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*.

$\large{ \delta_{\theta}^{\, \sf{a}} = +1 }$ | $\large { \delta_{\theta}^{\, \sf{a}} = -1 }$ |

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.

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Phase |

Summary |

Adjective | Definition | |

Phase | $\delta_{\theta}^{\, \sf{P}} \equiv \begin{cases} +1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \sf{\text{is spin-up}} \\ \ \ 0 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is not rotating} \\ -1 &\sf{\text{if}} \ \ \mathbf{\Theta} \ \ \text{is spin-down} \end{cases}$ | 6-22 |