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?

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?

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.)

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, like that for some reason it ends up disagreeing with experimental data - and if so, why does it end up disagreeing despite that it seems on the surface like a perfectly reasonable quantum model of electromagnetism? 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?

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?