DOCSLIB.ORG

Advanced Cosmology : Primordial Non-Gaussianities

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Cosmology : Primordial Non-Gaussianities Advanced Cosmology : Primordial non-Gaussianities Eugene A. Lim This is a set of lectures given to the Cambridge DAMTP Math Tripos 2012 Advanced Cosmology Class (Lent Term). What it really needs are figures right now. I. GAUSSIAN RANDOM FIELDS AND CHARACTERIZATION OF NON-GAUSSIAN FIELDS Consider a field f(x) living on some space S, where x is a coordinate on S. We can Fourier transform this Z d3k Z f(x) = f(k)eik·x ; f(k) = d3xf(x)e−ik·x (1) (2π)3 where, without loss of generality, we can parameterize the Fourier coefficients as f(k) = ak + ibk; (2) p 2 2 with amplitude jf(k)j = ak + bk. Reality of f(x) imposes ak = a−k , bk = −b−k. In words, for a given configuration in real space f(x), a set of real numbers (ak; bk), parameterized by the vector k, completely (and uniquely) describe it. Different configurations of f(x) are described by different set of numbers. Suppose now we want to randomly generate a field configuration f(x), then one way to do it is to prescribe a probability distribution function for the set (ak; bk). If, furthermore, we want f(x) to be a Random Gaussian Field configuration, then this put a tight constraint on what is the PDF for (ak; bk). Let's begin by studying the PDF for one particular mode k. Definition : A Gaussian distribution for a mode is one such that ak and bk is drawn from the following Gaussian distribution in k space with zero mean, and k-dependent variance σk 2 2 1 ak + bk P (ak; bk) = 2 exp − 2 : (3) πσk σk where we have normalized this distribution such that Z 1 Z 1 1 2 2 dak dbk 2 exp[−|f(k)j /σk] = 1: (4) −∞ −∞ πσk Note that the integral is over all possible values of ak and bk for the mode k. The double Gaussian integral results in 2 p 2 the normalization 1/πσk instead of 1/πσk. Generalizing to all the modes, then the formal definition of the PDF is via the functional 1 2 2 P [f(k)] = 2 exp[−|f(k)j /σk]: (5) πσk We have made quite a large leap between Eqn. (3) and Eqn. (5), so let's decode a bit. A functional, as you may have learned, is a map that eats a function, and spits out a scalar field. Eqn. (5) is the just high brow way of saying that all the fields f(k) are drawn from a distribution that obeys Eqn. (3). 2 2 Notice that, in general we can have a directional dependence in the variance, i.e. σk instead of σk. But if we assume statistical isotropy, then the variance becomes just a function of the amplitude k { this also of course means that we can have non-isotropic but still Random Gaussian fields. Now we encounter the first instance of the much abused angled brackets : h i. What is hQi? Here what we are doing is \taking the expectation value" of some observable Q over an ensemble of possible realizations of this observable. Consider the following simple example. A plane drops many parachutists from the sky, who will land on a line x on the ground. The parachutists are scattered around some normalized distribution ρ(x). The expectation value of x is then defined to be Z 1 hxi = xρ(x)dx: (6) −∞ 2 In words, the expectation value is the average of a large (formally infinite) number of drawings from this distribution, i.e. our \best" guess value. If you like, for example, ρ(x) could be some normalized Gaussian distribution with mean x0 and variance σ r 1 ρ(x) = exp[−(x − x )2/σ2]: (7) πσ2 0 We can also calculate the expectation value of any function of x, say g(x) i.e. Z 1 hg(x)i = g(x)ρ(x)dx: (8) −∞ Given a functional distribution P [φ(k)] for some scalar field φ(k), then we can generalize the idea : the expectation value of finding the functional of some field configuration φ(k), Q[φ(k)], is given by (remember, a functional spits out a scalar) Z hQ[φ(k)]i = Dφ Q[φ(k)]P [φ(k)] (9) where the integral Dφ is over all possible field configurations in k-space. (Of course, here we are working in fourier space { in general it can be any space that φ lives in, including real/configuration space x). With all these definitions, we can now define the notion of a Gaussian Random Field: Let P [f(k)] be a PDF of the form