"However, it’s still possible that photons have mass, and therefore (ironically) don’t move at the speed of light. That’s not quite as crazy as it sounds. As you’d expect, though, even a tiny photon mass would have important consequences for theory and experiment. Let’s list a few of them, then consider each in more detail.

While there’s still a special speed, it would no longer be the speed light travels. Technically, that would mean you could travel at the same speed a photon travels, though as a practical matter that would likely be unfeasible.

The electrical force binding atoms together would have a characteristic length scale, determined by the photon mass. Large photon mass means the force acts only over short distances, while a small mass would lead to influences over larger distance. This is what we see with the particles carrying the weak force (memorably known as the W and Z gauge bosons): they have very large masses, and the weak force consequently extends over very short distances. A massless photon, by contrast, leads to an inverse-square law, meaning that the electrical force technically extends to infinity".

https://galileospendulum.org/2013/07/26/what-if-photons-actually-have-mass/

If it is proved that the photon is not massless, then the universe will not change. Only we have to complete our theories, no more!!!

Nothing, since ``modern physics'' doesn't mean anything. Additionally, photons are massless, for reasons independent of Lorentz invariance. One shouldn't mix up Lorentz invariance-that can describe both massive and massless particles-and gauge invariance that implies that photons are massless.

There are techniques as the one due to Stuckelberg, to convert Proca theory into a gauge theory. That is to say, to have a massive photon in short ranged electrodynamics. What is important is that the Feynman propagator in this theory is also Lorentz covariant as Electrodynamics must be. Thus the massive photon is not a problem at all for considering Electrodynanamics at macroscopic or microscopic levels.

Once more-it's not Lorentz invariance that's the obstacle to having a mass term for the photon-it's gauge invariance. They are distinct symmetries.

There's no gauge invariant quantity that can describe the fact that the one-particle state, described by a free, ``massive photon'' field, is that of a free, massive, particle, in four dimensions. That's why appealing to the properties of the propagator is misleading. The photon propagator isn't a gauge invariant quantity. The statement that the photon is massless is meaningful because it has consequences for gauge-invariant quantities.

Once such a gauge invariant quantity is produced, one can ask, whether it can be identified with the value of one of the two Casimir operators of the Lorentz group and, therefore, be identified with the mass. In the usual formulation such a procedure-or any of its equivalents-does imply that the photon is massless.

For, if Lorentz invariance is, already, broken (which occurs, for instance, in a medium, that defines a privileged frame, or by boundary conditions) mass, in general, isn't a well-defined quantity, since it depends on the frame-that's why the statement that the photon, in particular, is (or isn't) massless isn't well-defined in that case at all.

Dear Stam,

You even enter in what is a massive theory as Proca's or Stuckelberg's theories. Try to solve the problem of the different degrees of freedom for the fields and you could understand better my comment.

Dear Hossein,

Let me try to answer your question explaining what are the main differences if you had a massive photon and which could lead to new Physics

1.The most direct consequence of a finite photon mass is a frequency dependence in the velocity of electromagnetic waves propagating in free space. Because of the nonzero photon mass, the dispersion produces a frequency dependence, and the group velocity will differ from the phase

velocity.

2. The exponential deviation from Coulomb’s law and its magnetic analogue in Ampere’s law provide many sensitive approaches to test for a photon rest mass in laboratory `experiments, because there is field decaying term with the distance (Yukawa potentials).

3. There is a longitudinal photon besides the two usual transverse.

4. Massive photon Electrodynamics contains a new characteristic length scale, the Compton wave length, which will lead to new physical effects over the length scale, including the Casimir effect.

Dear Daniel Baldomir

Thank you for your scientific and logical answer.

You wrote: "The most direct consequence of a finite photon mass is a frequency dependence in the velocity of electromagnetic waves..."

My answer: No, the photon velocity does not depend on frequency.

The behavior of a photon in a gravitational field shows that the photon frequency directly depends on its energy. This is also a direct result of Planck's formula E=hf.

Your approach comes from that in the standard model, the photon is an unstructured point-like particle. While the photon frequency is due to the interactions between its constituent members in the structure of photon.

Hossein Javadi, et, at., Generalization of the Dirac’s Equation and Sea, 2016

https://www.researchgate.net/publication/303988070_Generalization_of_the_Dirac's_Equation_and_Sea

When we describe the photon using its components, the Casimir effect, the Compton effect and other photon properties can be obtained directly from the photon structure.

Hossein Javadi, et, at. , Interactions Between Real and Virtual Spacetimes

https://www.researchgate.net/publication/270339919_Interactions_Between_Real_and_Virtual_Spacetimes

Hossein Javadi, et, at. , Sub quantum space and interactions properties from photon structure to fermions and bosons

https://www.researchgate.net/publication/237009789_Sub_quantum_space_and_interactions_properties_from_photon_structure_to_fermions_and_bosons

When we assume that the energy is massless, the existence and explosion of singularity with that specification (infinite density and zero volume) are also accepted. While not true.

For physicists of the early twentieth century, including Einstein and Planck and other physicists, the assumption of a massless photon help they put the physics in a way that could explain physical phenomena well. But physics has deviated from the path of reality, and nowadays there are problems that physicists are pursuing physically beyond the standard model.

Read more:

Why and how did modern physics go in the wrong way?

https://www.researchgate.net/publication/330985813_Why_and_how_did_modern_physics_go_in_the_wrong_way

The Greatest Unsolved Mystery in Physics

https://www.researchgate.net/publication/330366766_The_Greatest_Unsolved_Mystery_in_Physics

If we accept that energy has mass, then many physical problems, such as particles masses (eg, electron mass) will be solved. In addition, the properties of fermions can be found in photon structures.

Sub quantum space and interactions properties from photon structure to fermions and bosons

https://www.researchgate.net/publication/237009789_Sub_quantum_space_and_interactions_properties_from_photon_structure_to_fermions_and_bosons

Generalization of the Dirac’s Equation and Sea, 2016

https://www.researchgate.net/publication/303988070_Generalization_of_the_Dirac's_Equation_and_Sea

This approach guides us to describe the hometown and mechanism of production of fields and fundamental forces.

What is the Universe made of? And how it works? From the point of view of CPH theory

https://www.researchgate.net/publication/330853751_What_is_the_Universe_made_of_And_how_it_works_From_the_point_of_view_of_CPH_theory

Why and how did modern physics go in the wrong way?

https://www.researchgate.net/publication/330985813_Why_and_how_did_modern_physics_go_in_the_wrong_way