## Thursday, February 17, 2011

### Pointing in space and time or why one needs four numbers for a photon

In the previous discussion it was described how photons are described by fields, and that the fields are somehow like the surface of an ocean. The truth is, unfortunately a bit more complex. This can already be seen from the magnetic field. If you have a magnet, you cannot only feel its field in the same plane as where the magnet is, but also above and below it. Thus, the field is something which not only is like the surface of an ocean, but which is more like the ocean itself, it is above and below and all around. Well, this is not yet a problem, since one can imagine that, say, a subsurface explosion also can make a wave which has volume, and the analogy is only a bit more harder to imagine because of the third direction.

But things become still a bit more messy. Take the magnet and take a pretty hot flame, and place it under the magnet, not too close. If you now measure the magnetic field at some point in the space surrounding the magnet, you will notice that the magnetic field decreases over time. That is because when you heat a magnet sufficiently (a couple of hundred degrees), it will loose its magnetic properties. Thus, the field is not static, it changes with time, and can even vanish. Of course, you could have noticed the same feature by just moving the magnet far away, but then you could bring it back again. Thus, a field is something that tells something about a direction and a strength at some point in space and time.

But these seems a bit odd. To identify a position, you need four numbers, four coordinates. But the direction of the magnetic field you can enumerate with just three, two for the direction, one for its strength. There is nothing like a time direction to the magnetic field. Indeed, electric and magnetic fields are peculiar in this sense. As said before, they can be derived from a quantity which had four numbers, as the four coordinates just needed to characterize the evolution of the magnetic field. It is about time to tell what the four numbers are.

Indeed, it turns out that a field which describes a particle has four components, each of which depends on the space-time point one is looking at. So what is this fourth number? In a sense it is the direction of the field in time. That sounds a bit peculiar, and in fact it is. The reason for this is the arena in which physics takes place.

If one goes back to ones experience of reality, then there is the space with its three dimensions, and there is time, which appears to be just flowing along in the background. But in fact space and time are connected, and are not two independent entities. That has been an observation which has actually been made very early on in physics. However, it took a while to note that the structure is peculiar, but this will be discussed at a different time.

Again, it helps to make an analogy. Take a flat cylinder. Put in the cylinder a disc, which fits perfectly in it. Now, if you elevate the disc at a constant rate than everything on the disc can move freely on the disc, but there is a constant change in height, just as time changes constantly. In our world, the disc has one dimension more, and the changing height is the changing time, but otherwise it is the same concept. Somebody on the disc could even measure time by measuring height, because it is lifted constantly.

Now, of course, it is possible to give a direction which is entirely on the disc. But for us, which can see the cylinder as a whole, we can also give a direction which points upwards or downwards from the disc. In contrast to someone living on the disc, we need one quantity more to specify a direction. But if someone on the disc is very clever, he will notice that his space is larger, and then she can invent, at least as a mathematical concept, a direction off the disc, which will agree with our idea of direction. However, since she only knows the disc she has no intuition of what means 'off the disc', but has a mathematical grasp of it.

And so it is the case for us with time. We can mathematical describe our cylinder (though it actually looks very much different from a cylinder), and we can describe a direction off our three-dimensional world by giving it a direction in both time and space. Then, we notice that the field that describes a particle is actually requiring to have such an additional direction, and this is the reason why the photon field has four numbers at every space-time point: a magnitude and a direction in space and time. And the electric and magnetic field with only a direction in space are something like shadows of this object in time and space in a purely spatial world, in which we can move freely.

Of course, these four numbers are not independent, but this is because of the symmetry. Without the symmetry, they would be. The symmetry is something additional, and has nothing to do with space and time.