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Cause and Effect

Consider some particle P that is described by its lifetime  \tau, its orbital period  \widehat{\tau} and a chain of events that is written as

\Psi^{\mathsf{P}} = \left( \mathsf{P}_{i} \; \ldots \; \mathsf{P}_{f} \right)

Let these events be historically ordered. Then  \mathsf{P}_{i} and  \mathsf{P}_{f} are called the initial and final events of \Psi. And if the particle is perfectly isolated then  \mathsf{P}_{i} and  \mathsf{P}_{f} are established by the formation and decay of P. With an inertial frame of reference, the period does not vary. Then the elapsed time  \Delta t between initial and final events is P’s lifetime, and all these quantities are related as

\tau = t_{f} - t_{i} \equiv \Delta t = \left( f-i \rule{0px}{10px} \right) \widehat{\tau}

Let \Psi be a chain of very many events so that

i+1 \ll f

For this case the lifetime must be much greater than the orbital period

1 \ll f-i = \dfrac{\tau}{\widehat{\tau}}

and we say that P is stable. Then it is probable that P will not decay during any specific cycle. P is steady and consistent. One orbit will almost invariably follow another with dependable regularity. This predictability is useful, so we give the events of particles like this special names. If

\widehat{\tau} \ll \tau

then the initial event of \Psi is called the cause, and the final event is called the effect of the cause. We may say that P’s events are causally linked to each other. For example consider the proton in its ground-state. It has a period  \widehat{\tau} of about 10-24 seconds, so in principle we could assign a very precise time of occurrence to any events in the history of a proton. The lifetime  \tau is more than 1036 seconds, so  \widehat{\tau} \ll \tau for protons. And for electrons, the period is about 10-20 seconds with a lifetime of more than 1034 seconds. So both the proton and electron are extremely stable particles. This gives them starring roles in narratives connecting cause and effect.

Cause and effect are suggested by the images of life and death in this Indonesian weaving.
Tampan, Paminggir people. Lampung region of Sumatra, Kota Agung district, circa 1900, 38 x 37 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.