DOCSLIB.ORG

Extra Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Extra Dimensions Extra Dimensions The line has magnitude in one way, the plane in two ways, and the solids in three ways, and beyond these there is no other magnitude because the three are all. Aristotle, from On Heaven Hamidreza Sepangi ► The practicality of Aristotle’s observation is difficult to argue against; it seems to be our everyday experience that there are no more than three spatial dimensions. We only need three numbers to specify the location of a point in space. ► The possibility of extra dimensions is not even considered in Euclid’s Elements while Ptolemy went so far as to offer a proof of the non- existence of extra dimensions in his treatise on distance. ► Kepler entertained the idea that nature’s apparent preference for three dimensions had to do with the holy trinity, while more modern geometers have offered arguments in favor of what our intuition tells us, including the extra-ordinary stability of planetary orbits and atomic ground sates in 3-dimensional space. ► Yet, despite the reasoning of the common sense, many people have been interested in the idea that the world is a fundamentally higher- dimensional arena. ► Extra dimensions have even been considered, in the nineteenth century, to answer questions like “where do angels live,” and so on. A certain sect of the Bolshevik party even tried to use the idea of extra dimensions to present a novel brand of spiritualism and then use it to gradually convert Russian peasantry to the tenets of socialism. The 4th dimension ► The story of extra dimensions began, arguably, with Bernhard Riemann’s habilitation lecture on June 10th 1854. This was a theme that his supervisor, Carl Friedrich Gauss, had preferred amongst a number of other themes that Riemann had proposed. The lecture titled “On the hypothesis which lie at the bases of geometry.” It was in this lecture that the notion of a curved manifold, or “ many-fold,” made its debut. ► Despite the initially hesitant reception, Riemann’s ideas concerning n-dimensional manifolds gradually became accepted in scientific circles. Bernhard Riemann (1826-1866) ► Other important figures in presenting the notion of extra dimension were William Clifford and Hermann von Helmholtz. ► A gifted mathematician by the name of Charles Howard Hinton even tried to visualize a four- dimensional hypercube. A representation of Hinton's four- dimensional hypercube ► Hinton argued that a being that could move freely in the fourth dimension could appear or disappear at will by temporarily leaving our 3-dimensional world and re- entering at some other location, which was seen as a plausible explanation of the behavior of ghosts and other creatures!! ► Even legal consequences of extra dimensions surfaced in the nineteenth century. The London trial of Henry Slade is an example in which the existence of an extra dimension was an integral part of the defense. Testimony on Slade’s behalf was offered by William Webber, J. J. Thompson and Lord Rayleigh. Their efforts was in vein; Slade was eventually found guilty!! The unification of time and space ► The first popular novel concerning extra dimension was published in 1895; the “Time Machine” by H. G. Wells. Well’s conception of an extra dimension was closest to what was soon to become an accepted truth in physics. ► Einstein’s special theory of relativity, inspired by the structure of Maxwell’s theory of electromagnetism, was the undisputed proof of what we now consider as the fourth Well's imagination and extra dimension. dimensions notwithstanding, most ► This destroyed the Newtonian physicists still believe time travel to be notion of a universal time. either impossible or highly impractical ► In 1909, Hermann Mikowski put forth a geometrical interpretation of Einstein’s theory. He added a fourth dimension, ict, to the familiar three dimensions of Aristotle and Euclid, showing that the predictions of special relativity could be Various geometric properties of understood in terms of an Minkowski spac-etime are illustrated in this cartoon. Time runs vertically extended space-time upwards and the (two) spatial directions manifold. are orthogonal. The figure outlined in grey is the light cone of an observer in the center of the plot. ► It was Einstein who accepted the idea of space-time and constructed a generally covariant 4-dimensional theory of relativity. Fortunately, Einstein did not need to invent everything himself, for it was at this time that the idea of a n- manifold, put forward by Riemann, came out of the realm of pure mathematics. ► Einstein used it in 1915 to propose the general theory of relativity Einstein writing down the vacuum field equations of general relativity Unification of gravity and electromagnetism ► About the time when the general theory of relativity was introduced, there