Saturday, June 30, 2007

Impressions from the Loops '07

The Loops 2007 this year was held at the University in Morelia, Mexico, in a very nice building (see also Stefan's previous post). The auditorium had a stage with the speaker being in the spotlight, the seats were very comfortable to doze off, and the hallways had very picturesque pillars and arches (see photo to the left).

My head is still spinning a bit, trying to process all the information gathered here (well, it might also be the lack of oxygen in the air, it is rather polluted and the high altitude doesn't help either). It's been my first conference in this community, and I have to say from all the conferences where I've been this year, the Loops has been the nicest experience. The atmosphere has been very welcoming, openminded and constructive.

My talk yesterday on 'Phenomenological Quantum Gravity' (slides here) went well (that is to say, I stayed roughly in the time limit and didn't make any completely embarrasing jokes). Though I wasn't really aware that I would be the only one on this conference speaking about DSR. If I had known, I might have extended my summary of that topic (there was a DSR talk scheduled by Florian Girelli, see picture to the right, but he changed the topic shortly, which I didn't know).

However, after more than a month of traveling, I am sitting here in the inner yard of the hotel and try to remind myself why I go to conferences:

  • Because it's just such a nice experience to arrive in a foreign city where nobody understands your language, without any baggage, after a 36 hour trip, with an 8 hour jetlag, having just figured out that the credit card doesn't work, and the hotel doesn't have a reservation for you - and then to find the conference site with familiar faces, the air filled with words like 'propagator', 'manifold' and 'background independence'.
  • Because one meets old and new friends, because there's a conference dinner or reception, and plenty of free coffee and cookies.
  • Because the registration fee includes a welcome package that usually features a more or less useful bag, a notebook and a pen, and a significant amount of tourist information. Occasionally there are some surprises to that, e.g. this time the bag was a woven shoulder bag, or one of the previous SUSY conferences featured a squeezable brain...
  • Because some people present new and so far unpublished results.
  • To see and be seen.
  • To inspire and get inspired...

  • And of course to blog about it ;-)

There has been a significant amount of braiding on this conference (for program and abstracts see here), talks by Yidun Wan, Jonathan Hacket, Lee Smolin and Sundance Bilson-Thompson, the latter shown in the picture below with John Swain

Olaf Dreyer, preparing his talk

Some people from the blogosphere that I've meet in person here. Garrett Lisi and Frank Hellmann:

Alejandro Satz (from Reality Conditions) and Yidun Wan (from Road To Unification)

And here is a photo of Carlo Rovelli and Abhay Ashthekar - pen, notebook and coffee included:

Admittedly, I found the phenomenlogical part on this conference somewhat underrepresented. Indeed, I found myself joking I am the phenomenology of the conference! Likewise, Moshe Rozali (who you might also know from comments on this blog) has been the String Theory of the conference. He gave a very interesting talk about the meaning of background independence.

This afternoon, I am chilling out (okay, actually I am writing referee reports that have been due about a month ago). Below a picture of the hotel's inner yard where I am sitting (taken yesterday, right now it is raining). Tomorrow I am flying back to Canada, and I am really looking forward to sleeping in my own bed. A nice weekend to all of you!

[Try clicking on the photos to get a larger resolution.]
Updates: Chanda just sent me a photo she took yesterday evening. I am very pleased about the truly intellectual expression on my face, must be the glasses.

And Garrett took this nice photo.

Friday, June 29, 2007

Philosophia Naturalis Blog Carnival

I have to admit that it took some time until I understood the idea of the "Philosophia Naturalis Blog Carnival", even more so as the actual blog just contains links to posts at different other blogs. This idea of these "philosophia naturalis" contributions, hosted by a different blog every month, is to publish a collection of interesting posts on topics in the physical sciences that have appeared over the last month. Thus, they provide a selection of noteworthy reading out there, and help to give you an overview over interesting blogs dealing with physics and related topics.

This month's Philosophia Naturalis #11 is hosted by geologist Chris Rowan at Highly Allochthonous, and I am very proud to see this blog represented by both its contributors - with Bee on Kaluza-Klein, and myself about the Bouncing Neutrons!

If you have time to waste over the weekend, there may be worse possibilities to do so than reading some of the posts presented by Chris so cogently according to the 50 orders of magnitude of characteristic length scales they cover. I've enjoyed especially the writings from astronomy and the earth sciences. Indeed, if you are a wannabe amateur geologist like me, you will probably like Chris' blog anyway, and wonder why you have not chosen a subject where you can make cool field trips to Namibia...

Have a nice weekend!

Wednesday, June 27, 2007

Ever heard of of planet Eris?

This is planet Eris, in the outskirts of the Solar System, as seen through the eyes of the Hubble Space Telescope:

Hubble Telescope's Advanced Camera for Surveys has taken this image of Eris in December 2005. The analysis of several such photos yields a diameter of Eris of 2400±100 km, or 1500±60 miles. (Credits: HubbleSite News Release STScI-2006-16, M. Brown)

It's such a faint and blurred blob even in the Hubble Space Telescope because it is quite small, and far away from Earth: At the moment, Eris is close to its aphelion, at a distance of 97 astronomical units - that's 97 times the mean distance of the Earth from the Sun! You can explore its eccentric and highly inclined orbit with this applet from the JPL.

In fact, according to the new definition of the International Astronomical Union from last August, Eris is not a planet, but only a dwarf planet. However, it is larger than Pluto, and it has more mass than Pluto, as was reported in a beautiful short note in Science two weeks ago [Ref 5]! So, for all those who prefer to stick to the old definition of a planet, it may be the true ninth planet - unless some other, even larger guy shows up from out there in the Kuiper belt.

Eris was discovered in October 2003 by astronomers Michael Brown, Chad Trujillo, and David Rabinowitz [Ref 1]. It was given the provisional name 2003 UB313, or Xena for short. Soon after the discovery, its size could be measured by observations with a radio telescope of the Max-Planck Society [Ref 2] and the Hubble Space Telescope [Ref 3], and it came out that the new planet is larger than Pluto!

Moreover, there is a small moon in orbit around Eris, which was given the nickname Gabrielle. On September 14, 2006 the International Astronomical Union made official the names Eris for the planet and Dysnomia for its satellite.

A comparison of the sizes of (left to right) Eris, Pluto and Charon, the Moon, and the Earth. Eris' moon Dysnomia is not shown, but it is much smaller than Eris (diameter: 2400±100 km from HST, 3000±400 km from radio observation), Pluto (2300 km), Charon (1200 km), the Moon (3500 km), and the Earth (12800 km). (Credits: Max-Planck Gesellschaft Press Release News/SP/2006(10), Frank Bertoldi)

The discovery of Eris, and of several other large objects in orbits beyond Neptune, had spurred the hot debate about what is a planet, which then lead to the degradation of Pluto from planet to "dwarf planet" in August 2006. So it is fitting that Eris is the Greek goddess of strife and discord - Dysnomia, her daughter, is the goddess of lawlessness.

