Supplemental Lecture 1
How did Newton discover the inverse square law
of gravity?
Abstract
We show that Kepler’s Laws imply the inverse square character of Newton’s law of gravitation. This discussion reproduces Newton’s original derivation in his Principia Mathematica. The lecture is self-contained and begins with the kinematics of two bodies in nonrelativistic Newtonian mechanics, a statement of Kepler’s three laws, an introduction to plane polar coordinates, the interpretation of Kepler’s second law in polar coordinates, the derivation of the inverse square law of gravity and the derivation of the fact that the strength of the gravitation force is proportional to the product of the attracting masses.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Supplemental Lecture 1 The Inverse Square Law of Gravity
Supplemental Lecture 2
The Gauss Bonnet Theorem
Abstract
We derive the Gauss Bonnet theorem in the framework of classical differential geometry. The far reaching significance of the theorem is discussed.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Supplemental Lecture 2 Derivation of the Gauss Bonnet Theorem
Supplemental Lecture 3
Acceleration: Relativistic Rocket Dynamics and Accelerating Reference Frames (Rindler Coordinates)
Abstract
An accelerating rocket is studied in special relativity and its equation of motion is contrasted to its non-relativistic treatment. Uniformly accelerating reference frames are formulated and the transformation between measurements in this non-inertial reference frame and an inertial frame is obtained and applied. The space-time metric inside an accelerating rocket is derived and the accelerating frame is found to be locally equivalent to the environment in a uniform gravitational field. The global properties of the accelerating frame (Rindler Wedge) are different from those of a uniform gravitational field: there is a past and future horizon of the accelerating frame but not for the uniform gravitational field. Analogies to the Schwarzschild black hole are drawn.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Supplemental Lecture 4
Surfaces of Zero, Positive and Negative Gaussian Curvature.
Euclidean, Spherical and Hyperbolic Geometry.
Abstract
This lecture considers two dimensional surfaces embedded in three dimensional Euclidean space. The lecture begins by introducing familiar surfaces of revolution which are generated by rotating a profile curve around the z axis. Their various parametrizations, metrics and intrinsic properties such as their Gaussian curvatures are studied. Euclidean, spherical and hyperbolic geometries are introduced, contrasted and analyzed. In both spherical and hyperbolic geometries the “Parallel Axiom” of two dimensional Euclidean space is untrue. In both spherical and hyperbolic geometries their non-zero intrinsic curvatures sets a fundamental length scale. For distances small compared to the reciprocal of the square root of the curvature, the curved spaces are well approximated by a flat Euclidean metric but this is untrue for distances large compared to the reciprocal of the square root of the curvature, In addition, unlike Euclidean space, geodesic triangles in spherical and hyperbolic geometries which are similar are also congruent. Both the Poincare Disc and Upper-Half-Plane models of spaces of constant negative curvature are presented and studied. Their isometries, geodesics and hyperbolic triangles are introduced and analyzed.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Non-Euclidean geometry, Poincare disc, H model, surfaces of revolution, metric, Gaussian curvature, projective geometry, geodesic triangles, isometries, Mbius transformations
Supplemental Lecture 5
Thomas Precession and
Fermi-Walker Transport in Special Relativity
and Geodesic Precession in General Relativity
Abstract
This lecture considers two topics in the motion of tops, spins and gyroscopes: 1. Thomas Precession and Fermi-Walker Transport in Special Relativity, and 2. Geodesic Precession in General Relativity. A gyroscope (“spin”) attached to an accelerating particle traveling in Minkowski space-time is seen to precess even in a torque-free environment. The angular rate of the precession is given by the famous Thomas precession frequency. We introduce the notion of Fermi-Walker transport to discuss this phenomenon in a systematic fashion and apply it to a particle propagating on a circular orbit. In a separate discussion we consider the geodesic motion of a particle with spin in a curved space-time. The spin necessarily precesses as a consequence of the space-time curvature. We illustrate the phenomenon for a circular orbit in a Schwarzschild metric. Accurate experimental measurements of this effect have been accomplished using earth satellites carrying gyroscopes.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Fermi-Walker transport, Thomas precession, Geodesic precession, Schwarzschild metric, Gravity Probe B (GP-B).
Supplemental Lecture 5 Precession in Special and General Relativity
Lambare_2017_Eur._J._Phys._38_045602
Supplemental Lecture 6
Rotating Sources, Gravito-Magnetism and The Kerr Black Hole
Abstract
This lecture consists of several topics in general relativity dealing with rotating sources, gravito-magnetism and a rotating black hole described by the Kerr metric. We begin by studying slowly rotating sources, such as planets and stars where the gravitational fields are weak and linearized gravity applies. We find the metric for these sources which depends explicitly on their angular momentum J. We consider the motion
of gyroscopes and point particles in these spaces and discover “frame dragging” and Lense-Thirring precession. Gravito-magnetism is also discovered in the context of gravity’s version of the Lorentz force law. These weak field results can also be obtained directly from special relativity and are consequences of the transformation laws of forces under boosts. However, general relativity allows us to go beyond linear, weak field physics, to strong gravity in which space time is highly curved. We turn to rotating black holes and we review the phenomenology of the Kerr metric. The physics of the “ergosphere”, the space time region between a surface of infinite redshift and an event horizon, is discussed. Two appendices consider rocket motion in the vicinity of a black hole and the exact redshift in strong but time independent fields. Appendix B illustrates the close connection between symmetries and conservation laws in general relativity.
Keywords: Rotating Sources, Gravito-Magnetism, frame-dragging, Kerr Black Hole, linearized gravity, Einstein-Maxwell equations, Lense-Thirring precession, Gravity Probe B (GP-B).
