A build-it-yourself book

From Newton to M-Theory

Special relativity, general relativity, quantum mechanics, quantum field theory, the Standard Model and string theory — derived, not asserted.

Every theory in this book is the same three moves: pick a symmetry, write the most general action invariant under it, quantise. Nothing is stated that hasn't been built. Where a step is quoted rather than derived, it is marked ⚑ so you always know what you're standing on.

37 of 76 chapters built · 24 of them dedicated math chapters · Math Ledger

Part 0 · The Toolkit

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

0.1What a Derivative Really Ismath ready
0.2Integration and Accumulationmath ready
0.3Series, Approximation, Orders of Magnitudemath ready
0.4Vector Spaces and Linear Mapsmath ready
0.5Inner Products, Eigenvectors, Spectral Theoremmath ready
0.6Multivariable Calculusmath ready
0.7Fields, Flux, and the Big Theoremsmath ready
0.8Differential Equations and the Oscillatormath ready
0.9Fourier, Delta Functions, Probabilitymath ready

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.

2.1The Crisis of 1900 ready
2.2The Lorentz Transformation, Derived ready
2.3Minkowski Geometry ready
2.4Tensors, Honestlymath ready
2.5Relativistic Dynamics ready
2.6Electromagnetism Is Relativity ready

Part III · General Relativity

Geometry becomes dynamical.

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

Part IV · Quantum Mechanics

Linear algebra, taken absolutely seriously.

4.1What Classical Physics Cannot Do ready
4.2The Linear Algebra of Quantum Statesmath ready
4.3Function Spaces: Measure, L², and Completenessmath ready
4.4Domains, and the Adjoint's Domainmath ready
4.5The Spectral Theorem in Infinite Dimensionsmath ready
4.6The Schrödinger Equation ready
4.7Wells, Barriers, and Tunnelling ready
4.8The Oscillator, and the Ladder ready
4.9Commutators, Uncertainty, and Symmetry ready
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 Propagatormath
5.5The Dirac Equation
5.6The Path Integral
5.7Gaussian and Grassmann Integrationmath
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 Algebrasmath
6.2Representationsmath
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 Algebramath
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