The property of a particle is that it is discrete. But a particle can be discrete in many ways. It can be discrete from the viewpoint of energy; or it can be discrete from the viewpoint of space. A particle of energy does not necessarily imply that it is also a particle of space. A particle of energy can very well be continuous spatially.

The current idea of particles in Physics comes from quantum interaction that was first observed in black body radiation.

Einstein then came up with the idea of a “light particle” or photon from the discreteness of energy interactions as observed in black body radiation. Thus, the idea of quantum was born as “a discrete chunk of energy”.

But then, simultaneously, Einstein proposed his theory of special relativity that sees light as a continuous field of waves. Does this pose a contradiction?

No contradiction arises if the light wave is continuous in space but appears discrete only when interacting with matter. This raises the question, “Does discreteness of energy interactions mean that there must also be a discreteness in space?”

We seem to have an energy particle here. Do we then assume that such particles exist in space as discrete chunks? Can an energy particle remain continuous in space as a wave?

According to Disturbance Theory, Space is like a field of zero frequency. It may be compared to undisturbed surface of a pond.

When space is disturbed it splits into electrical and magnetic fields. These two fields seem to transmute back and forth at a frequency. This forms the disturbance.

This disturbance is transmitted to other parts of space as threedimensional waves. This is similar to the waves on the surface of a pond when it is disturbed.

The waveinterference in the doubleslit experiment is a spatial phenomenon and not an energy phenomenon. We are dealing here with a spatial wave and not with spatial particles.

Quantum is a particle of energy and not a particle of space. It seems to be continuous spatially.
NOTE: We are talking about photon here. The case of electron is a little more complex but the same argument is applicable.
Comments
Wikipedia: “A photon is an elementary particle, the quantum of light and all other forms of electromagnetic radiation. It is the force carrier for the electromagnetic force, even when static via virtual photons.”
According to the following argument no virtual photons are necessary.
(1) The electric field exerts a force on other electrically charged objects.
(2) The magnetic field exerts a force on other magnetized objects.
(3) A changing electric field produces a magnetic field of force.
(4) A changing magnetic field produces an electric field of force.
(5) Force is generated by change in electrical and magnetic fields relative to each other.
(6) The changes in fields spread through space like threedimensional waves.
(7) Force can then influence other things by bringing change in them.
(8) Force is thus carried by the electromagnetic field and not by photons.
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Wikipedia: “Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of waves and of particles. For example, a single photon may be refracted by a lens or exhibit wave interference with itself, but also act as a particle giving a definite result when its position is measured.”
(1) The duality is in terms “space wave – energy particle”.
(2) Photons are discrete as chunks of energy. Photons are not discrete in space.
(3) Detecting a proton in space requires an energy interaction with a detector, so we are not really measuring a photon as a particle in space.
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I see space as a “theoretical field” of zero frequency. It is analogous to “undisturbed surface of a pond”.To me the Higg’s field is this “theoretical field” of zero frequency.
When space is disturbed it splits into electrical and magnetic fields. These two fields seem to transmute back and forth at a frequency. This forms the disturbance.
This is analogous to disturbance in the surface of a pond, which splits into peaks and valleys. The peaks and valleys transmute back and forth at a frequency.
This disturbance is transmitted to other parts of space as threedimensional waves. This is analogous to twodimensional waves on the surface of the pond.
I see electromagnetic waves as this primary disturbance. Photon is discrete only in terms of energy. It is not discrete in terms of space. So it is a wave in space. It is discrete only in terms of energy when it interacts with other fields.
Photon is first degree of disturbance. Higher degrees of disturbances will be “disturbances within disturbance”. Those will be more complex fields. All fundamental particles with mass shall belong to these higher degrees of fields.
Right now I am focusing on photon, because it seems to be the simplest of all particles.
I want to encourage this direction of research. The study of space will be fruitful. Paradoxes may unravel.
You are doing a good job on this.