Geometric Algebras in Physics: Eigenspinors and Dirac Theory
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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. -
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. -
Clifford Algebras, Spinors and Supersymmetry. Francesco Toppan
IV Escola do CBPF – Rio de Janeiro, 15-26 de julho de 2002 Algebraic Structures and the Search for the Theory Of Everything: Clifford algebras, spinors and supersymmetry. Francesco Toppan CCP - CBPF, Rua Dr. Xavier Sigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil abstract These lectures notes are intended to cover a small part of the material discussed in the course “Estruturas algebricas na busca da Teoria do Todo”. The Clifford Algebras, necessary to introduce the Dirac’s equation for free spinors in any arbitrary signature space-time, are fully classified and explicitly constructed with the help of simple, but powerful, algorithms which are here presented. The notion of supersymmetry is introduced and discussed in the context of Clifford algebras. 1 Introduction The basic motivations of the course “Estruturas algebricas na busca da Teoria do Todo”consisted in familiarizing graduate students with some of the algebra- ic structures which are currently investigated by theoretical physicists in the attempt of finding a consistent and unified quantum theory of the four known interactions. Both from aesthetic and practical considerations, the classification of mathematical and algebraic structures is a preliminary and necessary require- ment. Indeed, a very ambitious, but conceivable hope for a unified theory, is that no free parameter (or, less ambitiously, just few) has to be fixed, as an external input, due to phenomenological requirement. Rather, all possible pa- rameters should be predicted by the stringent consistency requirements put on such a theory. An example of this can be immediately given. It concerns the dimensionality of the space-time. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory. -
A List of All the Errata Known As of 19 December 2013
14 Linear Algebra that is nonnegative when the matrices are the same N L N L † ∗ 2 (A, A)=TrA A = AijAij = |Aij| ≥ 0 (1.87) !i=1 !j=1 !i=1 !j=1 which is zero only when A = 0. So this inner product is positive definite. A vector space with a positive-definite inner product (1.73–1.76) is called an inner-product space,ametric space, or a pre-Hilbert space. A sequence of vectors fn is a Cauchy sequence if for every ϵ>0 there is an integer N(ϵ) such that ∥fn − fm∥ <ϵwhenever both n and m exceed N(ϵ). A sequence of vectors fn converges to a vector f if for every ϵ>0 there is an integer N(ϵ) such that ∥f −fn∥ <ϵwhenever n exceeds N(ϵ). An inner-product space with a norm defined as in (1.80) is complete if each of its Cauchy sequences converges to a vector in that space. A Hilbert space is a complete inner-product space. Every finite-dimensional inner-product space is complete and so is a Hilbert space. But the term Hilbert space more often is used to describe infinite-dimensional complete inner-product spaces, such as the space of all square-integrable functions (David Hilbert, 1862–1943). Example 1.17 (The Hilbert Space of Square-Integrable Functions) For the vector space of functions (1.55), a natural inner product is b (f,g)= dx f ∗(x)g(x). (1.88) "a The squared norm ∥ f ∥ of a function f(x)is b ∥ f ∥2= dx |f(x)|2. -
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. -
Special Relativity in Complex Space-Time. Part 2. Basic Problems of Electrodynamics
Special relativity in complex space-time. Part 2. Basic problems of electrodynamics. JÓZEF RADOMANSKI´ 48-300 Nysa, ul. Bohaterów Warszawy 9, Poland e-mail: [email protected] x Abstract This article discusses an electric field in complex space-time. Using an orthogonal paravector transformation that preserves the invariance of the wave equation and does not belong to the Lorentz group, the Gaussian equation has been transformed to obtain the relationships corresponding to the Maxwell equations. These equations are analysed for compliance with classic electrodynamics. Although, the Lorenz gauge condition has been abandoned and two of the modified Maxwell’s equations are different from the classical ones, the obtained results are not inconsistent with the experience because they preserve the classical laws of the theory of electricity and magnetism contained therein. In conjunction with the previous papers our purpose is to show that space-time of high velocity has a complex structure that differently orders the laws of classical physics but does not change