But in a time when ancient Greek gods and goddesses are reduced to large balls of rock and gas in space, even the goddess of lawlessness is subject to Newton's universal law of gravitational attraction. And this allows, in an elementary and elegant, classical way, to determine the mass of Eris.

The photo on the left is an image of Eris and its moon Dysnomia, the small spot left of Eris, taken with the Hubble Space Telescope on 30 August 2006. Dysnomia's projected orbit around Eris is superimposed on this photo in the image on the right (Credits: HubbleSite News Release STScI-2007-24, M. Brown)

In a first step, the orbit of Dysnomia around Eris has been determined from several observations with the Hubble Space Telescope and the Keck telescope in Hawaii.

The orbit of Dysnomia, the moon of Eris, as reconstructed from observations using the Keck telescope on 20, 21, 30, and 31 August 2006 and the Hubble Space Telescope on 3 December 2005 and 30 August 2006. Observations are show as crosses, the predicted positions at the time of observations are shown by circles. The solid circle in the center is 10 times the actual angular size of Eris. (Fig. 1 from Brown and Schaller, Science 316 1585 (2007 June 15), doi:10.1126/science.1139415. Reprinted with permission from AAAS.)

The observation of Dysnomia as shown in this figure reveals an apparent diameter of the orbit of roughly 1.1 arcsec, which, at the distance of 97 Astronomical units, or 97 × 150 million km ≈ 1.46·1010 km, corresponds to a diameter of 1.46·1010 × 1.1 × 2π / (360 × 3600) km ≈ 78000 km. A more detailed analysis yields an indeed circular orbit with a semimajor axis of r = 37500±200 km.

Now, this can be used to deduce the mass ME of Eris!

Using elementary Newtonian mechanics, we equal the gravitational attraction between Eris and Dysnomia at distance r with the centrifugal force,

GMEmD/r2 = mDrω2.

The frequency ω = 2π/T has to be calculated from the orbital period T of Dysnomia. With the equivalence principle at work, the mass of the satellite cancels out, and we can solve for the mass of Eris:

ME = r3ω2/G.

Now, the orbital period of Dysnomia has been determined quite precisely to 15.772±0.002 days. This yields ω = 2π/(15.77 d × 24 h/d × 3600 s/h) ≈ 4.63·10-6 s-1. Newtons constant is 6.673·10-11 m3 kg-1 s-2, and plugging everything together, the mass of Eris comes out as

ME = 1.7·1022 kg.

For comparison, Pluto has a mass of 1.3·1022 kg, the Moon of 7.35·1022 kg, and the Earth of 5.97·1024 kg. This means, the masses of Eris, Pluto, the Moon, and the Earth compare as 1.3 : 1 : 5.6 : 459.

And so, indeed, Eris and Pluto are on equal footing, revealed by the goddess of lawlessness and Newton's law!

Here are some "historical" references about Eris:

  1. Discovery of a Planetary-sized Object in the Scattered Kuiper Belt, by M.E. Brown, C.A. Trujillo, and D.L. Rabinowitz, The Astrophysical Journal 635 L97-L100 (2005 December 10) [PDF, ADS entry], arXiv:astro-ph/0508633 [see also: Eris webpage by Michael Brown]

  2. The trans-neptunian object UB313 is larger than Pluto, by F. Bertoldi, W. Altenhoff, A. Weiss, K.M. Menten and C. Thum, Nature 439 563-564 (2006 February 2) [PDF, ADS entry], doi:10.1038/nature04494 [see also: Press release by the Max-Planck Gesellschaft, comment by Frank Bertoldi]

  3. Direct Measurement of the Size of 2003 UB313 from the Hubble Space Telescope, by M.E. Brown, E.L. Schaller, H.G. Roe, D.L. Rabinowitz, and C.A. Trujillo, The Astrophysical Journal 643 L61-L63 (2006 May 20) [PDF, ADS entry] [see also: Press Release by the HubbleSite]

  4. Satellites of the Largest Kuiper Belt Objects by M.E. Brown et al., The Astrophysical Journal 639 L43-L46 (2006 March 1) [PDF, ADS entry], arXiv:astro-ph/0510029 [see also: Press Release by the Keck Observatory, News Item by Marcos van Dam, Dysnomia, the moon of Eris - webpage by Michael Brown]

  5. The Mass of Dwarf Planet Eris by Michael E. Brown and Emily L. Schaller, Science 316 1585 (2007 June 15), doi:10.1126/science.1139415 [see also: Press Release by the HubbleSite News item by the Planetary Society, Press Release by the Keck Observatory]

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Monday, June 25, 2007

Loops'07 in Morelia

The conference Loops'07 takes place this week in Morelia, Mexico.

The Colegio de San Nicolás in Morelia, where the Loops'07 takes place this week.
(Credits: Wikipedia)

Well, I can't tell you much more than what you can read on the web page. Bee could, as she has just arrived there, after a neat 36 hour trip (I guess she has wasted at least one week sitting around in airports and train stations over the last month or so...). But there seems to be no stable Internet connection at the conference venue, or at the hotel where she stays.

So, we will have to be a little patient to hear first-hand accounts about the latest news in "Foundational questions of quantum gravity, Loop quantum gravity, Spin foam models, Dynamical triangulations, Causal sets, String theory, Cosmology related to quantum gravity, Phenomenology of quantum gravity" (topics from the conference web page) - and the same for photos with loopy physicists in Mexico ;-)

Update: here is a short first report by Bee sent via BlackBerry (June 26, 2007 8:26:10 PM GMT+02:00):

Morelia is a charming city, though quite noisy, and it's hard to get around with my Spanish skills or rather absense thereof (my Spanish is even worse than my Italian and Polish). The trip (air france, Frankfurt-Paris-Mexico City-Morelia) was somewhat annoying. My-stupid-bag remained in Paris, the transatlantic flight was delayed and so I missed the onward flight. Since it turned out to have been the last flight to Morelia that evening I had to spend a very mediocre night in Mexico City, and arrived in Morelia yesterday morning just in time for the first talk. (The hotel people then insisted I had no reservation, and my bag still hasn't reappeared - so much about the joys of travelling).

The conference is very interesting so far. It is my first time at the Loops and since I'm new in the community I'm somewhat nervous about my talk on Friday. The topics are very diverse, ranging from foundations of quantum mechanics over loop quantum gravity to semiclassical quantum field theory in curved backgrounds etc. The Strings 07 is also this week in Madrid, so its unsurprising that there isn't much stringy stuff around here (e.g. I haven't seen any ducks, just chickens).

Being the phenomenologist I of course find the phenomenology somewhat underrepresented - the Trieste workshop was much better balanced for my interests.

