Group as a mathematical structure is used to study multi-dimensional symmetry. Since it is a collection of elements, different groups are described by different rules exclusively applicable to them. For example, when the product rule is commutative the group is called Abelian. When it is not then it is non-Abelian. Likewise, for a 4-dimensional continuous rotation group where the proper time is the imaginary part of the complex space-time interval, it is a Lorentz group http://en.wikipedia.org/wiki/Lorentz_group. If 4D translation is incorporated then it becomes a Poincare group http://en.wikipedia.org/wiki/Poincar%C3%A9_group. All these were used to describe elementary particles. However, none of these could completely and satisfactorily describe the differences in mass: inertial, gravitational, or relativistic; not even for some stable elementary particles like the protons and electrons.
For mass, a simpler mathematical structure is all that is required. This is the ring structure for all NxN real matrices. One of the infinite subsets of this ring contains all the Hadamard matrices. They are also a collection of elements but for each dimensional ring structure, it can only contain two elements. For the rule of addition, an identity exists. For multiplication rules, there is associativity, commutativity, and scalar products. The addition rule is applied for the physical descriptions of charge interactions. The multiplication rule is applied for the physical descriptions of mass differences. Moreover, a six-dimensional ring structure of Hadamard matrices could be used to derive from first principle the mass ratio of proton and electron within 10% of experimental value.
http://mathworld.wolfram.com/Ring.html
http://en.wikipedia.org/wiki/Ring_(mathematics)


LinkBack URL
About LinkBacks
Reply With Quote