were two well-founded theories in physics: gravity and electromagnetism. Strong and weak forces were still decades away. ► Attempts to unify gravity and electromagnetism bore fruit in 1919 through Gravity manifests itself as the curvature of a theory presented by Space-time induced by massive objects Theodore Kaluza. ► In 1919, Kaluza sent Einstein a preprint—later published in 1921– that considered the extension of general relativity to five dimensions. He assumed that if the space-time is endowed with an extra dimension, then electromagnetism and gravity could be unified in an extension of general relativity to 5 dimensions where the resulting 15 field equations naturally break into a set of 10 equations representing gravity, four describing electromagnetism, and one, the wave equation of a scalar field. Kaluza postulated that both effects could be understood in terms of the same 5- dimensional geometric framework. Kaluza-Klein theory ► The metric ► Field equations ► But there were problems with Kaluza’s theory, the least of which was the nature of the fifth dimension. Where is this extra dimension and why can’t we see it? Oscar Klein suggested that the extra dimension is compact, that is to say, it is as small as the Planck length, 1.6 x 10^-35 meters. This construction is called compactification of the fifth dimension. Theodor Kaluza (1885-1945) Dimensionality and unification of fundamental forces ► As the years passed, physicists realized that Kaluza-Klein theory cannot possibly be the theory of everything. The strong and weak interactions needed more degrees of The strong force is what holds freedom than Kaluza- together quarks in atomic nuclei Klein could offer. The weak force mediates the radioactive decay of various types of particles http://userweb. ► The question is: how many dimensions do we need to unify strong and weak forces with gravity and electromagnetism? The answer is at least eleven which was shown by E. Witten in 1981. ► There are two types of fields representing fundamental forces: fermions and bosons. Kaluza-Klein theory can only accommodate bosons and says nothing about fermions. Since both are needed, the idea of supersymmetry was introduced. This symmetry states that the fundamental theory of physics should be invariant under an exchange of identity between fermions and bosons. The resulting The weak force mediates the theory is called supergravity. But this theory is not free of difficulties! radioactive decay of various types of Particles. ► String theory gradually came about as the most promising theory which unifies the fundamental forces. It is a 10-dimensional theory where both bosons and fermions live together. In string theory, elementary particles and field are realized as the oscillations of fundamental one dimensional objects. The 5th dimension and Space-Time- Matter (STM) theory ► In the early 1990’s, space-time- matter theory started to appear. This is a 5-dimensional theory that attempts to realize Einstein’s old dream of transforming the “based wood” of the stress-energy tensor in his field equations into the “pure marble” of geometry. ► In conventional relativity, it is the distribution of matter in the form of the energy-momentum tensor that determines the geometry through the Einstein’s field equations: In STM theory, all the matter in the universe is viewed to be a manifestation of higher-dimensional geometry ►In STM theory, we postulate that the 4- dimensional universe is embedded in a higher dimensional vacuum manifold. ►An observer measuring the matter content of the universe using its curvature, geometrical artifacts from this embedding appear to be real matter. That is why this theory is sometimes called “Induced Matter Theory.” Brane-World Theory ► An extremely popular alternative theory involving extra dimensions is the brane-world scenario. ► Our 4-dimensional universe (brane) is embedded in a n- dimensional manifold (bulk). ► Ordinary matter cannot leave the brane and wander into the bulk space. ►Gravity however, can escape into the bulk. ►The theory has the Z2 symmetry, meaning that each half of the bulk is exactly like the other. ►This setup inspired Randall and Sundrum to construct a 5-dimensional model where there is only one extra dimension, which can be taken to be either non-compact or compact with macroscopic radius. Implications of the Randall-Sundrum model ►3-dimensional Newtonian gravity is recovered at large scales. ►The model explains the problem of hierarchy in particle physics. The huge disparity between the fundamental forces. ►Brane-world scenarios have left a great influence in the way people look at the universe and try to understand its origin and dynamics..
Load more

Recommended publications
  • Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
    Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings.