I've meet a lot of people here in Morelia I only knew by email so far, and my hair is standing upright - not only because I couldn't comb it for several days but because the atmosphere is very stimulating - there are so many interesting aspects I only begin to understand... (Plus the absence of a wireless significantly increases my attention span, I'll have to think about this...)

My name-tag says International Conference on Quantum Gravity and I can't avoid but think this is what I've always wanted to work on, and that I am very happy to be part of it - baggage or no baggage.

Rumors reached me that the hotel across the street from mine has internet access, so I'll try to check the arxiv in the evening...

Sent from a researcher in motion

Saturday, June 23, 2007

The GZK cutoff

The earth is hit by cosmic rays all the time. Those with the highest energies collide with atoms already in the upper atmosphere, and produce a cascade of secondary particles, a so-called cosmic ray shower. These secondary cosmic rays include pions (which quickly decay to produce muons, neutrinos and gamma rays), as well as electrons and positrons produced by muon decay and gamma ray interactions with atmospheric atoms.

Nowadays, the showers can be simulated with appropriate software. The picture below, from Hajo Drescher, illustrates such a cosmic ray shower

Here, the primary particle was a proton with an energy of 1019 eV, the colors indicate blue: electrons/positrons, cyan: photons, red: neutrons, orange: protons, gray: mesons, green: muons. (Unfortunately, one can't see the colors very clearly, you can decompose the shower into colors on the website. The incoming proton is the line from the upper left, the other upgoing line is cyan and a photon). If you have Quicktime installed, you can also look at this very illustrative movie, which shows the particles cascading down on earth. The above figure has be created using the software SENECA (down-loadable here), the competitor is AIRES, which has a somewhat more impressive advertisement movie (the exact differences between both codes elude me).

The number of particles reaching the earth's surface is related to the energy of the cosmic ray that struck the upper atmosphere. Cosmic rays with energies beyond 1014 eV are studied with large "air shower" arrays of detectors distributed over many square kilometers that sample the particles produced, e.g. at HiRes in Utah, AGASA in Japan and Pierre Auger in Agentinia, the latter has a very nice homepage, summarizing the mysteries that still need to be solved.

Energies over 1014 eV sounds extremely large. In comparison, the collision energy that the LHC will reach is 1013 eV. However, one has to keep in mind that in cosmic ray events the energy is typically that of the incoming particle in the earth rest frame and not actually the collision energy in the center of mass frame (LHC collides two beams head on, thus the lab frame is identical to the center of mass frame).

To give you an example, the energy in the center of mass frame of an incoming proton with an already extremely high (and rare) energy of 1017 eV hitting a proton in rest is roughly the square-root of 1017 eV times the proton rest-mass, 109 eV, which is approx 1013 eV and comparable to LHC energies. However, one has to keep in mind that cosmic ray events, despite their potentially large energy, are far less in control and attached with higher uncertainties than collider experiments. Most of the air showers are believed to be created by protons. Since the incoming directions are evenly distributed (and inside our galaxy no mechanism is known to accelerate them to these high energies) the proton's origin is most likely not in our galaxy. That means the protons must have travelled at least roughly 50 Mpc [1] before they reach earth.

Now, if the incoming proton's energy increases further, then eventually it will not only react with our atmosphere, but also with the photons in the cosmic microwave background (CMB). That is, for photons with sufficiently high energies, the universe will stop being transparent. The protons will start to scatter on the photons in the microwave background, loose energy and can't reach earth any more. The first reaction that can take place with increasing energy is photo-pion production which happens at a center of mass energy of roughly 200 MeV. This pion production is extremely well measured in earth's laboratories, where photons are scattered on nuclei in rest. If one sets the energy of the photon to be that of the CMB temperature (3 K is approximately 2.5 10-4 eV), one finds that the proton needs an energy of roughly 1021 eV to cross the threshold for pion production. (It is roughly (200 MeV)2 divided by the photon's energy).

The figure to the left (credits go to Stefan) shows the cross-section for photon-proton scattering in the laboratory (proton in rest), the blue dots are data from the particle data booklet. The red line indicates the initial threshold for the process to take place, the orange lines are the delta resonances where the cross-section has peaks.

However, what one actually wants to know is when the mean free path of the protons drops below typically 50 Mpc. To get a better result than the above estimate one has to take into account that the CMB has a small percentage of photons with larger energy than the temperature, the distribution given by the Planck spectrum. Such, the mean free path of the protons drops significantly already at a somewhat smaller energy than the above 1021 eV because the proton has a chance to hit the higher energetic photons.

My husband, as usual, has made a lot of effort to answer my yesterday's question and produced the figure to the right, which very nicely illustrates that indeed roughly 10% of the photons have energies five times larger than the background temperature.

The exact calculation for the cut-off has been done by Stecker (Effect of Photomeson Production by the Universal Radiation Field on High-Energy Cosmic Rays), and one finds the drop to happen at roughly 6 x 1019 eV. At this energy, one thus expects a cut-off in the cosmic ray spectrum because the sources for the showers can no longer reach earth. First pointed out by Greisen[2], Zatsepin and Kuzmin this is referred to as the GZK-cutoff. To summarize, this prediction relies on

a) The initial particle of the shower being a proton from outside our galaxy
b) The total cross-section of protons with photons, and
c) The assumption that the cross-section (a Lorentz scalar itself) can be boosted from the earth laboratory (proton in rest) into the rest-frame of the CMB (photon in rest).

Now AGASA claimed to have observed cosmic ray events with energies above this cut-off (see e.g. Has the GZK suppression been discovered, by Bahcall and Waxman). This has lead to a significant amount of speculation how this could be explained. One of the explanations for example is that a violation or deformation of Lorentz invariance might be the cause for a shift in this threshold, which has been argued to be a signature for quantum gravity (see e.g. Alfaro and Palma, Loop quantum gravity corrections and cosmic ray decays, hep-th/0208193).

I have explained previously that I find these explanations implausible - as mentioned above, the energy in the center of mass frame is somewhere around a GeV, now could please somebody explain me why on earth (pun intended) you'd expect quantum gravitational effects in that energy range?

One should also keep in mind, that HiRes on the other hand has observed the cutoff where it is supposed to be. Their new data analysis again confirms the cutoff. The recent paper is here, and last month there was a brief article in Physics Today "Fluorescence Telescopes Observe the Predicted Ultrahigh-Energy Cutoff of the Cosmic-Ray Spectrum" by Betram Schwarzschild.

It is expected that Pierre Auger will present first results at the 30iest International Cosmic Ray Conference, which will take place in Merida, Yucatan, Mexico from July 3 - 11, 2007. Hopefully, the situation will be clarified then.

So stay tuned...

I too am on my way to Mexico, to another conference, the Loops 2007.

[1]: Mpc means Mega parsec. Mega is 106, one parsec is approx 3 lightyears.
[2]: Kenneth I. Greisen, Cornell professor emeritus of physics and a pioneer in the study of cosmic rays, died March 17 2007 at age 89.

