# Plan for an intitiation to theoretical physics

This page is just the draft of personal notes listing the main points I consider to address.

### Geometric theories

geom = 1st theory of physics
theory =descrip of a system of objects
• logical framework
• types
• structures
• axioms
geom = desc "spaces" (syst of points)
few structures ; Galois connection between sets of structures and groups of transformations (closures : definable / generated).
List of possible structures (but these structures and the considered combinations are possible structures of different spaces, that cannot always be contained in the same space).
• Center
• affine structure : Alignment (3-ary/ straight lines), parallelism (thus, ratios of volumes)
• Sphericity / circularity / angles
• unit of volume

less than that : lines (topology)

### Vector spaces and duality

Vector space = center+parallelism

sum, multiplication (from a center). Linear transformations (one between any 2 basis).

Linear forms, geometrical construction of sum and multiplication
Abstract definition of 2 spaces in duality
Orthogonal subspaces, quotient by a subspace.

### Affine and projective spaces

Construction of an n-dim affine space from an (n+1)-dim vector space with a linear form u
u.x = 1 <=> x point
u.x= 0 <=> x vector
Barycenter.
Translations
Correspondance between affine and projective spaces (affine space = projective space - subspace of points at infinity).
Subspaces.
Description of projective transformations
Same structures = different geom in different contexts (local view without ratios of volumes, to be completed at infinity / global view not completed):
• alignment= affine or projective (horizon as a structure)
• circularity = affine euclidean or conformal
Affine space of units systems
product of signs (pairs)

mult of vectors by quantities

Definition
Quadratic form ("squared distance") ; 2 x.y= (x+y)2-x2-y2 thus nonzero ; def by differential
Euclidean subspace
Signature

### Special Relativity

...
Description in the 2-dimensional case : signature (1,1) = space-time ; circle = hyperbola ; rotation with Doppler effect - orthogonality
Horizon of a black hole

### notes on geometry

Possible combinations :
• Distances (or ratios of) = alignment + sphericity -> Spherical and other geometries with constant curvature
• Affine quadratic = Parallelism + sphericity (though the same as alignment + sphericity on a different space) - also obtained from (center + sphericity), with "center" seen at infinity
• Quadratic vector space = center + parallelism + sphericity

### Curved geometries

#### in 2 dimensions

Spherical geometry
Definition
Curvature: sum of angles of a triangle - parallel transport - relation to radius
Hyperbolic geometry
Curved geometry with (1,1) signature

#### In higher dimension

behavior depends on type of direction

### Conformal space

affine quadratic representation (paraboloid - subsphere) - spherical representation

### Signatures of permutations

the pair constructed from a finite set
cases n=3,4

### more on vector spaces and duality

linear = transposable
Infinite dim and divergence pb

Manifolds and distributions
tangent vector spaces

### Tensors

...
families of vectors, rank
inverse, dual basis

#### Symmetric and antisymmetric tensors

Spaces of symmetric and antisymmetric tensors
Representation of subspaces by antisymmetric tensors - duality
Deduction of double cross product formula
symplectic space

### Infinitesimal rotation

= antisymmetric tensor
Same in spherical and affine quadratic geometries
symplectic case : symmetric tensor.

### Least action principle

Statement of the principle
Structure of conserved quantities (force) : screw = antisymmetric tensor in n+1 dimension - antisymmetric product of point and force vector.

### Conservation laws

flows of any dimension = field of antisymmetric tensors - duality between subsurfaces and fields - exterior derivative and border
Phase space - symplectic geometry - case of the spin

### Energy-momentum / stress tensor

Proof of its symmetry
The rotation-invariant case
Interpretation by equilibrium with the metric
Geometric representation of such 2-dimensional field in 3 dimension - relation with 3-dim general relativity

### Electromagnetism

the electromagnetic field
the potential

### Introduction to General relativity

Tensorial form of the curvature
The 3-dimensional case
The simplest component of Einstein's equation : energy density = sum of curvatures on 3 orthogonal space-like planes
Computation of the cosmological models - dark energy.
tensorial form of Einstein's equation

### Quantum physics

Geometric expression of probabilistic evolutions : proba state ; evolution (affine) ; measurement (projective transf)
Principles of quantum physics
qbit and measurement
Decoherence
Statistical physics
formal analogy between statistical mechanics and quantum theory
Schrodinger equation spinors, clifford algebras
Dirac equation

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List of physical theories