Given the relationships between mass and energy in relativity, and given that particles with mass can be created given energy over the threshold energy, and vice-versa, can we say that mass is simply an extremely dense form of energy? Or is there a deceptive parallel between the two?
I'll offer my take on this, which is very specific to me. As an alternative to the vague notion that matter and energy are interchangeable I claim the following
To specify further, I will describe a process commonly thought as changing matter into energy, then a process commonly thought of as changing energy into mass, then show that neither of them "loose" either matter of energy in favor of creating the other.
Start off with a nuclear reactor. During operation the nuclear energy is changed into heat and electrical energy. Say that this is done in a system that is thermally isolated. It doesn't matter specifically what form that energy takes, because should it go into batteries, the weight of the batteries would increase, should be be stored as heat, the weight of the medium holding the heat increases. This energy exists either in chemical bonds and kinetic motion respectively. Both increase the mass of the system.
For changing energy into mass, we can look at particle creation as you mention. In order to create the particle, that energy had to exist previously, and sure enough, whatever reservoir held the energy before it was used experienced a decrease in mass corresponding with the movement of "energy".
A recent question I wrested with was Explain how (or if) a box full of photons would weigh more due to massless photons. Even though photons are massless, if they are somehow confined they will increase the mass of the system they are a part of. This is because no matter what transitions occur, the measured mass within a boundary that doesn't exchange mass or energy will remain constant. Likewise, the energy of that system will remain constant. This is in spite of the fact that matter-energy transitions are apparently occurring.
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Please spend some time reading about four vectors.
The concept of a four vector is an extrapolation to an extra dimension of the three vector.
Say that you have two three vectors, the three momenta of two particles. Each vector has a length: p1, p2. When you add them vectorially, the length of the vectors do not add linearly. It will depend at what angle they are added. For example, if p1=p2 and the addition is head on, the length of the resultant vector will be 0.
It is the same with four vectors where the mass is the measure, what you get from the dot product, of the four vector. If you read the link above you will see that a system of four vectors will have masses which will depend on the values of the summed energies and the summed three vector momenta, in no way linear. When $E^2=P^2$ (in a system where $c=1$) the mass is zero, as happens with the photons.
When you have a system of massive particles the lowest mass you can have is the sum of the rest masses, if all particles are at rest. These are the masses that enter the $E=mc^2$ (in a system where $c=1$) which says that masses have a minimum energy content. It comes naturally in the four vector representation, since at rest, the three momentum vectors are zero, and only energy exists, thus the "length" of this four vector, which is its mass, is also its energy.
We can say that mass is a form of energy, depending on the coordinate system where the particles are studied. There is no density concept, just "length".
Energy is a four vector component. Mass is a scalar identifying the length of the four vector.