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Thursday, June 21, 2007

Random Sampling: Scientific American, October 1960

These days I came by chance across some older issues of the Scientific American, out of a collection Bee had inherited some time ago from a former science professor. I find it always fascinating to browse through old, yellowed magazines. We can see what kept busy the minds of people at that time, and witness now well-established knowledge in the making.

The oldest number in the collection is the issue of October 1960, and the feature articles form a mix that is very similar to what we could find in the magazine today: Archaeology ("A Forgotten Civilization - Bahrein Island in the Persian gulf was a link between Sumer and ancient India"), Biology ("Electric Fishes - Not only the electric eel but also other fishes can generate a respectable charge"; "The Eradication of the Screw-worm Fly - A serious pest of cattle has been combatted by sterilizing its males with X-rays"), Geology ("The Rift in the Ocean Floor - The great ridge that bisects most of the oceans is split by a remarkable fissure"), History of Science ("Count Rumford - Born Benjamin Thompson in Massachusetts, he was a great investigator of heat"), and Physics ("Optical Pumping - Light is used to pump electrons to higher energies for spectroscopic purposes"; "The Physics of Wood Winds - Modern physics makes possible a closer analysis of how these instruments work"; "High-Speed Impact - Bodies collecting at speeds higher than 8000 feet per second behave like fluids").

Of course, there are some curiosities: the article on the mid-ocean ridges still predates the firm establishment of plate tectonics, and discusses the "Expanding-Earth Theory" based on a decrease of the Newtonian constant G over geologically relevant time-scales as a possible explanation of the data. Optical pumping is today usually referred to in connection with the laser - but the laser, whose first realisation by Theodore Maiman was just published two months earlier, in August 1960, is only mentioned in the very last paragraph of the article (... T. H. Maiman at the Hughes Aircraft Company has already observed some degree of coherent emission from a ruby, and he as well as workers at several other laboratories are trying to put R light to practical use. Although a working light amplifier may still be some time away, its prospects now seem excellent.)

But the really cool stuff which conveys the spirit of the time is not to be found in the editorial content of the magazine, it's in the advertisements! I was fascinated to see just how many advertisements there are - the index of advertisers lists more than 120 companies, and most of the advertisements cover at least one page, inflating the issue to 224 pages total. In these ads, we can learn that

  • Teflon was not a spin-off of the Flight to the Moon, as I was told when I was a kid - at least, as we can see, DuPont de Nemours was placing advertisements for this stuff before Kennedy's bold speech...

  • Tubes are still essential parts of electronic equipment - that's what we are told by RCA, whose tubes are used "to listen to the sound of colliding galaxies 300 million light-years away" and equip "satellites and space probes [...] gathering information about the van Allen belts and other hitherto unknown phenomena of outer space" ...

  • "At Raytheon, Scientific Imagination focusses on Ferromagnetics", to be used in microwave and radar equipment - and these guys are working with second-quantised Hamiltonians on the blackboard...

Science as showcased in these ads is an all-male business, and focusses to a large part on military applications: Electronic equipment to study what happens when launching rockets,

... steel to construct silos for housing ICBMs in the Plaines of Wyoming, ...

it's all "rocket science":

But wait: Not everything in the advertisements is manifestly military stuff: There is a group of ads scattered throughout the magazine around the ECHO satellite project, the first experiment to test telecommunication via satellite:

Are you also puzzled by that funny-shaped antenna displayed in the ads of the JPL and Bell Labs?

But yeah - that is the antenna which came to true scientific fame three years later, when Penzias and Wilson recognised that it was constantly registering some funny, isotropic microwave noise corresponding to a temperature of 3 Kelvin - the Cosmic Microwave Background!

Monday, June 18, 2007

Save Money on Health Insurance

What I just read in the newspaper, I thought it might amuse you.

Mumbai, India, June 17th 2007: Pradeep Hode, a 30 year-old from Diva in Thane, chronic patient of tuberculosis had to undergo an emergency surgery on Friday morning during which 117 coins where removed from his stomach. Based on hearsay and some bizarre logic of his own, the man started swallowing coins, hoping that the heavy metal would cure him.

(From the Times of India, In search of cure, tuberculosis patient swallows 117 coins.)

This must be a black day for those who think money cures everything...

Saturday, June 16, 2007

Trains and Airplanes

Taking the plane is usually a fast, and often a quite convenient way of travelling. But sometimes, thunderstorms interfere with the flight plan, and then it can happen that all flights but one from Munich to Frankfurt are cancelled. And if you have bad luck, you are sitting at the airport and wait and wait ... only to hear that you have either to spend the night there, or take a train, which, if you would have done that earlier, you have brought to your destination already since hours... That's what has happened to Bee yesterday - and that's why I'll meet her now this morning at the train station, instead of the airport yesterday evening.

Meanwhile her cellphone ran out of battery, so here is the last mail (rough translation):

    From: sabine[@]
    To: scherer[@]********.de
    Subject: still in munich
    I missed the onward flight to Frankfurt and was rebooked to the next flight. Right now there's a really impressive thunderstorm outside, one can't see anything except loads of water running down the window and an occasional lightning. Unfortunately, it seems our aircraft was struck by a lightning during touch down and it's not yet clear whether it can take off again [it turned out later it had to go out of service].... *argh* there was just an announcement that the airport is temporarily closed. Will try to find an outlet, I am running out of battery... say hello to the blogosphere ;-)



To avoid that I've forgotten how she looks like after all these delays, she has send me a recent photo taken at the Warsaw conference vine and cheese reception - one of the more important events at every conference:

Thanks to Akin Wingerter for the photo - and I am off for the train station.

Friday, June 15, 2007

Positively Crazy

Some inspiriation for your next seminar:

"better get used to staying up all night" - yo, man.

Wednesday, June 13, 2007


I meant to tell you something about the String Pheno, but the slides are still not online. So, instead just a lovely photo from the old part of the village, Frascati, where it took place. I am on my way back to Italy, this time to Trieste where I will attend the workshop From Quantum to Emergent Gravity: Theory and Phenomenology.

Tuesday, June 12, 2007

Blog Life

My husband just sent me an email, reporting that a former colleague pointed out this very interesting article at

which is part of a monthly column:

You see, we are in good company :-) I am very flattered by such honor and find the text remarkably accurate (well, I didn't know how to spell 'Landolt-Börnstein').

The mentioned Aero chocolate is here, and the 'humorous take on the recent debate over the status of string theory' is here. The 'lengthy and thoughtful posts' as well as the inpiration series you find in the sidebar.

Monday, June 11, 2007


Can you spot a fake smile? Take the test.

"Scientists distinguish between genuine and fake smiles by using a coding system called the Facial Action Coding System (FACS), which was devised by Professor Paul Ekman of the University of California and Dr Wallace V. Friesen of the University of Kentucky."

