Relation between quantum fluctuations and vacuum energy?

[bhobba]

In a set of notes I've read in the last couple of days the author refers to quantum fluctuations of a quantum field in terms of the variance of its Fourier modes $\delta\phi( t, \mathbf{k}) =\sqrt{\langle\phi^{2}\rangle}$ about its expectation value $\langle\phi\rangle =0$. I'm not saying I place any trust in this description, but I was just wondering what your thoughts are on it?

How can an operator vary? Obviously they are in some way referring to the variance of an observation of some sort.

Thanks
Bill

 

[A. Neumaier]

How can an operator vary?

In general, an operator is the noncommutative version of a random variable. (Probability theory is equivalent to the theory of operators acting by multiplication on a function space. The multiplication operators all commute.)

 

[Brage]

I think it is not quite right to say that vacuum fluctuations are not physically real (which is what it seems to me you are saying), as a lot of effects are very precisely described by them. Take for instance Hawking radiation and the lambda shift (which was not predicted by the Dirac equation). I believe I saw you (A. Neumaier) comment earlier on how (and I paraphrase) "Hawking radiation is produced from the gravitational field" (or so I gathered), and therefore not due to virtual particles. Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

 

[bhobba]

the lambda shift

Have you heard of Lattice Gauge Theory:
https://en.wikipedia.org/wiki/Lattice_gauge_theory

That predicts the Lamb shift and has no vacuum fluctuations or virtual particles. Nor is it the explanation for Hawking radiation as has been discussed many times on this forum.

All such are is intuitive cruxes for the math to aid intuition. That is not to belittle their importance - developing intuition is vital They get repeated in populist press and we get people who, for reasons that perplex me, won't accept what those that have studied the theory in detail tell them - namely they are fictions.

This isn't the only area it happens - it occurs all the time in advanced areas. I mentioned previously about the 'hole' issue in how transistors work. You will find electronics textbook after textbook that promulgates the same load of rubbish. But one must start somewhere. Feynman commented on it. He would prefer to tell the truth, the whole truth, and nothing but the truth, from the start. But being the great educator he was and having thought hard about it realized it would be very unwise doing that. So as you proceed from beginner to advanced you need to unlearn things. Its nothing to really worry about except when those at the beginner level do not believe what those at the more advanced level tell them. That is unbelievably frustrating as I know only too well.

Its one of the many myths of QM:
http://arxiv.org/abs/quant-ph/0609163

I have posted the above many times. It should really be the end of the matter but it isn't - people have so many ingrained ideas that are hard to shift.

Thanks
Bill

 

[A. Neumaier]

"Hawking radiation is produced from the gravitational field" (or so I gathered), and therefore not due to virtual particles. Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

If you insist on waiting for a consistent quantum theory of gravity one cannot discuss Hawking radiation at all - any statement about it would be ill-founded according to this overly stringent criterion!

The whole concept of Hawking radiation depends (at present) on semiclassical reasoning. And in semiclassical reasoning, Hawking radiation is produced from the gravitational field.

 

[bhobba]

Correct me if I am wrong on this recollection, but such a statement wouldn't be well founded as we have yet to formulate a consistent theory of quantum gravity that is validated by experiment.

Validated by experiment probably has some validity (although I would like an experimental type to comment) but its a, admittedly very common, misconception we don't have a quantum theory of gravity.

We indeed have one, and it elegantly explains Hawking radiation:
http://blogs.umass.edu/grqft/effective-field-theory/

What we don't have is one valid to beyond about the Plank scale - but gravity is not alone in that - the standard model is not trusted in that region either.

As I am won't to say it doesn't make developing a full quantum theory of gravity any easier, but is an interesting take that keeps what really is going on in perspective. Its not quite the mess some sources make it out to be.

Thanks
Bill

 

[vanhees71]

If you insist on waiting for a consistent quantum theory of gravity one cannot discuss Hawking radiation at all - any statement about it would be ill-founded according to this overly stringent criterion!

The whole concept of Hawking radiation depends (at present) on semiclassical reasoning. And in semiclassical reasoning, Hawking radiation is produced from the gravitational field.

Well, what's even more interesting is, whether there is any empirical evidence for Hawking radiation. I'm not aware of any...

 

[A. Neumaier]

Well, what's even more interesting is, whether there is any empirical evidence for Hawking radiation. I'm not aware of any...

We can be lucky, it would require a very big black hole close to us...

 

[vanhees71]

Well, whether this is a lucky situation, is another question ;-)).

 

[George Jones]

We can be lucky, it would require a very big black hole close to us...

Big?

 

[vanhees71]

Yeah, real big, because otherwise we have no chance to detect Hawking radiation. However, if we come too close (which I guess we should to detect the radiation), maybe we have no time to write the paper publishing the discovery ;-)). Anyway, I guess Hawking radiation is a safe place to make theoretical predictions, because it's very unlikely to be disproven empirically ;-).

 

[George Jones]

Yeah, real big, because otherwise we have no chance to detect Hawking radiation

But the Hawking temperature of a black hole is inversely proportion to mass, and power radiated goes as T^4. The temperature of a solar mass black hole is far, far lower than the 2.7 K CMB temperature, so something much, much less massive is needed in order to be detectable.

 

[A. Neumaier]

But the Hawking temperature of a black hole is inversely proportion to mass, and power radiated goes as T^4. The temperature of a solar mass black hole is far, far lower than the 2.7 K CMB temperature, so something much, much less massive is needed in order to be detectable.

Far away from a black hole of typical astrophysical size, Hawking radiation is indeed completely swamped by the microwave background radiation, while small black holes that have a higher Hawking temperature are not radiating enough to be detectable from far away. Since the intensity of Hawking radiation decays with distance like $r^{-2}$ it is not detectable unless one is reasonably close to the black hole and a lot of radiation is produced, which in turn requires a high gravitational field and hence a big black hole.

But one can study in the lab Hawking radiation in so-called analogue black holes, optical analogues where nonlinear optics simulates a quantum metric; see Hawking radiation from ultrashort laser pulse filaments:

the same physics that underlie black hole evaporation in the form of Hawking radiation may be found and studied in other, more accessible systems. Our measurements highlight spontaneous emission of Hawking radiation from an analogue event horizon generated by an “evaporating” refractive index perturbation and suggest a path towards the experimental study of phenomena traditionally relegated to the areas of quantum gravity and astrophysics.

There are also acoustical analogue black holes; see, e.g., http://arxiv.org/abs/1410.0238.

 

[vanhees71]

Well that are analogues but not the true thing. In this sense you could say that we have discovered all kinds of "exotic" particles like anyons, magnetic monopoles, Majorana and Weyl fermions,... However all these are, of course, no elementary particles but quasiparticles in condensed matter physics :-).

 

[Demystifier]

We can be lucky, it would require a very big black hole close to us...

Well, Hawking radiation is much easier to detect for a small black hole.

EDIT: Now I have seen that George Jones already said that.

 

[A. Neumaier]

Well, Hawking radiation is much easier to detect for a small black hole.

But only when you are very close by. And we don't have experimental evidence of small black holes close to us.