Supplemental Lecture 6 Rotating Sources, Gravito-Magnetism, the Kerr Metric
Supplemental Lecture 7
Light Cone Variables, Rapidity and High Multiplicity Collisions
Abstract
Light cone variables, are introduced to diagonalize Lorentz transformations (boosts) in the x (beam) direction. The “rapidity” of a boost is introduced and the rapidity is shown to transform additively under boosts, similar to the ordinary velocity in Newton’s world. The non-linear formula for the addition of velocities in Einstein’s world follows from the linear additivity of rapidities. The use of rapidity and light cone momenta in high energy collisions is introduced. Momentum space 
distributions of particles created in high energy proton-proton and nucleus-nucleus collisions are discussed. Feynman scaling is introduced. The reader is referred to the recent literature from the Large Hadron Collider (LHC) and the Relativistic Heavy Ion Collider (RHIC) for recent developments and phenomenological applications.
Supplemental Lecture 7 Light Cone Variable, Rapidity and High Multiplicity Collisions
Supplemental Lecture 8
Curves in 2-D, 3-D and on curved surfaces
Abstract
We develop the concepts of the curvature of a curve in two dimensions, curvature and torsion in three dimensions and the generalization and application of these ideas to curves constrained to curved surfaces. Parallel transportation and covariant differentiation are defined, developed and illustrated. The connection of the holonomy of closed, simple curves to the curvature of a surface is derived and illustrated. This lecture prefigures and provides the intuitive geometry behind more sophisticated, abstract developments in Riemannian geometry which are critical to General Relativity.
Supplement Lecture 8. Curves in 2-D, 3-D and on Curved Surfaces
Supplemental Lecture 9
Planes, Spheres and Surfaces:
From Straight Lines, Triangles and Differentiation to Geodesics, Covariant Differentiation, Curvature and Holonomy
Abstract
We consider the geometry of planes, spheres and surfaces in the context of classical differential geometry. Geometric concepts which generalize from planes to spheres to surfaces are emphasized. This includes straight lines in planes which become geodesics on spheres and surfaces. Triangles in planes which become geodesic triangles on spheres and surfaces. The properties of such triangles are sensitive to the curvature of the sphere and general curved surfaces. Differentiation of functions and vectors on a plane generalize to covariant differentiation on spheres and curved surfaces. Covariant differentiation lead to constructions of quantities which measure curvature locally on spheres and surfaces. The spatial distribution of geodesics is controlled by differential equations which are sensitive to the curvature of the surfaces they are embedded in. The equations of Jacobi fields, the two dimensional analog of geodesic deviation, are obtained and studied.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Differential Geometry, curvature, geodesic, covariant differentiation, Gaussian curvature, holonomy, Jacobi fields.
Contents
Planar Geometry: Setting the Stage. 2
Geometry of Curved Surfaces. 27
The Gauss Map and Its Derivative, the Weingarten Map. 28
Illustrations. Parametrizing Surfaces and Computing their Properties. 33
Geometric Interpretation of the Gauss Map. 39
Gauss’ Theorem Egregium. Mainardi-Codazzi Consistency Equations. 41
Parallel Transport, Covariant Differentiation and the Gauss-Bonnet Theorem.. 44
Parallel Transport and Covariant Differentiation. Illustrations. 44
The Turning angle for curves on Curved Surfaces and the Gauss-Bonnet Theorem.. 49
Geodesic Polar Coordinates and the Geodesic Deviation. 53
Parallel Transport, Curvature, Holonomy and Jacobi Fields. 61
Supplemental Lecture 10
Accelerating Clocks, Hyperbolic Trajectories and the Twins Yet Again.
Abstract
We consider a clock experiencing constant proper acceleration g and a clock held by an inertial observer. We derive the relationship between times recorded by each clock, the proper time of the accelerating clock and the time t of the inertial observer. This problem is solved using elementary relations of special relativity. Several results found in Supplemental Lecture 3 on Rindler space are re-derived. The resulting hyperbolic relation between and t is plotted and discussed. The twin paradox is reconsidered in this context. This problem illustrates the Equivalence Principle of General Relativity as discussed in Supplemental Lecture 3.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Special relativity, acceleration, inertial frames of reference, proper time, twin paradox, Equivalence Principle.
Supplementary Lecture 10 Accelerating Clocks in Special Relativity
General Relativity for Physics Students
Contents
The Video Lecture series begins with a discussion of the Equivalence Principle, gravity and apparent (“virtual”) forces. Concepts are illustrated using the metric of a relativistic rotating coordinate system. Tidal forces of Newtonian mechanics are reviewed and are seen to be an indicator of curved space time and non-Euclidean geometry. The idea of the tangent space, local inertial reference frames, in a curved four dimensional space time manifold is introduced. The natural physical limitations of the Equivalence Principle in environments of non-uniform gravitational fields are discussed. The gravitational redshift is introduced as an application of the Equivalence principle, and the Twin Paradox is resolved as a problem in accelerating reference frames using the Equivalence Principle. The similarities and differences in the theories of electromagnetism and gravity are discussed. Polarization properties of electromagnetic waves (photons) are contrasted with those of gravity waves (gravitons). The equation for a geodesic in Riemannian space time is developed and applied to massive point particles. The covariant derivative is introduced to realize a coordinate independent measure of the rate of change of vector and tensor fields. Christoffel symbols are introduced to describe the space time dependence of sets of basis vectors. Metric Compatibility and the Equivalence Principle are discussed and are used to find an expression for Christoffel coefficients in terms of the metric. The curvature of space time is discussed and the Einstein Field Equations are introduced. The geodesic equation of motion of point particles are rederived from the field equations. Linearized gravity is introduced to systematically study environments of weak gravitational fields. The Schwarzschild metric is derived and its black hole is studied. Orbital motion around a static spherical mass is explored and the non-linearities that distinguish General Relativity from Newtonian gravity are explored. Relativistic tidal effects are discussed in the context of the equation for the Geodesic Deviation. Gravitational waves are studied and the LIGO experiment is discussed and illustrated as a detector of traveling gravitational tidal effects. The Cosmological constant and Dark Energy are introduced in a brief look at modern puzzles of gravitational physics. In a set of “Special Topic” lectures, rotating stars, frame-dragging, and the Lense-Thirring effect are studied as manifestations of velocity dependent effects, gravito-magnetism, in General Relativity. The Kerr metric is introduced and its event horizons and surfaces of infinite redshift are presented. Frame-dragging is illustrated by considering light rays propagating inside a rotating star’s ergosphere. Symmetries and conservation laws are discussed and applied to an exact treatment of the gravitational redshift. Special Relativity problems involving accelerating clocks and reference frames are discussed. The twin paradox is reconsidered as a problem in observing accelerating clocks and their hyperbolic motion in Minkowski space time. The Rindler Wedge is introduced and its metric is obtained and discussed from the perspective of the Equivalence Principle. Geodesic precession of gyroscopes in orbits around static stars is studied and illustrated.