Especially recommendable when sitting in the afternoon session while the sun outside is shining brightly...

    Why Are There Always So Many Other Things To Do?
    Distractions, Like Butterflies Are Buzzing 'Round My Head...

~ Paul McCartney, Distractions

Saturday, June 09, 2007

Femme fatale, post mortale

Oh well, I promised you photos from Europe, didn't I? Here's how you look like if you hang above a door frame for too long ;-)

Seen somewhere in Rome around here, close by the Piazza del Quirinale. Still think you like buildings where you can feel the presence of the ancestors?

Friday, June 08, 2007

Hello from Warsaw!

And here I am in Warsaw for the next conference, the Planck 2007. I am very happy to report that this time my-stupid-bag arrived with me. However, at the airport I noticed that I managed to book what I thought was a hotel without having a street address. It took me some back and forth to find it out (since my Polish is even worse than my Italian). The taxi driver dropped me off and pointed vaguely into a direction since the address turned out to be in a pedestrian-only zone. After pulling my-stupid-bag through several cobblestone roads, I found the house. There I was explained that it's not an hotel at all. They rent apartments in 'the old town', and mine would actually be in another building, and also, would I please pay in advance, preferably in cash.

I probably didn't exactly contribute to Germany's good reputation by simply refusing to pay anything before I had seen the apartment. However, I shouldn't have worried. The apartment is great, larger than mine at home, has a living room as well as a completely equipped kitchen including washing machine and dishwasher. And is less expensive than every hotel I could find in this area (I was pretty late with booking).

The only drawback is the absence of any internet connection. (The reason being 'there are mostly older ladies staying with us'). Right now, I am sitting on a bench in the middle of Warsaw's old town blogging over an open access wireless that has a pretty good bandwidth. There is a small fountain in the middle and the place is surrounded by lovely colorful old houses, most of which have restaurants. Lots of people are sitting here, chatting, having dinner, drinking beer. Pigeons are hopping around, children are playing with the water in the fountain. A guy behind me just started playing the violin, its incredibly bad. Oh no! I recognize that song 'Und wenn wir alle Englein wären....'.

I spent a considerable amount of time trying to get some fruits or vegetables, but none of the grocery stores I found had any. On the other hand they had a truly impressive selection of sausages, bacon and all other kinds of very dead looking meat. I figure it's not a good place for a vegetarian. The only person who I could find on the street who spoke English was a German and explain he had only arrived two hours earlier, and whether I had an idea where to change Euro in Zloty (the stores still don't take Euro). Ah, the guy with the violin moved on. Now there's somebody with an accordion approaching...

I just love to sit an watch people. Meanwhile it has gotten dark and next to the fountain is a women juggling with torches. Unfortunately, I can see lots of clouds in the north and it looks like rain. Plus, there are plenty of mosquitoes, and the accordion isn't convincing either. So I'll pack my laptop and try to get some sleep to be fit for all the physicists tomorrow.

PS: Ach, und könnt mir vielleicht jemand sagen was 'Wo bitte bekomme ich Damenbinden?' auf Polnisch heisst und wie ich das ausspreche?

Thursday, June 07, 2007

The Early Extra Dimensions

I have been fascinated by the idea of extra dimensions (XDs) long before I finished high school. I, as apparently many other theoretical physicists, was a science fiction fan then. Besides randomly reading everything labeled with 'SF', I made my way through a considerable part of the Perry Rhodan series, which I loved for picking up political topics and projecting them into the future [1]. Admittedly, the technical details somehow escaped me, especially when it came to time travel and the hyperspace.

This is just to say that the topics of hyperspace and XDs have inspired generations of physicists. And whoever it was who first did the calculation that shows string theory needs extra dimensions to make sense, it must have been one of the most exciting moments I can imagine for a theoretical physicist.

But XDs have come a long way, and were around long before string theory. People sometimes ask me why my talks never mention the earlier works on the topic. The reason is that the theories with XDs proposed in the 1920ies by Theodor Kaluza and Otto Klein, are in their idea different to the 'modern' XDs. Yet, this usually takes too much time to clarify in a talk, so I rather skip it. However, since you - and yes, I mean YOU who you are just raising your eyebrows - are of course the most attentive reader there is, I want to elaborate somewhat on these 'early' XDs since I noticed very little people actually read the original works by Kaluza and Klein.

General Relativity

The first mentioning of adding another dimensions to our three space-like dimensions that we experience every day goes to my knowledge back to Nordström in 1913 [2]. He however did not yet use General Relativity (GR) to build his theory upon. Since we know today that the gravitational potential is not a scalar field, but described by the curvature of space time, let us skip to the next attempt which uses GR as we know it today.

GR couples the metric tensor (g) to a source term of matter fields, whose characteristics are encoded in the stress-energy tensor of the matter. All kind of energy and matter results in such a source term, and hence causes the metric to deviate from flat space. This theory does not say anything about the origin of the source terms. The matter and its properties have to be described by another theory - for example by electrodynamics. Electrodynamics on the other hand has a similar problem. The source for the electromagnetic field (charged particles) is not described by Maxwell's equations [5]. They need to be completed by further equations, e.g. the Dirac equations.

In the beginning of the last century, physicists had just understood gravity as a geometrical effect instead of a field in Minkowski space, so it was only natural to try the same for other fields as well, with the obvious next choice being the electric field. The idea of the early XDs is plain and simple. Einstein's field equations are a set of non-linear differential equations for the metric tensor. They are built up of the Ricci-tensor (two indices) which is a contraction over the full curvature tensor (four indices), and the curvature scalar - a further contraction of the Ricci-tensor. Such a contraction is basically a sum over two indices. The indices on these tensors label space-time directions - that is, in the standard case of GR with three space and one time dimensions, they run from 1 to 4 (or, depending on taste, sometimes from 0 to 3).

Now if one had an additional dimension, then two things happen with Einstein's field equation. First, one has more equations because there are more free indices. Since the Ricci tensor and the metric are symmetric, the number of independent equations is D(D+1)/2, here D is the total number of dimensions. The second thing happening is that the equations with the indices belonging to the 'usual' directions acquire additional terms since the sum runs over the additional indices as well. The trick is then to separate the usual part (sum from 1 to 4) from the additional part (sum over the extra dimension), shift the additional part to the other side of the equations, and read it as a source term. In such a way, one obtains a source term even if the higher dimensional field equations were source free.

Kaluza and Klein

The result is that components of the higher-dimensional metric tensor appear as source terms for the four-dimensional sub-sector that we observe. The first such approach was Theodor Kaluza's whose ansatz uses one additional dimension. In the remaining entries of the metric tensor (those with one index being a 5) he put the electromagnetic potential with a coupling constant alpha (since the metric tensor is dimensionless but the electromagnetic potential isn't)

(Here, the large Latin indices run over all dimensions, the small Greek indices over the usual four dimensions). Kaluza apparently sent a draft of his paper to Einstein in 1919, to ask for his opinion. It got published with a delay of two years [3].

