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The Nahm Transform for Calorons

Benoit Charbonneau and Jacques Hurtubise

in The Many Facets of Geometry: A Tribute to Nigel Hitchin

One mysterious feature of the self-duality equations on ℝ4 is the existence of a quite remarkable non-linear transform, the Nahm transform. It maps solutions to the self-duality equations on ℝ4 ... More

One mysterious feature of the self-duality equations on ℝ4 is the existence of a quite remarkable non-linear transform, the Nahm transform. It maps solutions to the self-duality equations on ℝ4 invariant under a closed translation group G ⊂ ℝ4 to solutions to the self-duality equations on (ℝ4)✱ invariant under the dual group G✱. This transform uses spaces of solutions to the Dirac equation, it is quite sensitive to boundary conditions, which must be defined with care, and it is not straightforward: for example, it tends to interchange rank and degree. This chapter is organized as follows. Section 4.2 summarizes the work of Nye and Singer towards showing that the Nahm transform is an equivalence between calorons and appropriate solutions to Nahm's equations. Section 4.3 describes the complex geometry (‘spectral data’) that encodes a caloron. Section 4.4 studies the process by which spectral data also correspond to solutions to Nahm's equations. Section 4.5 shows that the two Nahm transforms are inverses. Section 4.6 gives a description of moduli, expounded in Charbonneau and Hurtubise (2007).Less

Quantized Dirac Fields

Carlo Giunti and Chung W. Kim

in Fundamentals of Neutrino Physics and Astrophysics

This chapter discusses the physics of quantized Dirac fields with detailed treatment of Dirac equation, representations of gamma matrices, products of gamma matrices, relativistic covariance (boosts, ... More

This chapter discusses the physics of quantized Dirac fields with detailed treatment of Dirac equation, representations of gamma matrices, products of gamma matrices, relativistic covariance (boosts, rotations, and invariants), helicity, gauge transformations, chirality, solution of the Dirac equation (Dirac representation, chiral representation, two-component helicity eigenstate spinors, and massless field), quantization, symmetry transformation of states (space-time translations and Lorentz transformations), C, P, and T transformations, wave packets, and Fierz transformations.Less

Dirac Theory of the Electron

P.J.E. Peebles

in Quantum Mechanics

This chapter explores applications drawn from Dirac theory of the electron. In the treatment of electrons, it uses the following: an electron has spin 1/2; its magnetic dipole moment is very nearly ... More

This chapter explores applications drawn from Dirac theory of the electron. In the treatment of electrons, it uses the following: an electron has spin 1/2; its magnetic dipole moment is very nearly twice that of the orbital model in which charge and mass move together; and the spin-orbit interaction is a factor of two off the value arrived at by the heuristic argument in the Chapter 7. The factor of two in the last effect is recovered if one does the Lorentz transformations in a more careful (and correct) way, but it is easier to get it from the relativistic Dirac equation. This equation applied to an electron also says the particle has spin 1/2, as observed, and it says the gyromagnetic ratio in equation (23.11) is g = 2. The small difference from the observed value is accounted for by the quantum treatment of the electromagnetic field.Less

Relativistic wave equations

Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski, Jr.

in Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

This chapter deals with problems related to the Klein-Gordon equation and the Dirac equation.

Towards a Relativistic Quantum Mechanics

J. Iliopoulos and T.N. Tomaras

in Elementary Particle Physics: The Standard Theory

The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions ... More

The Klein–Gordon and the Dirac equations are studied as candidates for a relativistic generalisation of the Schrödinger equation. We show that the first is unacceptable because it admits solutions with arbitrarily large negative energy and has no conserved current with positive definite probability density. The Dirac equation on the other hand does have a physically acceptable conserved current, but it too suffers from the presence of negative energy solutions. We show that the latter can be interpreted as describing anti-particles. In either case there is no fully consistent interpretation as a single-particle wave equation and we are led to a formalism admitting an infinite number of degrees of freedom, that is a quantum field theory. We can still use the Dirac equation at low energies when the effects of anti-particles are negligible and we study applications in atomic physics.Less