Foreward. 2
1.The Equivalence Principle, Gravity, and Apparent Forces. 3
2.Motion in a Rotating, Relativistic Reference Frame. 11
3.Tidal Forces, non-Euclidean Geometry and “Local Inertial Reference Frames”. 19
4.Gravitational Red Shift 23
5.The Twins Again. 33
6.Similarities and Differences of Electromagnetism and Gravity. 36
7.The Equation of Motion of Particles in Curved Space-Time. 46
8.Covariant Derivatives and Covariant Vector Fields. 53
9.The Equivalence Principle, Metric Compatibility and Christoffel Symbols. 56
10.The Curvature of Space-Time. 60
11.The Einstein Field Equation: “ Curvature ~ Energy-Density”. 64
- Geodesic Equation of Motion as a Consequence of the Field Equations. 68
- Linearized Gravity. 71
12.The Schwarzschild Metric and Black Hole. 77
13.Circular Orbital Motion in the Schwarzschild metric. 94
14.Relativistic Tidal Forces. 100
15.The Discovery of Gravitational Waves. 103
16.Gravitational Radiation and Linearized General Relativity. 106
- Freely Propagating Gravity Waves in Minkowski Space Time. 107
- How to Detect Gravity Waves. 110
- The Radiation and Detection of Gravity Waves from Binary Systems. 112
17.Contrasting Special and General Relativity. The Cosmological Constant and Dark Energy. 120
Special Topic A. The Isotropic Metric and Linearized General Relativity. 126
Special Topic B. Slowly Rotating Stars, Frame-Dragging and Lense-Thirring Precession. 128
Special Topic C. Gravito-Magnetism.. 133
Special Topic D. Rotating Stars, Black Holes and an Introduction to the Kerr Metric. 136
Special Topic E. Newton in Orbit around a Schwarzschild Black Hole. 144
Special Topic F. Symmetries, Conservation Laws, Exact Redshift Formula for Static Gravity. 146
Special Topic G. An Accelerating Clock and Hyperbolic Motion. 150
Special Topic H. Uniform Proper Acceleration. 157
Special Topic I. An Accelerating Coordinate System: The Rindler Wedge. 158
Special Topic J. Geodesic Precession in General Relativity. 167
References. 170
Supplementary Lecture Series 11: General relativity for Physics Students
Differential Forms for Physics Students
1. Introduction
2. Special Relativity and Electrodynamics
Abstract
In our textbook and in various supplementary lectures, special relativity and electrodynamics have been presented using the mathematics of vector calculus and tensor analysis. More modern approaches to these subjects use differential forms. The aim of this lecture is to use differential forms to unify Maxwell’s equations into two quartets (Gauss’ Law and the Maxwell-Ampere equations in one case, and the absence of magnetic monopoles and Faraday’s equation in the other case) as a consequence of Lorentz covariance. We begin with an introduction to the mathematics of differential forms, exterior products, exterior differentiation, interior products, the Hodge star operator (duality), the Poincare Lemma and generalized Stoke’s Theorems. Examples are drawn from three dimensional Euclidean space and four dimensional Minkowski space time. The electromagnetic field strength differential form is introduced and the transformation properties of the electric and magnetic fields under Lorentz boosts are obtained. Maxwell’s equations are discussed in the language of differential forms and comparisons with tensor analyses are presented. The electromagnetic field strength differential form and the interior product lead to a formulation of the Lorentz force law which is a manifestly covariant.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Differential Forms, Exterior Algebra, Exterior Derivative, Poincare Lemma, Generalized Stoke’s Theorems, Special Relativity, Lorentz Invariance, Electrodynamics, Maxwell’s Equations, Lorentz Force Law.
