Reality Conditions

Sunday, April 02, 2006

My work (2): the technical details

This second post on my work is addressed primarly to readers with a background in physics. Read it at your own risk…

The transition rate at proper time t of a particle detector moving along the trajectory x(t) can be calculated by integrating over the detector’s trajectory the correlation function or Wightman two-point function of the field W(x(t), x(t-s)), times an oscillating function exp^(-i E s), where E is the energy gap of the transition. (t is proper time "tau", not coordinate time; it's just that Greek letters are not displayed). This integral done naively gives an infinite answer, as so many do in quantum field theory; the coincidence limit of the two-point function needs to be regularized to get a finite answer. The traditional way of doing this is adding a small imaginary part to time and the taking it to zero at the end of the calculation. However this procedure for regularization has been shown to give wrong answers in this paper by S. Schlicht. A different regularization procedure is introduced there, that uses a rigid spatially extended detector and "smeares" the field over it, taking the limit of a point-like detector at the end. This procedure, unlike the traditional one, preserves Lorentz covariance at each stage.

In Schlicht's paper a specific Lorentzian profile function is used to model the spatially extended detector. In my work I have adressed the question of whether the results obtained by this kind of regularization procedure are at the end independent of the profile function used or not. I have found that, under not very restrictive assumptions, they are indeed independent and there is a single expression for calculating the transition rate of a particle detector regularized with any profile function, and moving in any trajectory. (Perhaps I should say "we have found" because my supervisor Jorma Louko has played a very helpful role in the proof of this result). Our formula applies in particular to the uniformly accelerated case, in which it recovers the thermal result found by Unruh. It also applies to many "nonstationary" trajectories which can have arbitrarily varying acceleration, leaving out only a few trajectories that are
"too pathological" in a special sense that we define. In summary, we now have a crisp, simple expression for calculating how many particles from a quantum field in its vacuum state will be detected by an observer moving in an almost arbitrary way (in flat space; one of our main future goals is to extend this to curved spaces in the future).

And this is, more or less, what I shall say to the audience in the BritGrav 6 meeting this Wednesday, and what you shall be able to read in full detail when we have completed our paper on this subject, which hopefully shouldn’t take more than a few weeks from now.

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