Particle Physics Question

What is the highest possible velocity a particle can be accelerated to?


Before anyone says "Speed of Light in a vacuum," "c," or anything related, let me point out why I ask this question:

1)  Einstein's Theories of General and Special Relativity both say that no particle with a rest mass >0 can be accelerated to the speed of light in a vacuum.

2)  As energy is added to a particle, it's velocity, as well as it's mass, increases.  Up until roughly 70.7% of c, the velocity of the particle increases faster than the particle's mass.  After 70.&% of c, the opposite occurs.  This is seen in the Lorentz equation.

3)  So, hypothetically, an infinite amount of energy can be added to a particle (this would increase the particle's mass to infinity, as a consequence).

4)  The universe does not have an infinite energy supply, so there is a definite limit to how much energy a particle can carry.

Now if we take points 1, 2, and 3, we have a problem.  When E = inf, then the particle's velocity would be c (the .999999...=1 proof that any decent algebra student knows).  So, even in a universe that had an infinite energy supply, relativity would break down.

If we take all four points, then there is still a limit (a limit that is close to c, but is still less than c).

Now I know that there is the GZK limit that places a theoretical limit on how much energy a particle can carry (which direct observation has proven the calculations of the GZK limit to be wrong), but even when I look for values of the GZK limit, there are none to be found.  All I get is information about pions and other useless shit like that.

So my question is:  What is the highest velocity/energy a particle can achieve? or What is the highest lorentz factor a particle can have?

Let's face it, there has to be a limit.  And if there's a limit, then why are we pussy-footing around with inferior particle colliders when we could build the ultimate biggest and baddest collider possible?

Even if the universe had an infinite amount of energy, it would take an infinite amount of time to accelerate something to the speed of light (from the observer reference frame).  Thus, nothing special happens.

There are several problems with "energy of the universe"
a) It is not well defined, unless the metric has a time light killing vector field (energy only makes sense if it's conserved/independent of time)
b) Evidence suggests that the universe is flat, in which case it is infinite in size.

The idea of a changing mass (as a function of velocity) makes me queasy.  Almost all modern treatments of the subject just have the momentum equal to an invariant rest mass * gamma factors.

Ignoring all that, and assuming a finite amount of energy in a fixed Minkowski space, I'm pretty sure you can achieve optimal results by using half the available energy to accelerate the particle, and the other half to accelerate yourself in the other direction.

As for the actual limit, I have no idea how much energy is available.

>>3
Standard quantum mechanics is not a relativistic theory.  I have not learned QFT yet, so there's little I can say.