Contents
Introduction: Setting the Stage. 2
Mappings and Coordinate Changes. 15
The Poincare Lemma and its Converse. 16
Special Relativity: Inertial Reference Frames, the Minkowski Metric and Four Vectors. 17
Electromagnetism, Tensors and Forms. 19
The Lorentz Force Law and the Interior Product. 26
Generalized Stoke’s Theorem and Integral Theorems in R3. 28
Differential Forms for Physics Students
III. Classical and Riemannian Differential Geometry
Abstract
Differential forms are employed to develop the theory of surfaces and manifolds. The method of moving frames of E. Cartan is introduced and the fundamental formulas of differential geometry are obtained in a coordinate-free fashion. This perspective leads to remarkably simple derivations of Gauss’ Theorem Egregium and the Gauss-Bonnet Theorem. The Gauss-Bonnet Theorem is developed as a structural equation of differential geometry without the need for coordinate systems or detailed formulas for the Gaussian curvature of surfaces or the geodesic curvature of curves. These ideas are generalized to higher dimensional manifolds. Topics in Riemannian and non-Euclidean geometry are introduced and studied.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Differential Forms, Classical Differential Geometry, Riemannian Geometry, E.Cartan, Orthonormal Frames, Gaussian Curvature, Theorem Egregium, Gauss-Bonnet Theorem, Hypersurfaces, Manifolds, Non-Euclidean Geometry
Contents
Gaussian and Mean Curvatures. 7
Gauss-Bonnet Theorem: Global Version. 13
Gauss-Bonnet Theorem: Local Version. 14
Intrinsic Geometry of Manifolds. 20
Appendix: Making Contact with Classical Differential and Riemannian Geometry. 28
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Abstract
This Essay, “Particles and Anti-Particles” explains how locality, causality and special relativity imply, in the context of quantum field theory, that each charged particle must be accompanied by an anti-particle of opposite charge but equal mass. The Essay consists of two parts: “Background” which presents preliminary ideas on non-relativistic harmonic motion by introducing creation and annihilation operators, illustrating them with coherent states, and then presents “second quantization” in the context of non-relativistic many body quantum mechanics. These ideas set the stage for the next portion of the Essay which presents the primary result, that relativistic quantum field theory predicts the existence of anti-particles. We will see that current conservation, locality and causality are the crucial ingredients here.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
This Essay supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Harmonic Oscillator, Creation and Annihilation Operators, Heisenberg Uncertainty Relation, Coherent States, Second Quantization, Field Operators, Relativistic Quantum Field Theory, Charge Conservation, Current Conservation, Causality, Particles and Anti-Particles.
———————————————————————————————————————
Contents
One Dimensional Harmonic Oscillator 2
Creation and Annihilation Operators. 2
Relativistic Quantum Fields and Anti-Particles. 20
Relativistic Fields and Operators. 20
Lorentz Transformations of Fields, States and Operators. 27
Charged Relativistic Fields. 29
Introduction to the Foundations of Quantum Field Theory for Physics Students
II. The Unruh Effect
Abstract
We consider the vacuum state of a scalar quantum field theory in Minkowski space time from the perspective of an accelerating observer and find that she observes a thermal bath of scalar particles at a temperature proportional to her proper acceleration. This is the Unruh effect. The mechanics of Rindler space time are developed in order to obtain this effect. Similarities to Hawking radiation are drawn.
In preparation for this discussion we begin with a simpler problem: The quantum mechanics of a driven harmonic oscillator. We use coherent states to describe its scattering states and find the probability distribution of the excitations in the final state.
In order to discuss problems in quantum field theory, we include several appendices on background subjects. One appendix introduces canonical quantization and Lagrangian methods for field theory. Noether’s Theorem is introduced and the relation between symmetries and conservation laws is obtained. Another appendix introduces Bogoliubov transformations which are used in the body of this Essay to relate the vacuum states of Minkowski and Rindler space times and equip us to solve for the Unruh temperature. Bogoliubov transformations will be useful in later Essays on the Casimir effect, the Schwinger effect and spontaneous symmetry breaking. A final appendix introduces the Euler Gamma function which is used in the calculation of the Bogoliubov transformation used to derive the Unruh temperature.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
This Essay supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Harmonic Oscillator, Creation and Annihilation Operators, Coherent States, Field Operators, Relativistic Quantum Field Theory, Minkowski Space Time, Rindler Space Time, The Unruh Effect, Bogoliubov Transformations, Scattering States, Noether’s Theorem, Symmetries and Conservation Laws.
————————————————————————————————————
Contents
The Driven One Dimensional Harmonic Oscillator. 2
Simple Harmonic Oscillator in a Uniform Gravitational Field. 2
The Driven Harmonic Oscillator: A Model Scattering Experiment 5
The Vacuum of an Accelerating Reference Frame. The Unruh Effect. 7
Rindler Variables: The View of Minkowski Space Time from an Accelerating Observer. 8
Scalar Fields in Minkowski and Rindler Space Time. 15
Appendix A. Lagrangian Field Theory, Canonical Quantization, Symmetries and Conservation Laws. 22
Appendix B. Bogoliubov Transformations. 31
Appendix C. The Euler Gamma Function. 34
Introduction to the Foundations of Quantum Field Theory
For Physics Students
III. The Casimir Effect
Abstract
Uncharged grounded conducting parallel plates experience a mutual attractive force which is a quantum, relativistic effect: the force is proportional to Planck’s constant and the speed of light. It was calculated in the early days of relativistic quantum field theory (1948). In fact, it was originally analyzed as a limiting case of the retarded van der Waal’s force between dielectric plates. Casimir made the fascinating observation that if the plates had sufficiently high dielectric constants, then for some physical effects the plates’ effect on the electromagnetic field could be replaced by boundary conditions and the force between the plates could be calculated just by considering the quantum zero point fluctuations of the electromagnetic field in this environment. In this case the attractive force is called the Casimir effect. These considerations indicated that the force can be calculated either from 1. the direct electromagnetic interactions between the plates (van der Waals), or 2. the spatial dependence of the energy stored in the vacuum fluctuations of the electromagnetic field. The heuristic derivation of the Casimir effect is presented here. The derivation is analyzed, critically assessed and it’s physical and unphysical elements are discussed.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
This Essay supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Casimir effect, van der Waals forces, Casimir-Polder, F. London, polarizable materials, electromagnetic field, zero point fluctuations.
