The Solutions of Dirac Equation on the Hyperboloid Under Perpendicular Magnetic Fields
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1 Introduction 2 Klein Gordon Equation
Thimo Preis 1 1 Introduction Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations from the Hamiltonian formalism of non-relativistic Classical Mechanics by replacing dynamical variables by operators through the correspondence principle. Therefore, the physical properties predicted by the Schrödinger theory are invariant in a Galilean change of referential, but they do not have the invariance under a Lorentz change of referential required by the principle of relativity. In particular, all phenomena concerning the interaction between light and matter, such as emission, absorption or scattering of photons, is outside the framework of non-relativistic Quantum Mechanics. One of the main difficulties in elaborating relativistic Quantum Mechanics comes from the fact that the law of conservation of the number of particles ceases in general to be true. Due to the equivalence of mass and energy there can be creation or absorption of particles whenever the interactions give rise to energy transfers equal or superior to the rest massses of these particles. This theory, which i am about to present to you, is not exempt of difficulties, but it accounts for a very large body of experimental facts. We will base our deductions upon the six axioms of the Copenhagen Interpretation of quantum mechanics.Therefore, with the principles of quantum mechanics and of relativistic invariance at our disposal we will try to construct a Lorentz covariant wave equation. For this we need special relativity. 1.1 Special Relativity Special relativity states that the laws of nature are independent of the observer´s frame if it belongs to the class of frames obtained from each other by transformations of the Poincaré group. -
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. -
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 . -
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. -
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. -
2 Lecture 1: Spinors, Their Properties and Spinor Prodcuts
2 Lecture 1: spinors, their properties and spinor prodcuts Consider a theory of a single massless Dirac fermion . The Lagrangian is = ¯ i@ˆ . (2.1) L ⇣ ⌘ The Dirac equation is i@ˆ =0, (2.2) which, in momentum space becomes pUˆ (p)=0, pVˆ (p)=0, (2.3) depending on whether we take positive-energy(particle) or negative-energy (anti-particle) solutions of the Dirac equation. Therefore, in the massless case no di↵erence appears in equations for paprticles and anti-particles. Finding one solution is therefore sufficient. The algebra is simplified if we take γ matrices in Weyl repreentation where µ µ 0 σ γ = µ . (2.4) " σ¯ 0 # and σµ =(1,~σ) andσ ¯µ =(1, ~ ). The Pauli matrices are − 01 0 i 10 σ = ,σ= − ,σ= . (2.5) 1 10 2 i 0 3 0 1 " # " # " − # The matrix γ5 is taken to be 10 γ5 = − . (2.6) " 01# We can use the matrix γ5 to construct projection operators on to upper and lower parts of the four-component spinors U and V . The projection operators are 1 γ 1+γ Pˆ = − 5 , Pˆ = 5 . (2.7) L 2 R 2 Let us write u (p) U(p)= L , (2.8) uR(p) ! where uL(p) and uR(p) are two-component spinors. Since µ 0 pµσ pˆ = µ , (2.9) " pµσ¯ 0(p) # andpU ˆ (p) = 0, the two-component spinors satisfy the following (Weyl) equations µ µ pµσ uR(p)=0,pµσ¯ uL(p)=0. (2.10) –3– Suppose that we have a left handed spinor uL(p) that satisfies the Weyl equation. -
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). -
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). -
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. -
On the Schott Term in the Lorentz-Abraham-Dirac Equation
