Mass energy equivalence

Question

We know that in pair annihilation all of the mass is converted into energy. Is it possible to convert all energy into mass?

Answer 1

Yes you can convert all mass into energy. For example, in electron–positron annihilation the masses of the electron and the positron are completely converted into the energy of, in general, two photons.

Answer 2

Any process in quantum field theory that happens one way can theoretically happen in the reverse. But it's much more difficult to get two photons to collide with the appropriate amount of energy to create a pair of particles. It's easy to get electrons and positrons to collide because they are charged particles that are attracted to each other.

Answer 3

Yes, but with a great many caveats! It's not really correct to talk of converting energy to mass. One can say, however, that one can change a system's state so that the system has maximal rest mass. The system's energy (if the system is isolated) is constant. You can convert certainly convert a zero rest mass system into one where its rest mass is maximal and its momentum nought.

The situation is more pedantic, complicated and awkward and at the same time less interesting from a physics point of view than you probably think!

Mass is an awkward concept, falling further and further into disuse as physics advances, at least in the field of particle physics where your question makes sense (I hasten to add that the notion is still fundamental to Newtonian mechanics as the concept of inertia, though). The only remaining rigorous concept of mass that people find useful is that of rest mass, which is simply the total energy of a system (divided by $c^2$ if you're not using natural units) when you boost to an inertial frame that is at rest relative to the system (or at least momentarily so, if the system is accelerating). Otherwise put, it is defined the Minkowski length of a system's momentum four-vector:

$$m_0^2\,c^2=\frac{E^2}{c^2} - p^2\tag{1}$$

where $p$ is the magnitude of the 3-momentum, $E$ its total energy and $m_0$ its rest mass.

Let's look at some examples that will probably seem artificial to you, but will hopefully let you glimpse the awkwardness of the "mass" notion.

Think of a photon. The length of its four-momentum is nought. There is no inertial frame wherein it is at rest. It therefore has zero rest mass.

Now think of two photons, moving in opposite directions. The system has zero momentum, so, from (1), it has rest mass $E/c^2$, where $E$ is the sum of the photon energies. This probably seems artificial, but if the two photons are imagined in a box - a perfect optical resonator - then it can be shown that the box's Newtonian inertia indeed increases by $E/c^2$, as I show in my answer here. Indeed a lone photon in a perfectly reflecting box also raises the system's inertia by $E/c^2$.

So, in these cases, the system's energy is converted into a state that has maximal rest mass and zero momentum.

Pair Production is another mechanism whereby zero rest mass objects - photons - can be converted into particles with rest mass. Common pair production requires the presence of a nucleus to fulfill conservation of both energy and momentum, but my impression is that very few people in physics seriously dispute the possibility of $\gamma-\gamma$ pair production - the true inverse of matter - antimatter annihilation - even though it has not yet been observed experimentally.

These examples highlight the awkwardness of even the rest mass notion: it can seem "artificial" for multiparticle systems, it is not conserved and it is not linearly additive: the rest mass of the sum of systems is not the sum of the system rest masses. It is interesting that the next upgrade of SI, which focusses on the fundamental physical constants of Nature, makes the notion of mass rather secondary and consequential to the more fundamental definitions.