them. Keywords: Alternative special relativity, complex space-time, Maxwell equations, paravectors, wave equation 1 Introduction The classical special theory of relativity (STR) assumes that space-time is a 4-dimensional real structure, and Lorentz transformation is its automorphism which preserves the invariance of the wave equation. In the paravector formalism, the Lorentz transformation has a form X 0 = ΛX Λ∗, where Λ is a complex orthogonal paravector, and the asterisk means the conjugation [2]. The article [7] shows that transformation X 0 = ΛX preserves the invariance of the wave equation and also states the hypothesis that space-time is a complex structure C C 3, and it is real only locally in the observer’s rest frame. -
About the Inversion of the Dirac Operator
About the inversion of the Dirac operator Version 1.3 Abstract 5 In this note we would like to specify a bit what is done when we invert the Dirac operator in lattice QCD. Contents 1 Introductive remark 2 2 Notations 3 2.1 Thegaugefieldsandgaugeconfigurations . 4 5 2.2 General properties of the Dirac operator matrix . ... 5 2.3 Wilson-twistedDiracoperator. 6 3 Even-odd preconditionning 9 3.1 implementation of even-odd preconditionning . ... 10 3.1.1 ConjugateGradient . 11 10 4 Inexact deflation 15 4.1 Deflationmethod........................... 15 4.2 DeflationappliedtotheDiracOperator . 16 4.3 DeflationFAPP............................ 16 4.4 Deflation:theory........................... 18 15 4.4.1 Some properties of PR and PR ............... 18 4.4.2 TheLittleDiracOperator. 19 5 Mathematic tools 20 6 Samples from ETMC codes 22 1 Chapter 1 Introductive remark The inversion of the Dirac operator is an important step during the building of a statistical sample of gauge configurations: indeed in the HMC algorithm 5 it appears in the expression of what is called ”the fermionic force” used to update the momenta associated with the gauge fields along a trajectory. It is the most expensive part in overall computation time of a lattice simulation with dynamical fermions because it is done many times. Actually the computation of the quark propagator, i.e. the inverse of the Dirac 10 operator, is already necessary to compute a simple 2pts correlation function from which one extracts a hadron mass or its decay constant: indeed that correlation function is roughly the trace (in the matricial language) of the product of 2 such propagators. -
Chapter 2: Linear Algebra User's Manual
Preprint typeset in JHEP style - HYPER VERSION Chapter 2: Linear Algebra User's Manual Gregory W. Moore Abstract: An overview of some of the finer points of linear algebra usually omitted in physics courses. May 3, 2021 -TOC- Contents 1. Introduction 5 2. Basic Definitions Of Algebraic Structures: Rings, Fields, Modules, Vec- tor Spaces, And Algebras 6 2.1 Rings 6 2.2 Fields 7 2.2.1 Finite Fields 8 2.3 Modules 8 2.4 Vector Spaces 9 2.5 Algebras 10 3. Linear Transformations 14 4. Basis And Dimension 16 4.1 Linear Independence 16 4.2 Free Modules 16 4.3 Vector Spaces 17 4.4 Linear Operators And Matrices 20 4.5 Determinant And Trace 23 5. New Vector Spaces from Old Ones 24 5.1 Direct sum 24 5.2 Quotient Space 28 5.3 Tensor Product 30 5.4 Dual Space 34 6. Tensor spaces 38 6.1 Totally Symmetric And Antisymmetric Tensors 39 6.2 Algebraic structures associated with tensors 44 6.2.1 An Approach To Noncommutative Geometry 47 7. Kernel, Image, and Cokernel 47 7.1 The index of a linear operator 50 8. A Taste of Homological Algebra 51 8.1 The Euler-Poincar´eprinciple 54 8.2 Chain maps and chain homotopies 55 8.3 Exact sequences of complexes 56 8.4 Left- and right-exactness 56 { 1 { 9. Relations Between Real, Complex, And Quaternionic Vector Spaces 59 9.1 Complex structure on a real vector space 59 9.2 Real Structure On A Complex Vector Space 64 9.2.1 Complex Conjugate Of A Complex Vector Space 66 9.2.2 Complexification 67 9.3 The Quaternions 69 9.4 Quaternionic Structure On A Real Vector Space 79 9.5 Quaternionic Structure On Complex Vector Space 79 9.5.1 Complex Structure On Quaternionic Vector Space 81 9.5.2 Summary 81 9.6 Spaces Of Real, Complex, Quaternionic Structures 81 10. -
Exact Solutions of a Dirac Equation with a Varying CP-Violating Mass Profile and Coherent Quasiparticle Approximation
