The simplified Lorentz transformations are expressed using hyperbolic functions. These are necessary for describing boosts and rotations near the vacuum speed of light. However, the one real compelling reasons is really to breakaway the complex domain of imaginary linear independence in order to add up all the Hamiltonian energy contributions of all space-time points of the gravitational field, which is not necessarily quantized.
On the other hand, both quantum mechanics and any of the quantum field theories, for example, quantum electrodynamics (QED) or quantum chromodynamics (QCD), work extremely well when expressed in circular functions of ordinary trigonometry using Euler’s formula describing the imaginary phase factors of wavefunctions. In this case, the approach to real is by conjugate boost. However, it unavoidably makes the square absolute wavefunctions into proper fractions of probabilistic superposition requiring normalization. Whose absolute certainty can only be derived by using Feynman path integral and his space-time diagrams of particle interactions and taking the Lagrangians of energy differences (subtracting instead of adding as in a Hamiltonian mechanism).
Hyperbolic and conjugate boosts seem to work very well for all spin half fermions for finding solutions to Dirac equation using imaginary quantum vectors called spinors. However, extension to imaginary space-time vectors called twistors could not dissolve the inconvenient of the cosmological constant of general relativity field equations, unless, of course, by placing postulates for the existence of dark matter and energy analogous to the existence of antimatter in quantum mechanics. Nevertheless, the crucial critical indifference is that antimatter can be detected while dark matter and energy cannot.
Re: origin of hyperbolic boost -
02-26-2008, 12:14 PM
Linear Mode
