unique algebra

[AntonioLao]

The Hadamard algebra is unique simply because of a few operational symmetries. These operations are carried out using multiplication, addition, and subtraction. The fourth operation of division is not considered in this unique algebra. This is an algebra operated on Hadamard matrices. They are to be defined as matrices whose elements can only be 1’s and -1’s. However, in the events of carrying out operations, it can happen that the answers can contain the zero matrixes, whose elements are all 0. Although Hadamard matrices can be extended to infinite order and infinite dimension, in this short discussion, only 2 by 2 Hadamard matrices are considered.


At the outset, two distinct Hadamard matrices can be defined. These are square symmetric matrices such that elements made up of 1’s and -1’s are symmetrically distributed within each matrix. One has the main diagonal elements made up of all 1’s with a trace of 2 and the two off-diagonal elements are both -1. The other matrix has the main diagonal made up of all -1’s with a trace of -2 and the off-diagonal elements are both 1. Each unique matrix signifies a distinct Möbius topology describing an infinitesimal double rotation. Without a direction attached to each topology both are equivalent. On the other hand, attaching a direction makes them topologically non-equivalent and neither topology can be transformed onto the other by the strict rules of linear transformations or nonlinear transformations. One is called the H-plus topology and the other is called the H-minus topology. Three unique properties will now be mentioned. (A) The square (self Product) of either topology is equal to the sum of the same topology: Y =Y+Y= 2Y or F =F+F= 2F. In the real number domain, only the integer 2 can satisfy this unique property since the square of 2 is 4 and 2 plus 2 is also 4: 2=2+2. (B) The product of Y and F equals their differences: YF=Y-F=F-Y= 2Y= 2F. These two properties imply that Y =F =FY=YF= 2F= 2Y represent a mathematical paradox if direction is not introduced. (C) The product of D or -D and Y or F is the zero matrix where D is the matrix with all elements equal to 1 and -D is the matrix with all elements equal to -1. These properties give another option of removing infinities in non-renormalizable quantum field theories. For example, the quantum field theory of gravity.

[mkirkpatrick]

I will need to read through this a few times to get my head around it.


regards michael.

[AntonioLao]

All it means is that 2 times 2 is equal to 2 plus 2 or 2 minus 2. The last 2 must be a negative.

[mkirkpatrick]

All it means is that 2 times 2 is equal to 2 plus 2 or 2 minus 2. The last 2 must be a negative.

You make it sound so simple Antonio,that is your gift.


regards michael.

[AntonioLao]

It is now simple but it took me over 40 years to understand it.

[mkirkpatrick]

It is now simple but it took me over 40 years to understand it.

That's right,it's the understanding that counts.

regards michael.

[AntonioLao]

It can serve as a lemma to further understanding but never be a dilemma of misunderstanding.

[mkirkpatrick]

It can serve as a lemma to further understanding but never be a dilemma of misunderstanding.

You made that point very poetically,no misunderstanding at all.


regards michael.

[AntonioLao]

If the first lemma is misconceived what would become of all the succeeding lemmas?

[mkirkpatrick]

If the first lemma is misconceived what would become of all the succeeding lemmas?

That's a most interesting question,I have not the remotest idea.


regards michael.