Holographic Noise

Craig Hogan, from the University of Chicago, has written several papers predicting a noise that gravitational wave interferometers would be able to detect. This noise would be a signature of Planck-scale uncertainty if a certain type of holography was realized in Nature. The GEO600 interferometer near Hannover, Germany, would due to its construction details be particularly well suited to detect this noise.

And indeed, the experimentalists seem to be seeing such "holographic noise" in the frequency range between 300 and 1500 Hertz, even tough its detection is unpublished. Its occurrence is quoted in Hogan's paper (0806.0665) as “private communication,” but implicitly acknowledged on the website of the Astrophysics and Space Research Group at the University of Birmingham, a partner of the GEO600 collaboration: “To test the theory of holographic noise, scientists from Hannover and Birmingham will shift the frequency of GEO600's maximum sensitivity towards higher frequencies," and they carefully add “Even if it turns out that the mysterious noise is the same at high frequencies as at the lower ones, this will not constitute proof for Hogan's hypothesis. It would, however, provide a strong motivation for further study.”

Ununderstood noise in experiments is a good prediction to make, especially with large detectors. CERN's Large Electron Positron Collider (the tunnel of which is now reused for the LHC) was sensitive to the tides in Lake Geneva, and GEO600 is sensitive to the tides in the North Sea, and registers even smallest Earthquakes in the South Sea.

But still, we are all looking, waiting, hoping for signatures of Quantum Gravity.

Thus, NewScientist reported that our world may be a giant hologram, in a quite balanced article which quotes Karsten Danzmann of the Max Planck Institute for Gravitational Physics in Potsdam: “We work to identify [the] cause [of the noise], get rid of it and tackle the next source of excess noise. In this respect I would consider the present situation unpleasant, but not really worrying.”

Graviational Wave Interferometry

The idea underlying Hogan's prediction is that our world might have holographic properties, in which case not all three dimensions of our spacetime would encode really independent degrees of freedom. This conjectured property would become noticeable only at very large distances. A device that was able to measure distances in orthogonal directions at long distances and to high precision could be sensitive to this fundamental limit of encoding details, and be subject to a new kind of uncertainty. Gravitational wave interferometers provide exactly such a device. The holography would show up as noise in the detector.

Gravitational waves create distortions in our space-time that make themselves felt as tiny changes in lengths which are not the same for all three spatial dimensions. Interferometers lead a laser through a beam-splitter that splits the beam into two orthogonal directions into the “arms” of the interferometer, bounce the beam back on mirrors at the end of these arms, and compare the phases of the light when it comes back. This procedure can detect tiny deviations in the arm lengths which will change the phase shift. A common way to enhance the sensitivity of interferometers are “recycling techniques” that basically artificially increase arm lengths by reflecting the beam several times back and forth. GEO600 would be particularly sensitive to the holographic modification of quantum mechanics Hogan is proposing because the laser is reflected through both arms several times, whereas LIGO, VIRGO and TAMA use so called Fabry-Perot arms that reflect the beam in each arm separately. You find a very useful illustration of this difference between LIGO and GEO600 in Peter Shawhan's presentation, slide 19 and 20.

That is how the narrative goes.

Stefan and I then looked up Hogan's papers and tried to find out what the underlying model is. Since many points remained unclear to us, I wrote an email to the author who replied almost immediately and patiently answered my questions. He also agreed to be quoted here, which I hope will clarify some points better than Stefan and I could have done.

Holography

Hogan works with a modification of quantum mechanics in which position operators at different times fail to commute and instead the commutator is proportional to the Planck length l_p and a measure of distance between the positions

[x₁, x₂] = l_p L (1)

(Eq (3) from 0706.1999) In the cases considered in the paper x₁ and x₂ are taken for different events 1 and 2 on a lightcone, and denote the coordinates in a direction orthogonal to the lightpath connecting 1 and 2. While the proper distance between the events actually is null, L measures the spatial length of the connecting lightpath. In usual quantum mechanics, the commutator (1) vanishes. In Hogan's theory it can deviate arbitrarily much from the ordinary case, depending on how large L is, but this deviation would only become noticeable at large distances. One obtains from this commutator an uncertainty relation
Δ x₁ Δ x₂ > l_p L/2 (2)
(Eq (7) from 0706.1999), which is then basically the origin of the noise in the interferometer*.

Let me start with the question of motivation. The Holographic Principle is the conjecture that all the information about a volume of spacetime is actually encoded on its surface. This conjecture originated from black hole thermodynamics: the entropy of the black hole is proportional to its surface. The Holographic Principle is supported by string theoretical considerations. There is no experimental evidence for it. The Wikipedia entry on the Holographic Principle interestingly refers to the NewScientist article as an experimental test.

There is however no derivation of Hogan's modified commutator in any of his papers from the Holographic Principle, there are just many references to papers by Susskind, t'Hooft, Bekenstein, Busso and so on. We are thus actually dealing with an approach that is conjectured to be related to a conjecture. The motivation Hogan provides is from the black hole entropy. He is claiming that his modified version of position uncertainty is necessary for consistency in the black hole entropy and that of its radiation, a conclusion I could not follow (see estimate, comments are welcome).