Kaluza derived the higher dimensional field equations in the linear approximation. Generically, all the components of the metric tensor will be functions of all coordinates, including the additional one. This however is in conflict with what we observe. Kaluza therefore added what he called the 'cylinder condition' that set derivatives with respect to the additional coordinates to zero. In the linear approximation, he then found the ansatz to reproduce GR plus electrodynamics.

However, the use of this linear approximation is not necessary, as was shown by Oskar Klein five years later [4]. Klein used a different ansatz for the metric which has an additional quadratic term:

(Sorry, coupling constant is missing, my fault not Oskar's) And he assumed the additional coordinate is compactified on a circle. Then, one can expand all components in a Fourier-series and the zero mode will fulfill Kaluza's cylinder condition' that is, it is independent of the fifth coordinate. However, if you compare both ansätze [7], Klein's and Kaluza's, you will notice that Klein set the g55 component to be constant to one. This is an additional constraint that generally will not be fulfilled. In fact, the additional entry behaves like a scalar field and describes something like the radius of the XD. At this time however, people had little for additional scalar fields.

Klein's derivation is simply one of the most beautiful calculations I know. One just writes down the higher dimensional field equations, parametrizes the metric tensor according to Klein's ansatz, decomposes the equations - and what comes out is GR in four dimensions (in the Lagrangian formulation as well as the field equations), plus the free Maxwell equations.

(Here, the supscript (4) and (5) refer to the 4 and 5 dimensional part of the curvature/metric). Further, the geodesic equation gets an additional term which is just the Lorentz force term and thus describes a charged particle moving in a curved space with an electromagnetic field.

In the course of this derivation, one is lead to identify the momentum in the direction of the fifth coordinate as the ratio of charge over mass (q/m). It can be shown that this quantity is conserved as it should be. Klein concluded that this charge is quantized in discrete steps (this is a geometrical quantization), the first example of the Kaluza-Klein tower.

Extensions and Problems

To understand the excitement this derivation must have caused one has to keep in mind that this was 30 years before Yang and Mills, and the understanding of gauge theory was not on today's status. With today's knowledge, the argumentation appears somewhat trivial. One adds an additional dimension with U(1) symmetry, the compactified dimension. The resulting theory needs to show this symmetry that we know belongs to electrodynamics. From this point of view, it is only consequential to extend the Kaluza-Klein (KK) approach to other gauge symmetries, i.e. non-abelian groups. This was done in 1968 [6].

One has to note however that for non-abelian groups the curvature of the additional dimensions will not vanish, thus flat space is no longer a solution to the field equations. However, it turns out that the number of additional dimensions one needs for the gauge symmetries of the Standard Model U(1)xSU(2)xSU(3) is 1+2+4=7 [10]. Thus, together with our usual four dimensions, the total number of dimensions is D=11. Now exponentiate this finding by the fact that 11 is the favourite number for those working on supergravity, and you'll understand why KK was dealt as a hot canditate for unification.

But there are several problems with the traditional KK approach. First, meanwhile the age of quantum field theory had begun, and all these considerations have been purely classical and unquantized. Even more importantly, there are no fermions is this description - note that we have only talked about the free Maxwell equations. The reason is easy to see: fermions are spin 1/2 fields and unlike vector bosons one can not just write them into the metric tensor. One can of course add additional source terms, but this makes the idea somewhat less appealing [8]. The high hope had been to explain all matter and fields from a purely geometric approach.

If one thinks more about the fermions, one notices another problem: right- and left-handed fermions belong to different electroweak representations, a feature that is hard to include in a geometrical interpretation. Furthermore, there is the problem of stabilization of the compact extra dimensions (the sizes should not or only negligibly depend on the time-like coordinate), and the problem of singularity formation from GR persists in this approach. However. If I consider what landscape of problems other theories suffer from, it makes me wonder why the KK approach was so suddenly given up in the early 70ies. A big part of the reason might simply have been that the quark model got established, and it was the dawn of the particle-physics era.

The 'modern' extra dimensions differ from the KK approach by not attempting to explain the other standard model fields as components of the metric. Instead, fermionic- and gauge-fields are additional fields that are coupled to the metric. They are allowed to propagate into the extra dimensions, but are not themselves geometrical objects. Most features of the KK approach remain, most notably the geometrical quantization of the momenta into the extra dimensions and thus the KK-tower of excitations. So remains the problem of stabilization, singularities and quantization (for higher dimensional quantum field theories the coupling constants become dimensionful). However, for me this 'modern' approach is considerably less appealing as one has lost the possibility to describe gauge symmetries and standard model charges as arising from the same principle as GR.

But obviously, the largest problem with the KK approaches was - and still is - that it is not clear whether it is just a mathematical possibility or indeed a description of reality. As Oskar Klein put it in 1926:

    "Ob hinter diesen Andeutungen von Möglichkeiten etwas Wirkliches besteht, muss natürlich die Zukunft entscheiden."

    "Whether these indications of possibilities are built on reality has of course to be decided by the future." [9]


[1] And if you read the wikipedia entry on Perry Rhodan you find a use for the word 'Zeitgeist'...

G. Nordström, "Zur Theorie der Gravitation vom Standpunkt des Relativitätsprinzips" Annalen der Physik, vol. 347, Issue 13, pp.533-554 (1913); G. Nordström, "Über die Möglichkeit, das elektromagnetische Feld und das Gravitationsfeld zu vereinigen" ( "About the possibility to unify the electric field and the gravitational field" ) Physik. Zs. 15, 504-506 (1914) [Abstract]

[3] T. Kaluza, "Zum Unitätsproblem der Physik'' ("On the Problem of Unity in Physics, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) (1921) 966.

[4] O. Klein, "Quantentheorie und fünfdimensionale Relativitätstheory" ("Quantum Theory and fivedimensional General Relativity", Z. Phys. 37, 895 (1926).

[5] Unlike to what the Wikipedia entry states, the Lorentz force law can not be derived from the Maxwell equations without further assumptions (like a Lagrangian for the coupled sources). E.g. Maxwell's equations are perfectly consistent for a static superposition of two negativley charged objects, just that we know the charged particles would repel and the configuration can't be static.

[6] R. Kerner, "Generalization of the Kaluza-Klein theory for an arbitrary non-abelian gauge group", Ann. Inst. Henri Poincare, 9, 143-152 (1968)

[7] Contrary to the wide spread believe, the plural of the German word 'Ansatz' is not 'Ansatzes' but 'Ansätze' (pronounced 'unsetze'). 'Ansatz' could be roughly translated as 'a good point to start', or a preparation. E.g. the pre-stage for yeast dough is called 'Ansatz'...

