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Posts Tagged ‘quantum field theory’

Let’s play a little game, hopefully illustrating something about quantum field theory, quantum gravity, and all that other good stuff. We already know that particles can be viewed as excitations of fields, which are some continuous media filling up space. One way of illustrating this is by drawing a checkers board filled with black or white squares (on second thought, most of what is below might be more intuitive to the go player). We say that the square is white when the field is not excited, and black when it is excited at some location. In other words, it costs a fixed amount of energy to paint one of the squares black (actually these turn out to be dark brown, which is some kind of sad statement about my eyesight).

Obviously in the lowest energy state all the squares on the board are white, and the excitation costing least energy is obtained by painting one square black, and that could be anywhere.  This is what we can call a single particle:

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About a couple of years ago I received, to my surprise, an invitation to be the token string theorist in the annual conference “Loops 2007”. Over the years, especially during my long term stays at the Perimeter Institute, I befriended quite a few people from the loop community. Mexico is always a pleasure to visit, and there is practically no jet lag involved, so I could just fly in and out of there, in between getting a few days to go around with my camera and enjoy a picturesque colonial town. So, I thought to myself, why not?

Of course, there was also the small issue of what to talk about. Previous such talks involved string theorists reviewing recent developments in the field, and where progress has been made. Nothing is wrong with that, but I thought I’d go a different route. Years ago Lee Smolin asked me why I don’t work on LQG (Lee is a true believer, and a direct sort of guy, I respect that), so I thought I’d use my hour to give a detailed answer. Without being overly obnoxious, of course, I would not want people having the wrong impression of us string theorists…

Naturally, one hour is not enough for such a large topic, so I had to narrow it down to cover just one major issue, and I chose to talk about background independence. Among all the fundamental differences between different approaches to quantum gravity (including for example, what does it mean to quantize gravity), this to me seems like the one needing most clarification. It is the one expression most likely to be used as a mantra, or as the one answer fitting almost any question, so basic and primary that evidently no elaboration is needed.

So, I’ve done my part now, I have given the talk, which was a real pleasure, and I have now expanded it to this writeup. In the process I have learned that writing a polemic is not as easy as it looks…I hope someone enjoys reading this, and it provokes a few useful discussions. I’d be happy to answer questions if it is evident they are formed after reading what I wrote.

Update: I guess the link to the writeup is not going to be available till tomorrow, apologies for the bad planning. In the meantime anyone interested can play the traditional lunch-time game of trying to guess what is in a paper you haven’t read. Oftentimes you discover things more interesting than the actual paper, that may well be the case here.

Update (September 23rd): I am not sure how much I would trust someone who cannot manage to submit a simple PDF file to the arXiv, but here it is finally, knock yourselves out.

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This is the third part of a series of posts introducing the basics of quantum field theory, previous installments are here (part 1) and here (part 2).

Suppose we have a continuous medium, an elastic band or the surface of a pond, we can then have traveling waves propagating in the medium. Just throw a stone into a calm pond and observe those waves. As long as the equations governing the medium are linear, we can superpose solutions and create pulses: localized bundles of energy which look a lot like particles, almost. So, it looks like we can throw away the problematic point particle, replacing it by a special kind of field, the one that obeys linear equations. Those are called linear fields, or free fields. However, we are not yet there…

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(with apologies)

When I left things off last week, we were stuck with the somewhat schizophrenic viewpoint that we have two kinds of entities in the world, particles- pointlike masses – that interact with each other through an intermediate agent, the field. Each particle is capable of distorting the field configuration, creating then a potential for other particles to feel its presence through that distortion.

The particle concept starts falling apart when considering the issue of self-interaction: does the particle itself feel the distortion in the field configuration it creates? If so, how does that influence the particle motion? It seems impossible to keep track which particle created which distortion of the field, if particles respond to the fields it seems clear they ought to respond also to the distortions they create. However, that leads to all kinds of nonsense…

There are two sets of problems that arise once trying to deal with the issue of  self interaction. First, that force the particle exerts on itself tends to be very large, infinite in fact if we believe the particle is exactly a mathematical point. Even if we consider the particle to be a very small distribution of matter, but not exactly pointlike, that force is enormous, and tends to tear that matter distribution apart. Just think about an electron as a spherical shell of negatively charged matter – what holds that shell together?  recall that charges of the same type repel each other.

Second set of problems arises if we include the self-force in the equations that determine the motion of the particle. The equation then become higher order in derivatives. Newton’s laws give rise to second order differential equations, which means that two pieces of initial data, say the initial position and velocity, specify a solution. Not so for higher order equations, those have more solutions, and need more data to select a specific solution. Moreover, some of those extra solutions are really bizarre, they have non-local and non-causal effects, clearly there is something wrong…

So, small particle do not make sense in classical mechanics. This should not very surprising, we now know that the world of the small is governed by quantum mechanics, so we probably need to quantize something to get small distribution of matter such as particles, but what do we quantize exactly?

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