# Reality Conditions

## Saturday, April 01, 2006

### My work (1): Particle detectors and the Unruh effect

I had promised that I would post something on my work, and this is it. I decided to start with a general explanation of my area of research for nonphysicist (but patient and curious) readers, followed by a more precise explanation of my own work for the physicists among you.

The area in which I work is particle detectors in quantum field theory. What is this about? The basic entities that make up Nature (in our best current theories, setting aside General Relativity) are quantum fields. These fields, unlike classical fields, have a "complementary" (in the famously obscure words of Niels Bohr) particle description; the electromagnetic field, for example, can be described as well as a bunch of photons, which "carry the field" in discrete lumps. How are the two descriptions related? There are many ways of describing the relation, ranging from the very philosophical to the extremely mathematical and technical, some of them better for some situations and some for others. The "particle detector" approach is a simple, clear-cut, no-nonsense way of describing it: according to it, fields are the fundamental entities, and particles are defined in an operational way by the dictum "a particle is what a particle detector detects".

We start with the equation for a quantum field, which is actually the same as the equation for the corresponding classical field; for example, the Maxwell equations for the electromagnetic field. We add to the equation a small term representing a weak interaction between the field and a simple, well understood quantum system that has discrete energy levels: for example an hydrogen atom. This atom will be our "detector". We set the initial condition as the atom being in a particular energy level E0 and the field in some state [A> at some time t0. (For physicists: I apologize for this horrible way of representing a ket, and promise to try to get some math displaying package). Quantum mechanics allows us to calculate the probability that the atom would have, due to the interaction with the field, changed its energy level to E1 at some time t > t0. This change is interpreted as the atom having absorbed or emitted a "field quantum" or particle (e.g. a photon, if the field is the electromagnetic one). If E1 > E0 the atom has absorbed a particle of energy E1 - E0; if E1 < E0 it has emitted one of energy E0 - E1.

Suppose we have a very large number of identical atoms interacting with the field in this way (the technical word is "ensemble"). Then the probability of transition we have calculated is the fraction of atoms that have made the transition at time t. The time derivative of this probability is proportional to the number of atoms that are getting excited at t. (Suppose for simplicity that E0 is the lowest energy level so there is no de-excitation). We use this quantity to define the number of quanta (particles) present in the field at time t. Skipping over some proportionality constants and technicalities, the central idea is to define the answer to the question "how many particles does the field have at time t if it is in state [A>?" as being the same as the answer to the question "how many transitions per unit time are being registered in the particle detector (the ensemble of atoms) at time t when the atoms interact with with the field at state [A>?" This is a question quantum mechanics allow us to calculate an answer to.

This approach, though perhaps insufficiently fundamental and therefore insatisfactory from a philosophical point of view (after all, the atom with its discrete energy levels is taken by granted here) is very convinient in practice, gives sensible answers to many questions, and in particular can be used to shed light on an important and fascinating aspect of quantum field theory: the Unruh effect.

To explain the Unruh effect, I have to give some explanations on what are the states [A> a field can have. The lowest energy state of a quantum field is called its "vacuum" and represented by [0>; as you might have guessed, in the particle description this corresponds to "zero particles", and in the particle detector approach one can find that an atom interacting with a field in this state does not get excited (although more on this later!!) . Then there are states called [n>k, with n an integer, which are interpreted as "n particles with momentum k", and combinations of them like [n>k [m>q ("n particles with momentum k and m with momentum q"). But these states represent at the same time just certain wavelike vibrations of the field; the interpretation as particles comes (in the approach I am describing) via the interaction with particle detectors and the effect these vibrations produce on them. There are also approaches to quantum field theory in which particles are ontologically fundamental and fields are mathematical devices; whether the choice between both approaches is a matter of pragmatical convinience or has physical content is a philosophical question I shall not discuss in this post.

The relations described above between states of the field and particle content as measured by particle detectors, however, hold only when the detector is at rest or moving with uniform velocity (states physically indistinguishable according to the Special Theory of Relativity). If the detector is not moving uniformly, then a strange thing starts to happen: the detector starts to click even if the field is in the vacuum state! The detection of particles by an accelerating detector moving through a space that to an observer at rest appears as a perfect vacuum is one of the most surprising facts predicted by quantum field theory; it implies that contrary to all physical intuition, the notion of particles is relative to the observer. But what happens in the particular case of the detector moving with exactly constant acceleration is even more surprising. Bill Unruh proved in 1976 that in this case the particles detected will have a very peculiar pattern: the number of particles detected as a function of their energy is exactly that of a gas of normal, "real" particles at a certain temperature T equal to the acceleration of the detector divided by 2 Pi (in natural units). This is called the Unruh effect, and can be summarised as follows: the vacuum state of a quantum field appears to a uniformly accelerating observer as a thermal state with temperature proportional to the acceleration. For more info, see the Wikipedia entry.

I see that this post is already much too long, and I have not even started to describe my own work! (all the things I have explained are known since the mid 1970s). So I better stop here and leave the rest for one or two following posts. I just add here, to keep your interest alive, that the Unruh effect is intimately related to the famous discovery made by Stephen Hawking that black holes evaporate via a quantum radiation of particles. I will elucidate this on the following post on this topic. Meanwhile, I'll be more than happy to answer questions on the points covered on this post.

• I found article
http://arxiv.org/PS_cache/quant-ph/pdf/0401/0401170.pdf
which cleary describe Unruh effect. But I think that still clearer description exists, because thermal radiation is very elementary. Can anyone know how to get still simpler derivation?

• I think the paper you pointed to is about as simple as it can get. You might like to check the presentation using particle detectors in section 3.3 of the Birrel and Davies textbook, which is also quite elementary (although not completely rigorous).

• I do not only search simple explanation, but also thermal nature of Black Hole is something very basic, because
1.it appears also in Bekenstein's clasical explanation.
2. entropy is also element of informations.
3. Thermal nature is something elementary and statistical entropy is only help, it is not precise definer of thermal nature. (article in Physics today 2000, april, I will chech year).
4. Hawking radiation has added some other effect as recoil and dopler effect, but end result is not different from thermal radiation.
Explanation in above mentioned article is short, but it is not directly visible, why constant acceleration gives thermal specter, why not other sorts of acceleration.

Peoples search for explanations of thermal radiation with D-branes, but simpler explanation should be found also in Hawking radiation or in Unruh radiation.

P.S.1
Calculation in my above mentioned article is valid only for static black hole mass, but, rising of temperature of black hole (which emits radiation) is only approximation.
P.S.2
New article: gr-qc/9702022 v1: Black holes and thermodynamics.
regards

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