Relationship between a field and its quanta

[wotanub]

I just listened to a lecture on this, and I'm not sure I quite get the point.

What the lecturer did was examine coherent states in the simple harmonic oscillator then linked excitations of an oscillator to the number of field quanta. The goal of the lecture was to show how Maxwell's equations naturally fall out of attempting to link quantum mechanics with special relativity. We were not trying to go through second quantization.

Does this ring a bell to anyone? Maybe I'm not explaining it exactly right, but I'm looking for a resource (link or book) where I can read about this in detail to get a better grip on it. He was going quite fast and my lecture notes aren't that good because I was trying to pay attention to understand.

 

[wotanub]

I found the answer to my own question. I was looking to an intro to relativistic quantum mechanics.

 

[vanhees71]

Usually, when one says "relativistic quantum mechanics" one talks about a naive copy of the non-relativistic wave-mechanics picture a la Schrödinger ("first quantization"). This is, however, a not very good approach, because it simply doesn't work for relativistic quantum theory. The reason simply is that this formalism is for the case of a conserved number of particles, and in relativistic QT particles are always produced and destroyed again, if only the collision energy is high enough.

Historically there was Dirac's approach, now known as the "hole-theoretic formulation of QED", which uses the 1st-order quantization picture but artificially introduces a many-body concept by assuming the so-called "Dirac sea", i.e., one fills up the negative-energy states with electrons and declares this as the vacuum state of the theory, and then holes in this sea appear as positively charged particles of the same mass as electrons. In this way Dirac predicted anti-particles, and indeed the positive charged "holes in the Dirac sea" were indeed found. These are the positrons, i.e., the anti-particle of the electron. This approach leads to the same theory as the QFT approach, namely Quantum Electrodynamics (QED), but it's pretty cumbersome to learn about it, and thus nobody teaches this at the universities anymore.

So the right way to look at relativistic QT is to use relativistic QFT, i.e., "second quantization", which you left out in your study so far!

Also coherent states (of photons, e.g.) are many-body states, and you need QFT for them (although you can also look at the coherent states of the simple harmonic oscillator in non-relativistic QT as well, and it's a good model to learn about them, and it's pretty similar to the full story in QFT). The point is that these are superpositions of states of any photon number, leading to a state with an indefinite particle number. You can only define an average photon number. At large average photon number, a coherent state physically describes in a quantum-theoretical way classical electromagnetic fields like a laser field. The classical fields are a very good approximation for this case. For very low average photon numbers (even less than 1!) you get the utmost possible dimmed light in some sense, and then the coherent state describes something which still on the average behaves very much like a classical electromagnetic wave, but you have to wait a long time, until you have accumulated inough photon events (e.g., on a CCD screen or photo plate). This is, however NOT a single-photon state but still a superposition of all photon-number states. In this case it's mostly the vacuum state (no photons at all) and the single-photon state.