Can an energy-momentum four vector include the quantities of all objects in a closed system?

Say I have a particle moving along the $$x$$-axis in the Earth's reference frame. It decays into an upsilon and a proton, each of which has an energy of 60 GeV. They are traveling in opposite directions. The proton has a mass of 1 (or 1GeV/c^2) and the upsilon has a mass of 10 (or 10GeV/c^2).

My question is; can I set the four-vector of the original particle as:

$$(E, Px, Py, Pz)$$

And the four-vector of the decay particles as one general vector:

$$(E', Px', Py', Pz')$$

Such that $$E'=120$$GeV, the total energy of the two decay particles? Or, to find the energy and momentum of each particle, would I have to have two separate four-vectors and calculate them using the inner product?

1 Answer

Among the properties of vectors is that they have an addition operation, so you can certainly add two or more four-vectors together.

More over that is a useful operation: the result represents the total energy and momentum of the system.

But it goes one step further: the (invariant) mass of a system is found from the square of the system's four-momentum just like the (invariant) mass of a particle is found from the square of its four-momentum.