Article On the Schott Term in the Lorentz-Abraham-Dirac Equation Tatsufumi Nakamura Fukuoka Institute of Technology, Wajiro-Higashi, Higashi-ku, Fukuoka 811-0295, Japan; t-nakamura@fit.ac.jp Received: 06 August 2020; Accepted: 25 September 2020; Published: 28 September 2020 Abstract: The equation of motion for a radiating charged particle is known as the Lorentz–Abraham–Dirac (LAD) equation. The radiation reaction force in the LAD equation contains a third time-derivative term, called the Schott term, which leads to a runaway solution and a pre-acceleration solution. Since the Schott energy is the field energy confined to an area close to the particle and reversibly exchanged between particle and fields, the question of how it affects particle motion is of interest. In here we have obtained solutions for the LAD equation with and without the Schott term, and have compared them quantitatively. We have shown that the relative difference between the two solutions is quite small in the classical radiation reaction dominated regime. Keywords: radiation reaction; Loretnz–Abraham–Dirac equation; schott term 1. Introduction The laser’s focused intensity has been increasing since its invention, and is going to reach 1024 W/cm2 and beyond in the very near future [1]. These intense fields open up new research in high energy-density physics, and researchers are searching for laser-driven quantum beams with unique features [2–4]. One of the important issues in the interactions of these intense fields with matter is the radiation reaction, which is a back reaction of the radiation emission on the charged particle which emits a form of radiation. -
7 Quantized Free Dirac Fields
7 Quantized Free Dirac Fields 7.1 The Dirac Equation and Quantum Field Theory The Dirac equation is a relativistic wave equation which describes the quantum dynamics of spinors. We will see in this section that a consistent description of this theory cannot be done outside the framework of (local) relativistic Quantum Field Theory. The Dirac Equation (i∂/ m)ψ =0 ψ¯(i∂/ + m) = 0 (1) − can be regarded as the equations of motion of a complex field ψ. Much as in the case of the scalar field, and also in close analogy to the theory of non-relativistic many particle systems discussed in the last chapter, the Dirac field is an operator which acts on a Fock space. We have already discussed that the Dirac equation also follows from a least-action-principle. Indeed the Lagrangian i µ = [ψ¯∂ψ/ (∂µψ¯)γ ψ] mψψ¯ ψ¯(i∂/ m)ψ (2) L 2 − − ≡ − has the Dirac equation for its equation of motion. Also, the momentum Πα(x) canonically conjugate to ψα(x) is ψ δ † Πα(x)= L = iψα (3) δ∂0ψα(x) Thus, they obey the equal-time Poisson Brackets ψ 3 ψα(~x), Π (~y) P B = iδαβδ (~x ~y) (4) { β } − Thus † 3 ψα(~x), ψ (~y) P B = δαβδ (~x ~y) (5) { β } − † In other words the field ψα and its adjoint ψα are a canonical pair. This result follows from the fact that the Dirac Lagrangian is first order in time derivatives. Notice that the field theory of non-relativistic many-particle systems (for both fermions on bosons) also has a Lagrangian which is first order in time derivatives. -
Dirac Equation
Particle Physics Dr. Alexander Mitov µ+ µ+ e- e+ e- e+ µ- µ- µ+ µ+ e- e+ e- e+ µ- µ- Handout 2 : The Dirac Equation Dr. A. Mitov Particle Physics 45 Non-Relativistic QM (Revision) • For particle physics need a relativistic formulation of quantum mechanics. But first take a few moments to review the non-relativistic formulation QM • Take as the starting point non-relativistic energy: • In QM we identify the energy and momentum operators: which gives the time dependent Schrödinger equation (take V=0 for simplicity) (S1) with plane wave solutions: where •The SE is first order in the time derivatives and second order in spatial derivatives – and is manifestly not Lorentz invariant. •In what follows we will use probability density/current extensively. For the non-relativistic case these are derived as follows (S1)* (S2) Dr. A. Mitov Particle Physics 46 •Which by comparison with the continuity equation leads to the following expressions for probability density and current: •For a plane wave and «The number of particles per unit volume is « For particles per unit volume moving at velocity , have passing through a unit area per unit time (particle flux). Therefore is a vector in the particle’s direction with magnitude equal to the flux. Dr. A. Mitov Particle Physics 47 The Klein-Gordon Equation •Applying to the relativistic equation for energy: (KG1) gives the Klein-Gordon equation: (KG2) •Using KG can be expressed compactly as (KG3) •For plane wave solutions, , the KG equation gives: « Not surprisingly, the KG equation has negative energy solutions – this is just what we started with in eq.