DOCSLIB.ORG

What Is the Dirac Equation?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
What Is the Dirac Equation? What is the Dirac equation? M. Burak Erdo˘gan ∗ William R. Green y Ebru Toprak z July 15, 2021 at all times, and hence the model needs to be first or- der in time, [Tha92]. In addition, it should conserve In the early part of the 20th century huge advances the L2 norm of solutions. Dirac combined the quan- were made in theoretical physics that have led to tum mechanical notions of energy and momentum vast mathematical developments and exciting open operators E = i~@t, p = −i~rx with the relativis- 2 2 2 problems. Einstein's development of relativistic the- tic Pythagorean energy relation E = (cp) + (E0) 2 ory in the first decade was followed by Schr¨odinger's where E0 = mc is the rest energy. quantum mechanical theory in 1925. Einstein's the- Inserting the energy and momentum operators into ory could be used to describe bodies moving at great the energy relation leads to a Klein{Gordon equation speeds, while Schr¨odinger'stheory described the evo- 2 2 2 4 lution of very small particles. Both models break −~ tt = (−~ ∆x + m c ) : down when attempting to describe the evolution of The Klein{Gordon equation is second order, and does small particles moving at great speeds. In 1927, Paul not have an L2-conservation law. To remedy these Dirac sought to reconcile these theories and intro- shortcomings, Dirac sought to develop an operator1 duced the Dirac equation to describe relativitistic quantum mechanics. 2 Dm = −ic~α1@x1 − ic~α2@x2 − ic~α3@x3 + mc β Dirac's formulation of a hyperbolic system of par- tial differential equations has provided fundamental which could formally act as a square root of the Klein- 2 2 2 models and insights in a variety of fields from parti- Gordon operator, that is, satisfy Dm = −c ~ ∆ + 2 4 cle physics and quantum field theory to more recent m c . This is possible if the coefficients αj; β are 2 2 applications to nanotechnology, specifically the study 4 × 4 matrices satisfying αj = β = I and the anti- of graphene. commutation relationship (for j 6= k) αjαk = −αkαj; αjβ = −βαj: Dirac Equation Typically the Pauli matrices, To formulate the Dirac equation consistent with a quantum mechanical interpretation, the wave func- 0 −i 0 1 1 0 σ1 = ; σ2 = ; σ3 = ; tion at time t = 0 should determine the wave function i 0 1 0 0 −1 ∗The first author is a professor of mathematics at the Uni- are used to formulate the Dirac system versity of Illinois at Urbana-Champaign. His email address is 3 [email protected]. X yThe second author is an associate professor of mathematics 2 i~ t(x; t) = − ic~ αk@xk + mc β (x; t); (1) at Rose-Hulman Institute of Technology. His email address is k=1 [email protected]. zThe third author is a Hill Assistant Professor at Rutgers 1For concreteness, we only consider the case of three spatial 3 University. Her email address is [email protected]. dimensions, that is when x 2 R . 1 where A model dispersive PDE is the free Schr¨odinger equation: 2 IC 0 0 σi β = ; αi = : 0 −I 2 σ 0 C i i t = −∆ : (3) For notational convenience we choose physical A plane wave of the form (x; t) = exp(ik · x − i!t) units c = = 1, and denote P3 α @ = α · r, ~ k=1 k xk formally solves the equation provided that the \dis- to write the equation as persion relation", ! = jkj2, holds. Writing exp(ik · x − i!t) = exp(ik · (x − kt)), this relationship may be i (x; t) = (−iα · r + mβ) (x; t): (2) t interpreted as saying the speed of the wave is equal By the representation (2), it is clear that the Dirac to the frequency of vibration. operator does not act on a complex-valued wave func- Using Fourier transform techniques, we can repre- tion as the Schr¨odingeroperator does, but rather acts sent the initial data (x; 0) = f(x) as a superposition 4 of plane waves exp(ik · x) for a broad class of ini- on spinor fields. That is the solution to (2) is a C - valued function. tial data. Now, the dispersion relation dictates that In contrast to the Schr¨odingermodel, in which the higher frequency (jkj large) portions of the solution move more quickly than lower frequency portions of free operator H0 = −∆ is non-negative, the free Dirac the solution. A by-product is that a concentrated operator Dm = −iα·r+mβ is unbounded both above initial profile f(x) will spread out, becoming smaller and below. In particular, the spectrum is σ(Dm) = (−∞; −m] [ [m; 1). and smoother in some sense. The smoothing effect is These peculiarities lead to an ambiguity in the more subtle compared to parabolic equations such as physical interpretation of the solution . One inter- the heat equation since the linear evolution preserves 