    [Show full text]
  • Einstein's Mistakes
    Einstein’s Mistakes Einstein was the greatest genius of the Twentieth Century, but his discoveries were blighted with mistakes. The Human Failing of Genius. 1 PART 1 An evaluation of the man Here, Einstein grows up, his thinking evolves, and many quotations from him are listed. Albert Einstein (1879-1955) Einstein at 14 Einstein at 26 Einstein at 42 3 Albert Einstein (1879-1955) Einstein at age 61 (1940) 4 Albert Einstein (1879-1955) Born in Ulm, Swabian region of Southern Germany. From a Jewish merchant family. Had a sister Maja. Family rejected Jewish customs. Did not inherit any mathematical talent. Inherited stubbornness, Inherited a roguish sense of humor, An inclination to mysticism, And a habit of grüblen or protracted, agonizing “brooding” over whatever was on its mind. Leading to the thought experiment. 5 Portrait in 1947 – age 68, and his habit of agonizing brooding over whatever was on its mind. He was in Princeton, NJ, USA. 6 Einstein the mystic •“Everyone who is seriously involved in pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man..” •“When I assess a theory, I ask myself, if I was God, would I have arranged the universe that way?” •His roguish sense of humor was always there. •When asked what will be his reactions to observational evidence against the bending of light predicted by his general theory of relativity, he said: •”Then I would feel sorry for the Good Lord. The theory is correct anyway.” 7 Einstein: Mathematics •More quotations from Einstein: •“How it is possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” •Questions asked by many people and Einstein: •“Is God a mathematician?” •His conclusion: •“ The Lord is cunning, but not malicious.” 8 Einstein the Stubborn Mystic “What interests me is whether God had any choice in the creation of the world” Some broadcasters expunged the comment from the soundtrack because they thought it was blasphemous.
    [Show full text]
  • Black Hole Production and Graviton Emission in Models with Large Extra Dimensions
    Black hole production and graviton emission in models with large extra dimensions Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe-Universitat in Frankfurt am Main von Benjamin Koch aus Nordlingen Frankfurt am Main 2007 (D 30) vom Fachbereich Physik der Johann Wolfgang Goethe-Universitat als Dissertation angenommen Dekan Gutachter Datum der Disputation Zusammenfassung In dieser Arbeit wird die mogliche Produktion von mikroskopisch kleinen Schwarzen Lcchern und die Emission von Gravitationsstrahlung in Modellen mit grofien Extra-Dimensionen untersucht. Zunachst werden der theoretisch-physikalische Hintergrund und die speziel- len Modelle des behandelten Themas skizziert. Anschliefiend wird auf die durchgefuhrten Untersuchungen zur Erzeugung und zum Zerfall mikrosko ­ pisch kleiner Schwarzer Locher in modernen Beschleunigerexperimenten ein- gegangen und die wichtigsten Ergebnisse zusammengefasst. Im Anschluss daran wird die Produktion von Gravitationsstrahlung durch Teilchenkollisio- nen diskutiert. Die daraus resultierenden analytischen Ergebnisse werden auf hochenergetische kosmische Strahlung angewandt. Die Suche nach einer einheitlichen Theorie der Naturkrafte Eines der grofien Ziele der theoretischen Physik seit Einstein ist es, eine einheitliche und moglichst einfache Theorie zu entwickeln, die alle bekannten Naturkrafte beschreibt. Als grofier Erfolg auf diesem Wege kann es angese- hen werden, dass es gelang, drei 1 der vier bekannten Krafte mittels eines einzigen Modells, des Standardmodells (SM), zu beschreiben. Das Standardmodell der Elementarteilchenphysik ist eine Quantenfeldtheo- rie. In Quantenfeldtheorien werden Invarianten unter lokalen Symmetrie- transformationen betrachtet. Die Symmetriegruppen, die man fur das Stan­ dardmodell gefunden hat, sind die U(1), SU(2)L und die SU(3). Die Vorher- sagen des Standardmodells wurden durch eine Vielzahl von Experimenten mit hochster Genauigkeit bestatigt.
    [Show full text]
  • Albert Einstein
    THE COLLECTED PAPERS OF Albert Einstein VOLUME 15 THE BERLIN YEARS: WRITINGS & CORRESPONDENCE JUNE 1925–MAY 1927 Diana Kormos Buchwald, József Illy, A. J. Kox, Dennis Lehmkuhl, Ze’ev Rosenkranz, and Jennifer Nollar James EDITORS Anthony Duncan, Marco Giovanelli, Michel Janssen, Daniel J. Kennefick, and Issachar Unna ASSOCIATE & CONTRIBUTING EDITORS Emily de Araújo, Rudy Hirschmann, Nurit Lifshitz, and Barbara Wolff ASSISTANT EDITORS Princeton University Press 2018 Copyright © 2018 by The Hebrew University of Jerusalem Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA (Revised for volume 15) Einstein, Albert, 1879–1955. The collected papers of Albert Einstein. German, English, and French. Includes bibliographies and indexes. Contents: v. 1. The early years, 1879–1902 / John Stachel, editor — v. 2. The Swiss years, writings, 1900–1909 — — v. 15. The Berlin years, writings and correspondence, June 1925–May 1927 / Diana Kormos Buchwald... [et al.], editors. QC16.E5A2 1987 530 86-43132 ISBN 0-691-08407-6 (v.1) ISBN 978-0-691-17881-3 (v. 15) This book has been composed in Times. The publisher would like to acknowledge the editors of this volume for providing the camera-ready copy from which this book was printed. Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources. Printed in the United States of America 13579108642 INTRODUCTION TO VOLUME 15 The present volume covers a thrilling two-year period in twentieth-century physics, for during this time matrix mechanics—developed by Werner Heisenberg, Max Born, and Pascual Jordan—and wave mechanics, developed by Erwin Schrödinger, supplanted the earlier quantum theory.
