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

Quadratic vector space

Definition
Quadratic form ("squared distance") ; 2 x.y= (x+y)²-x²-y² thus nonzero ; def by differential
Quadratic subspace and orthogonality
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
link with galilean spacetime

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