Legendre transformation

Optimization is hard to understand.

H=H(q,p;t)=T+V

Hamiltonian represents the total energy of the system, which is the sum of kinetic (T) and potential energy (V). q is generalized coordinates, p is generalized momenta, and t is time.

The Hamiltonian is calculated using the usual definition of H as the Legendre transformation of L.

・ is differentiation.

The momenta are calculated by differentiating the Lagrangian with respect to the generalized velocities.

H=0. It is numerical vectors.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?
— Stephen Hawking


This is the Legendre transformation of L.

You see the bottom (A). y=0.

Fermi's golden rule

Planck length seems to be the golden ratio.

ρ is the density of final states (number of continuum states per unit of energy).

H is Hamiltonian mechanics which is based on gravity.

You need the light to keep going.

Event Horizon

S=k(4πR^2)/(2l)^2

This is Bekenstein–Hawking formula about black hole. 4πR^2 is event horizon which is the end of the universe. l is Planck length. It is about 1.616*10^-35(m).

They are symmetry. Entropy increases according to the length of black hole. It is chaotic.

v^2/2=GM/r

This is law of the conservation of energy. Kinetic energy is universal gravitation. If the horizon (r) were small, the gravity would increase. Therefore, the kinetic energy is faster to escape from the black hole. This is speed of light.

r=2GM/c^2

There is the horizon even in the speed of light. You can't escape from the black hole.

You need singularity for expansion of time and space.

Grand Unification Theory

G=SU(3)×SU(2)×U(1)

G is gauge group.

Grand unification theory is expansion of G.

We have 4 powers in our nature, which are weak interaction, strong interaction, electromagnetic interaction, and gravity.

The integration is so hard because of gravity. G may need super gravity for expansion. There is no space and time.

SU(5)⊃SU(3)×SU(2)×U(1)

This is corruption of everything, and it must be the beginning.

It is very difficult to capture Zero.

SO(10)⊃SU(5)×U(1)

Higgs field

Now I got φ. There is no matter in Gauge theory, but scalar potential is for gravity.

gσρ(x)→W(x)gσρ(x)

Gravity and Electromagnetic Field are integrated.

However, there are space and time between x and x0. This is against Gauge theory because of General theory of relativity.

Then, Higgs field solved this contradiction.

SU(2)×U(1)

This is called spontaneous symmetry breaking.

Gauge symmetry never be broken, and matter is not Zero.

Gauge Theory

I use fractal as art, but there are some relations with gauge theory.

E is electric field.

∇E=ρ/ε0

∇B=0

B is Magnetic field.

B=∇A

A is vector potential.

∇E=-∂B/∂t

∴

∇(E+∂A/∂t)=0

E+∂A/∂t=-∇φ

φ is scalar potential.

∴

B=∇A, E=-∇φ-∂A/∂t

This is Gauge symmetry, and the potential is chaotic.

φ(x)→φ'(x)=e^iθ*φ(x)

e is Electric charge.

Dark Pattern

This is a mathematical trick.

Every square is just the line

or

Every number is reduced to 1 except for 0

There is chaos, but it is in squares. You can also expand any dimensions, when you accept Zero.

Zn=6n

1^2=1

∴