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Contents
Heuristic Derivation of Casimir Effect for Parallel Conducting Plates. 8
Appendix. The Euler-Maclaurin Formula. 17
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Abstract
Quantum Electrodynamics predicts that a classical electric field of sufficient strength will produce pairs of electrons and positrons when applied to the vacuum. This is a tunneling effect, in the sense of ordinary quantum mechanics, and it illustrates the field theoretic nature of the vacuum state: its virtual fluctuations, electron-positron pairs, can be materialized by external classical sources. The critical electric field needed for the Schwinger effect is estimated and is found to be reachable with modern laser technology. Two Appendices in this Essay lay the groundwork for Schwinger’s prediction: The WKBJ approximate description of quantum tunneling and the Bogoliubov transformation that makes possible the construction of the vacuum state in the presence of the external electric field.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of quantum field theory will be developed within these Essays.
This Essay supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Schwinger effect, WKBJ approximation, tunneling, Bogoliubov transformation, critical field.
Contents
Strong Electric Fields and “Pulling Pairs out of the Vacuum”. 2
Appendix A. Tunneling in Quantum Mechanics. The WKBJ Approximation. 10
Appendix B. The Bogoliubov Transformation and Virtual Pairs in the Quantum Vacuum.. 15
Introduction to the Foundations of Quantum Field Theory
For Physics Students
- Statistical Field Theory, Landau-Ginzburg Theory, Spontaneous Symmetry Breaking, Symmetry and Conservations Laws, Goldstone Bosons and the Higgs Mechanism
Abstract
This lecture consists of two parts: An introduction to Statistical Field theory, Landau-Ginzburg Theory and the Goldstone Mechanism in condensed matter physics, and second, elementary particle field theory, symmetries and conservation laws, the Goldstone Theorem and the Higgs mechanism of the Standard Model. We begin with a description of Statistical Field Theory for condensed matter systems and work through various illustrations including elastic systems and magnetic materials. Phase transitions, phase diagrams and critical behavior are introduced and these concepts are illustrated with magnetic systems and the ferro-paramagnetic phase transition. The critical indices of second order phase transitions are defined and the correlation length is seen as the source of the singularities characterizing the bulk thermodynamics of continuous phase transitions. The Landau-Ginzburg Theory of Statistical Field Theory is introduced and illustrated for magnetic systems. Mean Field Theory is illustrated for magnetic systems and values of the critical indices are obtained. Spontaneous symmetry breaking of continuous symmetries is illustrated within the context of mean field theory. The Goldstone Theorem and Goldstone modes are illustrated for superfluidity. This exercise is done in d-dimensions and the Mermin-Wagner Theorem, the absence of spontaneous symmetry breaking and Goldstone modes in two dimensions, is illustrated. We then turn to relativistic field theories of elementary particles and find analogous phenomena. Continuous symmetries and conservation laws are studied in general by developing Noether’s Theorem. Symmetries and their realizations are contrasted in Quantum Mechanics and Quantum Field Theory. In field theory we find two natural modes of symmetry realization in a theory’s low energy spectrum: the Wigner-Weyl mode and the spontaneous symmetry breaking mode. The Goldstone Theorem is illustrated in theories with spontaneous symmetry breaking. The “Mexican Hat” potential is illustrated in several contexts. These considerations are generalized to Abelian Gauge Theories with charged scalar fields and the Higg’s mechanism is developed. The Higg’s mechanism leads to massive vector fields and massive scalar fields. The spontaneous symmetry breaking mechanism underlying the Higg’s mechanism is seen to alter the theory’s long-distance features, the theory’s spectroscopy, while leaving its high energy, short distance behavior unaffected. This leads to the great success of the Standard Model where the Higg’s mechanism gives masses to the vector mesons carrying the weak force as well as providing masses to other fundamental fields while leaving the photon massless. At the same time, the Higg’s mechanism leaves the short distance, gauge invariant behavior of the Standard Model unaffected, thus guaranteeing its renormalizability. This underlies the calculational success of the Standard Model.
The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the textbook, and 2. Undergraduate physics courses on quantum mechanics and statistical mechanics. The fundamentals of quantum field theory will be developed within these lectures.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Statistical Field Theory, Landau-Ginzburg Theory, Spontaneous Symmetry Breaking, Goldstone Theorem, Continuous Symmetries, Conservation Laws, Gauge Invariance, Gauge Theories, Electroweak Symmetry Breaking, Higg’s Mechanism.
Contents
An Example: Phonons in an Elastic Solid. 5
Phenomenological Field Descriptions. 8
The Partition Function and Mean Field Theory. 19
Spontaneous Symmetry Breaking. Goldstone Modes. 23
Continuous Symmetries and Conservation Laws. 30
Lagrangian Formalism of Classical Mechanics. 30
Lagrangian Formulation of Scalar Field Theory. 32
Symmetries and Conservation Laws. 33
Internal Symmetries and Conservation Laws. 36
Illustration: Relativistic Scalar Field in Minkowski Space Time. 37
Symmetries in Quantum Mechanics and Quantum Field Theory. 39
Nambu-Goldstone Realization. 41
Examples of Goldstone’s Theorem.. 44
The Dirac Equation, Anti-Particles and the Quantum Vacuum
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part VI
Abstract
This lecture introduces the Dirac equation, the relativistic equation for spin-1/2 point-like particles and fields. The Hamiltonian and the covariant forms of the equation are obtained and discussed. Dirac’s “Hole Theory” re-interpretation of the theory’s vacuum is introduced and discussed. The physical origins of pair production and charge renormalization are discussed from this perspective. The Charge Conjugation transformation is introduced to implement Dirac’s interpretation of positrons as the absence of negative energy electrons in the quantum vacuum. The theory is seen to be intrinsically multi-particle as a consequence of the unification of special relativity and quantum mechanics. The low energy properties of the Dirac equation are derived and the free field gyromagnetic ratio of the spin-1/2 electron is found and compared to the Pauli equation of non-relativistic quantum mechanics. The covariance of the equation is established and the implementation of Lorentz transformations in spinor space is developed.
