Gravity with Explicit Diffeomorphism Breaking
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Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
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. -
LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1. -
TASI 2008 Lectures: Introduction to Supersymmetry And
TASI 2008 Lectures: Introduction to Supersymmetry and Supersymmetry Breaking Yuri Shirman Department of Physics and Astronomy University of California, Irvine, CA 92697. [email protected] Abstract These lectures, presented at TASI 08 school, provide an introduction to supersymmetry and supersymmetry breaking. We present basic formalism of supersymmetry, super- symmetric non-renormalization theorems, and summarize non-perturbative dynamics of supersymmetric QCD. We then turn to discussion of tree level, non-perturbative, and metastable supersymmetry breaking. We introduce Minimal Supersymmetric Standard Model and discuss soft parameters in the Lagrangian. Finally we discuss several mech- anisms for communicating the supersymmetry breaking between the hidden and visible sectors. arXiv:0907.0039v1 [hep-ph] 1 Jul 2009 Contents 1 Introduction 2 1.1 Motivation..................................... 2 1.2 Weylfermions................................... 4 1.3 Afirstlookatsupersymmetry . .. 5 2 Constructing supersymmetric Lagrangians 6 2.1 Wess-ZuminoModel ............................... 6 2.2 Superfieldformalism .............................. 8 2.3 VectorSuperfield ................................. 12 2.4 Supersymmetric U(1)gaugetheory ....................... 13 2.5 Non-abeliangaugetheory . .. 15 3 Non-renormalization theorems 16 3.1 R-symmetry.................................... 17 3.2 Superpotentialterms . .. .. .. 17 3.3 Gaugecouplingrenormalization . ..... 19 3.4 D-termrenormalization. ... 20 4 Non-perturbative dynamics in SUSY QCD 20 4.1 Affleck-Dine-Seiberg -
Observability and Symmetries of Linear Control Systems
S S symmetry Article Observability and Symmetries of Linear Control Systems Víctor Ayala 1,∗, Heriberto Román-Flores 1, María Torreblanca Todco 2 and Erika Zapana 2 1 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile; [email protected] 2 Departamento Académico de Matemáticas, Universidad, Nacional de San Agustín de Arequipa, Calle Santa Catalina, Nro. 117, Arequipa 04000, Peru; [email protected] (M.T.T.); [email protected] (E.Z.) * Correspondence: [email protected] Received: 7 April 2020; Accepted: 6 May 2020; Published: 4 June 2020 Abstract: The goal of this article is to compare the observability properties of the class of linear control systems in two different manifolds: on the Euclidean space Rn and, in a more general setup, on a connected Lie group G. For that, we establish well-known results. The symmetries involved in this theory allow characterizing the observability property on Euclidean spaces and the local observability property on Lie groups. Keywords: observability; symmetries; Euclidean spaces; Lie groups 1. Introduction For general facts about control theory, we suggest the references, [1–5], and for general mathematical issues, the references [6–8]. In the context of this article, a general control system S can be modeled by the data S = (M, D, h, N). Here, M is the manifold of the space states, D is a family of differential equations on M, or if you wish, a family of vector fields on the manifold, h : M ! N is a differentiable observation map, and N is a manifold that contains all the known information that you can see of the system through h. -
Mixed States from Diffeomorphism Anomalies Arxiv:1109.5290V1
SU-4252-919 IMSc/2011/9/10 Quantum Gravity: Mixed States from Diffeomorphism Anomalies A. P. Balachandran∗ Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA and International Institute of Physics (IIP-UFRN) Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Amilcar R. de Queirozy Instituto de Fisica, Universidade de Brasilia, Caixa Postal 04455, 70919-970, Brasilia, DF, Brazil October 30, 2018 Abstract In a previous paper, we discussed simple examples like particle on a circle and molecules to argue that mixed states can arise from anoma- lous symmetries. This idea was applied to the breakdown (anomaly) of color SU(3) in the presence of non-abelian monopoles. Such mixed states create entropy as well. arXiv:1109.5290v1 [hep-th] 24 Sep 2011 In this article, we extend these ideas to the topological geons of Friedman and Sorkin in quantum gravity. The \large diffeos” or map- ping class groups can become anomalous in their quantum theory as we show. One way to eliminate these anomalies is to use mixed states, thereby creating entropy. These ideas may have something to do with black hole entropy as we speculate. ∗[email protected] [email protected] 1 1 Introduction Diffeomorphisms of space-time play the role of gauge transformations in grav- itational theories. Just as gauge invariance is basic in gauge theories, so too is diffeomorphism (diffeo) invariance in gravity theories. Diffeos can become anomalous on quantization of gravity models. If that happens, these models cannot serve as descriptions of quantum gravitating systems. There have been several investigations of diffeo anomalies in models of quantum gravity with matter in the past. -
