Modern Physics/Conservation of Four-momentum

We earlier introduced the ideas of energy and momentum conservation. In other words, if we have a number of particles isolated from the rest of the universe, each with momentum p_i and energy E_i, then particles may be created and destroyed and they may collide with each other.

In these interactions the energy and momentum of each particle may change, but the sum total of all the energy and the sum total of all the momentum remains constant with time:

$E=\sum _i{E}_i=\text{const}𝐩=\sum _i{𝐩}_i=\text{const}$

The expression is simpler in terms of four-momentum:

$\underset{\_}{p}=\sum _i{\underset{\_}{p}}_i=\text{const}$

At this point a statement such as the one above should ring alarm bells. Just what does it mean to say that the total energy and momentum remain constant with time in the context of relativity? Which time? The time in which reference frame?

Suppose two particles exchange four-momentum remotely at the time indicated by the fat horizontal bar in the left panel. Conservation of four-momentum implies that

${\underset{\_}{p}}_A+{\underset{\_}{p}}_B=\underset{\_}{p}{\text{'}}_A+\underset{\_}{p}{\text{'}}_B$

where the subscripted letters correspond to the particle labels in the figure. Primed values refer to the momentum after the exchange while no primes indicates values before the exchange.

Now view the exchange from the reference frame in the right panel. A problem with four-momentum conservation exists in the region between the thin horizontal lines. In this region particle B has already transferred its four-momentum, but it has yet to be received by particle A. In other words, four-momentum is not conserved in this reference frame!

This problem is so serious that we must eliminate the concept of action at a distance from the repertoire of physics. The only way to have particles interact remotely and still conserve four-momentum in all reference frames is to assume that all remote interactions are mediated by another particle, or by a field.

If the force is being mediated by a particle then first, particle A emits particle C in a manner which conserves the four-momentum. Second, particle C is absorbed by particle B in a similarly conservative interaction.

If the force is being mediated by a field then first, particle A emits wave C, with momentum proportional to its wavenumber in a manner, which conserves the four-momentum. Wave C then travels at c or less until it is absorbed by particle B in a similarly conservative interaction.

We'll see that in quantum theory the difference between a particle and a field vanishes, so these two pictures actually both describe the same mechanism a different perspective. Which every picture we use, four-momentum is conserved at all times in all reference frames.

In other words, momentum and energy are transferred from particle A to particle B in a two step process. In between the momentum resides in a particle or field