How does stuff scatter in de Sitter space? Its the next most important question to ask. Even though we have seen that we don’t have a great notion of a particle “coming in” and a particle “going out” we can still try our best to answer this question.
In physics, “scattering” problems, intuitively, refer to a class of problems where some particles come in from a long distance away where they cannot affect each other, approach each other where they get pretty close, interact strongly for some short period of time, then leave again, going their separate ways. Scattering problems are of great importance in quantum field theory, as most physical experiments are exactly that: scatter problems. In the LHC, particles are rammed into each other at very high speeds, interact, then leave. Of course, the particles that leave are not the same particles that enter. In the LHC we have no way to see ‘how’ the particles are interacting (and indeed, quantum mechanically, this is a very poorly defined notion anyway). All we can see is what goes out and what goes in.
So particles “going in” and “coming out” are not particularly well defined in de Sitter space, but that shouldn’t stop us cold in our tracks.
Just like in the last post, when I had to set up a calculation in de Sitter space based off of what I knew about flat space, here too I tried to come up with some analog of the “LSZ reduction formula” for de Sitter space. What is the LSZ reduction formula? Well, in quantum field theory, you don’t really start out with a notion for particles. You start out with a notion for fields, as I talked about in an early post. You can calculate these “correlation functions” that tell you what the field in one point of space has to do with the field in another point in space. The LSZ reduction formula tells you how to use these fundamentally important correlation functions to calculate scattering probabilities.
And indeed, that’s what I did. First I used what I knew (the derivation of the LSZ reduction formula) and some very slight guiding physical intuition to start finding an analog of it in de Sitter space.
Actually, the first time I got it wrong. My adviser could tell that my answers were wrong because they admitted oscillatory solutions for late times, when solutions should have been exponentially decreasing as we already discussed on this blog. Sort of interesting how people who do this stuff for a long time can use their hawk eyes to quickly tell if an equation is right or wrong…