The Chiral Anomaly and the Dirac Sea
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part VII
Abstract
We consider axial symmetries and the conservation of the axial current in 1+1 and 3+1 dimensional massless quantum electrodynamics. Although the axial vector current is predicted to be conserved in the classical versions of these theories as a consequence of continuous axial rotations of the fermion field, these symmetries and conservation laws are broken in the quantum theories. We find that the response of the Dirac sea to an external electric field leads to this “axial anomaly”. The exact expressions for the divergence of the axial currents in 1+1 and 3+1 dimensions are derived. In 3+1 dimensions the degeneracy of the theory’s Landau levels plays an important role in the final result. Other approaches to understanding and calculating this and other axial anomalies are presented and discussed briefly.
Prerequisite: This lecture continues topics started in Supplementary Lecture 19.
Bosonization in 1+1 Dimensions and Solving the Schwinger Model
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part VIII
Abstract
We derive the duality map between fermions and bosons in 1+1 dimensions and apply this method to solve the Schwinger model, quantum electrodynamics in 1+1 dimensions. For massless interacting fermions, the solution is a free but massive neutral boson. The physics of the model is discussed from the perspective of quantum choromodynamics and other non-abelian gauge theories. The Schwinger model illustrates the physics of flux tubes, confinement, the Higgs mechanism, the chiral anomaly, chiral symmetry breaking, mass generation and theta-vacua in a simple setting.
Prerequisite: This lecture continues topics started in Supplementary Lectures 19 and 20.
Dark Mysteries
- Dark Energy and the Cosmological Constant
- Dark Matter
Abstract
We discuss two of the most perplexing problems in physics today: Dark energy and Dark matter. Both forms of energy were discovered through their gravitational effects. Dark energy accounts for 69% of the energy of the universe and Dark matter accounts for 26%. The matter and energy that scientists have studied till now, the luminous matter and energy, only account for the remaining 5%. We have a lot to learn! We describe Dark energy through the cosmological constant. De Sitter space time is introduced to understand a universe whose only energy content comes from the cosmological constant. The metric of an expanding universe, the Friedmann-Robertson-Walker metric, is introduced and its evolution due to various forms of energy density is described by the Friedmann equation. We find that Dark energy causes the universe to expand at an ever- increasing rate. The universe is modeled as a multi-component fluid and the effects of various energy densities (radiation, non-relativistic matter, Dark energy and spatial curvature) are seen to influence its evolution in characteristic fashions. The Big Bang is found as the moment the scale factor of the universe vanished. The age of the universe is discussed and estimated by these models and equations of general relativity. The experimental evidence for Dark matter is reviewed and prospects for the discovery of its particle nature are discussed.
Quantum Electrodynamics in a Strong External Magnetic Field
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part IX
Abstract
We consider charged particles of mass and charge propagating in a uniform magnetic field . We begin with classical non-relativistic motion, then turn to non-relativistic quantum mechanics, then relativistic quantum mechanics and, finally, our goal: Quantum Electrodynamics (QED) in a strong, uniform external magnetic field, . The stationary states of these problems are Landau levels. Their energy spectra and degeneracies are computed. We argue that some aspects of the full 3+1 dimensional QED problem reduce to 1+1 dimensional QED (Schwinger model) which is soluble. The static potential between heavy charged impurities is strongly screened and the screening length can be computed from mass generation in 1+1 dimensional QED, in the idealized case of massless electrons. These considerations show that, for the limiting case of massless electrons, 3+1 dimensional QED in a strong external field exists in a Higgs phase where the electrostatic potential is screened to a Yukawa form.
Contents
- Motion of a Charged Non-Relativistic Classical Particle in a Uniform Magnetic Field. 3
- The Schrodinger Equation for a Charged Particle in an Electromagnetic Field. 8
- Non-relativistic Landau Levels. 9
- Relativistic Landau Levels. 13
- QED in a Strong Uniform External Magnetic Field, Dimensional Reduction and the Importance of the Schwinger Model. 15
How Hot is an Accelerating Spacecraft (and a Black Hole) ?
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part X
Abstract
In relativistic quantum field theory, the vacuum state contains quantum fluctuations which have important implications for atomic physics (the Lamb shift in hydrogen spectroscopy), elementary particle physics (electron-positron creation in quantum electrodynamics) and gravity (the Unruh effect and Hawking radiation). The Unruh effect was studied in Supplementary Lecture 15 and we found that the accelerated vacuum (Rindler vacuum) is a thermal ensemble of inertial states with a linear relation between the acceleration and temperature. In this lecture we consider a small accelerating spacecraft ( a “detector”) and find the relation between its state and that of an inertial detector in a thermal bath. The Unruh effect is rediscovered in a different and simpler fashion. The relation of these calculations to the Hawking effect is discussed and various related puzzles such as “firewalls” in black hole environments, the violation of conservation laws by black holes and black hole evaporation are introduced and commented upon.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Unruh effect, Hawking effect, Hawking temperature, Green functions, Thermal Green functions, Propagators, Gauge theories, Conservation laws, Firewalls in general relativity, Information loss paradox and black holes.
