Collisions and Explosions

Here is an archetypal vignette for Newtonian mechanics. Two compound atoms called $\mathbf{A}$ and $\mathbf{B}$ have an interaction with each other by swapping $\sf{X}$ another particle which is called the exchange particle. The interaction is caused when $\mathbf{A}$ emits $\sf{X}$ at event $\mathbf{A}_{\it{i}}$ which is called the initial event of the interaction. This is written as

$\mathbf{A}_{\it{i}\sf{-1}} \to \mathbf{A}_{\it{i}} + \sf{X}_{\it{i}}$

Particle $\sf{X}$ then has an effect on $\mathbf{B}$ by being absorbed at event $\mathbf{B}_{\it{f}}$ which is called the final event of the interaction. We express this by

$\mathbf{B}_{\it{f}} + \sf{X}_{\it{f}} \to \mathbf{B}_{\it{f}\sf{+1}}$

For WikiMechanics, the interaction is described using three repetitive chains of historically ordered events written as

$\Psi \left( \bar{r}, t \right) ^{\mathbf{A}} = \left( \mathbf{A}_{1}, \mathbf{A}_{2} \ldots \mathbf{A}_{\it{i}} \ldots \mathbf{A}_{\it{f}} \ldots \right)$

$\Psi ^{\sf{X}} = \left( \sf{X}_{\it{i}} \ldots \sf{X}_{\it{f}} \right)$

$\Psi \left( \bar{r}, t \right) ^{\mathbf{B}} = \left( \mathbf{B}_{1}, \mathbf{B}_{2} \ldots \mathbf{B}_{\it{i}} \ldots \mathbf{B}_{\it{f}} \ldots \right)$

Since $\mathbf{A}$ and $\mathbf{B}$ are composed from atoms, we assume that they can be described by space-time events with a position $\bar{r}$ and time of occurrence $t$. We do not assume that $\sf{X}$ is an atom, rather we often take it to be a photon or a graviton. So we cannot always describe $\sf{X}$ using a trajectory. And the position of $\sf{X}$ is well-defined only for the initial and final events where it is included as part of an atom. Overall, the interaction is characterized by the following quantities.

$\Delta \bar{p}^{ \mathbf{A}} = - \, \bar{p}^{ \sf{X}}$

$\Delta \bar{p}^{ \mathbf{B}} = \bar{p}^{ \sf{X}}$

$\begin{align} \Delta t = \frac { h \left( \, f-i \right) }{ E ^{\sf{X}} } \end{align}$

$\begin{align} \ell = \frac { h \left( \, f-i \right) }{ p ^{\sf{X}} } \end{align}$

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