Collision data and four-momentum conservation

Throughout, $(A,V,\eta ,C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and $c>0$.

(a) Collision datum and conservation. A collision datum is a triple $(p;(P_1,\dots ,P_N);(P_1^{\mathrm{\prime }},\dots ,P_K^{\mathrm{\prime }}))$ consisting of an event $p\in A$, a finite nonempty list $(P_1,\dots ,P_N)$, $N\ge 1$, of vectors of $V$, called the incoming legs, and a finite nonempty list $(P_1^{\mathrm{\prime }},\dots ,P_K^{\mathrm{\prime }})$, $K\ge 1$, of vectors of $V$, called the outgoing legs. Its side totals are the finite sums of A finite sum in a commutative monoid indexed by an arbitrary finite set in the additive group $(V,+)$, formed as in Total four-momentum, invariant mass and center-of-momentum frame of a finite system:

$Q_{\mathrm{i}\mathrm{n}}:={\sum }_{i=1}^N{P}_i,Q_{\mathrm{o}\mathrm{u}\mathrm{t}}:={\sum }_{j=1}^K{P}_j^{\mathrm{\prime }}\mathrm{.}$

The datum conserves when $Q_{\mathrm{i}\mathrm{n}}=Q_{\mathrm{o}\mathrm{u}\mathrm{t}}$. The two sides are compared only through these totals: if two data have the same incoming legs and the same outgoing legs, their conservation behaviour is the same, and for a datum with a single leg on each side conservation is exactly $P_1=P_1^{\mathrm{\prime }}$.

(b) Legs, mass data and mass sums. A leg $P$ is massive with mass $m>0$ if there is a future timelike $U\in V$ with $\eta (U,U)=-c^2$ (the four-velocity normalisation of Four-velocity and four-acceleration of a future timelike worldline) and $P=mU$; a leg is null if it is a nonzero future null vector (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones). For a massive leg both $U$ and $m$ are determined by $P$: $\eta (P,P)=-m^2{c}^2$ forces $m=\sqrt{-\eta (P,P)}\mathrm{/}c$, the invariant mass $m(P)$ of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum, and then $U=P\mathrm{/}m$; so the recorded mass of a massive leg is necessarily its invariant mass. A datum is on-shell if every leg is massive or null; a mass-specified datum is an on-shell datum together with the recorded mass $m>0$ of each massive leg and the recorded value $0$ for each null leg. The mass sum of a side of a mass-specified datum is the sum of the recorded positive masses of its massive legs, and the side is massive-only if all its legs are massive.

(c) Special classes of data. A datum is causal if every leg lies in $\mathrm{c}\mathrm{l}(C^{+})$; two-leg if $N=K=2$; and equal-mass if, in addition to being mass-specified and two-leg, the recorded masses agree pair by pair, that is the incoming masses $(m_1,m_2)$ equal the outgoing masses $(m_1^{\mathrm{\prime }},m_2^{\mathrm{\prime }})$ under the identification of the pairing.

(d) Scope. The definition is purely mathematical: it neither asserts that any datum describes a physical process, nor that a conserving datum exists for a prescribed list of legs, nor any law of nature. Position, simultaneity, mechanism and interaction strength are not part of a datum; only the event, the two finite lists and, for mass-specified data, their recorded masses are. In particular conservation here is an equation between two vectors of $V$ and is not the adopted conservation law of a physical model, which is a separate postulate; the existence question for prescribed masses is answered only in special cases by Invariant mass, center-of-momentum frame and necessary conditions for conserving collision data. No choice principle is used.