From Newton to M-Theory

Contents

37 of 76 chapters written, in 8 parts

Titles in italic are planned.

Part 0 · The Toolkit

Rebuild the floor. No physics yet — but every example comes from physics.

0.1What a Derivative Really IsMath
0.2Integration and AccumulationMath
0.3Series, Approximation, Orders of MagnitudeMath
0.4Vector Spaces and Linear MapsMath
0.5Inner Products, Eigenvectors, Spectral TheoremMath
0.6Multivariable CalculusMath
0.7Fields, Flux, and the Big TheoremsMath
0.8Differential Equations and the OscillatorMath
0.9Fourier, Delta Functions, ProbabilityMath

Part I · The Action Principle

The most-skipped prerequisite in physics — and the reason people wall out at GR.

Part II · Special Relativity

Geometry replaces mechanics.

Part III · General Relativity

Geometry becomes dynamical.

3.1The Equivalence Principle
3.2ManifoldsMath
3.3Metric and ConnectionMath
3.4CurvatureMath
3.5Forms, Lie Derivatives, Killing VectorsMath
3.6The Einstein Field Equations
3.7Schwarzschild: The Solution and Its Orbits
3.8Light, Redshift, and What a Horizon Is
3.9Cosmology, and a Loose Thread

Part IV · Quantum Mechanics

Linear algebra, taken absolutely seriously.

4.1What Classical Physics Cannot Do
4.2The Linear Algebra of Quantum StatesMath
4.3Function Spaces: Measure, L², and CompletenessMath
4.4Domains, and the Adjoint's DomainMath
4.5The Spectral Theorem in Infinite DimensionsMath
4.6The Schrödinger Equation
4.7Wells, Barriers, and Tunnelling
4.8The Oscillator, and the Ladder
4.9Commutators, Uncertainty, and Symmetry
4.10The Classical Limit
4.11The Angular Momentum Algebra
4.12Spin, Orbitals, and Adding Angular Momenta
4.13The Hydrogen Atom
4.14The Degeneracy, and SO(4)
4.15Perturbation Theory
4.16The Fine Structure of Hydrogen
4.17Transitions
4.18Identical Particles
4.19Density Matrices and Entanglement
4.20Bell, Decoherence, and What Is Settled

Part V · Quantum Field Theory

Particles stop being fundamental.

5.1Why Quantum Mechanics and Relativity Force Fields
5.2Classical Field Theory
5.3Quantising a Field
5.4Distributions, Contours, and the Propagator
5.5The Dirac Equation
5.6The Path Integral
5.7Gaussian and Grassmann Integration
5.8Interactions, Wick's Theorem, and the Feynman Rules
5.9A Real Calculation
5.10Loops, Divergences, and Regularisation
5.11Renormalisation and the Renormalisation Group

Part VI · Gauge Theory and the Standard Model

Symmetry, made local, generates every force.

6.1Lie Groups and Lie Algebras
6.2Representations
6.3The Gauge Principle
6.4Yang–Mills
6.5QCD and Asymptotic Freedom
6.6Spontaneous Symmetry Breaking and the Higgs Mechanism
6.7The Electroweak Theory
6.8The Standard Model, and What It Does Not Explain

Part VII · Strings and M-Theory

Where gravity stops being an add-on and starts being unavoidable.

7.1Why Quantum Gravity Is Hard
7.2The Bosonic String
7.3Conformal Symmetry and the Virasoro Algebra
7.4Quantising the String, and D = 26
7.5The Spectrum
7.6Superstrings and D = 10
7.7T-Duality and D-Branes
7.8Compactification, Dualities, M-Theory
7.9Black Hole Entropy, Holography, and the Accounting