Area-Preserving Diffeomorphisms of the Torus Whose Rotation Sets Have
Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior Salvador Addas-Zanata Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Rua do Mat˜ao 1010, Cidade Universit´aria, 05508-090 S˜ao Paulo, SP, Brazil Abstract ǫ In this paper we consider C1+ area-preserving diffeomorphisms of the torus f, either homotopic to the identity or to Dehn twists. We sup- e pose that f has a lift f to the plane such that its rotation set has in- terior and prove, among other things that if zero is an interior point of e e 2 the rotation set, then there exists a hyperbolic f-periodic point Q∈ IR such that W u(Qe) intersects W s(Qe +(a,b)) for all integers (a,b), which u e implies that W (Q) is invariant under integer translations. Moreover, u e s e e u e W (Q) = W (Q) and f restricted to W (Q) is invariant and topologi- u e cally mixing. Each connected component of the complement of W (Q) is a disk with diameter uniformly bounded from above. If f is transitive, u e 2 e then W (Q) =IR and f is topologically mixing in the whole plane. Key words: pseudo-Anosov maps, Pesin theory, periodic disks e-mail: [email protected] 2010 Mathematics Subject Classification: 37E30, 37E45, 37C25, 37C29, 37D25 The author is partially supported by CNPq, grant: 304803/06-5 1 Introduction and main results One of the most well understood chapters of dynamics of surface homeomor- phisms is the case of the torus. -
Right-Angled Artin Groups in the C Diffeomorphism Group of the Real Line
Right-angled Artin groups in the C∞ diffeomorphism group of the real line HYUNGRYUL BAIK, SANG-HYUN KIM, AND THOMAS KOBERDA Abstract. We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG ∞ group, embeds into the C diffeomorphism group of the real line. 1. Introduction The right-angled Artin group on a finite simplicial graph Γ is the following group presentation: A(Γ) = hV (Γ) | [vi, vj] = 1 if and only if {vi, vj}∈ E(Γ)i. Here, V (Γ) and E(Γ) denote the vertex set and the edge set of Γ, respec- tively. ∞ For a smooth oriented manifold X, we let Diff+ (X) denote the group of orientation preserving C∞ diffeomorphisms on X. A group G is said to embed into another group H if there is an injective group homormophism G → H. Our main result is the following. ∞ R Theorem 1. Every right-angled Artin group embeds into Diff+ ( ). Recall that a finitely generated group G is a limit group (or a fully resid- ually free group) if for each finite set F ⊂ G, there exists a homomorphism φF from G to a nonabelian free group such that φF is an injection when restricted to F . The class of limit groups fits into a larger class of groups, which we call residually RAAG groups. A group G is in this class if for each arXiv:1404.5559v4 [math.GT] 16 Feb 2015 1 6= g ∈ G, there exists a graph Γg and a homomorphism φg : G → A(Γg) such that φg(g) 6= 1. -
The Symmetry Group of a Finite Frame
Technical Report 3 January 2010 The symmetry group of a finite frame Richard Vale and Shayne Waldron Department of Mathematics, Cornell University, 583 Malott Hall, Ithaca, NY 14853-4201, USA e–mail: [email protected] Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand e–mail: [email protected] ABSTRACT We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of [VW05] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames. Key Words: finite frame, geometrically uniform frame, Gramian matrix, harmonic frame, maximally symmetric frames partition frames, symmetry group, tight frame, AMS (MOS) Subject Classifications: primary 42C15, 58D19, secondary 42C40, 52B15, 0 1. Introduction Over the past decade there has been a rapid development of the theory and application of finite frames to areas as diverse as signal processing, quantum information theory and multivariate orthogonal polynomials, see, e.g., [KC08], [RBSC04] and [W091]. Key to these applications is the construction of frames with desirable properties. These often include being tight, and having a high degree of symmetry. Important examples are the harmonic or geometrically uniform frames, i.e., tight frames which are the orbit of a single vector under an abelian group of unitary transformations (see [BE03] and [HW06]). -