Eqn. (5). Such a distribution is called Gaussian Random, and such a field f(k) is called a Gaussian Random Field. The expectation value for some functional Q[f(k)] is given by Z Z 1 2 2 2 hQ[f(k)]i = Πk dak dbkQ[f(k)] 2 exp[−(ak + bk)/σk]: (10) πσk R R Note that the integral over all possible configurations has become Df ! Πk dak dbk, since ak and bk parameterize the configurations. Sometimes you hear the words \both the amplitudes and the phases of f(k) are drawn from a 2 2 2 2 Gaussian PDF" with variance σk. Now you can see this is clearly wrong : jf(k)j = ak + bk, so for each mode in a GRF, the amplitude is drawn from a Gaussian PDF while the phase is essentially drawn from a flat distribution { this is kinda obvious since you can't really define a Gaussian (with infinite support) over a compact phase, but these words are uttered with such religious fervor that we tend to repeat it without thinking. Despite the scary looking equation Eqn. (10), with all the integrals and product sums and all, it is actually very easy to use it. Consider the simple example Q[f(k)] = akak0 , then Z Z 1 2 2 2 haqaq0 i = Πk dak dbk aqaq0 2 exp[−(ak + bk)/σk] πσk σ2 = k δ(q + q0) + q0 $ −q: (11) 2 The point is that since the distribution is even under both ak and bk, all odd products of a's and b's vanish while, except for k = q and k0 = q0, the rest are simply trivial Gaussian integrals. Even-ness means that q = q0 gives us the delta function (recall that ak = a−k by reality). Let's do a simple but important example : what is the expectation of a two-point correlation function1 Q[f(k)] = f(k)f(k0): 0 2 0 hf(k)f(k )i = hakak0 i − hbkbk0 i = σkδ(k + k ) (12) where we have used the fact that the cross terms vanishes and reality imposes bk = −b−k and Z 2 3 0 σk 0 ha a 0 i = hb b 0 i = d k a a P (a ; b )δ(k + k ) = δ(k + k ) (13) k k k −k k k k k 2 1 Recall that the expectation value of any variable x or any product of variables f(x), given its PDF P (x), is hf(x)i = R dxf(x)P (x). 3 This definition carries through to configuration space : to compute something like the power spectrum hf(x)f(x0)i, Fourier transform f(x), and plug into Eqn. (10). Let's do a famous example Z 3 3 d k1 d k1 hf(x )f(x )i = eik1·x1+ik2·x2 (ha a i − hb b i) (14) 1 2 (2π)3 (2π)3 k1 k2 k1 k2 Z d3k d3k 1 1 ik1·x1+ik2·x2 2 = e δ(k1 + k2)σ (15) (2π)3 (2π)3 k1 Z d3k σ2 = eik·(x1−x2) k (16) (2π)3 (2π)3 which is of course the two-point correlation function in real space. You probably have seen Eqn. (16) { it is the 2 3 definition of the power spectrum P (k) ≡ σk=(2π) or two-point correlation function. σk is a k dependent power spectrum, and tells us about the amplitude of the correlations at k. Let us state a few famous true-facts using our high-brow tool: • Scale Invariance :A rescaling is defined to be x ! λx where λ > 0 is some constant. Then a scale invariant power spectrum obeys the following relation hf(x)f(x0)i = hf(λx)f(λx0)i: (17) Using δ(λ(k + k0)) = λ−3δ(k + k0), then it follows that for a power spectrum to be scale invariant, it must obey σ2 1 P (k) = k / : (18) (2π)3 k3 This should be familiar to you. 2 • White Noise : On the other hand, if σk = const for all k, the spectrum is known as white noise. Note it is not scale-invariant! • Correlations vs Gaussianity : Notice that Gaussianity is not a statement about two-point correlations! For example, if I take a random Gaussian sky (say the CMB), and then rearrange the cold and hot spots in such a way that it looks like Mickey Mouse2, the sky will then be highly correlated in a very Mickey Mouse way, but it will still be completely Gaussian random { what I have changed in my magical rearrangment is the power 2 spectrum σk, not the underlying distribution P [f(k)]. Mickey Mouse is of course not very isotropic, so hence the directional dependent power spectrum. It is clear that since the Gaussian PDF Eqn. (3) is even under parity around the mean 0, any odd expectation value vanishes hf(k1)f(k2)f(k3)i = 0 (19) while any even expectation value is simply a product of the respective power spectra which we can compute, for example for a 4-pt function hf(k )f(k )f(k )f(k )i = σ2 σ2 δ(k + k )δ(k + k ) + 1 $ 3 + 1 $ 4: (20) 1 2 3 4 k1 k3 1 2 3 4 You can do the algebra here to convince yourself (Example Sheet), but this kind of \contraction" algebra occurs very often so it's good to practice doing the combinatorics.