2 pretation is that the solution couples the evolution the L -based Sobolev norms. Various quantifications of a quantum particle with its anti-particle, namely of this decay and smoothing for linear equations are coupling the evolution of electrons and positrons, useful in the study of nonlinear counterparts. Heuris- [Tha92]. This interpretation predicted the existence tically, for equations with power nonlinearities the of the positron before its discovery in 1932. nonlinear terms become smaller compared to the lin- The unboundedness in both directions creates chal- ear terms due to the decay. lenges in the analysis of perturbations. For non- The best way to measure this spread is to use the linear perturbations, the concentration-compactness Fourier transform techniques to represent the solu- n method fails for Dirac as it relies on the operator be- tion of (3) as a convolution integral. For x 2 R , ing bounded below. Moreover, one can not define the Friedrichs extension for symmetric perturbations Z − n ijx−yj2=4t of Dirac operators since they are not bounded below, (x; t) = (−4πit) 2 e f(y) dy: (4) n [RS79]. R Using (4), one can see the global dispersive estimate The Dirac equation as a dispersive PDE it∆ − n ke fkL1 ≤ Cnjtj 2 kfkL1 : The Dirac equation is an example of a dispersive par- x x tial differential equation. By dispersive, we mean This L1 ! L1 bound reflects the dispersion present equations for which different frequency components in the Schr¨odingerequation, since the sup norm de- of the initial data (wave) propagate with different ve- cays in time as the waves of varying frequencies locities. We can envision the solution as being made spread out in space. These estimates can be used up of wave packets, where the frequency of vibration to obtain Strichartz estimates of the form is directly related to the speed at which the wave it∆ packet moves. ke fk q r ≤ Ckfk 2 ; Lt Lx Lx 2 for admissible exponents q; r ≥ 2, (q; r; n) 6= [EGT19]. See [GM01] for a more thorough discus- 2 1 1 (2; 1; 2), with q = n( 2 − r ): sion of further spectral issues. Strichartz estimates were first obtained by Dispersive estimates discussed in the previous sec- Strichartz via Fourier restriction theory of Stein. tion for (5) do not hold in general if the opera- Later, it was established that they can be obtained tor H has eigenvalues. One can project onto the using L1 ! L1 estimates. These mixed space-time continuous spectrum to recover dispersive estimates, estimates provide another way to quantify the dis- though threshold eigenvalues or resonances (at ener- persive nature of the equation, and are ubiquitous in gies λ = ±m) are known to affect the time decay, the study of nonlinear equations. [EGT19]. One can also measure the dispersion via smooth- One can study the perturbed evolution through ing estimates. Since a wave packet with frequency a generalized eigenfunction expansion, or through k moves with speed ∼ jkj it spends at most ∼ jkj−1 the resolvent operators and functional calculus tech- units of time on a ball of radius one. Averaging in niques. The free resolvent operators are defined by −1 time implies the Kato smoothing estimate R0(z) = (Dm − z) for z 2 C n σ(Dm). Recall- − 1 − 1 it∆ ing that Dm = −iα · r + mβ, and the the anti- khxi 2 (−∆) 4 e fk 2 ≤ CkfkL2 : Lt;x x commutation relations of the Pauli matrices, we see that D2 = −∆ + m, and This estimate can be interpreted to say that on av- m erage the solution is half a derivative smoother than (D − λ)(D + λ) = −∆ + m2 − λ2: the initial data. m m The dispersion in the Dirac equation is weaker than Formally, this leads us to the relationship that in the Schr¨odingerdue to the hyperbolic nature of the equation. Since the square of the free Dirac R (λ) = (D + λ)R (λ2 − m2); (6) 2 2 0 m 0 operator Dm = −∆ + m is a diagonal system of Klein-Gordon operators, there is finite speed of prop- −1 where R0(z) = (−∆ − z) is the Schr¨odingerresol- agation and the large frequency behavior of solutions vent. By Agmon's limiting absorption principle for does not travel arbitrarily fast as in the Schr¨odinger Schr¨odingeroperators, we can define the limiting op- evolution. This leads to weaker time decay and loss erators as z approaches the essential spectrum from of derivatives in dispersive estimates. upper and lower half plane. In the massive case, when m > 0, for jλj close to Linear perturbations m, the dominant contribution of the Dirac resolvent behaves like the Schr¨odingerresolvent. This leads to To account for particle interactions or external elec- − 3 a generic time-decay rate of size jtj 2 in three spa- tromagnetic fields in three dimensions, one perturbs tial dimensions.
