Spacetime Algebra of Dirac Spinors
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Quaternions and Cli Ord Geometric Algebras
Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a . -
Clifford Algebra and the Interpretation of Quantum
In: J.S.R. Chisholm/A.K. Commons (Eds.), Cliord Algebras and their Applications in Mathematical Physics. Reidel, Dordrecht/Boston (1986), 321–346. CLIFFORD ALGEBRA AND THE INTERPRETATION OF QUANTUM MECHANICS David Hestenes ABSTRACT. The Dirac theory has a hidden geometric structure. This talk traces the concep- tual steps taken to uncover that structure and points out signicant implications for the interpre- tation of quantum mechanics. The unit imaginary in the Dirac equation is shown to represent the generator of rotations in a spacelike plane related to the spin. This implies a geometric interpreta- tion for the generator of electromagnetic gauge transformations as well as for the entire electroweak gauge group of the Weinberg-Salam model. The geometric structure also helps to reveal closer con- nections to classical theory than hitherto suspected, including exact classical solutions of the Dirac equation. 1. INTRODUCTION The interpretation of quantum mechanics has been vigorously and inconclusively debated since the inception of the theory. My purpose today is to call your attention to some crucial features of quantum mechanics which have been overlooked in the debate. I claim that the Pauli and Dirac algebras have a geometric interpretation which has been implicit in quantum mechanics all along. My aim will be to make that geometric interpretation explicit and show that it has nontrivial implications for the physical interpretation of quantum mechanics. Before getting started, I would like to apologize for what may appear to be excessive self-reference in this talk. I have been pursuing the theme of this talk for 25 years, but the road has been a lonely one where I have not met anyone travelling very far in the same direction. -
International Centre for Theoretical Physics
:L^ -^ . ::, i^C IC/89/126 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Eric Bergshoeff and Ergin Sezgin INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION 1989 MIRAMARE - TRIESTE IC/89/I2G It is well known that supcrslrings admit two different kinds of formulations as far as International Atomic Energy Agency llieir siipcrsytnmclry properties arc concerned. One of them is the Ncveu-Sdwarz-tUmond and (NSIt) formulation [I] in which the world-sheet supcrsymmclry is manifest but spacetimn United Nations Educational Scientific and Cultural Organization supcrsymmclry is not. In this formulation, spacctimc supcrsymmctry arises as a consequence of Glbzzi-Schcrk-Olive (GSO) projections (2J in the Hilbcrt space. The field equations of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS the world-sheet graviton and gravilino arc the constraints which obey the super-Virasoro algebra. In the other formulation, due to Green and Schwarz (GS) [3|, the spacrtimc super- symmetry is manifest but the world-sheet supersymmclry is not. In this case, the theory lias an interesting local world-sheet symmetry, known as K-symmetry, first discovered by NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Sicgcl [4,5) for the superparticlc, and later generalized by Green and Schwarz [3] for super- strings. In a light-cone gauge, a combination of the residual ic-symmetry and rigid spacetime supcrsyininetry fuse together to give rise to an (N=8 or 1G) rigid world-sheet supersymmclry. Eric BergflliocIT The two formulations are essentially equivalent in the light-cone gauge [6,7]. How- Theory Division, CERN, 1211 Geneva 23, Switzerland ever, for many purposes it is desirable to have a covariant quantization scheme. -
Dirac Equation - Wikipedia
Dirac equation - Wikipedia https://en.wikipedia.org/wiki/Dirac_equation Dirac equation From Wikipedia, the free encyclopedia In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it 1 describes all spin-2 massive particles such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved and which was experimentally confirmed several years later. It also provided a theoretical justification for the introduction of several component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represents one of the great triumphs of theoretical physics. -
Sedeonic Theory of Massive Fields
Sedeonic theory of massive fields Sergey V. Mironov and Victor L. Mironov∗ Institute for physics of microstructures RAS, Nizhniy Novgorod, Russia (Submitted on August 5, 2013) Abstract In the present paper we develop the description of massive fields on the basis of space-time algebra of sixteen-component sedeons. The generalized sedeonic second-order equation for the potential of baryon field is proposed. It is shown that this equation can be reformulated in the form of a system of Maxwell-like equations for the field intensities. We calculate the baryonic fields in the simple model of point baryon charge and obtain the expression for the baryon- baryon interaction energy. We also propose the generalized sedeonic first-order equation for the potential of lepton field and calculate the energies of lepton-lepton and lepton-baryon interactions. Introduction Historically, the first theory of nuclear forces based on hypothesis of one-meson exchange has been formulated by Hideki Yukawa in 1935 [1]. In fact, he postulated a nonhomogeneous equation