B2: Symmetry and Relativity
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Newtonian Gravity and Special Relativity 12.1 Newtonian Gravity
Physics 411 Lecture 12 Newtonian Gravity and Special Relativity Lecture 12 Physics 411 Classical Mechanics II Monday, September 24th, 2007 It is interesting to note that under Lorentz transformation, while electric and magnetic fields get mixed together, the force on a particle is identical in magnitude and direction in the two frames related by the transformation. Indeed, that was the motivation for looking at the manifestly relativistic structure of Maxwell's equations. The idea was that Maxwell's equations and the Lorentz force law are automatically in accord with the notion that observations made in inertial frames are physically equivalent, even though observers may disagree on the names of these forces (electric or magnetic). Today, we will look at a force (Newtonian gravity) that does not have the property that different inertial frames agree on the physics. That will lead us to an obvious correction that is, qualitatively, a prediction of (linearized) general relativity. 12.1 Newtonian Gravity We start with the experimental observation that for a particle of mass M and another of mass m, the force of gravitational attraction between them, according to Newton, is (see Figure 12.1): G M m F = − RR^ ≡ r − r 0: (12.1) r 2 From the force, we can, by analogy with electrostatics, construct the New- tonian gravitational field and its associated point potential: GM GM G = − R^ = −∇ − : (12.2) r 2 r | {z } ≡φ 1 of 7 12.2. LINES OF MASS Lecture 12 zˆ m !r M !r ! yˆ xˆ Figure 12.1: Two particles interacting via the Newtonian gravitational force. -
1-Crystal Symmetry and Classification-1.Pdf
R. I. Badran Solid State Physics Fundamental types of lattices and crystal symmetry Crystal symmetry: What is a symmetry operation? It is a physical operation that changes the positions of the lattice points at exactly the same places after and before the operation. In other words, it is an operation when applied to an object leaves it apparently unchanged. e.g. A translational symmetry is occurred, for example, when the function sin x has a translation through an interval x = 2 leaves it apparently unchanged. Otherwise a non-symmetric operation can be foreseen by the rotation of a rectangle through /2. There are two groups of symmetry operations represented by: a) The point groups. b) The space groups (these are a combination of point groups with translation symmetry elements). There are 230 space groups exhibited by crystals. a) When the symmetry operations in crystal lattice are applied about a lattice point, the point groups must be used. b) When the symmetry operations are performed about a point or a line in addition to symmetry operations performed by translations, these are called space group symmetry operations. Types of symmetry operations: There are five types of symmetry operations and their corresponding elements. 85 R. I. Badran Solid State Physics Note: The operation and its corresponding element are denoted by the same symbol. 1) The identity E: It consists of doing nothing. 2) Rotation Cn: It is the rotation about an axis of symmetry (which is called "element"). If the rotation through 2/n (where n is integer), and the lattice remains unchanged by this rotation, then it has an n-fold axis. -
Chapter 5 the Relativistic Point Particle
Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story. -
Arxiv:1910.10745V1 [Cond-Mat.Str-El] 23 Oct 2019 2.2 Symmetry-Protected Time Crystals
A Brief History of Time Crystals Vedika Khemania,b,∗, Roderich Moessnerc, S. L. Sondhid aDepartment of Physics, Harvard University, Cambridge, Massachusetts 02138, USA bDepartment of Physics, Stanford University, Stanford, California 94305, USA cMax-Planck-Institut f¨urPhysik komplexer Systeme, 01187 Dresden, Germany dDepartment of Physics, Princeton University, Princeton, New Jersey 08544, USA Abstract The idea of breaking time-translation symmetry has fascinated humanity at least since ancient proposals of the per- petuum mobile. Unlike the breaking of other symmetries, such as spatial translation in a crystal or spin rotation in a magnet, time translation symmetry breaking (TTSB) has been tantalisingly elusive. We review this history up to recent developments which have shown that discrete TTSB does takes place in periodically driven (Floquet) systems in the presence of many-body localization (MBL). Such Floquet time-crystals represent a new paradigm in quantum statistical mechanics — that of an intrinsically out-of-equilibrium many-body phase of matter with no equilibrium counterpart. We include a compendium of the necessary background on the statistical mechanics of phase structure in many- body systems, before specializing to a detailed discussion of the nature, and diagnostics, of TTSB. In particular, we provide precise definitions that formalize the notion of a time-crystal as a stable, macroscopic, conservative clock — explaining both the need for a many-body system in the infinite volume limit, and for a lack of net energy absorption or dissipation. Our discussion emphasizes that TTSB in a time-crystal is accompanied by the breaking of a spatial symmetry — so that time-crystals exhibit a novel form of spatiotemporal order. -
Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, Antonio Franchi
Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, Antonio Franchi To cite this version: Giulia Michieletto, Angelo Cenedese, Antonio Franchi. Bearing Rigidity Theory in SE(3). 55th IEEE Conference on Decision and Control, Dec 2016, Las Vegas, United States. hal-01371084 HAL Id: hal-01371084 https://hal.archives-ouvertes.fr/hal-01371084 Submitted on 23 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Preprint version, final version at http://ieeexplore.ieee.org/ 55th IEEE Conference on Decision and Control. Las Vegas, NV, 2016 Bearing Rigidity Theory in SE(3) Giulia Michieletto, Angelo Cenedese, and Antonio Franchi Abstract— Rigidity theory has recently emerged as an ef- determines the infinitesimal rigidity properties of the system, ficient tool in the control field of coordinated multi-agent providing a necessary and sufficient condition. In such a systems, such as multi-robot formations and UAVs swarms, that context, a framework is generally represented by means of are characterized by sensing, communication and movement capabilities. This work aims at describing the rigidity properties the bar-and-joint model where agents are represented as for frameworks embedded in the three-dimensional Special points joined by bars whose fixed lengths enforce the inter- Euclidean space SE(3) wherein each agent has 6DoF. -
Physics 325: General Relativity Spring 2019 Problem Set 2
Physics 325: General Relativity Spring 2019 Problem Set 2 Due: Fri 8 Feb 2019. Reading: Please skim Chapter 3 in Hartle. Much of this should be review, but probably not all of it|be sure to read Box 3.2 on Mach's principle. Then start on Chapter 6. Problems: 1. Spacetime interval. Hartle Problem 4.13. 2. Four-vectors. Hartle Problem 5.1. 3. Lorentz transformations and hyperbolic geometry. In class, we saw that a Lorentz α0 α β transformation in 2D can be written as a = L β(#)a , that is, 0 ! ! ! a0 cosh # − sinh # a0 = ; (1) a10 − sinh # cosh # a1 where a is spacetime vector. Here, the rapidity # is given by tanh # = β; cosh # = γ; sinh # = γβ; (2) where v = βc is the velocity of frame S0 relative to frame S. (a) Show that two successive Lorentz boosts of rapidity #1 and #2 are equivalent to a single α γ α Lorentz boost of rapidity #1 +#2. In other words, check that L γ(#1)L(#2) β = L β(#1 +#2), α where L β(#) is the matrix in Eq. (1). You will need the following hyperbolic trigonometry identities: cosh(#1 + #2) = cosh #1 cosh #2 + sinh #1 sinh #2; (3) sinh(#1 + #2) = sinh #1 cosh #2 + cosh #1 sinh #2: (b) From Eq. (3), deduce the formula for tanh(#1 + #2) in terms of tanh #1 and tanh #2. For the appropriate choice of #1 and #2, use this formula to derive the special relativistic velocity tranformation rule V − v V 0 = : (4) 1 − vV=c2 Physics 325, Spring 2019: Problem Set 2 p. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
For Deep Rotation Learning with Uncertainty
A Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty Valentin Peretroukhin,1;3 Matthew Giamou,1 David M. Rosen,2 W. Nicholas Greene,3 Nicholas Roy,3 and Jonathan Kelly1 1Institute for Aerospace Studies, University of Toronto; 2Laboratory for Information and Decision Systems, 3Computer Science & Artificial Intelligence Laboratory, Massachusetts Institute of Technology Abstract—Accurate rotation estimation is at the heart of unit robot perception tasks such as visual odometry and object pose quaternions ✓ ✓ q = q<latexit sha1_base64="BMaQsvdb47NYnqSxvFbzzwnHZhA=">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</latexit> ⇤ ✓ estimation. Deep neural networks have provided a new way to <latexit sha1_base64="1dHo/MB41p5RxdHfWHI4xsZNFIs=">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</latexit> -
Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: a View from the Surface
