Explorations in Mathematical Physics

Preface.- The Language of Physics.- A Trip Down Linear Lane: Vector Spaces and Matrices.- Inner Products.- Crystallography and the Cobasis.- Finding Areas and Volumes: The Use of Determinants.- Diagonalisation and Similar Matrices: Changing Spaces.- Dirac's Bracket Notation.- Brackets and Hermitian Operators.- Frequency and Wavenumber.- Deriving the Fourier Transform Using Brackets.- Commutators and the Indeterminacy Principle.- Evolving Wave Functions in Time.- The Transition to Quantum Mechanics.- The Natural Language of Random Processes: From Bar Graphs to Histograms.- The Privileged Sum of Squares.- Least Squares Analysis, Bayes' Theorem, and the Matrix Pseudo Inverse.- Time Constants to Describe Growth and Decay.- Logarithms and Exponentials in Statistical Mechanics.- Signal Processing and the z-Transform.- The Discrete Fourier Transform.- Correct and Convincing: Presenting Solutions to Problems.- A Roundabout Route to Geometric Algebra: Matrix Representation of an Orientation.- Calculating the Matrix for an Arbitrary Rotation.- Combining Two Rotations.- Rotations Lead to Complex Numbers and Quaternions.- Producing a "Geometric" Algebra.- Rotations in Popular Usage.- Special Relativity and the Lorentz Transform: Deriving the Doppler Shift from an Invariance.- The Postulates of Special Relativity.- The Lorentz Transform.- The Symmetry of the Lorentz Transform.- Using Radar to Derive Time Dilation.- Space-time Becomes Spacetime.- Spacetime Diagrams and Hyperbolic Geometry.- The Lorentz Transform in an Arbitrary Direction.- Energy and Momentum in Special Relativity.- Four-Vectors and the Road to Tensors: Number Density and Flux Density.- Running Nonrelativistically.- Running Relativistically.- Examples of Other Four-Vectors.- Introducing Covectors and Fully Covariant Notation.- Accelerated Frames: Onward to the Principle of Covariance: The Clock Postulate.- Coordinates for the Accelerated Frame.- The Twin Conundrum.- A Glance Ahead to Gauge Theory.- Covariant Notation and Generalising the Clock Postulate.- Appendix: Details of Setting Up Adam's and Eve's Coordinates.- The Elegance and Power of Tensor Notation: Back to Vectors, in a More General Way.- Vectors and Coordinate Changes.- Generalising the Idea of Vector Length.- A Basis for Covectors.- Tensor Components With More Than Two Indices.- The Gradient Operator and the Cobasis.- Normalised Basis Vectors.- Volume Elements, Determinants, and Cross Products Again.- From Vector Calculus to Tensor Calculus.- Exterior Calculus and the Theorems of Stokes and Gauss in Higher Dimensions.- Curvature and Differential Geometry: Curvature in the Plane.- Geodesics: Curves with No Geodesic Curvature.- The Curvature of a Surface.- Gauss's Extraordinary Theorem.- Translating Vectors by Parallel Transport.- Relating Parallel Transport to Curvature.- From Geometry to Topology: the Gauss-Bonnet Theorem in Euclidean 3-Space.- Variational Calculus and Field Theory: The Story of the Fly and the Train.- The Concept of a Field.- The Lagrangian Formalism.- Building a Lagrangian.- Producing the Schrodinger Equation.- Quantising Field Theory: Fields Describe Particles Too!.- Gauge Theory and Quantum Electrodynamics.- The Path Integral Approach to Quantum Mechanics.- Density Matrices: the Language of Decoherence.- The Green Function Approach to Solving Field Equations: The Idea of a Green Function.- Deriving the Green Function for the Laplacian Operator via Fourier Theory.- Solving Maxwell's Equations via the Green Function Approach.- Variations on the Green Function Solution of Maxwell's Equations.- Fluctuation-Dissipation and Time's Arrow.- Airliners, Black Holes, and Cosmology: The ABC of General Relativity: The Equivalence Principle.- The Pound-Rebka-Snider Experiments.- A Space or Spacetime Description of Gravity?.- A Path to Einstein's Equation.- Solving Einstein's Equation for an Empty Spacetime: The Schwarzschild Metric.- The Schwarzschild Black Hole.- Calculating Curvature More Efficiently: Cartan's Structural Equations.- The Variational Approach to Einstein's Equation.- A Metric for the Universe: Proper Distances in Cosmology.- Index