# ABAtanasov/Representations

An upcoming book on mathematical physics
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Chapters Oct 30, 2018
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# Representations of the Physical Universe

## Prerequisites

### 1) Old Notions Revisited

First draft completed, need to draw up figures.

#### Euclidean vs. Affine Space

The naive inner product is not an invariant

#### Einstein's Summation Convention

Make clean and short

### 2) New Horizons

Close to being done.

We want a section for defining Direct sums, Dual Spaces <-> functionals, and tensor product both INTUITIVELY and FORMALLY

#### Elementary Topology

Gaps in this section Add stereographic projection of sphere as example of coordinate charges

## Part 1: A Better Language

### 3) Differential Geometry

THINGS WE SHOULD HAVE DEFINED EARLIER ON:

C^\infty(M), That 1/k! on the k-form coefficient tensor


#### Stokes' Theorem

Finish the proof here

#### Movement: Lie's Ideas

Finish some Lie derivative summaries

#### Exercises

A functional-programming type exercise to get a really clean compact way to define the exterior derivative

### 4) Harmonics: Fourier Analysis

#### Discrete, Bounded: Eigenvalues

Functions are very clearly n-component tuples forming a vector space

#### Continuous, Bounded: Fourier Series

Differentiation as a linear operator, finding its eigenvalues gives the fourier transform

#### Harmonic Analysis on Higher Euclidean Space

Here we derive the Laplacian, study diffusion, image processing, convolution Functions on 2D space as a limit of a 2D grid Laplacian is never obviously derived in any good text so derive it.

#### Harmonic Analysis on Graphs

Functions on graphs, graph laplacian. Motivate expanders Conclude with spectral graph theory

#### Exercises

What about real-valued eigenvalues giving rise to exponential decay eigenforms: c.f. the positive reals

### 5) Beyond Harmonics: Representation Theory

We begin assuming knowledge of group theory/ intro-level abstract algebra. Begin with spectral graph theory motivating representations as generalizations of eigenvalues

Look at finite groups...
Build analogous results to fourier analysis
graphs -> quivers?
Representation theory of k[x] = Jordan Normal Form

Define topological group.
Some statements about extending this to compact groups


## Part 2: Physics (In order of increasing difficulty)

### 6) SL(2,C), SU(2), SL(2,R)... the principles of finite dimensional QM

Review how this ties together: rep theory of SO(3) leading to SO(3,1), and spinors (projective representations)
QI section

Tie this in to SU(n) but probably not past taht


### 7) Classical Mechanics and Symplectic Geometry

What is the difference between velocity and momentum
Legendre transform
Feynman Argument for conservation in the Lagrangian viewpoint

Intuition behind how the symplectic form bundles position and momentum together
Lets us go from a Hamiltonian H -> \partial / \partial t -> the arrow of time -> A vector field
All we need is infinitesimal hamiltonian information, so gives an equivalence 1-forms <-> vector fields on T^*M


### 8) Einstein's Gravity

Covariant Derivatives. Motivate Riemann Curvature
READ WEYLS BOOK (or GR nut) it literally derives GR from symmetry

And then:

Black holes, let them spin let them be charged. Let there be parallel universe, everything.
Penrose diagrams

Expanding universes, redshift, Hubble's thing.. ADM might be too much its ugly imo (but the preceeding chapter HAS developed the hamiltonian formalism enough)


## Part 3: More Advanced Topics (No preferred order)

### 8) Cohomology and Homology

Goal is a simple motivation of poincare duality and relating the space of harmonic p-forms to the (co)homology


### 9) An Introduction to Geometric Quantization

The idea should be that even though, physically, we can barely grasp quantization at first glance, the mathematics guides us and we shuold try to interpret THAT physically.


### 10) Classification of Simple Lie Algebras over C

You Qi's notes, together with Humphreys will be the guide
The goal is to introduce, with reasonable proof, the ABCDEFG classification of Lie Algebras

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