Can the technology behind particle accelerators be used for space propulsion? As I understand, the kinetic energy of the proton beam in a hadron collider is quite large. Can you build a space propulsion system that is based on accelerating a proton bean to relativistic speeds and then using the resulting kinetic energy to propel a space vehicle? 
Edit: In the large hadron collider, the kinetic energy of a proton reaches 7 Tev, which translate to about 6.71E+20 joules of kinetic energy per kg of accelerated protons.
If the Collider can accelerate 1 kg of proton every hour and point it downwards, then the resulting thrust could launch into orbit the entire collider structure.
Of course, the LHC cant accelerate 1 kg of protons in an hour, but maybe a derivative of it could and would be the basis of space propulsion system.
This is basically the idea. I know that it is enormously hard to build, but can it be done, in theory?
 A: It would be an extremely cumbersome and inefficient  way to do it.
Already one uses the acceleration  of ions in ion propulsion systems in space:

An ion thruster is a form of electric propulsion used for spacecraft propulsion that creates thrust by accelerating ions. Ion thrusters are categorized by how they accelerate the ions, using either electrostatic or electromagnetic force. Electrostatic ion thrusters use the Coulomb force and accelerate the ions in the direction of the electric field. Electromagnetic ion thrusters use the Lorentz force to accelerate the ions. The term "ion thruster" by itself usually denotes the electrostatic or gridded ion thrusters.[citation needed]

Reply to the edit:

Of course, the LHC cant accelerate 1 kg of protons in an hour, but maybe a derivative of it could and would be the basis of space propulsion system.

Contemplate the LHC system, http://en.wikipedia.org/wiki/LHC . The large circle is 27kimometers diameter. Thousands of magnets and cryogenic support give us a few horsepower in energy. How can this be amplified to the science fiction numbers you propose? Scientists are not crazy to use kilometers of land and enormous power were they able to get the same energy result in miniature. 
The technology yes, as the link shows, is usable, but multiplying the energies by enormous factors does not belong to the present technology or available energies. In addition how would one produce megawatts in space? If one can, one does not need an intermediate wasteful step of such magnitude.
A: Your 1kg beam carries 6E20J of energy.  That's your power output (and therefore it is also a lower bound for your power input too!).  What's your output (thrust, in this case)?
Three considerations relevant to reaction drives:


*

*The thrust you produce, is proportional to the momentum of the stream you accelerate (i.e. $m v$)

*the energy needed to produce this thrust, is related to the k.e. you apply to the stream ($ \frac{1}{2}mv^2$

*if you carry your reaction mass onboard and if it is a significant fraction of your overall mass, you'll also want to consider the tradeoff of propellant mass consumption vs. energy consumption


Driving particles at relativistic speeds will entail a larger power consumption per unit of thrust generated, than driving a more massive exhaust stream at a lower speed.
A: Yes, that's an excellent idea.
Propellant is mandatory for acceleration due to laws prohibiting violating the conservation of momentum.
So how to get to Alpha Centauri if you have to bring along enormous amounts of propellant? Even with an infinite energy source with a mass of only 1 gram, you need so much propellant that you can't build the ship because we only have 1 sun we can use for propellant.
Of course, if we can accelerate our propellant to ridiculous speeds, then we can use the propellant we have more efficiently. In fact, close to the speed of light, we can bring this efficiency up towards infinity, lowering the required amount of propellant down towards zero. After all, a mass at the speed of light has infinite energy and momentum (Einstein's words, not mine).
Now all we need is an energy source with low mass to energy ratio (limited by E = mc2) and off we go. Alternatively, we can grab energy from elsewhere, such as a laser feeding the energy to the ship from a home base.
Sure, that the thing is enormous, that it doesn't solve the energy source issue, and that output capacity is currently a bit too limited are all valid issues, but these issues do not detract from the fact that thrust requires propellant, that thrust = ejected_propellant_mass * propellant_velocity, acceleration = thrust / mass_to_accelerate and that the propellant is part of the mass we need to accelerate, thereby being limiting factor. The only thing we can do to fix it is to increase the propellant velocity. Ion drives may work as well. The point is to eject propellant at relativistic speeds. The closer to c, the more efficient and the (exponentially) less propellant needed.
Due to Einstein's cool relativity things we can reach arbitrary ship acceleration with any fixed amount of propellant which is awesome and eliminates the need to carry several stars in the trunk for propellant.
A: It could actually be easy to build; and it could be a small, lightweight spacecraft, too.  We'd have to make some concessions on your design though.
New technology like this single computer chip can be used to accelerate a particle to 0.94c .  The entire structure of 1000 chips would be about 1 inch long.
You could use this in a micro-spaceship, or duplicate it many times for larger craft.
However, it would be difficult to create something that could launch from Earth from this.  More likely, it would be deployed from a traditional rocket in orbit.
It might even be able to scoop the reaction mass (hydrogen particles) from space (ramjet style), meaning that it would not need to carry any fuel.