To be fair, this motivation from the black hole entropy does not appear in Hogan's later papers. And despite the missing relation to the usual Holographic Principle one might consider his particular sort of holography and its consequences.

Lorentz Invariance

But let us have a closer look at the modification of quantum mechanics Hogan is proposing. In his approach, the Planck length plays a special role; combined with the distance L it determines when the new effects would become important. One should be wary of any framework that does such since lengths are not invariant under Lorentz transformations - there is a restframe in which the Earth is of Planck length. Thus, by merely writing down an equation that renders the Planck length special, one creates a problem (see eg my post on The Minimal Length Scale). This is already a serious issue in “Deformed Special Relativity” (DSR) which claims to have found a way to leave the Planck length invariant under a modified sort of Lorentz transformations. Unfortunately, this doesn't work well in position space (see paper) and creates all kinds of unappealing secondary problems, such as the need for a modified addition law of momenta and a missing macroscopic and multi-particle limit. Hogan doesn't mention any of this in his papers.

The problems with Hogan's approach are actually worse than that of DSR, as one can see by looking at (1) and (2). In (1) we have two quantities of dimension length on the left side that undergo Lorentz-contraction, while on the right side there is only one, since l_p is supposedly invariant. But as becomes particularly clear from (2), there is an additional problem, since L is orthogonal to the x axis - recall that it was a projection down along this axis. Thus, if one performs a boost in direction x, the left side of the inequality (2) can become arbitrarily small, but the right side remains as it is. This equation thus can clearly either not hold in all reference frames or requires some serious modification of Lorentz transformations. The former would imply Lorentz invariance is broken, and this case is very closely studied and tightly constrained by experiment. The latter implies all the problems of DSR and more, since there needs to be taken care of the unusual way orthogonal directions are treated.

Hogan clarifies in an email that he is considering the latter. With regard to the question of the invariance of the Planck length, he writes “The theory itself does not violate Lorentz invariance, but a particular apparatus or measurement does, by singling out a particular frame. There is no preferred direction in the theory.” And “I am working with the idea is that the Planck length really is fundamental and the same to all observers. An object can't Lorentz-contract below that; instead, some other physics kicks in, such as Matrix degrees of freedom in M theory.”

(Which I believe might refer to this recent paper.) With regard to the question of the relation of transverse boosts, he adds:

“In a holographic theory, you can't boost transversely; the wavefront is already moving at c. A boost would change the direction of the wavefront, and the observables.

I admit to writing down the relations in an arbitrary lab frame. The relationships of the position states depend on the frame; if L is Lorentz contracted by a longitudinal boost, the indeterminacy is less. In a highly boosted frame, L becomes the Planck length, and the indeterminacy is also a Planck length, reducing to the Planck scale noncommutativity. You can't boost more than that; in a holographic theory, once you have boosted that much, you have reached the 2D dual description, essentially living on the light sheet.”

Incidentally, Giovanni Amelino-Camelia considered that gravitational wave interferometers might detect Planck scale noise already in a 2003 paper “Quantum-gravity-motivated Lorentz-symmetry tests with laser interferometers". Without the holographic twist however, he concluded that the necessary sensitivity is out of reach.

Bottomline

Hogan is proposing an interesting modification of quantum mechanics that is holographic in the sense that it constrains the precision of measurements connected by lightpaths into directions orthogonal to each other. It is expressed through a modified commutation relation for position operators. The relation to the common Holographic Principle is not entirely clear to me, but it is an approach one can consider nevertheless. It does however necessitate a modification of Special Relativity in order to accommodate the invariance of the Planck length, and an appropriate transformation behavior of orthogonal directions. So far, Hogan's model has not addressed these issues that I believe pose significant challenges as to its consistency. Though I find the possibility that Planck scale physics might already have been detected exciting, I would appreciate an exposition of the framework that clarifies these points. It is certainly a long shot but, you see, if your shot is long enough, you might reach a stable orbit.

In reply to my suggestion to blog about it, Hogan wrote “On the blog, you are certainly welcome to post a piece on this --- indeed I hope it will help raise more discussion about it. Up to now I am interacting more with the experimental community, who seem to have a more urgent need to understand it!” Thus, having done my part, your comments are welcome.

Update: Thomas Dent points out that unlike what I wrote the GEO600 noise was published in The status of GEO 600, H Grote et al 2008 Class. Quantum Grav. 25 114043, and plots can be found online here and here. It is unclear to my why Hogan's papers do not refer to these publications.

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* Let me add here that it is not clear to me what L is in the general case, since we are talking about a "distance" between wave-functions whose position is an operator, whereas L is a c-number. If you look at the definition of L in Fig (2) of Hogan's paper you find that the operators x₁ and x₂ have suddenly lost their operator-hats. I am further not sure how the approach of this paper corresponds to that of later papers. Thus, despite this pictorially making sense, I am missing an operationally well-defined explanation what L is in the general case of this modification of quantum mechanics.