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

∴

Cellular Automaton

Sierpinski gasket is fractal, which is the concept of my web site.

This is a part of quantum theories, although it is chaotic.

Sierpinski gasket is in rule 90.

There are 3 cells, and you pick up the center.
There are 2^3=8 patterns, which are just on and off.

(0,0,0)→0

(0,0,1)→1

(0.1,0)→0

(0,1,1)→1

(1,0,0)→1

(1,0,1)→0

(1,1,0)→1

(1,1,1)→0

It is called exclusive disjunction.

I chose (1,0,0)→1, because a part includes everything in fractal.

You can put the part in a spreadsheet infinitely.

This isn't Тетрис.

Hilbert space

Hilbert space is R^3. It seems to be the normal world, although it is very complicated.

(X1,X2,X3)*(Y1,Y2,Y3)=X1Y1+X2Y2+X3Y3

X*Y=Y*X ―①

(AX1+BX2)*Y=AX1Y+BX2Y

X*X≧0, X=0

X and Y are vectors. ∥X∥ is the length which is called norm.

X*Y=∥X∥∥Y∥cosθ

When X and Y are the same direction, θ is Zero.

This is cooling the temperature for the order.

Absolute convergence ignore infinity.

When you think about noncommutative algebra, the temperature increase.

PQ-QP=h/2πi

This includes imaginary spaces.

PQ-QP is trace.

When PQ-QP=0, (h/2πi)^n must be zero.
It is impossible, and you need infinite Hilbert spaces. This is chaotic.

Therefore, PQ-QP　―① is commutative in R^3, which is cooling the temperature.

Entropy

This is high temperature and chaotic, which is called Ising model.

Jij>0

This is like Big Bang.

You can also write this.

Expansion is high temperature.

Therefore, you can divide it.

The division is cooling.

Tr is trace.

∴

Exterior Product

Is God a Mathematician?

Descartes ignored God, and he is still influential in this century.

There is Technological Singularity which nobody knows, although we can perceive it through mathematics.
I don't know whether the decision is based on scientific truth or not. It may be democratic or your preference.
It is better to keep good impression like politicians or movie stars. I don't know who is perfect.

Hermann Grassmann knows the infinity.

AB+BC=AC

BA=-AB and CB=-BC

AB=AC+CB＝AC-BC

∴AB+BC=AC

Then you can expand dimensions.

(1)

(2)

(3)


All vectors in any spaces are reduced to the same line.

Soddy's hexlet

Soddy's hexlet is the mirror image.

Two spheres which are green and red are connected.

There are 6 spheres which are also connected each other around the two of them.

OP*OP'=R^2

OP and OP' are spheres, and R is the radius of OP'.
Therefore, OP=OP'=R
This is absurd, but P' is point at infinity.

The same spheres are connected, and they are mirror image.
Moreover, 6 spheres around OP and OP' are also mirror image.

∴

1/r1+1/r4=1/r2+1/r5=1/r3+1/r6