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
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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.
Interesting videos. What tools you use to create the content like how you make animation and which AI voice you use ?
Cool man, keep going with your take on Tensors and General relativity?
Formulas in your blog post did NOT RENDER well
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?
One and a half = 3/2
Right, 1/2 is pronounced 'one half' or 'a half'