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Phase Angle

Let particle P be described by an ordered chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{P}_{1}, \mathsf{P}_{2}, \mathsf{P}_{3} \, \ldots \, \mathsf{P}_{k} \, \ldots \, \right)

that is repetitive so that  \Psi may also be written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \mathsf{\Omega}_{2}, \mathsf{\Omega}_{3} \, \ldots \, \right)

where each orbital cycle  \mathsf{\Omega} is composed of  N sub-orbital events

\mathsf{\Omega} = \left( \mathsf{P}_{1}, \mathsf{P}_{2}, \mathsf{P}_{3} \, \ldots \, \mathsf{P}_{N} \right)

If P contains just a few quarks then the time of occurrence may not be a useful parameter for describing these events because Planck’s postulate is plausibly justified on a statistical basis. So we also discuss the order of events using a phase angle defined by

\theta_{k} \equiv   \theta_{\mathsf{o}} - \delta_{z} \dfrac{2\pi k}{N}

where  \theta_{\mathsf{o}} is arbitrary and  \delta_{z} is the helicity of P. The ground-states and many excited-states of atoms have spin-down orientations, then  \delta_{z}=-1. Also, the change in  \theta during one sub-orbital event is called the phase angle increment. It is defined by  d \! \theta \equiv 2 \pi \! / \! N, so usually

\theta_{k} = \theta_{\mathsf{o}} + k \hspace{1px}   d \! \theta

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events  N may be large but not infinite. This requirement can be relaxed later to make a continuous approximation, thereby allowing the use of calculus. But in principle  N is finite and accordingly changes in  \theta may be small but not infinitesimal. For isolated particles the increment in the phase angle does not vary and so there is an equipartition of  \theta between sub-orbital events regardless of their quark content.

Phase angle dependence on quark distributions is suggested by this colorful German engraving of algae.
Melethallia (detail), Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.