Now that I've done some maths on it, the lost mass into too much of a problem. To accelerate a space ship you have to throw stuff out the back. More precisely, you have to throw stuff with momentum out the back and, to get maximum thrust from minimum expelled material, you want to throw it out as fast as possible.
So this engine which hasn't been invented yet expels matter at near light speed and the momentum of each particle is, as near as damn it, given by P = E/c where E is its energy. To put it another way, you are throwing out an amount of energy E = cP and, to produce that energy, you must consume an amount of mass m = P/c.
To accelerate each kilogram of your space ship at a modest 1 m/s/s, you have to expel 1kg.m/s of momentum each second and, whichever way you try to do it, you will use up mass at the rate of 1/c kg/s. That doesn't look like a lot but it will take you 100,000,000 seconds to get up to your target velocity of c/3 - so that's one third of your mass gone.
Actually it's not that bad because, as you point out, you need less thrust as the mass goes down. In theory you can get your ship up to c/3 with the loss of less than a third of your initial mass. I also forgot that, once up to cruising speed, you can switch the engine off and drift through interstellar space with only minimal fuel consumption for life support and the occasional directional nudge.
So far so good but here comes the real snag; what are you using for fuel?
My initial assumption was hydrogen but it's no good. You need to convert all the mass in your fuel into energy but nuclear fusion as we know it leaves you with a lot of dead weight in helium. It makes no difference whether you keep it in the ship, throw it out the back or let it spill into space. The hard fact is that only 0.8% of your hydrogen mass can be converted. Instead of converting a third of your initial mass, you can only use about 0.3% of it. With luck, that will get you up to a pathetic 1,000,000 m/s - and so the nearest star is not 12 years away but 1200!
The answer, as every trekkie knows, is to use fuel that converts entirely into energy, namely anti-matter which will most likely be in the form of anti-protons. These have been made in small numbers at CERN but you'll need them by the ton and of course you'll need a reliable container too. It looks like we won't be heading off to Proxima Centauri quite yet.