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • An Extension of the Dirac and Gotay-Nester Theories of Constraints for Dirac Dynamical Systems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Servicio de Difusión de la Creación Intelectual An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems Hern´anCendraa;1, Mar´ıaEtchechouryb and Sebasti´anJ. Ferraroc a Departamento de Matem´atica Universidad Nacional del Sur, Av. Alem 1253 8000 Bah´ıaBlanca and CONICET, Argentina. [email protected] b Laboratorio de Electr´onicaIndustrial, Control e Instrumentaci´on, Facultad de Ingenier´ıa,Universidad Nacional de La Plata and Departamento de Matem´atica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata. CC 172, 1900 La Plata, Argentina. [email protected] c Departamento de Matem´aticaand Instituto de Matem´aticaBah´ıaBlanca Universidad Nacional del Sur, Av. Alem 1253 8000 Bah´ıaBlanca, and CONICET, Argentina [email protected] Abstract This paper extends the Gotay-Nester and the Dirac theories of constrained systems in order to deal with Dirac dynamical systems in the integrable case. In- tegrable Dirac dynamical systems are viewed as constrained systems where the constraint submanifolds are foliated, the case considered in Gotay-Nester theory being the particular case where the foliation has only one leaf. A Constraint Al- gorithm for Dirac dynamical systems (CAD), which extends the Gotay-Nester algorithm, is developed. Evolution equations are written using a Dirac bracket adapted to the foliations and an abridged total energy which coincides with the total Hamiltonian in the particular case considered by Dirac. The interesting example of LC circuits is developed in detail. The paper emphasizes the point of view that Dirac and Gotay-Nester theories are dual and that using a com- bination of results from both theories may have advantages in dealing with a given example, rather than using systematically one or the other.
    [Show full text]
  • REFER Hivu:- IC/89/61 INTERNATIONAL CENTRE for THEORETICAL PHYSICS
    REFER hivU:- IC/89/61 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THEORY OF SINGULAR LAGRANGIANS Jose F. Carinena INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/89/61 1. INTRODUCTION International Atomic Energy Agency There are different approaches in Classical Mechanics and one usually speaks of New- tonian, Lagrangian and Hamiltonian Mechanics. The main guide for Newtonian mechan- and ics is the determinism principle, according to which the knowledge of the positions and United Nations Educational Scientific and Cultural Organization velocities of the points of a mechanical system at a fixed time determines their future positions and velocities [1]. The idea is then to use a system of second order differential INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS equations in normal form i- = /U*J>), (l.i) because the theorem of existence and uniqueness of the solution of such systems fits well to the deterrninism principle. In the following, for the sake of simplicity, we will restrict ourselves to the study of cases in which the functions /' do not depend on the time t. Such a study may be reduced to that of an associated Erst order system THEORY OF SINGULAR LAGRANGIANS • (1.2) Jose F. Carinena ** International Centre for Theoretical Physics, Trieste, Italy. and these equations can be geometrically interpreted as a set of differential equations locally determining the integral curves of a vector field (1.3) which is a special kind of vector field in the tangent bundle of the configuration space, to be studied later on. The introduction of the Lagrangian mechanics is based on the observation that, at least for conservative systems, the equations of motion may also be Been as the Euler equations determining the solution curves of a variational problem [1-3]; the action J Ldi, with L = T ~ V, is extremal for the actual path when compared with other fixed-endpoint paths.
