Is it just a co-incidence that the interference pattern of light can both be explained using a classical wave and using a probability function? Approach 1-Light can be thought of as a vector electric field wave. To explain the interference pattern, one can just add the electric fields like vectors and calculate the intensities.
Approach 2- On the other hand, one can also think it in terms of a burst of independent non-interacting photons, each with a probability wave associated with them.
I agree that the second approach is more general, as the first approach can't explain the experiment of firing photons one at a time.
Is it just a co-incidence that the first approach, of visualising light as oscillatins in a vector field, also explains the interference pattern perfectly?
I wonder if the unification of these approaches forms the basis of quantum field theory (which I don't know much about yet). But I do know that QFT is kinda similar to the first approach as it describes particles like photons as bulges (excitations) in a field (so kind of similar to oscillations in the electromagnetic field descibed by Maxwell). Is it true that the unification of these approaches was the idea behind QFT ? Probably my intuition is wrong.
 A: "Approach 2" is actually incorrect - it is based on intuitive thinking of photons, as if they were fermions and then applying logic of a two-slit experiment: as if they were point-like particles with probability waves associated with them.
Interference arizes whenever we deal with wave-likes solutions, whatever is the nature of waves:

*

*waves in water

*sound waves

*elastic waves in bulk materials

*electromagnetic waves (solutions of Maxwell equations)

*matter waves - solutions of Schrödinger equation
and so on.

Solutions of Schrödinger equation are wave-like - that is we attribute wave properties to electrons, which are normally thought of as point-like particles (which is what we call quantization). Thus, they exhibit interference patterns.
Solutions of classical Maxwell equations are also wave-like and exhibit interference behavior. Quantizing electromagnetic field in terms of photons does not reduce any of these wave-like properties - in fact, the field mode structure (one could say photon wave function) is still the solution of Maxwell equations, but the amplitudes of the electric and magnetic field obey the uncertainty relation. What makes electromagnetic field particle-like, is the descreteness of excitations, but they never become point-like, or wave packets spreading in space.
See also here on how "first quantization" of electromagnetic field is mathematically the same as "second quantization" of electrons/fermions.
A: No, this is certainly not a coincidence! After all the quantum theory of electromagnetism has to be designed in such a way as to exactly reproduce the classical theory of electromagnetism as otherwise it would conflict with 300 years of experimental data concerning light, electricity and magnetism. So no matter what theory we use to describe the world at small scales, in the end it must reproduce the same interference pattern as classical electromagnetism for all problems where the classical theory still works.
Concerning your QFT question I think that the honest answer is "no". Fundamentally the origin of QFT is the quest of unifying quantum mechanics with special relativity i.e. formulate a lorentzcovariant version of quantum mechanics. But it is true that QFT is necessary to really understand photons - afterall photons are of a very relativistic nature! In QFT Photons arise as certain excitations of the quantum field corresponding to the electromagnetic four potential. Less technically formulated: In quantum electrodynamics you do indeed take the classical theory of electromagnetism as a starting point and then use a certain mathematical apparatus to "quantize" the electromagnetic fields in a way that transforms correctly under lorentztransformations.
A: Maxwell's equations in vacuum are equivalent to a massless relativistic wave equation for the four vector potential. This wave equation produces a wave function, which has to be interpreted in a probabilistic way. It describes, simply put, the average number of photons that you can expect in a certain location. The observed number is a poissonian distribution with this average, which for large photon numbers reduces to a Gaussian distribution with average N and standard deviation ✓N. In this sense it is a wave function like Schrödinger's or Klein-Gordon's.
A: It is not a coincidence, it is due to the fact that Maxwell equations' solutions describe classical light, and a quantized maxwell equation is used to get the wavefunction of photons. See for example this

James Clerk Maxwell unknowingly discovered a correct relativistic, quantum theory for the light quantum, forty-three years before Einstein postulated the photon's existence. In this theory, the usual Maxwell field is the quantum wave function for a single photon. When the non-operator Maxwell field of a single photon is second quantized, the standard Dirac theory of quantum optics is obtained. Recently, quantum-state tomography has been applied to experimentally determine photon wave functions.

See this blog post of "How classical fields, particles emerge from quantum theory"