    [Show full text]
  • Loop Quantum Cosmology, Modified Gravity and Extra Dimensions
    universe Review Loop Quantum Cosmology, Modified Gravity and Extra Dimensions Xiangdong Zhang Department of Physics, South China University of Technology, Guangzhou 510641, China; [email protected] Academic Editor: Jaume Haro Received: 24 May 2016; Accepted: 2 August 2016; Published: 10 August 2016 Abstract: Loop quantum cosmology (LQC) is a framework of quantum cosmology based on the quantization of symmetry reduced models following the quantization techniques of loop quantum gravity (LQG). This paper is devoted to reviewing LQC as well as its various extensions including modified gravity and higher dimensions. For simplicity considerations, we mainly focus on the effective theory, which captures main quantum corrections at the cosmological level. We set up the basic structure of Brans–Dicke (BD) and higher dimensional LQC. The effective dynamical equations of these theories are also obtained, which lay a foundation for the future phenomenological investigations to probe possible quantum gravity effects in cosmology. Some outlooks and future extensions are also discussed. Keywords: loop quantum cosmology; singularity resolution; effective equation 1. Introduction Loop quantum gravity (LQG) is a quantum gravity scheme that tries to quantize general relativity (GR) with the nonperturbative techniques consistently [1–4]. Many issues of LQG have been carried out in the past thirty years. In particular, among these issues, loop quantum cosmology (LQC), which is the cosmological sector of LQG has received increasing interest and has become one of the most thriving and fruitful directions of LQG [5–9]. It is well known that GR suffers singularity problems and this, in turn, implies that our universe also has an infinitely dense singularity point that is highly unphysical.
    [Show full text]
  • Aspects of Loop Quantum Gravity
    Aspects of loop quantum gravity Alexander Nagen 23 September 2020 Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London 1 Contents 1 Introduction 4 2 Classical theory 12 2.1 The ADM / initial-value formulation of GR . 12 2.2 Hamiltonian GR . 14 2.3 Ashtekar variables . 18 2.4 Reality conditions . 22 3 Quantisation 23 3.1 Holonomies . 23 3.2 The connection representation . 25 3.3 The loop representation . 25 3.4 Constraints and Hilbert spaces in canonical quantisation . 27 3.4.1 The kinematical Hilbert space . 27 3.4.2 Imposing the Gauss constraint . 29 3.4.3 Imposing the diffeomorphism constraint . 29 3.4.4 Imposing the Hamiltonian constraint . 31 3.4.5 The master constraint . 32 4 Aspects of canonical loop quantum gravity 35 4.1 Properties of spin networks . 35 4.2 The area operator . 36 4.3 The volume operator . 43 2 4.4 Geometry in loop quantum gravity . 46 5 Spin foams 48 5.1 The nature and origin of spin foams . 48 5.2 Spin foam models . 49 5.3 The BF model . 50 5.4 The Barrett-Crane model . 53 5.5 The EPRL model . 57 5.6 The spin foam - GFT correspondence . 59 6 Applications to black holes 61 6.1 Black hole entropy . 61 6.2 Hawking radiation . 65 7 Current topics 69 7.1 Fractal horizons . 69 7.2 Quantum-corrected black hole . 70 7.3 A model for Hawking radiation . 73 7.4 Effective spin-foam models .
