Does the electro-dynamical lagrangian contain a (Dirac) wave-function?

Question

Consider a lagrangian for quantum electro-dynamics. It contains the two fields: the vector $A$-potential inside $F_{\mu\nu}$ and the matter field $\psi$ (Dirac's spinor). A series of questions arise for me.

1) This lagrangian looks like an inconsistent mixture of entities: the classical field $F$ (or potential $A$) and probabilistically treated (quantum!) $\psi$. Is it ok? The lagrangian itself is a classical object, so it must not contain anything quantum. It will have to be subjected to quantization in future, not now. Some people - Weinberg, Hobson etc - instruct us that the second quantization of the $\psi$-construction should be banned in physics at all. If the $\psi$ is a quantum $\psi$ then it is principally not observable, so I may not require a relativistic invariance to an equation it satisfies. Where is a bug in the reasoning above?

2) The possible way out here is to treat this $\psi$ in any way except (!) for the probabilistic one mentioned above. Is it true? Say, let us look at Dirac's equation as a classical field equation for a certain field function $\psi$ while observational quantities (corresponding to this classical field) come from the $\psi$ through the quadratic combinations like current $j=\bar\psi\gamma\psi$? The key point in my question is to forbid the quantum (hence statistical) wave-function interpretation to this classical $\psi$ and to the current $j$! I do not ask here where we've taken the field equation of such a special (Dirac's) form; it is a separate question. Instead, we could be based upon a gauge invariance of a complex scalar field $\phi$ and derive again the same (usual) electro-dynamics for the $F$-tensor plus some currents. Also, we do go to the new function $\psi$ instead of old (current) $j$ and, moreover, imply a quadratic (why?) shape $j=\bar\psi\gamma\psi$.

3) Well, let the $\psi$ have not had such a (quantum) treatment. What about eigenvalue problem for the stationary Dirac eq written in terms of this very letter $\psi$? For the hydrogen atom we arrive at a discrete spectrum. But equation and the problem itself are classical! Should I then forbid these spectral values to be thought of as quantum levels? It is assumed, according to the previous point, that their probabilistic distribution has already not been allowed. That is, these levels are just spectral levels of a certain spectral problem but they are not some quantum levels. There is a parallel here. The classical string equation does also yield a discrete spectrum in an appropriate spectral problem but we do not consider this spectrum as a quantum one. I consider a mixing the two entities of different nature - quantum wave function and some classical field - as illegal action; despite the fact that we may consider the eigen-value problem for both of them.

My guess here is this. Consider the electro-dynamics lagrangian with use of (classical) letter $\psi$. Then, through a quantization procedure, we introduce a notion of the quantum state - call it letter $\Psi$ - and derive in some way (I dont know how) a spectral problem for this quantum state. As a result, we get a (new) equation for this quantum $\Psi$ and it turns out to be the same form as the Dirac equation for the (classical) letter $\psi$. Equations formally coincide but these are DEs for different entities. Is this correct?

4) Addendum to 2). It seems I should also prohibit bilinear constructions like $\langle \psi_1|\psi_2\rangle$ coming from the Hilbert space(!) add-ons; no anything quantum I may have at the moment (inside lagrangian). By anything I mean here: Hilbert, bases of observables, operators etc. At the same time, the quadratic constructions $j=\bar\psi\gamma\psi$ mentioned above are the legal ones; the classical current densities. ... ?

Answer 1

Even in Classical Mechanics (CM) the variables in the Lagrangian are unknown functions, not the solutions to a particular problem.

In QM, apart from equations for unknown variables, there are some commutation relationships; the latter make the solutions of particular problems "quantized".

Normally, the new equations of motion are guessed first, and only then Lagrangians are constructed to "derive" them ;-) No wonder Lagrangians contain the corresponding variables to be found.

Answer 2

You can of course mix quantum and classical degrees of freedom in your theory, just think of a non-relativistic quantum particle subjected to a classical electromagnetic field. The corresponding hamiltonian $\hat{H} = \frac{1}{2m} \bigl (- \mathrm{i} \hbar \nabla - A(x) \bigr )^2 + V(x)$ contains scalar and vector potentials to the classical electromagnetic field.

Now specifically to your questions. I am assuming you are quantizing the Abraham model, which gives you the Pauli-Fierz hamiltonian. You seem unsettled by the distinction between classical and quantum waves. IMHO a lot of your confusion can be avoided if you kept track of the spaces, which is something that is not commonly done in the physics literature. If you want a presentation that is a bit more systematic, have a look at Chapter 13 of Spohn's book “Dynamics of Charged Particles and Their Radiation Field” (Cambridge University Press, 2004).

But there are two other points: you seem reluctant to use mathematical constructs such as scalar products in the context of classical fields. Don't. Hilbert space ≠ quantum mechanics. In fact, Hilbert spaces appear naturally in the context of purely classical electromagnetism. Moreover, the structural similarity between Maxwell's equations and the Dirac equation for a massless particle is by no means accidental and has been known since the advent of modern quantum mechanics: both are the relativistically covariant equations for a massless particle of spin-1 (Maxwell) and 1/2 (Dirac), respectively. Wigner mentioned this explicitly in his article where he analyzes the Poincaré group, and attributes this insight to Dirac. Other colleagues of mine, though, insist that other authors did this earlier, but I was unable to find evidence in publications. Note, however, that this does not mean you should interpret Maxwell's equations probabilistically! Indeed, Maxwell's equations are obtained from QED in the many photon limit (which is the classical limit).

And as Maxwell's equations are partial differential equations, you can of course use a Hilbert space structure and eigenvalue equations to your advantage. In fact, many modern insights to electromagnetism (e. g. photonic crystals and topological phenomena) rest on the fact that you can define a Hilbert space structure also in the context of electromagnetism and rewrite Maxwell's equations with a hermitian operator. None of that is “forbidden” by some obscure physical principle.