Lets start with main stream physics. Photons are elementary particles in the standard model of particle physics and have to follow the rules of special relativity and quantum mechanics.
Elementary particles in the table are point particles, and have the masses and quantum numbers given in the table in the link. The photon has mass zero, spin + or -1 projected on its direction of motion and carries energy and momentum according to the special relativity rules, they are described by four vectors:
The energy momentum four vector has a "length" :
The length of this 4-vector is the rest energy of the particle. The invariance is associated with the fact that the rest mass is the same in any inertial frame of reference.
As the photon has zero mass, and it is always traveling with velocity $c$ ,it only has energy and momentum, which have to be conserved. Quantum mechanical studies have determined that the energy of the photon is equal to $h.ν$ where $ν$ is the frequency of a possible classical light beam made up of zillions of photons. This $ν$ is not seen as a waving photon, because the photon is a point particle.
BUT a photon is also a quantum mechanical entity, and the probability of finding a photon at an (x,y,z,t) point is given by a complex number wavefunction ( a solution of a wave equation) which is related to the classical wave that can be built up by a superposition of such photons.
It is the probability of detecting a photon that "waves" i.e. displays interference patterns.
Here is an experiment with single photons at a time
. Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.
when there are only few photons they look random dots, when they become thousands the classical interference pattern of light appears for the frequency $ν$ of the laser used in the experiment.
The mathematics of both the quantum for photons and the classical for light depend on the Maxwell equations in different ways, and the classical is a superposition of the quantum states so as to give a unified beautiful whole explanation which needs mathematics to be understood.
Wave is a disturbance in particles. But in case of light in what particles is the disturbance caused?
Wave is a solution of wave equations, and classical electromagnetic waves are described very well by Maxwell's wave equations without the need of a medium. When the underlying quantum nature of photons was discovered, one can show mathematically that classical electromagnetic waves emerge from the underlying quantum level, but the proof needs studies in mathematical physics.
