(X1,X2,X3)*(Y1,Y2,Y3)=X1Y1+X2Y2+X3Y3
X*Y=Y*X ―①
(AX1+BX2)*Y=AX1Y+BX2Y
X*X≧0, X=0
(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.