# Vacuum fluctuations and temperature

1. Sep 24, 2011

### tom.stoer

Usually when studying thermodynamics and statistical mechanics of a macroscopic body one uses a "heat bath model" to define a temperature. In quantum mechanics one can assume that the heat bath has arbitrary low temperature.

When studying quantum electrodynamics one has vacuum fluctuations of the electromagnetic field (we know from the Casimir effect that these may have measurable consequences).

Is it possible and does it make sense to introduce a kind of "heat bath generated from these vacuum fluctuations"? Or ist it standard to regularize the vacuum fluctuation such that one always has T=0 in the vacuum? Does that mean that zero point energy never contributes to temperature? (like in a Fermi gas where at T=0 the energy is huge).

I do not even have a reasonable starting point; let's consider

$Z = \text{tr}\,e^{-\beta H}$

Again the temperature is introduced by hand and not generated by fluctuations.

Last edited: Sep 24, 2011
2. Sep 25, 2011

### tom.stoer

Let me ask a different question: is it possible to interpret the difference in vacuum energy density in the Casimir experiment as a difference in the temperature?

3. Sep 27, 2011

### tom.stoer

no idea?

4. Sep 27, 2011

### TrickyDicky

It has been interpreted as being the result of a negative pressure gradient in vacuum, the only reference I have of Casimir effect in relation with heat bath is from those that relate the dynamical Casimir effect (converting virtual photons into real photons) http://www.nature.com/news/2011/110603/full/news.2011.346.html with the Unruh effect: the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none and thus an Unruh temperature.

5. Sep 27, 2011

### tom.stoer

I know about these calculations. Especially the Casimir effect can be interpreted as van-der-Waals force w/o any reference to vacuum fluctuations at all (Jaffe).

I was just thinking that there could be a attempt no to introdce T by hand but to "derive" it from the theory. At least a difference in the vacuum could contribute to a difference in the temperature scale. But I haven't seen such an interpretation.

6. Sep 27, 2011

### Finbar

Isn't temperature necessarily related to a thermal partition function. In the sense that its really associated to an ignorance of the state of the system not arising from purely quantum effects?

So if one is doing standard QFT with real time in flat spacetime for non-accelerating observers one cannot really speak of temperature. So at least from this perspective the casimir effect isn't associated to a temperature.

For the unruh effect the additional ignorance comes from the presence of a horizon.

I think on a deep level the Unruh effect really tells us that, in order to combine GR with QM so that it we have a unified description for all observers, statistical and quantum fluctuations must some how be unified. But really this is well beyond what is need to understand the casimir effect.

7. Sep 28, 2011

### tom.stoer

I agree.

General question: is there a mathematical way to derive the partition function and let temperature emerge instead of introducing it by hand? I haven't seen such a construction, afaik T and Z are always introduced via definitions, never derived from a "deeper structure + approximation".

Would that make sense?

8. Sep 29, 2011

### Finbar

The derivation of the classical statistical partition function comes from considering an ensemble of possible microscopic states. So in the classical case its just related to the ignorance of which of the states is actually the real one.

At the quantum level there may be some kind of derivation along the lines you are thinking.
But I have never seen it. However I think that thermal field theory is usually assumed to be created by hand since one has to use imaginary time in order to do it.

I think also such a derivation would be very hard since it would really involve asking questions about states in quantum field theory very far from those considered by particle physics. The beauty of physics is one does not have to go from one approximation to another in a mathematically rigours way. One can just write down a particular formalism which captures the physics we are interested in and then compute things. If we agree with experiment then we know the assumptions we made which led us to such a formalism, e.g. thermal equilibrium at T, were justified.

9. Sep 29, 2011

### tom.stoer

I think we agree that this could be an interesting speculation ...

10. Sep 29, 2011

### Finbar

Yes, one could perhaps attack the problem from the other side. Since we know that statistical and quantum fluctuations are formally related by a wick rotation perhaps there could be a physical meaning to "imaginary time". I think understanding the unruh effect in detail and how we should properly transform from an accelerating frame to an inertial one. Then we might see how the unruh temperature is transformed into some other effect in the inertial frame.

Although you would think this is well understood by atleast some experts in QFT i don't think it can be since it is related so closely to qft in desitter space and the black hole paradox. Both of these are controversial subjects.