Load more

Recommended publications
  • Introductory Lectures on Quantum Field Theory
    Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties.
    [Show full text]
  • 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.
    [Show full text]
  • Transverse Instability of Line Solitary Waves in Massive Dirac Equations
    J Nonlinear Sci (2016) 26:365–403 DOI 10.1007/s00332-015-9278-1 Transverse Instability of Line Solitary Waves in Massive Dirac Equations Dmitry Pelinovsky1 · Yusuke Shimabukuro1 Received: 7 November 2014 / Accepted: 23 October 2015 / Published online: 6 November 2015 © Springer Science+Business Media New York 2015 Abstract Working in the context of localized modes in periodic potentials, we con- sider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe poten- tials. The second one, a generalized massive Gross–Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitary waves are spectrally unstable with respect to periodic transverse perturbations of large periods. The spectral instability is induced by the spatial translation for the general- ized massive Thirring model and by the gauge rotation for the generalized massive Gross–Neveu model. We also observe numerically that the spectral instability holds for the transverse perturbations of any period in the generalized massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the generalized massive Gross–Neveu model. Keywords Massive Thirring model · Massive Gross-Neveu equation · Line solitary waves · Transverse instability · Lyapunov-Schmidt reduction method · Chebyshev interpolation method Mathematics Subject Classification 35Q41 · 37K40 · 37K45 Communicated by Michael I. Weinstein. B Dmitry Pelinovsky [email protected] 1 Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada 123 366 J Nonlinear Sci (2016) 26:365–403 1 Introduction Starting with pioneer contributions of V.E.
    [Show full text]
  • 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.
    [Show full text]
  • 13 the Dirac Equation
    13 The Dirac Equation A two-component spinor a χ = b transforms under rotations as iθn J χ e− · χ; ! with the angular momentum operators, Ji given by: 1 Ji = σi; 2 where σ are the Pauli matrices, n is the unit vector along the axis of rotation and θ is the angle of rotation. For a relativistic description we must also describe Lorentz boosts generated by the operators Ki. Together Ji and Ki form the algebra (set of commutation relations) Ki;Kj = iεi jkJk − ε Ji;Kj = i i jkKk ε Ji;Jj = i i jkJk 1 For a spin- 2 particle Ki are represented as i Ki = σi; 2 giving us two inequivalent representations. 1 χ Starting with a spin- 2 particle at rest, described by a spinor (0), we can boost to give two possible spinors α=2n σ χR(p) = e · χ(0) = (cosh(α=2) + n σsinh(α=2))χ(0) · or α=2n σ χL(p) = e− · χ(0) = (cosh(α=2) n σsinh(α=2))χ(0) − · where p sinh(α) = j j m and Ep cosh(α) = m so that (Ep + m + σ p) χR(p) = · χ(0) 2m(Ep + m) σ (pEp + m p) χL(p) = − · χ(0) 2m(Ep + m) p 57 Under the parity operator the three-moment is reversed p p so that χL χR. Therefore if we 1 $ − $ require a Lorentz description of a spin- 2 particles to be a proper representation of parity, we must include both χL and χR in one spinor (note that for massive particles the transformation p p $ − can be achieved by a Lorentz boost).
    [Show full text]
  • THE DIRAC OPERATOR 1. First Properties 1.1. Definition. Let X Be A
    THE DIRAC OPERATOR AKHIL MATHEW 1. First properties 1.1. Definition. Let X be a Riemannian manifold. Then the tangent bundle TX is a bundle of real inner product spaces, and we can form the corresponding Clifford bundle Cliff(TX). By definition, the bundle Cliff(TX) is a vector bundle of Z=2-graded R-algebras such that the fiber Cliff(TX)x is just the Clifford algebra Cliff(TxX) (for the inner product structure on TxX). Definition 1. A Clifford module on X is a vector bundle V over X together with an action Cliff(TX) ⊗ V ! V which makes each fiber Vx into a module over the Clifford algebra Cliff(TxX). Suppose now that V is given a connection r. Definition 2. The Dirac operator D on V is defined in local coordinates by sending a section s of V to X Ds = ei:rei s; for feig a local orthonormal frame for TX and the multiplication being Clifford multiplication. It is easy to check that this definition is independent of the coordinate system. A more invariant way of defining it is to consider the composite of differential operators (1) V !r T ∗X ⊗ V ' TX ⊗ V,! Cliff(TX) ⊗ V ! V; where the first map is the connection, the second map is the isomorphism T ∗X ' TX determined by the Riemannian structure, the third map is the natural inclusion, and the fourth map is Clifford multiplication. The Dirac operator is clearly a first-order differential operator. We can work out its symbol fairly easily. The symbol1 of the connection r is ∗ Sym(r)(v; t) = iv ⊗ t; v 2 Vx; t 2 Tx X: All the other differential operators in (1) are just morphisms of vector bundles.
