop Quantm Gravity (LQG) [1; 2; 3], the problem of the dynamics is surely one of the most important. The regularization of the Hamiltonian constraint proposed by Thiemann [4] was a first promising attempt towards a solution of that problem. However, the technical dif- ficulties are such that this approach has not given a solution yet. Spin Foam mod- els [5] is an alternative way to explore the question: they are supposed to give a combinatorial expression of the Path integral of gravity, and should allow one to compute transition amplitudes between states of quantum gravity, or equivalently to compute the dynamics of a state. -
Hamiltonian Constraint Analysis of Vector Field Theories with Spontaneous Lorentz Symmetry Breaking
Colby College Digital Commons @ Colby Honors Theses Student Research 2008 Hamiltonian constraint analysis of vector field theories with spontaneous Lorentz symmetry breaking Nolan L. Gagne Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Physics Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Gagne, Nolan L., "Hamiltonian constraint analysis of vector field theories with spontaneous Lorentz symmetry breaking" (2008). Honors Theses. Paper 92. https://digitalcommons.colby.edu/honorstheses/92 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. 1 Hamiltonian Constraint Analysis of Vector Field Theories with Spontaneous Lorentz Symmetry Breaking Nolan L. Gagne May 17, 2008 Department of Physics and Astronomy Colby College 2008 1 Abstract Recent investigations of various quantum-gravity theories have revealed a variety of possible mechanisms that lead to Lorentz violation. One of the more elegant of these mechanisms is known as Spontaneous Lorentz Symmetry Breaking (SLSB), where a vector or tensor field acquires a nonzero vacuum expectation value. As a consequence of this symmetry breaking, massless Nambu-Goldstone modes appear with properties similar to the photon in Electromagnetism. This thesis considers the most general class of vector field theo- ries that exhibit spontaneous Lorentz violation{known as bumblebee models{and examines their candidacy as potential alternative explanations of E&M, offering the possibility that Einstein-Maxwell theory could emerge as a result of SLSB rather than of local U(1) gauge invariance. -
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. -
Characteristic Hypersurfaces and Constraint Theory
Characteristic Hypersurfaces and Constraint Theory Thesis submitted for the Degree of MSc by Patrik Omland Under the supervision of Prof. Stefan Hofmann Arnold Sommerfeld Center for Theoretical Physics Chair on Cosmology Abstract In this thesis I investigate the occurrence of additional constraints in a field theory, when formulated in characteristic coordinates. More specifically, the following setup is considered: Given the Lagrangian of a field theory, I formulate the associated (instantaneous) Hamiltonian problem on a characteristic hypersurface (w.r.t. the Euler-Lagrange equations) and find that there exist new constraints. I then present conditions under which these constraints lead to symplectic submanifolds of phase space. Symplecticity is desirable, because it renders Hamiltonian vector fields well-defined. The upshot is that symplecticity comes down to analytic rather than algebraic conditions. Acknowledgements After five years of study, there are many people I feel very much indebted to. Foremost, without the continuous support of my parents and grandparents, sister and aunt, for what is by now a quarter of a century I would not be writing these lines at all. Had Anja not been there to show me how to get to my first lecture (and in fact all subsequent ones), lord knows where I would have ended up. It was through Prof. Cieliebaks inspiring lectures and help that I did end up in the TMP program. Thank you, Robert, for providing peculiar students with an environment, where they could forget about life for a while and collectively find their limits, respectively. In particular, I would like to thank the TMP lonely island faction. -
Structure of Constrained Systems in Lagrangian Formalism and Degree of Freedom Count
Structure of Constrained Systems in Lagrangian Formalism and Degree of Freedom Count Mohammad Javad Heidaria,∗ Ahmad Shirzada;b† a Department of Physics, Isfahan University of Technology b School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5531, Tehran, Iran March 31, 2020 Abstract A detailed program is proposed in the Lagrangian formalism to investigate the dynam- ical behavior of a theory with singular Lagrangian. This program goes on, at different levels, parallel to the Hamiltonian analysis. In particular, we introduce the notions of first class and second class Lagrangian constraints. We show each sequence of first class constraints leads to a Neother identity and consequently to a gauge transformation. We give a general formula for counting the dynamical variables in Lagrangian formalism. As the main advantage of Lagrangian approach, we show the whole procedure can also be per- formed covariantly. Several examples are given to make our Lagrangian approach clear. 1 Introduction Since the pioneer work of Dirac [1] and subsequent forerunner papers (see Refs. [2,3,4] arXiv:2003.13269v1 [physics.class-ph] 30 Mar 2020 for a comprehensive review), people are mostly familiar with the constrained systems in the framework of Hamiltonian formulation. The powerful tool in this framework is the algebra of Poisson brackets of the constraints. As is well-known, the first class constraints, which have weakly vanishing Poisson brackets with all constraints, generate gauge transformations. ∗[email protected] †[email protected] 1 However, there is no direct relation, in the general case, to show how they do this job.