Tuesday, June 24, 2008

Relativity and quantum mechanics

In my previous post on this topic I have shown that while faster that light travel is impossible, it is possible for the electrons to move from one energy level to another in zero time. The distance it travels is like the distance between planets on our scale, however the time is zero.

The reason for such a result is very simple - relativity and quantum mechanics cannot be used together. It is not possible to apply relativity where one should use quantum mechanics. When we discussed individual electrons the relativity theory simply stopped working. The results that were correct for large scale become wrong on this scale.

But why is it so? There is after all a general agreement that a theory that works only under specific conditions should transform gradually to a different theory when the conditions it requires are changed. This part probably sounds a bit confusing, so here is a simple example:

Photo by wili_hybrid

A long time before Einstein, people noticed that for two system that are moving with a constant speed compared to each other, the system of coordinates has to be transformed when you move from one system to another. If, for example, you are on a train that moves with 50 kph east relative to the earth and you see someone who is sitting on the field outside, than from his system of coordinates you are moving with speed 50 kph east, but from your point of view he is moving with the same speed to the west. The transformation used to move from one system to another is the Galileo transformation. If your coordinates in one system are (x,y,z,t) than your coordinates in a system moving away from you with a constant speed v are (x',y',z',t'). If at the moment t=o the both observers where in the same place and the movement is only on the x axis we get that:

x'=x-vt
y'=y
z'=z
t'=t

However, according to relativity this is not correct when v is big enough. In relativity we use Laplace transformation instead of Galileo's. Under the same condition we will get:

$x'=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(x-\frac{v}{c}t)$

y'=y
z'=z

$t'=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(t-\frac{v}{c}x)$

The formulas look very different. Partially this is because the of units used. However, if we will go to the limit were c is significantly large than v (that is c is regarded as infinite), they will turn into the Galileo transformation. It is very easy to see - the only impostarnt part is to notice that the units need to be balanced, after this it is trivial. We can say therefore that relativity turns into classical mechanics when the speeds are low in comparison to the speed of light.

However, this doesn't happen with quantum mechanics. It is divided from relativity by a scale barrier, and when this barrier is approached the two theories start to contradict each other. A lot of work have been done to solve this problem. The main approach is to try to unify all the basic forces. Those forces are - Electricity, Magnetism, Gravitation, Strong and Weak. The first two are already unified for a lot of time. The weak force also can be unified with them. I also heard that the strong force was unified with the weak force, but I don't know the details. Gravitation is a problem however. For the other forces particle carriers where found - but not for gravitation. In fact, the question what gravitation really is, is still without answer. It is a mystery waiting to be solved...

A bit of trivia - It is a surprising fact that Einstein contributed a lot to both of these theories, but while he helped quantum mechanics to take roots he wasn't happy with the result. He was the one who proposed the duality of the photon, and he was the one who helped to promote the understanding that all particles have this duality.

I ended my previous post with a question - Was Einstein wrong? The answer should be clear, but I will say it anyway. He wasn't wrong. It is simply that by going to this very small scale I left the domain of relativity and there the rules are different.

You probably noticed that this post raises a very interesting question. Since faster than light travel is equivalent to time travel as I have shown, does the fact that the electron can move such great distance (on his scale) in zero time means time travel is possible? Nope. There is no time travel in this case. The reason for this is simple, but it requires getting used to. The electron belongs to the "quantum world". We can think about this world as being separated from our world by a "shield". This shield is called The Heisenberg Uncertainty principal. What it says is very simple - the uncertainty in the location and energy are always bigger than some constant number. It means that we simply cannot see too well what is going on in this "quantum world".
In fact the way I used to show that faster than light travel is equivalent to time travel depends on accurate measuring of distance and time. Since we cannot do this with the electrons, even this general result just doesn't apply.

By the way, if you want to be remembered forever in the history of science, finding a way to unify electricity and gravitation will surely achieve this goal...