Hamilton v Riemann

[AntonioLao]

Quantum mechanics was well understood at the beginning of WWII. On the other hand Einstein’s theory of general relativity was at the height of its success since Sir Arthur Eddington verified the prediction of gravitational bending of starlight in 1919. Although Einstein failed to formulate a combined theory of general relativity and electromagnetism before his untimely death in 1955, others have already started to formulate a combined theory of quantum mechanics (QM) and general relativity (GR) after WWII. Today, these efforts remain fruitless. These theoretical failures can be blamed on the underlying mathematical disparity between QM and GR. This disparity appeared superficial to the untrained eyes of most scientific investigators. But deeper analysis would uncover that the distinctive differences can be attributed to the founders of these mathematical physics between the mathematics of Hamilton and Riemann.

To be continued…

[AntonioLao]

Hamilton was the older. He was born in 1805 and died in 1865 at the young age of 60. Riemann was born in 1826 and died in 1866 at the younger age of 40. Clearly, geniuses are not measured by how long or by how healthy they lived. Riemann died of tuberculosis. They are noted by the uniqueness of their discoveries. Hamilton discovered the 1st–order equivalence of Lagrange’s 2nd-order differential equations and the Hamiltonian, total energy function, is expressed using generalized coordinates and momenta, a precursor to one out of three forms of the uncertainty principle of QM and phase space approach to specify the quantum states of a system in six-dimension. Riemann discovered the generalization of Gauss’s non-Euclidean geometry into a theory of manifolds. This particular generalization was written by Riemann using merely two-dimension and ideas from the theory of surfaces (the idea of two orthogonal principal curvatures and their sum of square the total curvature). Consequently, these led him to the discovery of minimal surfaces embedded in higher dimensional spaces, namely multiples of 2D inside 3D. For Hamilton, it seems he was embedding 1D inside 3D. Early on he realized this limited construct, which prompted him to invent the new mathematics of quaternions. Both Riemann and Hamilton knew that the mathematical tools they invented works well in 1D but they wanted these to work in 3D as well. As a compromise both settle to use the 2D field of complex analysis and unavoidably introduced the idea of an imaginary axis for the complex plane. The flexibility of these imaginary axes provided Riemann the tools to construct multi-dimensional surfaces. Two other interdisciplinary geniuses who were able to exploit these imaginary landscapes are found in the arts of Escher and the architectures of Frank Lloyd Wright. The latter was able to design structures that minimize the distortion of the natural topography. The idea of quaternions was used by Maxwell to construct his original theory of electromagnetism. However, two later investigators (namely, Heaviside and Gibbs) split the concept of quaternions into the scalar part and the vector part and discovered the theory of vector analysis. To be continued…