Overview

These are my maths and physics notes.

graph TB
    subgraph Maths
    N((<a href='/maths/sets' style=color:green >  Sets </a>)) --> LA

    LA((<a href='/maths/linearalgebra' style=color:green >Linear Algebra </a>))
    FA((<a href='/maths/fourier' style=color:green >Fourier Analysis </a>))
    LA --> C((<a href='/maths/calculus' style=color:green > Calculus </a>))
    DE((<a href='/maths/diffeq' style=color:green > ODEs </a>))
    LA --> RT((<a href='/maths/representations' style=color:green >Continuous groups</a>))
    DG((<a href='/maths/differential_geometry' style=color:green >Manifolds</a>))
    MT((<a href='/maths/measuretheory' style=color:green >Measure theory </a>))
    GT((<a href='/maths/groups' style=color:green >Group theory </a>)) --> RT
    CT((<a href='/maths/categories' style=color:green >Categories</a>)) --> LA
    CA((<a href='/maths/complexanalysis' style=color:green >Complex Analysis</a>))
    DG --> P((<a href='/maths/probability' style=color:green >Probability</a>))
    TP((<a href='/maths/topology' style=color:green >Topology</a>))
    HM((<a href='/maths/homology' style=color:green >Homology</a>))
    CHM((<a href='/maths/cohomology' style=color:green >Cohomology</a>))
    TP --> DG
    HM --> CHM
    C --> MT
    C --> DE
    DE --> P
    DE --> DG


    end

    subgraph Physics
    GAT((<a href='/physics/gaugetheory' style=color:green >Gauge Theory</a>)) --> QFT
    Q((<a href='/physics/quantum' style=color:green >Quantum</a>))
    MT --> QFT
    DG --> D
    LA --> DIM((<a href='/physics/dimensions' style=color:green >Dimensions </a>))
    DIM --> QFT
    DG --> GR((<a href='/physics/relativity' style=color:green >Relativity </a>))
    GR --> QFT((<a href='/physics/qft'>Quantum Field Theory</a>))
    D((<a href='/physics/classical' style=color:green >Classical physics</a>))
    TM((<a href='/physics/statisticalphysics' style=color:green >Statistical Physics</a>)) --> QFT
    Stats((<a href='/physics/statistics' style=color:green >Statistics</a>))

    end
    D --> TM
    D --> QFT
    D --> GR
    GT --> HM
    GT --> GAT
    TP --> GAT
    TP --> HM
    DG --> GAT
    DG --> CHM
    CT --> TP
    RT --> GR
    Q --> TM
    Q --> QFT
    FA --> Q
    CT --> GT
    C --> CA
    FA --> D
    CA --> QFT
    C --> FA

    Stats --> TM
    CHM --> GAT
    MT
    P --> Stats
    P --> Q

Caveat

These notes are all written by me - you therefore shouldn't assume they are either reliable or comprehensive.

Notation

While I mostly use notation that is standard in maths and physics, I do make use of some computer science notation in places where I feel it is useful. See here for a glossary.

FAQ

Why did you structure the notes in these way?

See here for more information

What material is included?

The premise of the notes is that it contains everything you need in order to understand quantum field theory, which happens to be most of the nice undergraduate mathematics and physics.

In general, I try to be sparse, given only the bare bones. For example, I state the spectral theorem in the notes on linear algebra, but not the proof. The goal is to give an outline of each subject, with clear technical detail.

The level of the material should be roughly what a smart theoretical physics undergraduate would know before grad school.

Is the material standard?

Mostly, but it is opinionated. A good example are the notes on statistical physics, which are presented from the perspective of Bayesian probability, information theory and geometry. Here the difference to the standard language is laid out in a table.

When it is possible to make connections between different fields that simplify ideas or reduce the need to repeat material, I do. For example, I present a number of related theorems (Green's, Stokes', divergence, even the fundamental theorem of calculus) as a special case of Stokes' general theorem.

Relatedly, I tried very hard to avoid ideas or equations that aren't clearly motivated. Everything mathematical derivation should be part of a story, and follow from some simple ideas. As an example, see the notes on complex analysis, where the Cauchy-Riemann equations are motivated, rather than postulated. Similarly for introducing matrix multiplication as a consequence of the composition of linear transforms.

What order should these notes be read in?

This is indicated by the graph above. For example, the notes on statistical physics assume as given all the material in the notes on statistics, quantum and classical physics, and their respective dependencies.

There's an error - can I fix it?

Yes, please let me know, I'm sure there are many errors. Or fix it yourself if you prefer - that's even better. You'll see an edit button on the top right of each page.

Why did you write these notes?

Partly to help myself learn and for my own reference. Partly to experiment with ideas for how to present technical material more effectively.

Use of AI tools

Pretty minimal, but I write these notes in VSCode, so copilot autocomplete sometimes generates Latex for me which I then check.