Nope, "Quantum electrodynamics has to be seen not as a more fundamental theory, but as an upgrade of classical electrodynamics, which permits an extension of classical theory to the description of phenomena that, while being related to the conceptual framework of the classical theory, cannot be addressed from the classical theory. "

See, https://arxiv.org/pdf/1201.5536.pdf

and 

http://www.cft.edu.pl/~birula/publ/ClassLimit.pdf

Maxwell’s electrodynamics, unifies electric and magnetic forces. Quantum Mechanics mainly deals with the motion of electron due to absorbtion and ejection of quanta of energy. There are certain things that Dirac’s quantum mechanics cannot explain, like the Lamb Shift, which is explained by QED (quantum electrodynamics). But there is no single formula in Dirac’s quantum mechanics that can explain fine structure of hydrogen, Lamb Shift and 1s hyperfine structure. Such a formula is recently proposed which is derived using Einstein’s field equations.

https://www.researchgate.net/publication/319291737_Hydrogen_spectra_using_Einstein%27s_field_equations

Maxwell’s electrodynamics is incorporated into electroweak theory which unifies three forces. Electric force, Magnetic force and the weak force. So you may find answer to your question if you study  the electroweak theory.

Article Hydrogen spectra using Einstein's field equations

According to correspondence principle we have some sort of correspondence between different good theories, which describe the same physics. In is not just a limit, it is a correspondence.

Classical and quantum theories have different kinematics, so the low h limit ("classical" limit) of quantum theory could not be compared with classical theory. Actually one has to compare predictions of experimental results for two theories.

E.g. quantum correction to classical prediction could be large, but oscillating, to see the correspondence one has to average the oscillations, after the averaging the quantum  correction becomes small.

So to compare the predictions of two theories one has to interpret the predictions of some experiment. This interpretation involve some analysis of measurement process in both theories.

In a typical QED scattering calculation we only use perturbation around non interactiong field where initial and final states are eigenstates of the non interacting Hamiltonian and actual interaction occurs for small interval of time in between. This is achied by considering the initial state at t=-inf (free particle) and final state at t=+inf (free particle). So we have no picture of interaction in finite time like we do have in classical electrodynamics.

"Can we get classical maxwell electrodynamics as a similar limiting case from quantum electrodynamics?"

I understand that the quoted question is about Maxwell's laws of electromagnetism.

Electromagnetic (e.m.) waves consist in a huge number of photons prepared according to a given preparation mode. For example coherent waves have an almost classical behavior. The bigger the average number of photons, the better is the coherence of the wave and the classicality of the behavior.

The difference between quantum e.m. field, and the e.m. field described by Maxwell's eqs. appears when we can deal with a small number of particles, s.t. we can measure all sort of eigenvalues. With a big number of particles we can measure averages, loosing the fluctuations observable at quantum level. For instance, under a very weak stream of x-polarized photons, an extremely light disk would rotate for each photon either clockwise or counter clockwise, indicating that the photon has an angular momentum 1 (in units of ħ). But, under a very strong stream of photons the disk won't have the time to rotate and would remain fixed. That may lead to the wrong conclusion that the photon anguklar momentum of the x-polarized photon is zero. 

Maxwell’s equations describe the coupling of “classical” electromagnetism to the charges and currents of classical “matter”. They are Euler-Lagrange equations, including coupling terms. This prescription, with “matter” described by Dirac’s electon equation, is the starting point for the development of Feynman’s QED. Thus QED is an extension of the classical theory obtained by interpeting energy and momentum as operators (“observables”) and reverts to it in situations where quantum-mechanical “uncertainties” cease to be relevant.

You can see a procedure in the classical book of Sakurai "Advanced Quantum Mechanics".  The method is known  as the "one photon exchange potential" (one boson exchange potential for other field theories).

The more general QFT was initially developed to reconcile classical electromagnetism with the idea that elementary electromagnetic particles such as the electron emerge as local excited states of an underlying neutral quantum energy field, thus introducing the concept of energy quantization. This is what gave rise to QED, which allows the description of interactions between elementary particles as quantized virtual exchange photons from the idea conceived of by Feynman as described mainly in his 1949 seminal paper, often viewed as spontaneous fluctuations of the quantum field.

The ease with which infinitesimally progressive sequences of motion can be mathematically represented by means of an indefinite number of instantaneous momentary excited states with the Lagrangian makes it easy to calculate energy/force levels at any distance between charges.

The Coulomb force which is in action between charged particles as a function of the inverse square of the distance separating them seems to have progressively become invisible in the background of the QED method, even if the Coulomb equation is an integral part of Maxwell's first equation, that is, Gauss' equation for the electric field, from which it can easily be derived.

The Coulomb force is indeed a critically important component of every "virtual photon" in QED, but metaphorically cut into so many little pieces that it now attracts little attention.  Metaphorically speaking, QED causes us to pay attention to every individual pixel in a metaphorical 4K screen that would represent the submicroscopic level, but if we mentally pull back sufficiently, its infinitesimally progressive action can again be observed.

So yes, you certainly get classical electrodynamics as the limiting case of quantum electrodynamics, since QED was developed to remain in agreement with classical electrodynamics.

Another issue that I don't understand  is the way we describe matter as particles and EM waves as field in classical theory. Is there anyway to describe classical light-matter interaction purely as field-field interaction (theoretically). Becoz particle picture itself is problematic. Here I am referring to particle as localised object with definite momentum.