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Contents
- Free Scalar Field in Minkowski Spacetime. 2
- Green Functions. 6
- Thermal Green Functions. 10
- Acceleration and temperature. 14
- Implications for Black Hole Physics. 18
- Rindler spacetime and a “firewall”. 18
- Evaporation of Black Holes. 21
- Black Holes and Conservation Laws. 22
- The specific heat of a black hole. 23
Nuclear Physics Theory and Spontaneously Broken Chiral Symmetry
Introduction to the Foundations of Quantum Field Theory
For Physics Students
Part XI
Abstract
Low energy experimental nuclear physics involves mesons, baryons and their interactions which determine the properties of nuclei. Low energy theoretical nuclear physics involves the vacuum state, the realization of vector and axial symmetries, mesons, baryons and their interactions which determine the properties of nuclei. The symmetries of low energy, few-body nuclear physics include isospin and axial isospin. The mode of symmetry realization in the low energy spectrum is argued to be algebraic for the isospin symmetry but the axial symmetry is argued to be spontaneously broken by the vacuum state. A triplet of pions emerges as Nambu-Goldstone bosons and their interactions with nucleons are described by Partial Conservation of Axial Current (PCAC) and the Goldberger-Treiman relation.. These physical concepts are captured in the linear and non-linear sigma models which are introduced and developed in this lecture. The mass of the nucleon is seen to be fundamentally nonperturbative and is a consequence of spontaneous axial symmetry breaking. The construction of the linear and non-linear sigma models is guided by properties of the underlying theory of quarks and gluons, Quantum Chromodynamics (QCD). Further developments in the modern era of nuclear physics include heavy ion collisions and hot nuclear matter
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Effective Field Theory, Nuclear Physics, Symmetry, Conservation Law, Chiral Symmetry, Spontaneous Symmetry Breaking, Partially Conserved Axial Current, Nambu-Goldstone Boson, Goldberger-Treiman Relation
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Contents
- Introduction. 2
- Field theory Background. 4
- Chiral Symmetry and PCAC. 12
- Spontaneous Breakdown of Chiral Symmetry. 18
- Introduction to the Nonlinear Sigma Model and Chiral Perturbation Theory. 28
- Concluding Remarks. 31
Appendix . Dirac equation, Gamma-ology, Spin, Helicity and Chirality. 31
Statistical Physics I: Fundamentals: Entropy and the Microcanonical Ensemble
Abstract
This is the first of several lectures on Statistical Physics. The series begins with discussions of isolated ensembles of systems of particles and the calculation of their mean properties. The ensemble’s entropy is seen to be the critical, central concept in this formulation of statistical physics. The calculation of the entropy reduces to a counting problem: in classical physics, one needs the volume of phase space accessible to the system, and in quantum mechanics one needs the number of states that are accessible. Liouville’s Theorem is derived and is seen to play a central role. The Laws of Thermodynamics are reviewed and the entropy of systems in equilibrium and the evolution of systems out of equilibrium are discussed. Simple physical systems, like the one-dimensional random walk and the ideal gas, illustrate the general concepts and assumptions of statistical physics. This lecture concentrates on the microcanonical ensemble, one that is isolated from the outside. The canonical and grand canonical ensembles are introduced and developed in the second lecture where the focus turns to quantum statistical physics.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Statistical Mechanics, Thermodynamics, Entropy, Microcanonical Ensemble, Equilibrium, Phase Space, Partition Function, Free Energy. Stirling’s Formula
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Contents
- Introduction and Overview. 2
- Brief Review of Classical Mechanics. 3
- Statistical Ensembles. 6
- The Liouville Theorem.. 6
- Microcanonical Ensemble. 8
- Entropy. 9
- Probability Distribution and Entropy. 11
- Equilibrium.. 13
- Thermal Equilibrium.. 14
- Mechanical Equilibrium.. 16
- Particle Equilibrium.. 18
- Connections between statistical and thermodynamic quantities. 19
- Entropy of an Ideal Gas in the Microcanonical Ensemble. 21
- Initial Look at Quantum Statistical Physics. 24
- What have we learned so far and what’s in the next lecture?. 26
Statistical Physics II: Canonical and Grand Canonical Ensembles. Introduction to Quantum Statistical Physics
Abstract
This is the second of several lectures on Statistical Physics. The Canonical and Grand Canonical ensembles are introduced and applications to ideal gases and chemical reactions illustrate many body dynamics. Quantum statistical physics is introduced with discussions of Fermi-Dirac and Bose-Einstein statistics. Applications to the free electron gas, Bose-Einstein condensation and black body radiation are presented. The density matrix approach to Quantum statistical physics is presented and calculations of the partition function, free energy and other thermodynamic quantities are illustrated.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Statistical Mechanics, Thermodynamics, Entropy, Free energy, Partition function, Canonical Ensemble, Grand Canonical Ensemble, Pauli exclusion principle, Fermi-Dirac statistics, Bose-Einstein statistics, Bose-Einstein condensation, Density Matrix
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Contents
Entropy in the Canonical Ensemble. 9
- Velocity Distribution and Equipartition of Energy in the Canonical Ensemble. 9
- The Grand Canonical Ensemble. 11
Grand Canonical Functions and Thermodynamics. 13
Ideal Gas in the Grand Canonical Ensemble. 14
- Chemical Potential in an External Field. 16
- Chemical Reactions. 18
- Fermi-Dirac Distribution. 19
- Heat Capacity of the Free Electron Gas at Low Temperatures. 22
- Bose-Einstein Distribution. 24
Bose-Einstein Condensation. 27
An illustration: Polarized Light 33
Time Dependence of the Density Matrix. 34
Density Matrix in Statistical Physics. 34
Statistical Physics III: Fluctuations, Noise, Brownian Motion and Diffusion
Abstract
This is the third in a series of lectures on Statistical Physics. The lecture begins with a discussion of fluctuations, deviations from average quantities, in systems at equilibrium. Statistical fluctuations have some universal features which impact a large number of physical systems. Energy fluctuations are seen to be proportional to a system’s specific heat and are therefore sensitive to a system’s phase transitions. Particle number and order parameter fluctuations are also analyzed. Random processes are defined, discussed and illustrated through random walks, diffusion and Markov processes. The central limit theorem is presented and illustrated. The power spectrum and correlation functions of a random process are introduced and related. Brownian motion and diffusion are also considered, and aspects of the fluctuation-dissipation theorem are discussed and illustrated. Kinetic theory is applied to particle conductivity and related phenomena in gaseous systems.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Fluctuations. noise, random processes, stochastic variables, central limit theorem, correlation functions, power spectrum, Brownian motion, diffusion, Fokker-Planck equation, fluctuation-dissipation theorem, particle and thermal conductivity.
