Massless Particles: why they must move at the speed of light

Relativistic-Quantum approach.

From the very beginning of my courses of Physics as an hobbyist, I was wondering why a particle of mass = 0 has to move to the speed of light. Why cannot it move at a lower speed?
Finally after reviewing Professor Barton Zwiebach’s first lessons of his course in the 2016 about Quantum Mechanics (available on OpenCourseWare site) and being inspired by it, I think I have found the answer

First I started from the relativistic energy equation:

and if m0= 0, then E = pc.

For simplicity I will limit this reasoning to the case of a particle moving in one dimension. For any particle moving freely (and this is the case, because no force can act on a massless particle) the wave function associated can be expressed as the sum of momentum eigenstates, each of type:

where I ignore the complex amplitude for simplicity and k and ω are associated with momentum and energy respectively by the well known relationships:

Which is the speed of this component? It’s as follows:

This is true independently of k and ω, so the entire wave packet associated to the particle moves with speed c.

Also reasoning in terms of group velocity, we come obviously to the same conclusions:

Pure relativistic approach

This is the most simple approach. The energy of such a particle would be:
E = pc
If we suppose that it has a speed v < c, then we could choose a reference frame in which the particle has speed zero. So in that frame its momentum would be null and consequently its energy too (because E=pc). In that reference frame that particle would not exist! But how can a particle not exist in a frame and exist instead in another? It’s not possible.
Equivalently if a particle exists for an observer it exists for any other.

An intuitive approach.

Suppose that a particle with a very very small mass appears in the space at an instant; the force needed to accelerate so mutch to take it at a velocity next to speed of light must not be so great. Obviously there is the limit of speed of light and, as its speed approaches that of light, its relativistic mass increases so that it cannot overcome it.
But if we choose a mass gradually lower and approaching zero, the force necessary to bring it ever closer to the speed of light, allowing it to maintain a relativistic mass however small, is gradually less and less.
The same considerations can be applied to the interval of time during which the force is present.
So, if the mass of the particle approaches zero, the force required to push it at speeds close to that of light and the time interval in which this force must be present, go down and approach zero themselves.

We know that in the vacuum, as predicted by Quantum Mechanics, particles and antiparticles appear continuously, so they would be what is needed to boost our particle with an infinitesimal mass!

The objection to this reasoning is that a particle of zero mass could not be pushed by any force. My answer is: probably Nature works in a continuous way…