    [Show full text]
  • P-Dirac Operators Craig A. Nolder Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510, USA
    p-Dirac Operators Craig A. Nolder Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510, USA and John Ryan Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA Abstract We introduce non-linear Dirac operators in Rn associated to the p-harmonic equation and we extend to other contexts including spin manifolds and the sphere. 1 Introduction Associated to each type of Laplacian one usually sees a first order lineariza- tion, to a Dirac operator. For instance associated to the Laplacian in Rn is the euclidean Dirac operator arising in Clifford analysis. For the Laplace- Beltrami operator associated to a Riemannian manifold there is the Hodge- Dirac operator d + d!, where d is the exterior derivative and d! is the Hodge codifferential which is the formal adjoint to d. Further, in reverse order, to the Atiyah-Singer-Dirac operator on a spin manifold there is the spinorial Laplacian. Also on Sn one has a conformal Dirac operator and the conformal Laplacian. See for instance [3, 5, 9, 11] for details. Besides the Laplacian in Rn there are also the non-linear operators re- ferred to as p-Laplacians. See for instance [7, 8, 10]. Despite being non-linear these second order operators posses properties very similar to the usual Lapla- cian in euclidean space. Further when p = 2 this operator corresponds to the usual Laplacian in euclidean space. Here we shall introduce a first order nonlinear differential operator which in the case p = 2 coincides with the euclidean Dirac operator. The confor- mal covariance of these operators are established.
    [Show full text]
  • Solvable Two-Body Dirac Equation As a Potential Model of Light Mesons?
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 048, 19 pages Solvable Two-Body Dirac Equation as a Potential Model of Light Mesons? Askold DUVIRYAK Institute for Condensed Matter Physics of National Academy of Sciences of Ukraine, 1 Svientsitskii Str., UA–79011 Lviv, Ukraine E-mail: [email protected] Received October 29, 2007, in final form May 07, 2008; Published online May 30, 2008 Original article is available at http://www.emis.de/journals/SIGMA/2008/048/ Abstract. The two-body Dirac equation with general local potential is reduced to the pair of ordinary second-order differential equations for radial components of a wave function. The class of linear + Coulomb potentials with complicated spin-angular structure is found, for which the equation is exactly solvable. On this ground a relativistic potential model of light mesons is constructed and the mass spectrum is calculated. It is compared with experimental data. Key words: two body Dirac equation; Dirac oscillator; solvable model; Regge trajectories 2000 Mathematics Subject Classification: 81Q05; 34A05 1 Introduction It is well known that light meson spectra are structured into Regge trajectories which are approximately linear and degenerated due to a weak dependence of masses of resonances on their spin state. These features are reproduced well within the exactly solvable simple relativistic oscillator model (SROM) [1, 2, 3] describing two Klein–Gordon particles harmonically bound. Actually, mesons consist of quarks, i.e., spin-1/2 constituents. Thus potential models based on the Dirac equation are expected to be more appropriate for mesons spectroscopy. In recent years the two body Dirac equations (2BDE) in different formulations1 with various confining potentials are used as a relativistic constituent quark models [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].
    [Show full text]
  • Dirac Operator in Matrix Geometry
    New Mexico Tech (January 2005) Dirac Operator in Matrix Geometry Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: [email protected] Dedicated to the Memory of Vladimir Fock and Dmitri Ivanenko We review the construction of the Dirac operator and its proper- ties in Riemannian geometry and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein- Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on arXiv:math-ph/0502001v1 31 Jan 2005 our previous attempts to generalize the notion of the Dirac opera- tor to the case of Matrix Geometry, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a “non-commutative” metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential op- erators, that can be called “non-commutative” Dirac operator and non-commutative Laplace operator. We construct the corresponding heat kernel for the non-commutative Laplace type operator and com- pute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein-Hilbert action. 1 1 Introduction Dirac operator was discovered by Dirac in 1928 as a “square root” of the D’Alambert operator in a flat Minkowskian space in an attempt to develop a relativistic theory of the electron.
