Quote:
Originally Posted by AntonioLao  The key to background independence or dependence is singularity. In one sense, it is equivalent to a singular square symmetric Hadamard matrix. That is a matrix without an inverse matrix. In another sense, it is a rational function of two integer variables X and Y such that X=Y/(1-Y) and Y=X/(1+Y). Their product is simply XY=X-Y. If an inverse function is defined as symmetric switching of X and Y, X«Y then the resulting graph is equivalent to a point reflection about the origin of a coordinate system: (X,Y)®(-X,-Y). This is also equivalent to orthogonal reflection about each axis or a 180° rotation about the origin.
In a polar-trigonometric coordinate system of (R,f), where X=Rcos(f) and Y=Rsin(f) then tan(f)+Rsin(f)=cot(f)-Rcos(f)=1 and Rsin(f)cos(f)=cos(f)-sin(f) or ½Rsin(2f)=cos(f)-sin(f) or R=csc(f)-sec(f). This appears as a dimensionless quantity of pure numbers: real, complex, imaginary, or hypercomplex depending on its usage in a particular situation. So, it is possible for R to be a singular square symmetric Hadamard matrix. |
What really goes on behind the scene?what links all this together,is the singularity the
provider of the "field"?
regards michael.