theory =descrip of a system of objects

- logical framework
- types
- structures
- axioms

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)

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.

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

product of signs (pairs)

mult of vectors by quantities

Quadratic form ("squared distance") ; 2 x.y= (x+y)

Quadratic subspace and orthogonality

Euclidean subspace

Signature

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

- 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

Definition

Curvature: sum of angles of a triangle - parallel transport - relation to radius

Hyperbolic geometry

Curved geometry with (1,1) signature

the pair constructed from a finite set

cases n=3,4

linear = transposable

Infinite dim and divergence pb

Manifolds and distributions

tangent vector spaces

...

families of vectors, rank

inverse, dual basis

Spaces of symmetric and antisymmetric tensors

Representation of subspaces by antisymmetric tensors - duality

Deduction of double cross product formula

symplectic space

Same in spherical and affine quadratic geometries

symplectic case : symmetric tensor.

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

Phase space - symplectic geometry - case of the spin

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

the electromagnetic field

the potential

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

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