PHYSICAL REVIEW X 6, 041068 (2016) Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface Meng Cheng,1 Michael Zaletel,1 Maissam Barkeshli,1 Ashvin Vishwanath,2 and Parsa Bonderson1 1Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA 2Department of Physics, University of California, Berkeley, California 94720, USA (Received 15 August 2016; revised manuscript received 26 November 2016; published 29 December 2016) The Lieb-Schultz-Mattis theorem and its higher-dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases with on-site unitary symmetries, enables us to develop a framework for understanding the structure of symmetry-enriched topological phases with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a “spinon” excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of “anyonic spin-orbit coupling,” which may arise from the interplay of translational and on-site symmetries. -
The Floating Body in Real Space Forms
THE FLOATING BODY IN REAL SPACE FORMS Florian Besau & Elisabeth M. Werner Abstract We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies. 1. Introduction Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant δ. Taking the right-derivative of the volume of the floating body gives rise to the affine surface area. This was established for all convex bodies in all dimensions by Schütt and Werner in [62]. The affine surface area was introduced by Blaschke in 1923 [8]. Due to its important properties, which make it an effective and powerful tool, it is omnipresent in geometry. The affine surface area and its generalizations in the rapidly developing Lp and Orlicz Brunn–Minkowski theory are the focus of intensive investigations (see e.g. [14,18, 20,21, 45, 46,65, 67,68, 70,71]). -
About Symmetries in Physics
LYCEN 9754 December 1997 ABOUT SYMMETRIES IN PHYSICS Dedicated to H. Reeh and R. Stora1 Fran¸cois Gieres Institut de Physique Nucl´eaire de Lyon, IN2P3/CNRS, Universit´eClaude Bernard 43, boulevard du 11 novembre 1918, F - 69622 - Villeurbanne CEDEX Abstract. The goal of this introduction to symmetries is to present some general ideas, to outline the fundamental concepts and results of the subject and to situate a bit the following arXiv:hep-th/9712154v1 16 Dec 1997 lectures of this school. [These notes represent the write-up of a lecture presented at the fifth S´eminaire Rhodanien de Physique “Sur les Sym´etries en Physique” held at Dolomieu (France), 17-21 March 1997. Up to the appendix and the graphics, it is to be published in Symmetries in Physics, F. Gieres, M. Kibler, C. Lucchesi and O. Piguet, eds. (Editions Fronti`eres, 1998).] 1I wish to dedicate these notes to my diploma and Ph.D. supervisors H. Reeh and R. Stora who devoted a major part of their scientific work to the understanding, description and explo- ration of symmetries in physics. Contents 1 Introduction ................................................... .......1 2 Symmetries of geometric objects ...................................2 3 Symmetries of the laws of nature ..................................5 1 Geometric (space-time) symmetries .............................6 2 Internal symmetries .............................................10 3 From global to local symmetries ...............................11 4 Combining geometric and internal symmetries ...............14 -
Introduction to SU(N) Group Theory in the Context of The
215c, 1/30/21 Lecture outline. c Kenneth Intriligator 2021. ? Week 1 recommended reading: Schwartz sections 25.1, 28.2.2. 28.2.3. You can find more about group theory (and some of my notation) also here https://keni.ucsd.edu/s10/ • At the end of 215b, I discussed spontaneous breaking of global symmetries. Some students might not have taken 215b last quarter, so I will briefly review some of that material. Also, this is an opportunity to introduce or review some group theory. • Physics students first learn about continuous, non-Abelian Lie groups in the context of the 3d rotation group. Let ~v and ~w be N-component vectors (we'll set N = 3 soon). We can rotate the vectors, or leave the vectors alone and rotate our coordinate system backwards { these are active vs passive equivalent perspectives { and scalar quantities like ~v · ~w are invariant. This is because the inner product is via δij and the rotation is a similarity transformation that preserves this. Any rotation matrix R is an orthogonal N × N matrix, RT = R−1, and the space of all such matrices is the O(N) group manifold. Note that det R = ±1 has two components, and we can restrict to the component with det R = 1, which is connected to the identity and called SO(N). The case SO(2), rotations in a plane, is parameterized by an angle θ =∼ θ + 2π, so the group manifold is a circle. If we use z = x + iy, we see that SO(2) =∼ U(1), the group of unitary 1 × 1 matrices eiθ.