If all particles are fields, why does first quantization work for some particles?

Question

After a lot of Google and asking professors about the two quantization methods, I have learned that first quantization is what you use to quantize classical particles, while second quantization is what you use to quantize classical fields or a large number of classical particles.

If, according to QFT, every particle is an excitation of a field, then why does the first quantization method work for some particles? Is it an approximation that works because the wavelengths involved are much smaller for, say, an electron than a photon?

If that's not the right question because I'm wrongly assuming that you shouldn't be able to use 1st quantization for fields, then why can't you use it for the electromagnetic field? I mean, you can to an extent, but it feels forced, the whole reason being Gauss' law forbidding a position eigenstate, but why does something similar not happen for other particles' fields?

I'm interested in why the localization problem applies to the photon and not to, say, the electron: Why can you (in principle) exactly locate an electron in space, if you can't do it for the photon? They are both fields, shouldn't they follow similar rules? Obviously they don't, so what is the difference?

Answer 1

First quantisation cannot accommodate pair-production, which is non-negligible at energies comparable to the mass of the particle. Therefore,

First quantisation only works for massive particles, and only in the range of energies that make motion non-relativistic.

Of course, this doesn't mean that a massive particle can always be modelled in a first quantisation scheme: for example, another requirement is that the particle is stable, and unaffected by confinement, etc.

Towards the edit, let me stress that the localisation of particles in a relativistic context is a very subtle issue (cf. this and this PSE questions). A nice reference for this problem is No place for particles in relativistic quantum theories?, by H. Halvorson and R. Clifton.

Answer 2

AccidentalFourierTransform gives you physical reasons but first quantization also came about partially for both historical and pedagogical reasons.

If you take the electron, a hundred years ago it was cutting edge science to match up the predictions of the "weird" quantum world to what was "obvious" in the classical world.

In something similiar to the simple (but misleading) Bohr semi-classical picture of the atom, first quantization makes certain simplifying assumptions and mixes classical and quantum descriptions.

In first quantization, some physical properties, such as electric or magnetic fields, and the potential wells associated with them, are treated classically, but their effects are seen as changes in the quantum description (utilising wave functions and matrices) of particles, especially the electron.

Actually, imo there is no other real alternative to learning QM basics, except by treating quantum systems as they are done in the first quantization method. As John says in the comments, otherwise you would need to know more about QED and QFT. If you were treating a system of say, an electron in a magnetic field without the use of first quantization, it would be a confusing chicken and egg like affair, with the need to explain the $E $ and $B $ fields in field terms.

Its much easier to learn the basics of QM this way, and allows for the particle's motion and spin to be dealt with in the first "explanation" of how a particle behaves quantum mechanically and without the distraction of having to explain non essential "external" effects.

Second quantization then appears after you upgrade from the Schroedinger equation to the Dirac equation, with it's built in description of spin, particle production and the later introduction of the field concept as the basis for modern particle physics.

Answer 3

Localisation of electrons and photons are based on very different concepts. In fact, it is possible to construct a wave-function for an electron (under the caveats mentioned below) because even with spinors (what is used to represent electrons) you can construct a quantity which behaves under the Lorentz transformation as the zeroth component of a 4-vector (a necessary condition to fulfill the continuity equation).

But for a photon, which is described by a 4-vector potential $A_\mu$, such a quantity cannot be found. Partly because they are bosons, but mainly because they always move at the speed of light (you cannot choose an system of reference in which the photon is at rest, while for the electron you always can), which makes their localisation impossible (except perhaps in geometrical optics, but that's a very coarse theory, so the localisation is also very coarse). Not to mention that photons can only be described in a relativistic quantum theory, where the concept of wave function has lost its meaning because it violates causality. That problem does not occur in non-relativistic QM where instant interaction is assumed.

Finally, the quick summary of the difference between the quantization methods is that in 1st quantization you replace the real 4-vector $p^{\mu}$ of a particle with a differential operator $i\nabla^{\mu}$, which acts on the one-particle wave function; whereas in 2nd quantisation, the momentum and energy of your system are contained in operators on the Fock space (suited for dealing with many particles). They consist of creation and annihilation operators, so there aren't any more differential operators.

{\bf Edit:} I've a couple of precisions to make. Actually, being a boson does not prevent the particle from being localized, that's not fully true. In a limited sense they can be still localized. Charged pions, for instance, are bosons, and a probability current $j^{\mu}$ can be constructed which fulfills the continuity equation $\partial_\mu j^\mu=0$. But already in this case appear problems, the zeroth component $j^0$ is not positive definite. Whereas in case of the photon field this current cannot be constructed at all, shortly there is no photon current at all.