[8] Which finally brings us to the topic on which I lost two years during my Ph.D. time, namely the question whether one can built up the metric tensor from spin 1/2 fields. I only learned considerably later that most of this approach had been worked out in the mid 1980ies, see e.g. hep-th/0307109 and references therein.

[9] He indeed writes it has to be decided 'by' the future not 'in' the future. Quotation from Ref. [4]

Further Litarature

Hello From Rome!

On the weekend, I flew to Rome to addend the String Pheno 2007. Meanwhile, my baggage decided to have a vacation in Palermo. It arrived with four days delay yesterday evening. I've been wearing the same clothes since the weekend, but this morning I saw myself faced with an incredible selection! A second jeans! Two T-shirts! A dress!

However, despite these inconveniences, I had a so far very pleasant stay since it turned out that Amara Graps (you might know her from the blogosphere) lives nearby. She was so kind to borrow me some clothes, and yesterday we spent a very nice afternoon in Rome. Since I am currently sitting in mentioned conference (and should at least pretend to listen) let me instead show you a photo.

Okay, now I have to prepare my talk... later more...

Wednesday, June 06, 2007

arXiv User Survey

Just a quick link: in case you ever wanted to express your opinion about the arXiv, take their poll (from June 4-8)

It will take approximately 20 minutes, and has plenty of comment options to complain about the new arXiv listing (or the eternal bug in the search field if you search for a tag containing the word 'not').

Other points that I find worth mentioning: the arxiv should allow comments on papers, and a ranking (different from times cited). Comments would be helpful to avoid the increasing amount of 'reply-to-reply-to-reply-to's, ranking I would find a good idea because it's become almost impossible to find a good review or lecture notes if one doesn't know the author (and lecture notes don't usually become top-cites).

Monday, June 04, 2007

The Teddy Factory

A somewhat belated 'Hello' from Germany!

What is new here? Well, the whole country is upset about the G8 summit in a city that nobody had ever heard of before. Border security has been enhanced, and Lufthansa had all passengers sniffed at by a huge dog. I can also report that Lufthansa meal sizes have increased again, they still serve liquor for free, and my flight was otherwise a classical chicken-or-pasta event. Besides this: congratulations America! When I was here last time late 2006, newspapers were praising the USA for remembering democracy, now you've succesfully regained global-asshole status (I am currently not in the mood to elaborate on global energy scarcity, for an extended version of my opinion, see Global Warming.)

Sure sure, America is still faster, bigger, better: Germany still doesn't have penny trays (I consider that to be one of the most important advantages of the USA), they still don't know what 'cash back' is, and shops are still closed when I finally find the time to go there.

Something completely different: since all-my-mother's-children have moved out and the cat died, the house gets populated by an ever increasing amount of handmade Teddy bears.

[Figure: Representavive sample of the German population
-- Can you find out which item does not fit in?]

Friday, June 01, 2007

Bouncing Neutrons in the Gravitational Field

I remember a moment of excitement and puzzlement early on in my first class in quantum mechanics, when our professor announced that now, he would discuss the "freier Fall" in quantum mechanics. I was excited, because it seemed great to me to transfer such an elementary situation as the free fall of a stone into the realm of quantum mechanics, and puzzled, because I knew that the gravitational potential is so extremely weak that it can be safely ignored on scales where quantum mechanics comes into play - at least, in most cases. Alas, that lecture was quite a disappointment, because of an ambiguity of the German wording "freier Fall": it can mean both the free "fall", and the free "case" (as in Wittgenstein's famous dictum "Die Welt ist alles, was der Fall ist.") - and what we learned in our lecture was just about the free case, the quite boring plane wave motion of a quantum particle subject to no potential whatsoever.

What we did not learn in our class was that, even back at that time, there had been several clever experiments with neutrons which demonstrate the influence of the gravitational potential on the phase of the neutron wave function using interferometers. Neutrons, of course, are ideal particles to perform such experiments, since they have no electric charge and are not subject to the influence of the ubiquitous electromagnetic fields.

But only over the last few years, new experiments have been realised that show directly the quantisation of the vertical "free fall" motion of neutrons in the gravitational field of the Earth. I had heard about them some time ago in connection with their possible role for the detection of Non-Newtonian forces, or the modifications of Newtonian gravity at short distances. Then, earlier this year, I heard a talk by one of the experimenters at Frankfurt University, and I was quote fascinated when I followed the papers describing the experiments.

The essential point of these experiments is the following: If you prepare a beam of very slow neutrons - with velocities about 10 m/s - you can make them hop over a reflecting plane much like you can let hop a pebble over the surface of a lake. Then, you can observe that the vertical part of the motion of the neutrons - with velocities smaller than 5 cm/s - is quantised. In fact, one can detect the quantum states of neutrons in the gravitational field of the Earth! Let me explain in more detail...

Free Fall in Classical Mechanics ...

In order to better understand the experiment, let's go back one step and consider the very simple motion of an elastic ball which is dropped on the ground. If the ground is plane and reflecting, and the ball is ideally elastic such that there is no dissipation of energy, the ball will jump back to the height of where it was dropped from, fall down again, jump back, fall, and so on. The height of the ball over ground as a function of time is shown as the blue curve in the left of this figure: it is simply a sequence of nice parabolas.

We can now ask, What is the probability to find the bouncing ball in a certain height above the floor? For example, we could make a movie of the bouncing ball, take a still at some random time, and check the distribution of the height of the ball if we repeat this for many random stills. The result of this random sampling of the bouncing motion of the ball is the probability distribution shown in red on the right-hand side of figure. The probability to find the ball at a certain height in this idealised, "stationary" situation, where the elastic ball is bouncing forever, is highest at the upper turning point of the motion, and lowest at the bottom, where the ball is reflected.

... and in Quantum Mechanics

So much for classical mechanics, as we know it from every-day life. In quantum mechanics, unfortunately, there is not anymore such a thing as the path of a particle, with position and velocity as well-defined quantities at any instant in time. However, it still makes sense to speak of stationary states, and of the probability distribution to find a particle at a certain position. In quantum mechanics, it is the wave function which provides us with this probability distribution by calculating its square. And the law of nature determining the wave function is encoded in the famous Schrödinger equation. The Schrödinger equation for a stationary state is an "eigenvalue equation", whose solution yields, at the same time, the wave function and the value of the energy of the corresponding state. For the motion of a particle in a linear potential - such as the potential energy mgx of a particle with mass m at height x above ground in the gravitational field with acceleration g at the surface of the Earth - it reads

In some cases, there are so-called "exact solutions" to the Schrödinger equation - wave functions that are given by certain functions one can look up in thick compendia, or at MathWorld. These functions usually are some beautiful beasts out of the zoo of so-called "special functions". Such is the case for the motion of a particle in a linear potential, where the solution of the Schrödinger equation is given by the Airy function Ai(x). Interestingly, this function first showed up in physics when the British astronomer George Airy applied the wave theory of light to the phenomenon of the rainbow...

