Real is dual to imaginary -- complex numbers are dual. The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics. Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality. Bosons are dual to Fermions -- atomic duality! Commutators (Fermions) are dual to anti-commutators (Bosons). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation). The big bang is an infinite negative curvature singularity -- non null homotopic. Gaussian negative curvature is defined using two dual points. Singularities are dual:- Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry. Same (symmetry, summations) is dual to difference (anti-symmetry, differences). Bosons like to be in the same state, Fermions like to be in different states. "Always two there are" -- Yoda. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero -- non null homotopic. Points are dual to lines -- the principle of duality in geometry!
I noticed that if a tensor has higher order than 2 and the relationship between every pair of indices is either symmetric OR antisymmetric, there is a simple way to determine how many "matchings" there are. For example the tensor T_(ijkl) has 4 indices. If every index is symmetric with respect to every other one, then there are (#indices - 1)! = 6 matchings to consider
More: en.fufaev.org/tensors
Real is dual to imaginary -- complex numbers are dual.
The Schrodinger representation is dual to the Heisenberg representation -- Quantum mechanics.
Symmetric wave functions (Bosons, waves) are dual to anti-symmetric wave functions (Fermions, particles) -- the spin statistics theorem or quantum duality.
Bosons are dual to Fermions -- atomic duality!
Commutators (Fermions) are dual to anti-commutators (Bosons).
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Dark energy is repulsive gravity, negative curvature or hyperbolic space (inflation).
The big bang is an infinite negative curvature singularity -- non null homotopic.
Gaussian negative curvature is defined using two dual points.
Singularities are dual:-
Positive curvature (synchronic points) is dual to negative curvature (enchronic points) -- Gauss, Riemann geometry.
Same (symmetry, summations) is dual to difference (anti-symmetry, differences).
Bosons like to be in the same state, Fermions like to be in different states.
"Always two there are" -- Yoda.
The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry!
That was the most compact lesson I could imagine. Very easy to follow.
Formulas in your blog post did NOT RENDER well
Cool man, keep going with your take on Tensors and General relativity?
Interesting videos. What tools you use to create the content like how you make animation and which AI voice you use ?
One and a half = 3/2
Right, 1/2 is pronounced 'one half' or 'a half'
Thank you!
I noticed that if a tensor has higher order than 2 and the relationship between every pair of indices is either symmetric OR antisymmetric, there is a simple way to determine how many "matchings" there are.
For example the tensor T_(ijkl) has 4 indices. If every index is symmetric with respect to every other one, then there are (#indices - 1)! = 6 matchings to consider
And for higher order tensors like M_(ijkl), it is possible to have both symmetric and antisymmetric index pairs... right?
U need to replace your commentary of " 1 and a ½" by "½" !