@Rahul, you ask:

==> "Is there anyway to describe classical light-matter interaction purely as field-field interaction (theoretically). Becoz particle picture itself is problematic. Here I am referring to particle as localised object with definite momentum."

As you say a particle is a localized object. However, a field is a distributed object. It has a value at every point in the space. How can you represent in classical physics the field-particle interaction as field-field interaction? If you admit/assume that the particle is localized, you have to stay with this assumption.

Yes sure, but as you pointed out previously that when large number of photons are present we loose the quantum fluctuations and result is classical field. Now is this similar procedure works for electrons or more generally fermions?  

@Rahul,

you ask,

"when large number of photons are present we loose the quantum fluctuations and result is classical field. Now is this similar procedure works for electrons or more generally fermions?"

I don't want to speak of fermions, it's more complicated, but I can speak of bosons. We have coherent beams of bosons (atoms) - we do with them atom optics. The bigger the number of atoms, the bigger is the similarity with a classical beam.

But you posed the problem of the interaction of field with matter. What is that matter, how many particles? One particle? A cloud of particles? A coherent beam of particles? You have to define the problem, for us to be able to answer. I never heard such a question as a coherent beam of photons interacting with a coherent beam of, say, ions. But, of course, the problem may be posed. So, please define what you ask.

Kind regards!

Dear Rahul,

Your write:

"Is there anyway to describe classical light-matter interaction purely as field-field interaction (theoretically). Becoz particle picture itself is problematic. Here I am referring to particle as localised object with definite momentum."

You raise a very good point here.

You do have an obvious fields vs localized EM particle interaction with the Lorentz equation F= q(E + v x B).

This equation allows establishing a precise trajectory for a single localized electron in straight line when the density of both ambient fields are equal, or on a curved trajectory if the B field is more intense than the ambient E field.

This is often given as an example in intro textbooks to explain the triple orthogonal electromagnetic relation between both fields with respect to the direction of motion of a charge.

Considering that the electron is an electromagnetic particle, it must by definition also have internal electric and magnetic fields corresponding to its mass, and that it would be these internal fields that interact with the ambient fields that can be calculated with the Lorentz equation.

Yes, indeed this is what i was thinking. These might be complete nonsense but can we treat charge-EM field interaction by replacing localized charge by its extended EM field? 

Dear Rahul,

You write:

"These might be complete nonsense but can we treat charge-EM field interaction by replacing localized charge by its extended EM field? "

We certainly can. This is definitely not nonsense, quite the contrary.

Indeed, invariant electric and magnetic fields could be defined in 2007 for the rest mass of the electron, thanks to a discovery made by Paul Marmet, which allowed defining them separately from the fields of its carrying energy.

For example, for an electron moving in straight line, we know that the Lorentz equation resolves to v=E/B.

By defining the combined B field of the electron charge separately from that of its ambient carrying energy, you get equation (49) in the below paper, published in 2007 in International IFNA-ANS Journal, No. 2 (28), Vol. 13, 2007, p. 123-140, Kazan State University, Kazan, Russia.

For the combined E field, you get equation (58)

If you then resolve v=E/B with these combined fields, you get equation (59) to (61)

You can try it with any value of carrying energy and get the right relativistic velocity for the electron.

To my knowledge, this is the first time that the invariant  E and B fields of the invariant rest mass of the electron have been defined.

These definitions allow revisiting all of fundamental physics from the electromagnetic perspective.

Article Field Equations for Localized Photons and Relativistic Field...

If I calculate electromagnetic mass of the stationary electron by taking the lower limit of radius to be classical radius what i get is mass of the electron. What does it imply?. 

Dear Rahul,

It implies only that the so-called classical radius of the electron is the transverse amplitude of the electromagnetic oscillation of the electron mass energy:

Since the electron Compton wavelengh is the EM wavelength of the electromagnetic quantum corresponding to the rest mass of the electron, then the lower limit of integration can be calculated from the transverse amplitude of the electron Compton wavelength, which happens to be the so-called classical electron radius.

re=(Lamba_C alpha)/2 pi

The total complement of energy of any elementary electromagnetic particle can be calculated by integrating to a lower limit of r=(Lamba alpha)/2 pi

This is what is revealed by the method that Marmet used to calculate the magnetic mass of a moving electron and the related magnetic field.

Apparently, the mathematical amplitude of electromagnetic transverse oscillation of an electromagnetic quantum is traditionally calculated as obeying the mathematical sinewave function, but its physical amplitude physically obeys the sinewave function multiplied by alpha.

That's what I observed.

It seems there is no definite answer to this question. There is a paper in the arXiv on this topic:

The relation between classical and quantum electrodynamics; Mario Bacelar Valente;

arXiv:1201.5536

maybe by using coherent states, as defined by roy glauber.

It can be done. Use the functional derivative Schroedinger equation for quantum electromagnetic field as described e.g. in K. Hwang QUARKS, LEPTONS AND GAUGE FIELDS or B. Hatfield "Quantum Field Theory of Point Particles and Strings" and then generalize the Ehrenfest theorem from quantum mechanics textbooks to this infinite-dimensional setting. I don't know if it has been published somewhere but it seems pretty straightforward to do at least in the Coulomb gauge. So the Maxwell equations will be reproduced as equations on expectation values derived from quantum equations.