Quantum States of Particles in the Gravitational Field

As a result of solving the Schrödinger equation, there is a stationary state with a minimal energy - the ground state - and a series of excited states with higher energies. Here is how the wave function of the second excited state of a particle in the gravitational field looks like as a function of the height above ground:

The wave function, shown on the left in magenta, oscillates through two nodes, and goes down to zero exponentially above the classical limiting height, which corresponds to the upper turning point of the parabola of a classical particle with the same energy. For neutrons in this state, this height is 32.4 µm above the plane. The green curve on the right shows the probability density corresponding to the wave function. It is quite different from the classical probability density, shown in red. As a characteristical property of a quantum system, there is, besides the two nodes, a certain probability to find the particle above the classical turning point. This is an example of the tunnel effect: there is a chance to find a quantum particle in regions where by the laws of classical physics, it would not be allowed to be because of unsufficient energy.

However, going from the ground state to ever higher excited states eventually reproduces the probability distribtion of classical physics. This is what is called the correspondence principle, and you can see what it means if you have a look at the wave function for 60th excited state: Here, the probability distribution derived from the quantum wave function follows already very closely the classical distribution.

So far, we have been talking about theory: the Schrödinger equation and its solutions in guise of the Airy function. There is no reason at all to doubt the validity of the Schrödinger equation: it has been thoroughly tested in innumerable situations, from the hydrogen atom to solid state physics. However, in all these situations, the interaction of the particles involved is electromagnetic, and not by gravitation. For this reason, it is extremely interesting to think about ways to check the solution of the Schrödinger equation for particles in the gravitational field. As we have seen before, the best way to do this is to work with neutrons, in order to avoid spurious electromagnetic effects.

Bouncing Neutrons in the Gravitational Field

Unfortunately, it is so far not possible to scan directly the probability distribution of neutrons in the gravitational field. However, in a clever experimental setup, one can look at the transmission of neutrons through a channel between with a horizontal reflecting surface where they can bounce like pebbles over a lake, and an absorber ahead. This is a rough sketch of the setup:

The decisive idea of the experiment is to vary the height of the absorber above the reflecting plane, and to monitor the transmission of neutrons as a function of this height. If the height of the absorber is to low, the ground state for the vertical motion of the neutrons does not fit into the channel, and no neutrons will pass the channel. Transmission sets in once the height of the channel is sufficiently large to accommodate the ground state wave function of the vertical motion of the neutron. Moreover, whenever with increasing height of the channel, one more of the excited wave function fits in, the transmission should increase. The first of these steps, and the corresponding wave functions and probability densities, are shown in this figure:

The interesting point now is, can this stepwise increase of transmission be observed in actual experimental data? Here are measured data, and indeed - the first step is clearly visible, and the second and third step can be identified:

Adapted by permission from Macmillan Publishers Ltd: Nature (doi:10.1038/415297a), copyright 2002.

This has been the first verification of quantised states of particles in the gravitational field!

What can be learned

You may wonder if the experiment may not have shown just some "particle in a box" quantisation, since the channel for the neutrons formed by the reflecting plane and the absorber may make up such a box. This objection has been raised, indeed, in a comment paper, and has been answered by detailed calculations, and improved experiments: the conclusion about quantisation in the gravitational field remains fully valid!

However, limits about modifications of Newtonian gravity from this experiment remain restricted. Such a modification would change the potential the neutrons are moving in. For example, a short-range force caused by the matter of reflecting plane could contribute to the potential of the neutrons. However, as comes out, such an additional potential would be very weak and have nearly no influence at all on the overall wave function of the neutron.

Moreover, it is clear that in this experiment, the gravitational field is always a classical background field, which itself is not quantised at all. There may be the possibility that a neutron undergoes a transition from, say, the second to the first quantised state, thereby emitting a graviton - similar to the electron in an atom, which emits a photon when the electron makes a transition. Unfortunately, this probability is so low that it is not reasonable to expect that it may ever be measured....

But all these restrictions do not change at all the main point that this a very exciting, elementary experiment, which could find its way into textbooks of quantum mechanics!

Here are some papers about the "bouncing neutron" experiment:

Quantum states of neutrons in the Earth's gravitational field by V.V. Nesvizhevsky, H.G. Boerner, A.K. Petoukhov, H. Abele, S. Baessler, F. Ruess, Th. Stoeferle, A. Westphal, A.M. Gagarski, G.A. Petrov, and A.V. Strelkov; Nature 415 (2002) 297-299 (doi: 10.1038/415297a) - The first description of the result.

Measurement of quantum states of neutrons in the Earth's gravitational field by V.V. Nesvizhevsky, H.G. Boerner, A.M. Gagarsky, A.K. Petoukhov, G.A. Petrov, H.Abele, S. Baessler, G. Divkovic, F.J. Ruess, Th. Stoeferle, A. Westphal, A.V. Strelkov, K.V. Protasov, A.Yu. Voronin; Phys.Rev. D 67 (2003) 102002 (doi: 10.1103/PhysRevD.67.102002 | arXiv: hep-ph/0306198v1) - A more detailed description of the experimental setup and the first results.

Study of the neutron quantum states in the gravity field by V.V. Nesvizhevsky, A.K. Petukhov, H.G. Boerner, T.A. Baranova, A.M. Gagarski, G.A. Petrov, K.V. Protasov, A.Yu. Voronin, S. Baessler, H. Abele, A. Westphal, L. Lucovac; Eur.Phys.J. C 40 (2005) 479-491 (doi: 10.1140/epjc/s2005-02135-y | arXiv: hep-ph/0502081v2) - Another more detailed discussion of the experimental setup, possible sources of error, and the first results.

Quantum motion of a neutron in a wave-guide in the gravitational field by A.Yu. Voronin, H. Abele, S. Baessler, V.V. Nesvizhevsky, A.K. Petukhov, K.V. Protasov, A. Westphal; Phys.Rev. D 73 (2006) 044029 (doi: 10.1103/PhysRevD.73.044029 | arXiv: quant-ph/0512129v2) - A long and detailed discussion of point such as the "particle in the box" ambiguity and the role of the absorber.

Constrains on non-Newtonian gravity from the experiment on neutron quantum states in the Earth's gravitational field by V.V. Nesvizhevsky, K.V. Protasov; Class.Quant.Grav. 21 (2004) 4557-4566 (doi: 10.1088/0264-9381/21/19/005 | arXiv: hep-ph/0401179v1) - As the title says: a discussion of the constraints for Non-Newtonian forces.

Spontaneous emission of graviton by a quantum bouncer by G. Pignol, K.V. Protasov, V.V. Nesvizhevsky; Class.Quant.Grav. 24 (2007) 2439-2441 (doi: 10.1088/0264-9381/24/9/N02 | arXiv: quant-ph/07702256v1) - As the title suggests: the estimate for the emission of a graviton from the neutron in the gravitational field.

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