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.