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Contents
Energy Fluctuations in the Canonical Ensemble. 3
Fluctuations in Concentrations in the Grand Canonical Ensemble. 8
Fluctuations and Droplet Formation. 11
Fluctuations in the Microcanonical Ensemble. 12
Fourier Analysis and Transforms. 13
The Central Limit Theorem.. 14
Wiener-Khintchine Theorem.. 17
Applications of the Nyquist Theorem.. 24
Microscopic View of the Pressure in a Gas. 29
The Compressibility of Gases and Radiation. 30
Molecular Collisions, Mean Free Path and Ionic Conductivity. 31
Statistical Physics IV: Kinetics, Einstein and Black Body Radiation, Detailed Balance, The H Theorem, and The Boltzmann Transport Equation
Abstract
This is the fourth in a series of lectures on Statistical Physics. The lecture begins with a discussion of kinetics and works out illustrations that include evaporation, ionization and chemical reactions. Einstein’s work on Black Body radiation is presented. The ultra-violet catastrophe of classical physics is derived and Planck’s argument showing how discrete quantum levels solve the catastrophe is discussed. The Boltzmann H theorem is presented. It states that the entropy in the microcanonical ensemble increases in time. The principle of detailed balance is introduced and applied to several quantum processes including nuclear reactions and photoionization. The Boltzmann transport equation is derived and is applied to the conductivity of the electron gas. Future topics are considered.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Kinetic methods, Black Body Radiation, The Ultra-Violet Catastrophe, Detailed Balance, The H theorem. Chemical Reactions, Boltzmann Transport Equation.
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Contents
Einstein’s Derivation of Planck’s Black Body Formula. 6
The Ultra-Violet Catastrophe and The End of Classical Physics. 8
Max Planck and the Quantization of Energy Levels. 9
Detailed Balance and the H Theorem of Boltzmann. 12
Principle of Detailed Balance. 13
Photoionization of an Atom.. 16
- Kinetics in Relaxation Processes. 19
- Boltzmann Transport Equation. 21
- Electrical and Thermal Conductivity of the Electron Gas. 22
Thermal Conductivity for the Maxwell Distribution. 25
Statistical Physics V: Non-Equilibrium Statistical Mechanics:
The Langevin and Fokker-Planck Equations
Abstract
This is the fifth in a series of lectures on Statistical Physics. We discuss statistical systems near but not necessarily at thermal equilibrium. Our focus is on how these systems evolve toward an equilibrium state and how their parameters are impacted by that limiting behavior. The lecture begins with an introduction to the Langevin equation and derives the Fluctuation-Dissipation Theorem which is a foundational result in the field of non-equilibrium statistical mechanics. Brownian particles and time correlation functions are emphasized. Diffusion and the physics of random walks are illustrated. The Fokker-Planck equation which describes how distribution functions evolve from near thermal equilibrium to thermal equilibrium is derived in the context of Markovian processes such as Brownian motion.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work.
Keywords: Langevin equation, Fluctuation-Dissipation Theorem, Brownian Motion, Diffusion, Random Walks, Fokker-Planck equation.
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Contents
- Langevin Equation and The Fluctuation-Dissipation Theorem. 2
- Time Correlation Functions. 6
- Brownian Motion and Correlation Functions. 9
- Correlation Functions and Fluctuating Dipoles. 11
- Fokker-Planck Equation and Distribution Functions near Equilibrium. 13
- Fokker-Planck Equation in Phase Space. 17
- Where are we going and what’s next. 23
Statistical Physics VI: Non-Equilibrium Statistical Mechanics:
Irreversible Statistical Mechanics from Reversible Mechanics?
Abstract
This is the sixth in a series of lectures on Statistical Physics. We revisit the question: Can statistical mechanics be irreversible if its underlying Newtonian mechanics is reversible? In this context, we review the original ideas of Boltzmann who introduced entropy and probabilities into statistical physics. The connection between susceptibilities and correlation functions is presented in the context of solving linear, causal differential equations. The Kramers-Kronig relations for susceptibilities are derived and applied. Linear response theory is discussed and illustrated with a study of currents in a wire which results in Ohm’s law. A soluble model of Brownian motion in a heat bath is analyzed and the importance of disparate time scales, a small time scale governing microscopic dynamics and a large time scale governing macroscopic dynamics, is emphasized. The analysis uncovers both viscosity in the motion of the Brownian particle and the fluctuation-dissipation theorem.
This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures refers to that work.
Keywords: Entropy, The Second Law of Thermodynamics, Brownian Motion, Correlation Functions, Fluctuation-Dissipation Theorem, Viscosity, Diffusion, Heat Baths.
Contents