373. Some musings about Unruh effect
ASTRO, HEP-TH/PH — By Dmitry Podolsky on April 27, 2009 at 5:17 pm
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Good new long work week, science geeks! I’ve just finished reading a recent paper by Ugo Moschella and Richard Schaeffer “Quantum fields on curved space times and a new look at the Unruh effect” and wanted to share some of my thoughts with you…
1. Unruh effect. Gravitation and thermodynamics
The subject of the paper – Unruh effect in particular and relation between gravitation and thermodynamics in general is not, I think, yet really well understood by us at the fundamental level (albeit it is for more than 20 years on the market). Probably, it would be fair to say that fresh graduate students facing with this subject on the path of their “scientific career” always find it somewhat puzzling and counter-intuitive 
, so before turning to the paper let me discuss basics a bit…
What is Unruh effect? We live in the Universe filled with matter: scalars, spinors, gauge fields, etc. Suppose for fun that our home is Apollo spacecraft with its engine turned off. Somehow we managed to fly to the point somewhere in intergalactic space. The gravitational field of Sun, other stars and planets is negligibly small – we can say that the metric of the spacetime around us is approximately Minkowski metric of the flat spacetime. We might start trying to detect particles from space: quanta of scalar fields, for simplicity. If we are really far from the strong sources of radiation (AGNs, stars, etc.), we won’t detect any and conclude that matter fields are in the vacuum state.
Imagine now that we turn the engine of the spacecraft on and start moving with constant acceleration 
. If we again start looking for particles coming from the outer space, we find that there are two major new effects compared to the previous situation. First of all, we find that spacetime around us now has an (apparent) horizon. The reason for its appearance is simple: since we are moving with constant acceleration now, it becomes harder and harder for distant particles travelling somewhere in the space to reach us. Actually, if some particles are relatively far from us, they will never be able to reach us even if they have an infinite time to do so..
The second effect is that we can observe a thermal radiation of quanta of various fields from the horizon, with the temperature of the thermal flux being related to the acceleration: 
– that is, the larger is our acceleration, the warmer is radiation we detect.
These two facts are combined into surprising relation of gravity (because accelerated frame of reference is equivalent to the presence of constant gravitational field!) and thermodynamics (see for example recent paper by Brustein and Hadad), that can be formulated as follows. The presence of apparent horizon means that there are regions of the global spacetime which will never be observer by us, so there should be a certain entropy 
associated to the lack of our knowledge about the physics behind the apparent horizon. If we calculate the energy flux dE from the horizon, we find that
,
i.e., the second law of thermodynamics that appears in a rather interesting context: we need a) a curved spacetime (or an accelerated observer) and b) quantum fields living in this curved spacetime. I don’t know about you, but I certainly feel that something really deep is covered behind these formal considerations.
2. What’s the paper about?
If we are trying to construct quantum field theory in a curved spacetime (with invariance 
) from the scratch, what you should do in the first place is to pick a coordinate patch. Typically, this patch won’t cover the whole spacetime, so the Cauchy surface in the patch under consideration will not necessarily correspond to a Cauchy surface for the whole spacetime. If so, after contructing the appropriate Fock space you’ll find that the vacuum state is not necessarily invariant w.r.t. the transformations from \Gamma: take for example Rindler patch of Minkowski space or planar patch of de Sitter space – corresponding vacua are not Minkowski- and de Sitter-invariant. In we want to find a -invarant vacuum, we need to find a coordinate system that covers the whole manifold. This can be quite hard – for example, it took about 50 years to discover Kruskal coordinate system covering the whole Schwartschild spacetime.
What authors suggest is a relatively simple algorithm for constructing an invariant vacuum starting from a QFT defined in the patch that does not cover the whole spacetime. What one has to do is to find the complete set of modes u_{i}(x) as usual and then combine them into generic quadratic form
. (1)
where 
is a Hermitean matrix, while 
is a complex matrix satisfying the condition 
. This form defines the Wightman pair Green fuction in a quantum state (not necessarily vacuum state for the modes 
). Only the first term in the r.h.s. of (1) contributes to the uniqual time commutator 
, the other terms basically determine how much is the given state deviates from the vacuum state (or the states related to vacuum by Bogolyubov transformations).
After you write this form, you need to put additional requirement that the form should be invariant w.r.t. the group and tune the free functions B and C correspondingly. The resulting W will give you the desired Wightman function for the -invarant state.