    [Show full text]
  • Black Hole Production and Graviton Emission in Models with Large Extra Dimensions
    Black hole production and graviton emission in models with large extra dimensions Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften vorgelegt beim Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at in Frankfurt am Main von Benjamin Koch aus N¨ordlingen Frankfurt am Main 2007 (D 30) vom Fachbereich Physik der Johann Wolfgang Goethe–Universit¨at als Dissertation angenommen Dekan ........................................ Gutachter ........................................ Datum der Disputation ................................ ........ Zusammenfassung In dieser Arbeit wird die m¨ogliche Produktion von mikroskopisch kleinen Schwarzen L¨ochern und die Emission von Gravitationsstrahlung in Modellen mit großen Extra-Dimensionen untersucht. Zun¨achst werden der theoretisch-physikalische Hintergrund und die speziel- len Modelle des behandelten Themas skizziert. Anschließend wird auf die durchgefuhrten¨ Untersuchungen zur Erzeugung und zum Zerfall mikrosko- pisch kleiner Schwarzer L¨ocher in modernen Beschleunigerexperimenten ein- gegangen und die wichtigsten Ergebnisse zusammengefasst. Im Anschluss daran wird die Produktion von Gravitationsstrahlung durch Teilchenkollisio- nen diskutiert. Die daraus resultierenden analytischen Ergebnisse werden auf hochenergetische kosmische Strahlung angewandt. Die Suche nach einer einheitlichen Theorie der Naturkr¨afte Eines der großen Ziele der theoretischen Physik seit Einstein ist es, eine einheitliche und m¨oglichst einfache Theorie zu entwickeln, die alle bekannten Naturkr¨afte beschreibt.
    [Show full text]
  • Abdus Salam United Nations Educational, Scientific and Cultural International XA0101583 Organization Centre
    the 1(72001/34 abdus salam united nations educational, scientific and cultural international XA0101583 organization centre international atomic energy agency for theoretical physics NEW DIMENSIONS NEW HOPES Utpal Sarkar Available at: http://www.ictp.trieste.it/-pub-off IC/2001/34 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NEW DIMENSIONS NEW HOPES Utpal Sarkar1 Physics Department, Visva Bharati University, Santiniketan 731235, India and The Abdus Salam Insternational Centre for Theoretical Physics, Trieste, Italy. Abstract We live in a four dimensional world. But the idea of unification of fundamental interactions lead us to higher dimensional theories. Recently a new theory with extra dimensions has emerged, where only gravity propagates in the extra dimension and all other interactions are confined in only four dimensions. This theory gives us many new hopes. In earlier theories unification of strong, weak and the electromagnetic forces was possible at around 1016 GeV in a grand unified theory (GUT) and it could get unified with gravity at around the Planck scale of 1019 GeV. With this new idea it is possible to bring down all unification scales within the reach of the next generation accelerators, i.e., around 104 GeV. MIRAMARE - TRIESTE May 2001 1 Regular Associate of the Abdus Salam ICTP. E-mail: [email protected] 1 Introduction In particle physics we try to find out what are the fundamental particles and how they interact. This is motivated from the belief that there must be some fundamental law that governs ev- erything.
    [Show full text]
  • Sacred Rhetorical Invention in the String Theory Movement
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Communication Studies Theses, Dissertations, and Student Research Communication Studies, Department of Spring 4-12-2011 Secular Salvation: Sacred Rhetorical Invention in the String Theory Movement Brent Yergensen University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/commstuddiss Part of the Speech and Rhetorical Studies Commons Yergensen, Brent, "Secular Salvation: Sacred Rhetorical Invention in the String Theory Movement" (2011). Communication Studies Theses, Dissertations, and Student Research. 6. https://digitalcommons.unl.edu/commstuddiss/6 This Article is brought to you for free and open access by the Communication Studies, Department of at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Communication Studies Theses, Dissertations, and Student Research by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. SECULAR SALVATION: SACRED RHETORICAL INVENTION IN THE STRING THEORY MOVEMENT by Brent Yergensen A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Communication Studies Under the Supervision of Dr. Ronald Lee Lincoln, Nebraska April, 2011 ii SECULAR SALVATION: SACRED RHETORICAL INVENTION IN THE STRING THEORY MOVEMENT Brent Yergensen, Ph.D. University of Nebraska, 2011 Advisor: Ronald Lee String theory is argued by its proponents to be the Theory of Everything. It achieves this status in physics because it provides unification for contradictory laws of physics, namely quantum mechanics and general relativity. While based on advanced theoretical mathematics, its public discourse is growing in prevalence and its rhetorical power is leading to a scientific revolution, even among the public.