Differential Topology: Exercise Sheet 3
Prof. Dr. M. Wolf WS 2018/19 M. Heinze Sheet 3 Differential Topology: Exercise Sheet 3 Exercises (for Nov. 21th and 22th) 3.1 Smooth maps Prove the following: (a) A map f : M ! N between smooth manifolds (M; A); (N; B) is smooth if and only if for all x 2 M there are pairs (U; φ) 2 A; (V; ) 2 B such that x 2 U, f(U) ⊂ V and ◦ f ◦ φ−1 : φ(U) ! (V ) is smooth. (b) Compositions of smooth maps between subsets of smooth manifolds are smooth. Solution: (a) As the charts of a smooth structure cover the manifold the above property is certainly necessary for smoothness of f : M ! N. To show that it is also sufficient, consider two arbitrary charts U;~ φ~ 2 A and V;~ ~ 2 B. We need to prove that the map ~ ◦ f ◦ φ~−1 : φ~(U~) ! ~(V~ ) is smooth. Therefore consider an arbitrary y 2 φ~(U~) such that there is some x 2 M such that x = φ~−1(y). By assumption there are charts (U; φ) 2 A; (V; ) 2 B such that x 2 U, −1 f(U) ⊂ V and ◦ f ◦ φ : φ(U) ! (V ) is smooth. Now we choose U0;V0 such ~ ~ that x 2 U0 ⊂ U \ U and f(x) 2 V0 ⊂ V \ V and therefore we can write ~ ~−1 ~ −1 −1 ~−1 ◦ f ◦ φ j = ◦ ◦ ◦ f ◦ φ ◦ φ ◦ φ : (1) φ~(U0) As a composition of smooth maps, we showed that ~ ◦ f ◦ φ~−1j is smooth. As φ~(U0) x 2 M was chosen arbitrarily we proved that ~ ◦ f ◦ φ~−1 is a smooth map. -
Classification of Partially Hyperbolic Diffeomorphisms Under Some Rigid
Classification of partially hyperbolic diffeomorphisms under some rigid conditions Pablo D. Carrasco ∗1, Enrique Pujals y2, and Federico Rodriguez-Hertz z3 1ICEx-UFMG, Avda. Presidente Antonio Carlos 6627, Belo Horizonte-MG, BR31270-90 2CUNY Graduate Center, 365 Fifth Avenue, Room 4208, New York, NY10016 3Penn State, 227 McAllister Building, University Park, State College, PA16802 June 30, 2020 Abstract Consider a three dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) either an Anosov diffeomorphism, a (gener- alized) skew-product or the time-one map of an Anosov flow, thus recovering a well known classification conjecture of the second author to this restricted setting. 1 Introduction and Main Results Let M be a manifold. One of the central tasks in global analysis is to understand the structure of Diffr(M), the group of diffeomorphisms of M. This is of course a very com- plicated matter, so to be able to make progress it is necessary to impose some reduc- tions. Typically, as we do in this article, the reduction consists of studying meaningful subsets in Diffr(M), and try to classify or characterize elements on them. We will consider partially hyperbolic diffeomorphisms acting on three manifolds. We choose to do so due to their flexibility (linking naturally algebraic, geometric and dynamical aspects), and because of the large amount of activity that this particular research topic has nowadays. Let us spell the precise definition that we adopt here, and refer the reader to [CRHRHU17, HP16] for recent surveys. -
Introduction (Lecture 1)
Introduction (Lecture 1) February 3, 2009 One of the basic problems of manifold topology is to give a classification for manifolds (of some fixed dimension n) up to diffeomorphism. In the best of all possible worlds, a solution to this problem would provide the following: (i) A list of n-manifolds fMαg, containing one representative from each diffeomorphism class. (ii) A procedure which determines, for each n-manifold M, the unique index α such that M ' Mα. In the case n = 2, it is possible to address these problems completely: a connected oriented surface Σ is classified up to homeomorphism by a single integer g, called the genus of Σ. For each g ≥ 0, there is precisely one connected surface Σg of genus g up to diffeomorphism, which provides a solution to (i). Given an arbitrary connected oriented surface Σ, we can determine its genus simply by computing its Euler characteristic χ(Σ), which is given by the formula χ(Σ) = 2 − 2g: this provides the procedure required by (ii). Given a solution to the classification problem satisfying the demands of (i) and (ii), we can extract an algorithm for determining whether two n-manifolds M and N are diffeomorphic. Namely, we apply the procedure (ii) to extract indices α and β such that M ' Mα and N ' Mβ: then M ' N if and only if α = β. For example, suppose that n = 2 and that M and N are connected oriented surfaces with the same Euler characteristic. Then the classification of surfaces tells us that there is a diffeomorphism φ from M to N.