3. The question of interaction
Typically, when we discuss the Unruh effect and relation between gravity and thermodynamics, we stay focused on a free field theory. While it would be fair to say that physics of free field theories in curved spacetimes is understood at the formal level very well, what remains to be understood is the physics of interacting field theories in curved spacetimes.That’s where a lot of interesting things remains to be discovered… As an example, take de Sitter space. De Sitter invariant Bunch-Davies state for massive free scalar field is known for decades, and I don’t think that the author’s construction does not really bring anything new for deeper understanding of physics of quantum field theory in dS space (basically, following authors’ prescription you can start from QFT in planar patch and find dS-invariant BD state, that’s all). What’s interesting is that turning interactions on will actually show you which states are relevant for physics, and which are not. For example, as it turns out, if the BD state is the initial state of your scalar field, you are not able to turn the interaction term adiabatically – i.e., BD state is associated with adiabatic catastrophe in the theory… In other words, if we turn ineraction, dynamics on, the prefactors B and C in the expression (1) above will become the function of time (and coordinate) – it is clear from the fact that the expression (1) describes Wightman function for a generic mixed state, i.e., carries information about occupation numbers, and occupation numbers are the functions of time if interaction is turned on. What happens with 
and 
in the dynamical, ineracting problem? Does dynamics decide which and I can actually choose? This is a problem for the future…
Some literature
1. N. Birrell, P. Davies, Quantum fields in curved space. Very good (if not the best) introduction into the subject. Everything you need to know about QFTs in curved spacetimes.
2. J. Kapusta. Finite temperature field theory. A fundamental textbook on Matsubara diagrammatics as well as canonical treatment of finite temperature QFTs.
3. R. Wald, Quantum field theory in curved spacetime and black hole thermodynamics. Much shorter than Birrell and Davies, nice discussion of the Unruh effect.
4 Comments
“carries information about occupation numbers, and occupation numbers are the functions of time if interaction is turned on.”
I’m a little skeptical about how well defined occupation number really is in these sorts of contexts. Its already mathematically challenging at the interacting level in flat space, but when you add in curved spacetimes to the mix it really becomes quite opaque.
Birrel and Davies make a good case throughout the book that the particle concept doesn’t really hold (for instance the moving mirror setup or discussions in chapter 3), which seems a little at odds with the attempt to make that wightman function globally well defined.
Dear Haelfix,
you are right, but one is still allowed to talk about particles even in this context since this is what is seen by a general detector associated with an observer. It is true that inertial and accelerated detectors see different numbers of particles. Moreover, it is known that particles of different modes can be in a coherent state with each other and, say, anomalous correlators
are not zero in a general situation.
Still, one ultimately wants to make a connection to the situation where adiabaticity holds and vacuum is well defined (in and out asymptotic states), that’s why I still prefer to talk in terms of particles.
You are also right saying that description in terms of just Wightman function may turn out to be irrelevant. What still survives even on this level of discussion is description in terms of Schwinger-Keldysh Green functions (Wightman function + imaginary part corresponding to nonzero number of particles is
in the Keldysh language). If you’ll carefully dig through Moschella-Schaeffer paper, you’ll find that they are aware of this issue.
Cheers,
Dmitry.
I want to ask the experts about an apparent confusion, that appears in many papers on the Unruh effect. As you stated clearly, the accelerated observer sees thermal radiation that comes from the horizon: (for uniform acceleration, this will be an infinite plane at some distance ahead). However, the picture above shows thermal radiation, coming from all directions, as if the observer is placed in a cavity filled with radiation. A third statement often seen is that in every infinitesimal volume dV, vacuum fluctuations, instead being described by virtual particles, become real particles with a thermal spectrum. All three cases are sometimes clumped together, by the authors saying that the observer is in a heath bath. I believe that such a statement is inaccurate. We can call the second (thermal radiation coming from the walls of a cavity) and the third case (interacting system at finite temperature) heath bath, but I believe that the real situation with the Unruh effect is that the observer sees a thermal (black body) source, localized in space, the same way as he sees Hawking radiation from a black hole’s horizon.
This distinction makes doubtful if the de-polarization of particle beams in storage rings can be attributed to the Unruh effect. Note, that if the radiation comes from a source, this source can be screened, by putting something between it and the observer. For polarized particles in a circular ring, it is not hard to see that the source of Unruh radiation due to the centripetal acceleration will be far enough to be screened by the beam pipe.
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