How can we determine the energy scale of a physics theory?

Question

It is very common in physics, when we refer to the most diverse theories, on the most diverse length scales, we also refer to their energy scale. It is through the energy scale that we classify a theory as being classical, quantum, or relativistic, for example.

Likewise, on such a scale, we know whether we are dealing with sub-atomic or sub-nuclear particles. Thus, when constructing a theory, it seems to be of vital importance for the physicist to understand where such a theory lies on the underlying energy scale.

However, for a young physicist student, it is not always easy to make such a correlation and find available material that deals with the subject in a didactic way. So, the question of how to determine the energy scale of a physical theory and classify it in the energy scale of physical phenomena remains somewhat nebulous, without many students realizing the importance of knowing how to classify the energy regime of a given theory.

From this, as the question in the title of this topic already delivers, I would like to know how, given any physical theory, one can determine its energy and classify it according to the cosmic energy scale (I don't know if I can call it so: "cosmic energy" scale.). I would like, if possible, indications of references that deal with the subject.

Answer 1

There are various systems of units used by physicists to avoid things like feet, pounds, hours and whatnot by juggling everything so (for example) the speed of light is equal to one, and then deriving/converting everything else to conform to that. These are called natural units and different branches of physics have different natural unit definitions, which are chosen to make setting up equations as straightforward as possible.

In the system used by particle physicists, energy turns out to have units of 1/(length), which naturally associates high energies with short distances. See the opening chapters of Voit's book Not Even Wrong for a good explanation of this. This means to probe the physics of things that are extremely tiny, you need a probe that packs extremely high energy. This is why the electrons that were used to probe what's inside a proton (see deep inelastic scattering) had to be revved up to ~giga electronvolt energies.

There are other approximate "rules of thumb" that physics geeks fall back on: chemistry problems (dealing with the outermost electrons surrounding an atom) are concerned with energies of order ~a few electron volts. The innermost electrons in an element like copper live in a smaller, ~kiloelectron volt world. When dealing with the still-smaller world of the nucleus of a copper atom, ~millions of electron volts are the customary currency and when dealing with the world of protons and neutrons that are in close proximity inside the nucleus, the bills are denominated in ~hundreds of millions of electron volts or more.

Answer 2

It depends. If your energy dependence is explicitly simple, well then it's just arithmetics. This is the case of special relativity. Expanding the energy around "small'' velocities $\beta=\frac{v}{c} \ll 1$ gives $$E=\gamma mc^2=\frac{mc^2}{\sqrt{1-\beta^2}}\sim mc^2(1+\frac{1}{2}\beta^2)=mc^2+\frac{1}{2}mv^2$$ which is the rest mass plus the classical kinetic term. Because $\beta=\frac{|p|c}{\sqrt{m^2c^4+|p|^2c^2}}$, you can infer that the energy scales at which special relativistic effects start "dominating'' is $|p|^2>m^2c^2$.
In quantum theory, where it is standard to think of energy as the inverse of length, high energies can be talked about in terms of short distances. This is why you need huge particle accelerators (high energy) to probe very tiny stuff (sub-atomic scales). In this case, energy appears in scattering amplitudes, so it is not as trivial. A nice example you can work out yourself is with the Klein-Nishina formula for photon-electron scattering.
As anna points out in her comment, the main goal at the end of the day is that your theory correctly describes what you see at a certain energy scale, and this is usually set at the start rather than by looking at final equations.
If your issue is numerical estimates, you will have plenty of time to involuntarily practice during your studies

Answer 3

The tag dimensional-analysis says it all. Given some constants relevant to your problem, you can usually construct some natural units of lengths and energy.

Well-known examples of the procedure are taken from the hydrogen atom. The relevant constants are the mass $m$ of the electron, the charge $e$ of the electron, $\hbar$, $\epsilon_0$ since the interaction is electromagnetic. You may or may not need the speed of light $c$. Given these, it's easy to show that the combination $$ \frac{\hbar^2\,\epsilon_0}{e^2 m} $$ has units of length, and this becomes your length scale.

Likewise, the combination $$ \frac{me^4}{\hbar^2\epsilon^2_0} $$ has units of energy, so this becomes your energy scale. If you are looking at muonic atoms, the mass of the electron should be replaced by the mass of the muon.

I did mention you may or may not have to use $c$. If you use CGS units, there is no $\epsilon_0$ and you'd then have to use $c$. From this perspective the CGS units are a little more natural than MKSA units and tends to be preferred in theoretical physics.

Other examples of this approach can be found to find various quantities. There's a nice discussion of the "philosophy" behind this in the book

Barrow and Tipler, The anthropic cosmological principle,

wherein there is a nice chapter on scaling of natural phenomena.