Phase
//Usap//, Sasak people. Lombok, circa 1900, 49 x 52 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
Usap, Sasak people. Lombok, circa 1900, 49 x 52 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.

Consider a frame of reference F that is employed to describe some generic particle P so that the events of F and P are associated in pairs

$\left\{ \sf{P}_{\it{k}}, \sf{F}_{\it{k}} \vphantom{{\Sigma^{2}}} \right\}$

Let the frame of reference include a clock noted by $\mathbf{\Theta} ^{\sf{F}}$ that is used to represent F so that events of P and $\mathbf{\Theta}$ are associated in pairs like

$\left\{ \sf{P}_{\it{k}}, \mathbf{\Theta}^{\sf{F}}_{ \it{k}} \vphantom{{\Sigma^{2}}^{2}} \right\}$

Finally let the frame's clock be characterized by its spin. To develop a sensory account of time we make a description of P, that is relative to the reference frame F, by defining a number called the phase of P

$\delta_{\theta}^{\, \sf{P}} \equiv \begin{cases} +1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \sf{\text{is spin-up}} \\ \ \ 0 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is not rotating} \\ -1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is spin-down} \end{cases}$

If two particles share the same value of $\delta _{\theta} =\pm1$ then we say that they are in phase with each other. If not, they are said to be out of phase.

achromatic.jpg
Sensory interpretation: The spin can be explained as a representation of achromatic visual sensation. So the phase depends on the brightness of the sensations objectifed as $\mathbf{\Theta}$. If the Earth is taken as a reference frame, and the brightness of daily events on Earth are used for a simple clock, then the phase indicates whether the background provided by terrestrial events is bright or dark. That is, the phase depends on if events occur in the day or at night. We exploit this ancient notion of phase when using quark icons. In the movies that follow, the phase of a quark is represented by the brightness of the background. For example, here are two austral quarks that are out-of-phase with each other
austral-bright.png austral-dark.png
$\large{ \delta_{\theta}^{\, \sf{a}} = +1 }$ $\large { \delta_{\theta}^{\, \sf{a}} = -1 }$

These images emphasize that $\delta_{\theta}$ is determined by the reference frame, then attributed to an associated particle. The phase is a very relative characteristic. Whenever we use the phase, we imply that there is a frame of reference somewhere in the description.

Right.png Next step: other clocks.
Summary
Adjective Definition
Phase $\delta_{\theta}^{\, \sf{P}} \equiv \begin{cases} +1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \sf{\text{is spin-up}} \\ \ \ 0 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is not rotating} \\ -1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is spin-down} \end{cases}$ 6-21
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