the unruh effect

This post is intended to introduce in a popular way the effect from quantum field theory to those, who are not even familiar with theoretical physics. I will try to demonstrate the physical beauty of the so-called Unruh effect and will try to give an idea how this effect is related to, say, the Sokolov-Ternov effect from quantum electrodynamics and the Hawking radiation of black holes.

So firstly, as always, I’ll need to do a small off-topic to make several important notes about related topics, that I’ll use later.

What is vacuum? I mean, of course, there’s no exact definition of this sort of environment. But what do you usually imply with this word. Does ‘vacuum’ mean the media with nothing in it? If so, what do you call at least something? Of course, even if you take a space far away from interstellar plasma of our solar system, the dust of our galaxy etc., you’ll still have an extremely tiny number of particles, i.e., photons of cosmic microwave background, that make your TV screen look ‘snowy’ when you grab off the cable. Ok, but what if we somehow separate some space far-far away from all the sources of particles, so that no particle can enter our chamber. Will that be a vacuum? Well, not really. You see, by definition from the quantum field theory, particles, which everything in our Universe is made of, are nothing but some sort of excitations of a strange physical phenomena called ‘field’. For example, the electron and positron are the excitations of the Dirac field; photons are the excitations of the electromagnetic field, etc. To understand this, one can imagine a chessboard with paper number notes on each square. Once all the numbers are the same, this is called the ground state of our field, or the unexcited state. If you change the number on, say, E5 square, you’ll have an excitation of the field, hence the particle. Minutephysics has a 2m video on this, which I strongly advice to watch before moving forward.

The possible ground states, particle creation, annihilation and interaction, their evolution and so on, are described with one simple function called the Lagrangian of the field and the principle of the least action.

The important thing here is, that all the theories made are only working on a finite segment of time and space, the lower boundaries are given by the so-called Planck time ( \(5.4\times10^{-44}\) sec) and Plank length ( \(1.6\times10^{-35}\) m). With the help of quantum field theory, one cannot tell exactly what’s going on on time scales less than Planck time, and spatial scales less than the Planck length. So it’s considered, that there is a sort of uncertainty in the form of fluctuations of the field appearing on that scales, which is shown on a picture above.

Coming back to our ‘vacuum’ chamber, if even we prevent any outer particle from entering inside, the quantum fluctuations of a field on Planck scales will always create particles from the seeming vacuum. Of course, the conservation laws should be taken into account in these processes, so the charge, momentum and lepton number is being conserved. Some of you may be familiar with the electron-positron creation and annihilation during the infinitesimal fluctuations of the Dirac field on Planck scales (the charges and lepton numbers are opposite, so the conservation laws hold). The thing is that these fluctuations are so weak and the time scales are so small, that normally we cannot detect them.

However, there were carried out a bunch of experiments with lasers and cold atoms intended to calculate the interaction of the bound electrons in atom with the electrons and positrons from of vacuum. This effect is well known as the polarization of the vacuum, but it is another sort of story.

Now we’ve understood that normally, our usual vacuum in real is simply boiling with quantum fluctuations everywhere in our Universe, creating and annihilating particles, but on the average this effects have a random nature. These infinitesimal fluctuations play a significant role in the atomic physics, as they serve as some sort vacuum media, exciting the bound electrons to spontaneously decrease energy state in an atom (if there is a free level, of course).

Another important topic to discuss, before going forward to our main subject of interest, is the temperature. What is the temperature? From the high school you may probably remember, that the temperature is just an easy measure of the particles’ velocity in some gas or anything. Or the measure of the intensity of atoms’ jiggle in a solid body. The higher the temperature, the more harshly atoms jiggle and hence make the neighboring atoms jiggle also. Well, this concept is quite familiar to all of us, but I want to tell you, it is а little bit perverted interpretation of the temperature. For example, what is the temperature of a fluid with all the particles moving in the same direction with the same speed? Those who know a little bit of school physics may say, well it’s about the formula \(3/2 kT=mv^2/2\) .

So we can calculate the \(T\) from here. And that’s wrong, of course! No matter what the velocity of fluid particles is, but the temperature of this system is not defined. You see, the temperature itself is just a parameter from the so-called Maxwell-Boltzmann distribution of particles, or in a more general way, the Gibbs distribution from statistical mechanics. The formula above is only applicable, if the particles have a Maxwell Boltzmann distribution (with 3 degrees of freedom).

And below is the formula and the plot representing the Maxwell-Boltzmann distribution. As you can see, the temperature is just a parameter in the exponential term, indicating the position of the distribution maximum \(n_{\epsilon}\propto \exp^{-\epsilon/kT}\) .

Speaking more physically, the temperature is a factor showing how much work dA you should do in the adiabatic process ( \(\mathrm{d}Q=0\) , without energy inflow or outflow) to increase the entropy by \(\mathrm{d}S\) , so that \(T\mathrm{d}S = -\mathrm{d}A\) . The most important keyword in all this stuff is the equilibrium state of our system, which is strongly required for the distribution and all the following parameters like temperature, pressure etc. to be defined.

The distribution of an ideal gas, without any sort of interaction between particles, in the potential well is called the Maxwell-Boltzmann distribution. However, in the real world on low energies, hence temperatures, quantum effects start playing significant role and the spin-statistics is required to be considered. So the fermions (spin-half particles) are considered to have a Fermi-Dirac distribution, and the bosons (integer-spin particles) have a Bose-Einstein distribution. What we are interested in, is the so-called Planck distribution, which is the Bose-Einstein distribution for photons. Yes, photons also can be in the equilibrium state and can also be represented as a photon gas with a given temperature and energy distribution^[1]. The plot and the formula for this distribution (with the comparison to Maxwell-Boltzmann distribution for energy) is given below. One can see, that for extremely high temperatures, the 1 in the denominator can be neglected with only the exponential term leaving, which is exactly the Maxwell-Boltzmann distribution.

So concluding, the temperature is simply a parameter of an energy distribution of our system in a thermal equilibrium. For Bose-Einstein (or Planck) distribution it shows how spread are the particles from the zero energy state. The larger the temperature of the photon gas, the more particles appear in high energy levels.

We’ve reviewed the most important moments from general physics that will be required further in our discussion. In the second part of my post, I’ll introduce the main topic of our interest, the Unruh effect.

[1] So is the cosmic microwave background, which is calculated to have a very accurate Planck-spectra with a characteristic temperature of \(2.7\) K.↩