Why can't we "simply" quantize Maxwell's equations without a Lagrangian to create a quantum theory of electrodynamics?

Question

Useful quantum field theories like quantum electrodynamics (QED) suffer from a litany of problems related to the fact that, at least in their usual Lagrangian formulation, interactions between the fields, here a matter field and electromagnetic field, must involve problematic products of operator distributions, which are not mathematically well-defined.

But what if we just admit the Lagrangian is junk and pitch it? How far can we go without it? What I'm thinking is that Maxwell's equations don't involve any product of fields. What stops us from just "quantizing" them directly, by promoting the various terms involved to quantum operators?

What I mean by that is this. A compact relativistic formulation of Maxwell's equations is

$$\partial_\nu \partial^\nu A^\mu = \mu_0 J^\mu$$

where $A^\mu$ is the space-time potential field and $J^\mu$ is the space-time current. What is stopping us from "putting hats" on those like thus:

$$\partial_\nu \partial^\nu \hat{A}^\mu = \mu_0 \hat{J}^\mu$$

that is to say, promoting the fields directly to quantum operators, just like that, and using this equation to do physics?

I suppose one problem is we cannot then account for back-reaction of the EM fields against the charge field, viz. pair-production of electrons and positrons from photons, which is one of the phenomena we would ideally like to be able to account for in the same framework. But as said, I'm not claiming this a complete replacement for regular QED, simply asking "how far can you go?" this way. Can we at least get some low-energy phenomena, i.e. where the energies involved are much less than the rest energy of an electron or positron, out of this?

Is the failure point something else? Is the equation inherently ill-defined? I.e. the partial derivative on the left cannot be had of a distributional field, so we're back in the same boat as with the Lagrangian? Or what? Or is it perfectly consistent, but ends up disagreeing with experimental data (more than "standard" QED) - and if so, why does it end up disagreeing despite that it seems on the surface like a perfectly reasonable quantum model of electromagnetism, following the same recipe you get in your intro QM textbook?

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ADD: Now that I see the comments about the gauge, what about if we use the EMF tensor instead? Viz.

$$\partial_\mu \hat{F}^{\mu \nu} = \mu_0 \hat{J}^\nu$$

And then enforce the constraints for space-like position at least,

$$[\hat{E}^i(^{(4)} X), \hat{E}'^i(^{(4)}Y)] = i\hbar \delta({}^{(4)}Y - {}^{(4)} X)$$ $$[\hat{B}^i(^{(4)} X), \hat{B}'^i(^{(4)}Y)] = i\hbar \delta({}^{(4)}Y - {}^{(4)}X)$$

where $\hat{E}'^i$ and $\hat{B}'^i$ are the components of the conjugate electric and magnetic field, and $^{(4)}X$ and $^{(4)}Y$ is how I denote a space-time (viz. four-)point, with these commutator relations chosen on the basis that a free space wave should look like a free linear field? (Note that I'm unsure what to do about the diagonal components viz. $\hat{F}^{\mu \mu}$ because they're all zero in the Lorentz coordinates. Perhaps we can just assume the same, as well as that the state should always suitably render them constant and trivial.)

Answer 1

What is stopping us from "putting hats" on those like thus: $$\partial_\nu \partial^\nu \hat{A}^\mu = \mu_0 \hat{J}^\mu$$ that is to say, promoting the fields directly to quantum operators, just like that, and using this equation to do physics?

That's not how quantum physics works. You've just written down an equation for some operator $\hat{A}^\mu$, but you can't really do any physics with this yet! Where do the states come from? How would you even start computing a scattering amplitude from just this?

Remember, on an abstract level, we get states as representations of canonical (anti-)commutation relations, which in turn are closely related to Lagrangians/Hamiltonians. You can't even get to what a wavefunction has to look like if you try to start normal QM from some equation like $\ddot{\hat{x}} = 0$. I think you can only claim that you can "do physics" with this if you can answer how we get to "states are wavefunctions $\psi(x)$" from that and show e.g. how an initial Gaussian $\psi(x)$ spreads with time.

Fundamentally, what we want from a QED-like theory - or any quantum theory - is to give us a notion of states, some of which we want to interpret at photons and (anti-)electrons, and a rule for how these states evolve in time and interact with each other. This is what quantum physics usually realizes via the time-evolution operator, which is formally something like $U(t) = \mathrm{e}^{\mathrm{i}\int H(t)\mathrm{d}t}$.

Phrased differently, a fundamental notion of quantum physics is time translation acting on arbitrary physical states, and the generator of time translation is the Hamiltonian. The "equation of motion" for the operators alone is pretty worthless since the solutions don't tell you anything about the states.

While the Heisenberg and the Schrödinger picture are equivalent, the Heisenberg picture needs a general equation that tells you the evolution equation for all operators - via the commutator with the Hamiltonian - not just the equation of motion for one particular operator.

Answer 2

What is really missing in your description is the Lorentz force. Note that by writing a Lagrangian you get both Maxwell's equation and the Lorentz force. So even without considering any quantum effects, such as pair productions, there is important physics that is not described by this equation.

One option is consider the currents as static, i.e. neglect the Lorentz force, and just quantize the electric and magnetic fields. This is a widespread procedure that leads to useful results.

Another option is to try writing the Lorentz force using the quantum operators you proposed. I do not see any fundamental reason why this could not work. However, it will be quite messy and one will have to work hard to get things that comes free with the Lagrangian formulation, such as Lorenz covariance and conservation laws.