for the scalar field similar to the Klein-Gordon equation, whose solution is an effective short-range potential (so-called Yukawa potential). This theory clarified the short-ranged character of nuclear forces and was successfully applied to the explaining of low-energy nucleon-nucleon scattering data and properties of deuteron [2]. Afterwards during 1950s - 1990s many-meson exchange theories and phenomenological approach based on the effective nucleon potentials were proposed for the description of few-nucleon systems and intermediate-range nucleon interactions [3-11]. Nowadays the main fundamental concepts of nuclear force theory are developed in the frame of quantum chromodynamics (so-called effective field theory [13-15], see also reviews [16,17]) however, the meson theories and the effective potential approaches still play an important role for experimental data analysis in nuclear physics [18, 19]. -
Should Metric Signature Matter in Clifford Algebra Formulations Of
Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories? ∗ William M. Pezzaglia Jr. † Department of Physics Santa Clara University Santa Clara, CA 95053 John J. Adams ‡ P.O. Box 808, L-399 Lawrence Livermore National Laboratory Livermore, CA 94550 February 4, 2008 Abstract Standard formulation is unable to distinguish between the (+++−) and (−−−+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformu- lations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a tilt transformation[8]. The broader question is that if the universe is arXiv:gr-qc/9704048v1 17 Apr 1997 signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms. Summary of talk at the Special Session on Octonions and Clifford Algebras, at the 1997 Spring Western Sectional Meeting of the American Mathematical Society, Oregon State University, Corvallis, OR, 19-20 April 1997. ∗anonymous ftp://www.clifford.org/clf-alg/preprints/1995/pezz9502.latex †Email: [email protected] or [email protected] ‡Email: [email protected] 1 I. -
Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime
HWM–09–5 EMPG–09–10 Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime1 Richard J. Szabo Department of Mathematics Heriot-Watt University Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K. and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K. Email: [email protected] Abstract A pedagogical introduction to some of the main ideas and results of field theories on quantized spacetimes is presented, with emphasis on what such field theories may teach us about the problem of quantizing gravity. We examine to what extent noncommutative gauge theories may be regarded as gauge theories of gravity. UV/IR mixing is explained in detail and we describe its relations to renormalization, to gravitational dynamics, and to deformed dispersion relations arXiv:0906.2913v3 [hep-th] 5 Oct 2009 in models of quantum spacetime of interest in string theory and in doubly special relativity. We also discuss some potential experimental probes of spacetime noncommutativity. 1Based on Plenary Lecture delivered at the XXIX Encontro Nacional de F´ısica de Part´ıculas e Campos, S˜ao Louren¸co, Brasil, September 22–26, 2008. Contents 1 Introduction 1 2 Spacetime quantization 3 2.1 Snyder’sspacetime ............................... ...... 3 2.2 κ-Minkowskispacetime. .. .. .. .. .. .. .. .. .. ... 4 2.3 Three-dimensional quantum gravity . .......... 5 2.4 Spacetime uncertainty principle . ........... 6 2.5 Physicsinstrongmagneticfields . ......... 7 2.6 Noncommutative geometry in string theory . ........... 8 3 Field theory on quantized spacetimes 9 3.1 Formalism....................................... ... 9 3.2 UV/IRmixing ..................................... 10 3.3 Renormalization ................................. ..... 12 4 Noncommutative gauge theory of gravity 14 4.1 Gaugeinteractions ............................... ...... 14 4.2 Gravity in noncommutative gauge theories . -
Relativistic Inversion, Invariance and Inter-Action
S S symmetry Article Relativistic Inversion, Invariance and Inter-Action Martin B. van der Mark †,‡ and John G. Williamson *,‡ The Quantum Bicycle Society, 12 Crossburn Terrace, Troon KA1 07HB, Scotland, UK; [email protected] * Correspondence: [email protected] † Formerly of Philips Research, 5656 AE Eindhoven, The Netherlands. ‡ These authors contributed equally to this work. Abstract: A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter. Keywords: invariants; inversion; division; non-division algebra; Dirac algebra; Clifford algebra; geometric algebra; special relativity; photon interaction Citation: van der Mark, M.B.; 1. Introduction Williamson, J.G. Relativistic Inversion, Invariance and Inter-Action. -
Quantum/Classical Interface: Fermion Spin