    [Show full text]
  • Symmetry, Ontology and the Problem of Time on the Interpretation and Quantisation of Canonical Gravity
    UNIVERSITY OF SYDNEY DOCTORAL THESIS Symmetry, Ontology and the Problem of Time On the Interpretation and Quantisation of Canonical Gravity Karim P.Y. THEBAULT´ Declaration and Copyright This thesis is an account of research undertaken initially undertaken between August 2008 and August 2011 at the Centre for Time, Department of Philosophy, School of Philosoph- ical and Historical Inquiry, University of Sydney, Australia. Subsequent changes have been in this the final text in line with recommendations of the examiners. Except where acknowledged in the customary manner, the material presented in this thesis is, to the best of my knowledge, original and has not been submitted in whole or part for a degree in any university. Sections of this thesis have been reproduced in papers for The British Journal for the Philosophy of Science (Thebault´ (2012)), Symmetry (Thebault´ (2011)), and Foundations of Physics (Gryb and Thebault´ (2012)). Where relevant, the copyright rests with the journal publishers. Otherwise it is retained by the author. Karim P. Y. Thebault´ July, 2012 i Preface The true path is along a rope, not a rope suspended way up in the air, but rather only just over the ground. It seems more like a tripwire than a tightrope. Franz Kafka The cause of our confusion is often that the solution to the problem at hand is so near to us that we cannot help but trip over it, and then assume it to be part of the problem itself. So, it will be argued, is the case of the problem of time in canonical gravity. Moreover, as we shall see, the realisation essential to understanding the true role of time in canonical gravity is that we have failed to notice that the solutions – in the sense of the true dynamical paths – lie in precisely the direction that is perpendicular to where we are accustomed to looking.
    [Show full text]
  • A BRST Primer*
    SLAC-PUB-4422 October 1987 T ” .. -. A BRST Primer* DENNIS NEMESXIAN~KY’ Department of Physics University of Southern California Los Angeles, California 90089 - and Deparment of Physics University of Maryland College Park, Maryland, 20’742 - and MARVIN WEINSTEIN Stanford Linear Accelerator Center _n. Stanford University, Stanford, California, 94sO5 -.._ - ;. Submitted to Annals of Physics _ I* - - _ * Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Supported by the Department of Energy under grant DE-FG03-84ER-40168 $ Supported by the National Science Foundation under grant NSF-PHY-86-05207 . ABSTRACT We develop a Hamiltonian formulation of the BRST method for quantizing constrained systems. The rigid rotor is studied in detail and the similarity of this simple quantum system to a gauge theory is explicitly demonstrated. The system is quantized as a gauge-theory and then the similarity between BRST and Gupta- Bleuler approach is displayed. We also apply our formalism to true gauge theories. Both Abelian and non-Abelian gauge theories are studied in detail. Finally, the Hamiltonian treatment of the relativistic and the spinning relativistic particle is presented. L. 2 1. INTRODUCTION .. _ Ever since the standard model’became the accepted candidate for a theory of the strong, weak and electromagnetic interactions, it has become customary to assume that the fundamental theory of everything must be a gauge theory. Unfortunately the theories one will have to deal with are considerably more com- plicated than QED, our prototypical example of a successful gauge theory, and the tools for dealing with the additional complexity have become more difficult to understand.
    [Show full text]
  • The Open Mathematics, Statistics and Probability Journal
    2666-1489/20 Send Orders for Reprints to [email protected] 8 The Open Mathematics, Statistics and Probability Journal Content list available at: https://openstatisticsandprobabilityjournal.com RESEARCH ARTICLE A More Suitable Definition of Quadratic Surds Orchidea Maria Lecian1,* 1Sapienza University of Rome, Via dei Marsi, 78- 00185 Rome, Italy Abstract: Background: The properties of the group PGL(2,C) on the Upper Poincar´e Half Plane have been analyzed. Objective: In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem. Methods: The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found. Results: A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)). Conclusion: The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori. Keywords: 02.20.-a Group theory mathematics, 43-02 Abstract harmonic analysis 02.20.Rt Discrete subgroups of Lie groups, 00A69 General applied mathematics, 70S05 Lagrangian formalism, Hamiltonian formalism, Upper Poincar´e Half Plane (UPHP). Article History Received: December 21, 2019 Revised: May 02, 2020 Accepted: August 20, 2020 1. INTRODUCTION tessellating the Hyperbolic Plane. The analysis of the solutions of Hamiltonian systems on Considering (the solutions of) a Hamiltonian system allows the Upper Poincar´e Half Plane (UPHP) allows to outline the one to investigate the properties of the action of the PGL(2,C) differences between the possible tessellation groups of the group on the Poincar´e Plane and needs the construction system and the folding groups for · chaotic systems.
    [Show full text]