Last time in my post I introduced two basic ideas that are strongly required to know in order to understand my further review. Firstly, we understood that our ordinary concept of vacuum is not quite correct, as basic vacuum is full of pairs of creating and annihilating particles (likewise to electron-positron pairs) on extremely tiny scales, caused by the quantum fluctuations of some hypothetic field. Secondly, it was important to grasp that the temperature is a statistical characteristic of a distribution, not a property of a single particle. And actually the temperature shows how far particles can get from the zero energy level.
Now we’re ready to turn directly to the Unruh effect. But first, we need to discuss how we’re going to detect the affect of this phenomena. Further, you will understand how important it is.
Detector
Imagine, you’re an experimenter and you have to think of some hypothetic physical system to describe the properties of vacuum. You could take, for example, a hydrogen atom and watch the interaction of creating and annihilating particles with the bound electrons in the atom. But, as we are hypothetic experimenters, in order to avoid complicated calculations with atom energy levels and wave functions, spin-spin interactions etc., we will simply take a double-level system. It is a widely used demonstrative system (detector) that consists of $$N$$ particles and two possible energy states. On the picture below we have a $$E_2$$ and $$E_1$$ levels (let $$E_2>E_1$$). The number of particles on each energy state are $$N_1$$ and $$N_2$$. The probabilities for a particle to spontaneously get from $$E_1$$ to $$E_2$$ is $$W_{12}$$, and for the opposite process - $$W_{21}$$.
To understand how this detector works, let’s consider a simple vacuum with no particles and no quantum fluctuations. In this case, the media can be reviewed as being in a thermal equilibrium with $$0$$ temperature. So, as our detector should be in a thermal equilibrium with the media (vacuum), thus, we got a $$0$$ temperature for particles in this system either, and all $$N$$ particles are on a $$E_1$$ energy level. If we raise the temperature of a vacuum (by pouring in some other particles or turning the fields on) we will get a thermal equilibrium with a nonzero temperature, hence the particles in the detector will have a nontrivial distribution (specifically Maxwell-Boltzmann distribution if quantum interactions between particles are neglected). So in this case most of them will be on the level $$E_1$$, but a tiny amount (less by the factor $$e^{-(E_2-E_1)/T}$$ will be on the level with energy $$E_2$$. The media with a constant temperature is called a thermostat. Placed in thermostat, any physical object with thermodynamic properties (like our double-level detector) will get a distribution with the same temperature as thermostat.
In the physical vacuum with random quantum fluctuations we have a thermostat with zero temperature, and with a nonzero probability for a particle of our detector to jump from $$E_1$$ to $$E_2$$, or backwards. But however, the majority of particles will still be on the 0-th level. There will be completely no distribution, so the population (or polarization) which is $$P=(n_+-n_-)⁄(n_++n_-)$$ ($$n_+$$ is the number of particles on the upper level $$E_2$$, $$n_-$$ - on the $$E_1$$ level) will be $$1$$, as the probabilities $$W_{21}=W_{12}$$.
[Note for geeks] In real physical calculations the scalar field is considered with a linear interaction Hamiltonian in the form $$H_{\rm int} = \mu \varphi$$, where $$\mu$$ is the moment of our detector. Just in sake of simplicity.
If we now consider a moving detector with a constant speed, we will see that the population doesn’t change, so nothing like a distribution appears. And that is pretty intuitive, because from school physics we do know, that the equivalence principle tells us that all the physical processes are the same in relatively inertial reference frames. The reference frame, where the detector is in the rest, and the one where it is moving with a constant velocity are relatively inertial, so nothing’s wrong here.
[Note for geeks] Of course, the transition to another inertial reference frame is being done by a Lorentz transformation for a Green function that appears in propagators like $$\langle 0\mid H_{\rm int}\mid 1 \rangle$$ (Wightmann functions).
Unruh effect
However, the most interesting thing appears, if we consider a non-inertial reference frame. It turns out, that if we take a detector that moves with a constant acceleration (so-called Rindler reference frame) the excitations of the particles in the detector, because of vacuum fluctuations, will start to form a distribution. This distribution will be very likewise to Planck distribution. So now we will have a temperature, which in case if our detector is accelerating with a constant acceleration $$a$$, will be equal to $$T = a / 2\pi$$ (of course, $$\hbar=c=1$$). And a characteristic population will be (if $$\delta E \gg T$$) $$P = 1 - \exp{(-\delta E / T)}$$).