Elementary relativistic quantum mechanics

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the ... More

This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.Less

Relativistic Wave Equations

J. Iliopoulos and T.N. Tomaras

in Elementary Particle Physics: The Standard Theory

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian ... More

We derive the most general relativistically covariant linear differential equations, having at most two derivatives, for scalar, spinor and vector fields. We introduce the corresponding Lagrangian and Hamiltonian formalisms and present the expansion of the solutions in terms of plane waves. In each case, we study the propagation properties of the corresponding Green functions. We start with the simplest example of the Klein–Gordon equation for a real field and generalise it to that of N real, or complex fields. As a next step we derive the Weyl, Majorana and Dirac equations for spinor fields. They are first order differential equations and we show how to adapt to them the canonical formalism. We end with the Proca and Maxwell equations for massive and massless spin-one fields and, in each case, we determine the physical degrees of freedom.Less

Relativistic square-root spaces∗

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £  Lorentz spacetime and momentum-energy vectors is readily ... More

Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £  Lorentz spacetime and momentum-energy vectors is readily applied to practical problems by using their scalar products that are Lorentz-invariant, the same in every Lorentz frame. Dirac went the other way by expanding the Lorentz quadratic invariant operator in the linear wave equation back into their original square-root vectors. The resulting Dirac spinor wave functions give full access to the rich spacetime properties of the wave motion in the square-root space. These properties include their spacetime symmetries, and the breaking of these symmetries by using other types of spinor wave functions. The connection is also made with Cartan's complexified square-root coordinates. Multidimensional arrays of spatial and spacetime vectors can be constructed and used. These objects include dyadics, Cartesian tensors and general tensors. Some of their useful properties are described.Less

Relativistic Mechanics

Oliver Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

It was apparent from its beginning that special relativity developed as the invariance theory of electrodynamics would require a modification of Newton’s three laws of motion. This chapter discusses ... More

It was apparent from its beginning that special relativity developed as the invariance theory of electrodynamics would require a modification of Newton’s three laws of motion. This chapter discusses that modified theory. The relativistically modified mechanics is presented and then recast into a fourvector form that demonstrates its consistency with special relativity. Traditional Lagrangian and Hamiltonian mechanics can incorporate these modifications. This chapter also discusses the momentum fourvector, fourvector form of Newton’s second law, conservation of fourvector momentum, particles of zero mass, traditional Lagrangian and traditional Hamiltonian, invariant Lagrangian, manifestly covariant Lagrange equations, momentum fourvectors and canonical momenta, extended and invariant Hamiltonian, manifestly covariant Hamilton equations, the Klein-Gordon equation, the Dirac equation, the manifestly covariant N-body problem, covariant Serret-Frenet theory, Fermi-Walker transport, and example of Fermi-Walker transport.Less

Relativistic quantum mechanics

G. Barr, R. Devenish, R. Walczak, and T. Weidberg

in Particle Physics in the LHC Era

This chapter starts with a brief summary of special relativity, describing how 4-vectors and the Lorentz transformation can be represented by complex matrices. Spinors are introduced as basic ... More

This chapter starts with a brief summary of special relativity, describing how 4-vectors and the Lorentz transformation can be represented by complex matrices. Spinors are introduced as basic building blocks of special relativity, allowing a demonstration of how the Weyl equation and the Dirac equation emerge from the Lorentz transformation of spinors. The Klein–Gordon equation is discussed to show the problems and applicability of relativistic quantum mechanics. The Dirac equation for a free particle is then discussed, including the use of different representations (including the Dirac and Weyl representations), discrete symmetries, and the non-relativistic limit. Interactions with the classical electromagnetic field are introduced, demanding a corresponding gauge symmetry. The chapter concludes by extending the gauge symmetry to account for weak and strong interactions.Less

First steps in classical field theory

Andrew M. Steane

in Relativity Made Relatively Easy Volume 2: General Relativity and Cosmology

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some ... More

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.Less

From Schrödinger to Einstein and Dirac: Relativistic Effects

Jochen Autschbach

in Quantum Theory for Chemical Applications: From Basic Concepts to Advanced Topics