    [Show full text]
  • A Spinorial Proof of the Rigidity of the Riemannian Schwarzschild Manifold
    A SPINORIAL PROOF OF THE RIGIDITY OF THE RIEMANNIAN SCHWARZSCHILD MANIFOLD SIMON RAULOT Abstract. We revisit and generalize a recent result of Cederbaum [C2, C3] concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems [BM, GIS, Hw] as well as the more recent uniqueness theorems for pho- ton spheres [C1, CG1, CG2]. 1. Introduction An (n+1)-dimensional vacuum spacetime is a Lorentzian manifold ( n+1, g) satisfying the Einstein field equations Ric = 0, where Ric is the RicciL tensor of the metric g. The vacuum is said to be static when n+1 = R M n, g = N 2 dt2 + g, L × − where (M n, g) is an n-dimensional connected smooth Riemannian manifold, that we will take to be orientable, standing for the unchanging slices of constant time and N C∞(M n) is a non-trivial smooth function on M n. To model the exterior∈ of an isolated system, it seems physically natural to require asymptotic flatness, that is, the Cauchy hypersurface M n is usually taken to be asymptotically flat. The vacuum Einstein field equations can be translated into the following two conditions on (M n, g) and the lapse function N: 2N = N Ric, ∆N = 0, (1.1) ∇ where Ric, and ∆ are respectively the Ricci tensor, the covariant deriva- ∇ n arXiv:2009.09652v1 [math.DG] 21 Sep 2020 tive and the Laplace operator of the Riemannian manifold (M , g). Taking traces in the first of these two equations and taking into account the second one, we conclude immediately that the scalar curvature of (M n, g) is zero.
    [Show full text]
  • Relativistic Quantum Mechanics 1
    Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).
    [Show full text]
  • 1. Dirac Equation for Spin ½ Particles 2
    Advanced Particle Physics: III. QED III. QED for “pedestrians” 1. Dirac equation for spin ½ particles 2. Quantum-Electrodynamics and Feynman rules 3. Fermion-fermion scattering 4. Higher orders Literature: F. Halzen, A.D. Martin, “Quarks and Leptons” O. Nachtmann, “Elementarteilchenphysik” 1. Dirac Equation for spin ½ particles ∂ Idea: Linear ansatz to obtain E → i a relativistic wave equation w/ “ E = p + m ” ∂t r linear time derivatives (remove pr = −i ∇ negative energy solutions). Eψ = (αr ⋅ pr + β ⋅ m)ψ ∂ ⎛ ∂ ∂ ∂ ⎞ ⎜ ⎟ i ψ = −i⎜α1 ψ +α2 ψ +α3 ψ ⎟ + β mψ ∂t ⎝ ∂x1 ∂x2 ∂x3 ⎠ Solutions should also satisfy the relativistic energy momentum relation: E 2ψ = (pr 2 + m2 )ψ (Klein-Gordon Eq.) U. Uwer 1 Advanced Particle Physics: III. QED This is only the case if coefficients fulfill the relations: αiα j + α jαi = 2δij αi β + βα j = 0 β 2 = 1 Cannot be satisfied by scalar coefficients: Dirac proposed αi and β being 4×4 matrices working on 4 dim. vectors: ⎛ 0 σ ⎞ ⎛ 1 0 ⎞ σ are Pauli α = ⎜ i ⎟ and β = ⎜ ⎟ i 4×4 martices: i ⎜ ⎟ ⎜ ⎟ matrices ⎝σ i 0 ⎠ ⎝0 − 1⎠ ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ ⎟ ψ = 2 ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ ⎛ ∂ r ⎞ i⎜ β ψ + βαr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ ⎛ 0 ∂ r ⎞ i⎜γ ψ + γr ⋅ ∇ψ ⎟ − m ⋅1⋅ψ = 0 ⎝ ∂t ⎠ 0 i where γ = β and γ = βα i , i = 1,2,3 Dirac Equation: µ i γ ∂µψ − mψ = 0 ⎛ψ ⎞ ⎜ 1 ⎟ ⎜ψ 2 ⎟ Solutions ψ describe spin ½ (anti) particles: ψ = ⎜ψ ⎟ ⎜ 3 ⎟ ⎜ ⎟ ⎝ψ 4 ⎠ Extremely 4 ⎛ µ ∂ ⎞ compressed j = 1...4 : ∑ ⎜∑ i ⋅ (γ ) jk µ − mδ jk ⎟ψ k description k =1⎝ µ ∂x ⎠ U.
    [Show full text]