Quantum/Classical Interface: Fermion Spin W. E. Baylis, R. Cabrera, and D. Keselica Department of Physics, University of Windsor Although intrinsic spin is usually viewed as a purely quantum property with no classical ana- log, we present evidence here that fermion spin has a classical origin rooted in the geometry of three-dimensional physical space. Our approach to the quantum/classical interface is based on a formulation of relativistic classical mechanics that uses spinors. Spinors and projectors arise natu- rally in the Clifford’s geometric algebra of physical space and not only provide powerful tools for solving problems in classical electrodynamics, but also reproduce a number of quantum results. In particular, many properites of elementary fermions, as spin-1/2 particles, are obtained classically and relate spin, the associated g-factor, its coupling to an external magnetic field, and its magnetic moment to Zitterbewegung and de Broglie waves. The relationship is further strengthened by the fact that physical space and its geometric algebra can be derived from fermion annihilation and creation operators. The approach resolves Pauli’s argument against treating time as an operator by recognizing phase factors as projected rotation operators. I. INTRODUCTION The intrinsic spin of elementary fermions like the electron is traditionally viewed as an essentially quantum property with no classical counterpart.[1, 2] Its two-valued nature, the fact that any measurement of the spin in an arbitrary direction gives a statistical distribution of either “spin up” or “spin down” and nothing in between, together with the commutation relation between orthogonal components, is responsible for many of its “nonclassical” properties. -
Spacetime Structuralism
Philosophy and Foundations of Physics 37 The Ontology of Spacetime D. Dieks (Editor) r 2006 Elsevier B.V. All rights reserved DOI 10.1016/S1871-1774(06)01003-5 Chapter 3 Spacetime Structuralism Jonathan Bain Humanities and Social Sciences, Polytechnic University, Brooklyn, NY 11201, USA Abstract In this essay, I consider the ontological status of spacetime from the points of view of the standard tensor formalism and three alternatives: twistor theory, Einstein algebras, and geometric algebra. I briefly review how classical field theories can be formulated in each of these formalisms, and indicate how this suggests a structural realist interpre- tation of spacetime. 1. Introduction This essay is concerned with the following question: If it is possible to do classical field theory without a 4-dimensional differentiable manifold, what does this suggest about the ontological status of spacetime from the point of view of a semantic realist? In Section 2, I indicate why a semantic realist would want to do classical field theory without a manifold. In Sections 3–5, I indicate the extent to which such a feat is possible. Finally, in Section 6, I indicate the type of spacetime realism this feat suggests. 2. Manifolds and manifold substantivalism In classical field theories presented in the standard tensor formalism, spacetime is represented by a differentiable manifold M and physical fields are represented by tensor fields that quantify over the points of M. To some authors, this has 38 J. Bain suggested an ontological commitment to spacetime points (e.g., Field, 1989; Earman, 1989). This inclination might be seen as being motivated by a general semantic realist desire to take successful theories at their face value, a desire for a literal interpretation of the claims such theories make (Earman, 1993; Horwich, 1982). -
Spacetime Algebra As a Powerful Tool for Electromagnetism
Spacetime algebra as a powerful tool for electromagnetism Justin Dressela,b, Konstantin Y. Bliokhb,c, Franco Norib,d aDepartment of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA bCenter for Emergent Matter Science (CEMS), RIKEN, Wako-shi, Saitama, 351-0198, Japan cInterdisciplinary Theoretical Science Research Group (iTHES), RIKEN, Wako-shi, Saitama, 351-0198, Japan dPhysics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA Abstract We present a comprehensive introduction to spacetime algebra that emphasizes its prac- ticality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in stud- ies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. -
Clifford Algebras, Algebraic Spinors, Quantum Information and Applications
Clifford algebras, algebraic spinors, quantum information and applications Marco A. S. Trindade1,∗, Sergio Floquet2 and J. David M. Vianna3,4 1Departamento de Ciˆencias Exatas e da Terra, Universidade do Estado da Bahia, Colegiado de F´ısica, Bahia, Brazil 2Colegiado de Engenharia Civil, Universidade Federal do Vale do S˜ao Francisco, Brazil 3Instituto de F´ısica, Universidade Federal da Bahia, Brazil 4Instituto de F´ısica, Universidade de Bras´ılia, Brazil Abstract We give an algebraic formulation based on Clifford algebras and algebraic spinors for quantum information. In this context, logic gates and concepts such as chirality, charge conjugation, parity and time re- versal are introduced and explored in connection with states of qubits. Supersymmetry and M-superalgebra are also analysed with our for- malism. Specifically we use extensively the algebras Cl3,0 and Cl1,3 as well as tensor products of Clifford algebras. Keywords: Clifford algebras; Algebraic Spinors, Qubits; Supersymmetry. PACS: 03.65.Fd; 03.67.Lx; 02.10.Hh arXiv:2005.04231v1 [quant-ph] 8 May 2020 ∗[email protected] 1 Introduction In relativistic quantum mechanics,1, 2 Clifford algebras naturally appear through Dirac matrices. Covariant bilinears, chirality, CPT symmetries are some of the mathematical objects that play a fundamental role in the theory, built in terms of the spinors and generators of Dirac algebra. The ubiquitous character of Clifford algebras suggests the possibility of using them as a link between quantum computation3, 4 and high energy physics. In fact, recently Martinez et al 5 performed an experimental demonstration of a simulation of a network gauge theory using a low-q-trapped quantum ion computer.