The implications of Einstein’s special relativity in chemistry are discussed. It is shown that relativistic effects on the electronic structure of an atom or molecule scales in leading order as Z2, ... More

The implications of Einstein’s special relativity in chemistry are discussed. It is shown that relativistic effects on the electronic structure of an atom or molecule scales in leading order as Z2, where Z is the charge number of the heaviest nucleus in the system. Well-known heavy atom effects in chemistry are discussed: The color of gold, the liquid state of mercury, the inert pair effect of heavy p-block elements, and more. Spin-orbit coupling (SOC) is also a relativistic effect and plays a big role in spectroscopy and chemistry. The Dirac equation (DE) replaces the electronic Schrodinger equation in relativistic quantum chemistry. The Dirac wavefunctions have 4 components. It is shown how an ‘exact 2-component’ (X2C) Hamiltonian can be constructed. X2C based all-electron calculations are becoming increasingly popular in quantum chemical applications. Molecular properties may undergo a picture-change effect when going from a 4-component to a 2-component framework.Less

Fermions and the Dirac equation

Michael Kachelriess

in Quantum Fields: From the Hubble to the Planck Scale

Starting from the spinor representation of the Lorentz group,Weyl spinors and their transformation properties are derived. The Dirac equation and the properties of its solutions are discussed. ... More

Starting from the spinor representation of the Lorentz group,Weyl spinors and their transformation properties are derived. The Dirac equation and the properties of its solutions are discussed. Graßmann numbers and the gener-ating functional for fermions are introduced. Weyl and Majorana fermions are examined.Less

Spin Separation and the Modified Dirac Equation

Kenneth G. Dyall and Knut Faegri

in Introduction to Relativistic Quantum Chemistry

In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent ... More

In the preceding chapters, the theory for calculations based on the Dirac equation has been laid out in some detail. The discussion of the methods included a comparison with equivalent nonrelativistic methods, from which it is apparent that four-component calculations will be considerably more expensive than the corresponding nonrelativistic calculations—perhaps two orders of magnitude more expensive. For this reason, there have been many methods developed that make approximations to the Dirac equation, and it is to these that we turn in this part of the book. There are two elements of the Dirac equation that contribute to the large amount of work: the presence of the small component of the wave function and the spin dependence of the Hamiltonian. The small component is primarily responsible for the large number of two-electron integrals which, as will be seen later, contain all the lowest-order relativistic corrections to the electron–electron interaction. The spin dependence is incorporated through the kinetic energy operator and the correction to the electronic Coulomb interaction, and also through the coupling of the spin and orbital angular momenta in the atomic 2-spinors, which form a natural basis set for the solution of the Dirac equation. Spin separation has obvious advantages from a computational perspective. As we will show for several spin-free approaches below, a spin-free Hamiltonian is generally real, and therefore real spin–orbitals may be employed for the large and small components. The spin can then be factorized out and spin-restricted Hartree–Fock methods used to generate the one-electron functions. In the post-SCF stage, where the no-pair approximation is invoked, the transformation of the integrals from the atomic to the molecular basis produces a set of real molecular integrals that are indistinguishable from a set of nonrelativistic MO integrals, and therefore all the nonrelativistic correlation methods may be employed without modification to obtain relativistic spin-free correlated wave functions. In most cases, spin–free relativistic effects dominate the relativistic corrections to electronic structure. We will show later that in a perturbation expansion based on the nonrelativistic wave function, the spin-free effects for a closed-shell system enter in first order, whereas the spin-dependent effects make their first contribution in second order. Less

Relativistic Wave Equations

Michael E. Peskin

in Concepts of Elementary Particle Physics

This chapter presents the wave equations that govern the behavior of quantum mechanical particles with spin 0, 1/2, and 1 in relativistic theories. These equations are the Klein-Gordon equation, the ... More