    [Show full text]
  • Arxiv:1308.1092V3 [Hep-Th] 28 Oct 2014
    Noname manuscript No. (will be inserted by the editor) Unifying Geometrical Representations of Gauge Theory Scott Alsid · Mario Serna Received: date / Accepted: date Abstract We unify three approaches within the vast body of gauge-theory research that have independently developed distinct representations of a geometrical surface- like structure underlying the vector-potential. The three approaches that we unify are: those who use the compactified dimensions of Kaluza-Klein theory, those who use Grassmannian models (also called gauge theory embedding or CPN−1 models) to rep- resent gauge fields, and those who use a hidden spatial metric to replace the gauge fields. In this paper we identify a correspondence between the geometrical represen- tations of the three schools. Each school was mostly independently developed, does not compete with other schools, and attempts to isolate the gauge-invariant geometri- cal surface-like structures that are responsible for the resulting physics. By providing a mapping between geometrical representations, we hope physicists can now isolate representation-dependent physics from gauge-invariant physical results and share re- sults between each school. We provide visual examples of the geometrical relation- ships between each school for U(1) electric and magnetic fields. We highlight a first new result: in all three representations a static electric field (electric field from a fixed ring of charge or a sphere of charge) has a hidden gauge-invariant time dependent surface that is underlying the vector potential. Keywords Kaluza Klein · Gauge field theory: Composite · Field theoretical model: CPN−1 · Gauge Geometry Embedding · Grassmannian Models · Hidden-spatial geometry PACS 04.20.Cv · 11.15.-q · 04.20.-q · 12.38.Aw S.
    [Show full text]
  • Quantum Gravity, Effective Fields and String Theory
    Quantum gravity, effective fields and string theory Niels Emil Jannik Bjerrum-Bohr The Niels Bohr Institute University of Copenhagen arXiv:hep-th/0410097v1 10 Oct 2004 Thesis submitted for the degree of Doctor of Philosophy in Physics at the Niels Bohr Institute, University of Copenhagen. 28th July 2 Abstract In this thesis we will look into some of the various aspects of treating general relativity as a quantum theory. The thesis falls in three parts. First we briefly study how gen- eral relativity can be consistently quantized as an effective field theory, and we focus on the concrete results of such a treatment. As a key achievement of the investigations we present our calculations of the long-range low-energy leading quantum corrections to both the Schwarzschild and Kerr metrics. The leading quantum corrections to the pure gravitational potential between two sources are also calculated, both in the mixed theory of scalar QED and quantum gravity and in the pure gravitational theory. Another part of the thesis deals with the (Kawai-Lewellen-Tye) string theory gauge/gravity relations. Both theories are treated as effective field theories, and we investigate if the KLT oper- ator mapping is extendable to the case of higher derivative operators. The constraints, imposed by the KLT-mapping on the effective coupling constants, are also investigated. The KLT relations are generalized, taking the effective field theory viewpoint, and it is noticed that some remarkable tree-level amplitude relations exist between the field the- ory operators. Finally we look at effective quantum gravity treated from the perspective of taking the limit of infinitely many spatial dimensions.
    [Show full text]
  • Einstein and the Kaluza-Klein Particle
    ITFA-99-28 Einstein and the Kaluza-Klein particle Jeroen van Dongen1 Institute for Theoretical Physics, University of Amsterdam Valckeniersstraat 65 1018 XE Amsterdam and Joseph Henry Laboratories, Princeton University Princeton NJ 08544 Abstract In his search for a unified field theory that could undercut quantum mechanics, Einstein considered five dimensional classical Kaluza-Klein theory. He studied this theory most intensively during the years 1938-1943. One of his primary objectives was finding a non-singular particle solution. In the full theory this search got frustrated and in the x5-independent theory Einstein, together with Pauli, argued it would be impossible to find these structures. Keywords: Einstein; Unified Field Theory; Kaluza-Klein Theory; Quantization; Soli- tons 1 Introduction After having formulated general relativity Albert Einstein did not immediately focus on the unification of electromagnetism and gravity in a classical field theory - the issue that would characterize much of his later work. It was still an open question to him whether arXiv:gr-qc/0009087v1 26 Sep 2000 relativity and electrodynamics together would cast light on the problem of the structure of matter [18]. Rather, in a 1916 paper on gravitational waves he anticipated a different development: since the electron in its atomic orbit would radiate gravitationally, something that cannot “occur in reality”, he expected quantum theory would have to change not only the ”Maxwellian electrodynamics, but also the new theory of gravitation” [19]2. Einstein’s position, however, gradually changed. From about 1919 onwards, he took a strong interest in the unification programme3. In later years, after about 1926, he hoped that he would find a particular classical unified field theory that could undercut quantum theory.
    [Show full text]