[Note for geeks] This problem, of course, cannot be solved precisely, as we didn’t define any boundary conditions. During the real calculations diverging integrals are being considered, which is not quite strict.
So we see, that if we consider a constantly accelerating double-level detector in the vacuum with N particles, the vacuum will start to behave like a thermostat with a temperature $$T = a / 2\pi$$, and we’ll have a non-trivial distribution of particles. Speaking more visually, the vacuum starts emitting particles like a blackbody radiation with some definite temperature, when considered from a reference frame of an accelerating detector. This effect is called the Unruh effect and is purely the consequence of classical QFT.
Hawking radiation
Separately, Unruh effect is not very interesting, because the approximation of a double-level detector, constant acceleration and linearly interacting scalar field is not quite physical. However, as we remember from school or any kind of sci-fi TV shows, acceleration (according to the principle of general covariance) is the same as the gravity. Everybody should memorize the famous thought experiment with an elevator. In a closed elevator, which is in rest on Earth (pic. 2) you’ll feel the same thing, if you consider the same elevator, but without Earth or any gravitational object, constantly accelerating upward (pic. 1) (which should be equal to a freefall acceleration $$g = 9.8$$ $$\text{m}/\text{s}^2$$).
So back to our Unruh effect, if we place the detector to a gravitational field, mathematically, this would be the same as the constant acceleration, and the media will again play a role of a thermostat with a temperature $$T = g / 2\pi$$. But of course, if a gravitational acceleration $$g$$ is not large enough, this temperature will be very low. So one should consider a system, where there is a large gravitational acceleration… like black holes. Close to the surface of black holes, where $$g$$ has it’s maximum value, vacuum emits particles with a temperature, mentioned above. This result was first obtained in 1975 by S. Hawking independently from Unruh effect. Further this radiation was called the Hawking radiation.
[Note for geeks] In his 1975 article Hawking considered a standard quantum field theory in the Penrose diagrams for a forming black hole. The article is pretty complicated, but yet not very strict. The Hawking radiation was later obtained using various methods and boundary/initial conditions.
The most impressive thing here is that the fact from quantum field theory (Unruh effect) gives a very intuitive explanation to the phenomenon from hypothetic quantum gravity (Hawking radiation), which is only mathematically understandable.
Sokolov-Ternov effect
However, in my opinion, the most impressive demonstration of the Unruh effect in real life can be shown with the help of another effect, that was discovered experimentally much earlier. The so-called Sokolov-Ternov effect was first noticed in the accelerator experiments in 1971. A beam of electrons with randomly directed spins is launched into the tube of an accelerator, where there is a nearly uniform magnetic field directed, say, upward.
In this case, of course, we have two energy levels (the same as in the double-level detector): spin along field and inversely. Here we can consider that a spin along the field is the level with higher energy $$E_2$$, so if we keep spinning our particles in the accelerator we expect all of them to finally get down to the level where spins of all particles point opposite the field. The spin-flip process of transition from one energy state to another is followed by a synchrotron radiation.
Literally the same thing as in case of the double-level detector in rest or in a constant speed movement. But when done an experiment in 1971, they found that not all the electron spins are pointing opposite to the field, but asymptotically, after a large period of time there still is a small fraction of particles that is on the upper energy level. Which means, as it was in case of accelerating double-level detector, there is a nonzero vacuum temperature!
Experiment results for a lepton beam polarization in HERA accelerator is shown below:
As one can see, the beam tends to become asymptotically polarized, with a maximum polarization level less than 100%.
Again, the thing is, that we have here a double-level system (detector) with a lot of particles (electrons), that are accelerating (the velocity is constant, but the motion is circular), and thus we have our favorite Unruh effect in action that gives us a nonzero temperature of vacuum, and hence a nontrivial particle population.
This effect was first predicted theoretically in 1963 by I.Ternov and A.Sokolov with the help of QED consideration of an electron synchrotron radiation in the magnetic field. It had nothing to do with the actual Unruh effect, which was predicted in 1970’s. However, the connection of these two phenomena became a matter of interest among physicists pretty recently. One should consider reading the following article about it.
The whole theoretical physics is full of these fine connections and relations between two separate and seemingly independent areas and phenomena. However, as we make sure by the time, these connections expose that we are on a right way, and they constantly lead us to more and more curious and beautiful laws of nature that we’ve never think of.
I hope you enjoyed reading this post, and if you have questions or objections, or found mistakes, feel free to contact me!
cd ~