Exact solutions of a Dirac equation with a varying CP-violating mass profile and coherent quasiparticle approximation Olli Koskivaara Master’s thesis Supervisor: Kimmo Kainulainen November 30, 2015 Preface This thesis was written during the year 2015 at the Department of Physics at the University of Jyväskylä. I would like to express my utmost grat- itude to my supervisor Professor Kimmo Kainulainen, who guided me through the subtle path laid out by this intriguing subject, and provided me with my first glimpses of research in theoretical physics. I would also like to thank my fellow students for the countless number of versatile conversations on physics and beyond. Finally, I wish to thank my family and Ida-Maria for their enduring and invaluable support during the entire time of my studies in physics. i Abstract In this master’s thesis the phase space structure of a Wightman function is studied for a temporally varying background. First we introduce coherent quasiparticle approximation (cQPA), an approximation scheme in finite temperature field theory that enables studying non-equilibrium phenom- ena. Using cQPA we show that in non-translation invariant systems the phase space of the Wightman function has in general structure beyond the traditional mass-shell particle- and antiparticle-solutions. This novel structure, describing nonlocal quantum coherence between particles, is found to be living on a zero-momentum k0 = 0 -shell. With the knowledge of these coherence solutions in hand, we take on a problem in which such quantum coherence effects should be promi- nent. The problem, a Dirac equation with a complex and time-dependent mass profile, is manifestly non-translation invariant due to the temporal variance of the mass, and is in addition motivated by electroweak baryo- genesis. -
Conformal Structures and Twistors in the Paravector Model of Spacetime
Conformal structures and twistors in the paravector model of spacetime Rold˜ao da Rocha∗ Instituto de F´ısica Te´orica Universidade Estadual Paulista Rua Pamplona 145 01405-900 S˜ao Paulo, SP, Brazil and DRCC - Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas CP 6165, 13083-970 Campinas, SP, Brazil J. Vaz, Jr. Departamento de Matem´atica Aplicada, IMECC, Unicamp, CP 6065, 13083-859, Campinas, SP, Brazil.† Some properties of the Clifford algebras Cℓ3,0, Cℓ1,3, Cℓ4,1 ≃ C⊗Cℓ1,3 and Cℓ2,4 are presented, and three isomorphisms between the Dirac-Clifford algebra C ⊗Cℓ1,3 and Cℓ4,1 are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin+(2,4) is also investigated, in the light of a suitable isomorphism between C ⊗Cℓ1,3 and Cℓ4,1. After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin+(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian R4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac-Clifford algebra C ⊗ Cℓ1,3 using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose the Clifford algebra over R4,1 is also used to describe conformal maps, instead of R2,4. Our formalism sheds some new light on the use of the paravector model and generalizations. -
Maxima by Example: Ch. 12, Dirac Algebra and Quantum Electrodynamics ∗
Maxima by Example: Ch. 12, Dirac Algebra and Quantum Electrodynamics ∗ Edwin L. Woollett June 8, 2019 Contents 1 Introduction 5 2 Kinematics and Mandelstam Variables 5 2.1 1 + 2 3 + 4 Scattering for Non-Equal Masses . ....... 6 2.2 Mandelstam→ Variable Tools at Work . .................. 7 2.3 CM Frame Scattering Description . .................. 9 2.4 CM Frame Differential Scattering Cross Section for (AB) (CD) .................... 13 → 2.5 Lab Frame Differential Scattering Cross Section for Elastic Scattering (AB) (AB) .......... 13 2.6 DecayRateforTwoBodyDecay. .→............... 14 3 SpinSums 14 1 4 Spin 2 Examples 15 4.1 e e+ µ µ+,ee-mumu-AE.wxm .................................... 15 − → − 4.2 e µ e µ ,e-mu-scatt.wxm .................................... 16 − − → − − 4.3 e− e− e− e− (Moller Scattering), em-em-scatt-AE.wxm . .......... 16 + → + 4.4 e e e e (Bhabha Scattering), em-ep-scatt.AE.wxm . .......... 17 − → − 5 Theorems for Physical External Photons 17 5.1 Photon Real Linear Polarization 3-Vector Theorems: photon1.wxm .................... 17 5.2 Photon Circular Polarization 4-Vectors, photon2.wxm . ........................... 18 6 Examples for Spin 1/2 and External Photons 18 6.1 e e+ γ γ,ee-gaga-AE.wxm .................................... 18 − → 6.2 γ e− γ e−, compton1.wxm, compton-CMS-HE.wxm . ........ 19 6.3 Covariant→ Physical External Photon Polarization 4-Vectors-AReview................... 20 7 Examples Which Involve Scalar Particles 20 7.1 π+K+ π+K+ .............................................. 21 → 7.2 π+π+ π+π+ .............................................. 21 → 7.3 e π+ e π+ ............................................... 22 − → − 7.4 γ π+ γ π+ ................................................ 23 → 8 Electroweak Unification Examples 23 8.1 W-bosonDecay...................................