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Unity Root Matrix Theory (URMT)

Book 4 : Unity Root Matrix Theory

This second volume naturally follows on from its predecessor in detailing the recent mathematical developments and physical applications of what is a relatively new subject area known as Unity Root Matrix Theory (URMT). Indeed, URMT was first established by the author in two earlier publications that provided the mathematical foundations and tentative physical interpretation under the banner of 'Physics in Integers', in the first book, with extensions to higher dimensions in the second book 'Higher Dimensional Extensions'.

With Volume I and the two earlier founding publications addressing URMT's physical application to such areas as Newtonian calculus, linear dynamics, harmonic oscillators and Special Relativity, this second volume widens the applications to angular dynamics, spin and electromagnetism. Most importantly, the mathematical advances also provide a more unified picture of all URMT's algebraic aspects, and with it a more recognisable connection to traditional mathematical physics by way of symmetric and skew-symmetric matrix operators and representation theory. In turn, these connections pave the way for future, closer development of URMT to existing theories - the ultimate goal being that of formulating the laws of nature in a simpler, discrete form, and thus reducing nature to the legal combination of integers.

A bit more detail. Volume II starts with an extension of URMT’s method of arbitrary vector embedding to enable two, completely arbitrary, n-dimensional vectors to be embedded as eigenvectors in an n+1 dimensional URMT scheme, where both eigenvectors are Pythagorean n+1 tuples. The complete eigenvector solution retains all the existing URMT eigenvector relations and conservation equations, including the energy conservation equation in the invariant eigenvalue. The four-dimensional variant is later used in the treatment of the electric and magnetic field vectors in the form of the electromagnetic field tensor for a plane wave in free space. In this case, the dynamical conservation equation (matrix characteristic equation) is a conservation in energy density.

Following this is a general, n-dimensional formulation of all three URMT A matrices in terms of projection operators and exterior products, leading to a complete reformulation of the founding theory URM3. As before, the three A matrices are physically associated with derivative, proportional and integral operators. The projection operators are, themselves, formed from pre-defined eigenvectors of the unity root matrix. This exterior product formulation is then used in a second, Poynting vector formulation of the electromagnetic plane wave, where the dynamical conservation equation is now a conservation of power flow.

The aforementioned electromagnetic plane wave solution also has a quaternionic representation of its Poynting eigenvector and, indeed, quaternions form a large part of Volume II by virtue of the similarity of URMT’s skew-symmetric A matrices to a quaternion representation matrix. Such similarity enables a URM3 (3x3) and URM5 (5x5) treatment of rotations in one and three dimensions respectively. This then naturally leads to angular dynamics and spin. In particular, with URMT fundamentally formulated in integers, the solution possesses integer and half-integer spin states.

Mathematical and Physical Advances Volume II