This chapter presents the wave equations that govern the behavior of quantum mechanical particles with spin 0, 1/2, and 1 in relativistic theories. These equations are the Klein-Gordon equation, the Dirac equation, and Maxwell’s equations.Less

The Schrödinger Equation in the Presence of Fields

Stephan P. A. Sauer

in Molecular Electromagnetism: A Computational Chemistry Approach

This chapter derives the Hamiltonian for the electronic Schrödinger equation of a molecule in the presence of external and internal static electric or magnetic fields. It starts off with the ... More

This chapter derives the Hamiltonian for the electronic Schrödinger equation of a molecule in the presence of external and internal static electric or magnetic fields. It starts off with the Born-Oppenheimer approximation and discusses in detail the minimal coupling approach to the interaction of molecules with fields from a non-relativistic as well as relativistic point of view. In this context, the Dirac equation of an electron is derived and reduced to the Schrödinger-Pauli equation via the elimination of the small component approach. It also considers the relation between electric and magnetic fields and their scalar and vector potentials through Maxwell's equations, and introduces the problems related to transformations of the gauge of these potentials.Less

Quantum Lattice Boltzmann (QLB)

Sauro Succi

in The Lattice Boltzmann Equation: For Complex States of Flowing Matter

The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed ... More

The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!Less

Gauge Field Theory and Gravitation

Daniel Canarutto

in Gauge Field Theory in Natural Geometric Language: A revisitation of mathematical notions of quantum physics

The notion of 2-spinor soldering form allows a neat formulation, called the ‘tetrad-affine setting’, of a theory of matter and gauge fields interacting with the gravitational field. The latter is ... More

The notion of 2-spinor soldering form allows a neat formulation, called the ‘tetrad-affine setting’, of a theory of matter and gauge fields interacting with the gravitational field. The latter is represented by a couple constituted by the soldering form and a 2-spinor connection. This approach is suited to describe matter fields with arbitrary spin and generic further internal structure. In particular one gets an approach to interacting Einstein-Cartan-Maxwell-Dirac fields, in which the only assumption is a complex bundle with 2-dimensional fibers: the needed bundles are obtained from it by natural geometric contructions, and any object which is not determined from these ‘minimal geometric data’ is assumed to be a dynamical field.Less

Density Functional Approaches to Relativistic Quantum Mechanics

Kenneth G. Dyall and Knut Faegri

in Introduction to Relativistic Quantum Chemistry

The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly: rather, its existence is inferred from the various experiments whose outcome is ... More

The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly: rather, its existence is inferred from the various experiments whose outcome is most rationally explained using a wave function interpretation of quantum mechanics. Further, the N-particle wave function is a rather complicated construction, depending on 3N spatial coordinates as well as N spin coordinates, correlated in a manner that almost defies description. By contrast, the electron density of an N-electron system is a much simpler quantity, described by three spatial coordinates and even accessible to experiment. In terms of the wave function, the electron density is expressed as . . . ρ(r) = N ∫ Ψ* (r1,r2,...,rN)Ψ (r1,r2,...,rN)dr2dr3 ...drN (14.1) . . . where the sum over spin coordinates is implicit. It might be much more convenient to have a theory based on the electron density rather than the wave function. The description would be much simpler, and with a greatly reduced (and constant) number of variables, the calculation of the electron density would hopefully be faster and less demanding. We also note that given the correct ground state density, we should be able to calculate any observable quantity of a stationary system. The answer to these hopes is density functional theory, or DFT. Over the past decade, DFT has become one of the most widely used tools of the computational chemist, and in particular for systems of some size. This success has come despite complaints about arbitrary parametrization of potentials, and laments about the absence of a universal principle (other than comparison with experiment) that can guide improvements in the way the variational principle has led the development of wave-function-based methods. We do not intend to pursue that particular discussion, but we note as a historical fact that many important early contributions to relativistic quantum chemistry were made using DFT-like methods. Furthermore, there is every reason to try to extend the success of nonrelativistic DFT methods to the relativistic domain. We suspect that their potential for conquering a sizable part of this field is at least as large as it has been in the nonrelativistic domain. Less