Wednesday, January 25, 2012

The Planck length as a minimal length

The best scientific arguments are those that are surprising at first sight, yet at second sight they make perfect sense. The following argument, which goes back to Mead's 1964 paper "Possible Connection Between Gravitation and Fundamental Length," is of this type. Look at the abstract and note that it took more than 5 years from submission to publication of the paper. Clearely, Mead's argument seemed controversial at this time, even though all he did was to study the resolution of a microscope taking into account gravity.

For all practical purposes, the gravitational interaction is far too weak to be of relevance for microscopy. Normally, we can neglect gravity, in which case we can use Heisenberg's argument that I first want to remind you of before adding gravity. In the following, the speed of light c and Planck's constant ℏ are equal to one, unless they are not. If you don't know how natural units work, you should watch this video, or scroll down past the equations and just read the conclusion.

Consider a photon with frequency ω, moving in direction x, which scatters on a particle whose position on the x-axis we want to measure (see image below). The scattered photons that reach the lens (red) of the microscope have to lie within an angle ε to produces an image from which we want to infer the position of the particle.

According to classical optics, the wavelength of the photon sets a limit to the possible resolution Δx But the photon used to measure the position of the particle has a recoil when it scatters and transfers a momentum to the particle. Since one does not know the direction of the photon to better than ε, this results in an uncertainty for the momentum of the particle in direction xTaken together one obtains Heisenberg's uncertainty principle
We know today that Heisenberg's uncertainty principle is more than a limit on the resolution of microscopes; up to a factor of order one, the above inequality is a fundamental principle of quantum mechanics.

Now we repeat this little exercise by taking into account gravity.

Since we know that Heisenberg's uncertainty principle is a fundamental property of nature, it does not make sense, strictly speaking, to speak of the position and momentum of the particle at the same time. Consequently, instead of speaking about the photon scattering off the particle as if that would happen in one particular point, we should speak of the photon having a strong interaction with the particle in some region of size R (shown in the above image).

With gravity, the relevant question now will be what happens with the measured particle due to the gravitational attraction of the test particle.

For any interaction to take place and subsequent measurement to be possible, the time elapsed between the interaction and measurement has to be at least of the order of the time, τ, the photon needs to travel the distance R, so that τ is larger than R. (The blogger editor has an issue with the "larger than" and "smaller than" signs, which is why I avoid using them.) The photon carries an energy that, though in general tiny, exerts a gravitational pull on the particle whose position we wish to measure. The gravitational acceleration acting on the particle is at least of the orderwhere G is Newton's constant which is, in natural units, the square of the Planck length lPl. Assuming that the particle is non-relativistic and much slower than the photon, the acceleration lasts about the duration the photon is in the region of strong interaction. From this, the particle acquires a velocity of vaRThus, in the time R, the aquired velocity allows the particle to travels a distance of LGω.

Since the direction of the photon was unknown to within ε, the direction of the acceleration and the motion of the is also unknown. Projection on the x-axis then yields the additional uncertainty ofCombining this with the usual uncertainty (multiply both, then take the square root), one obtainsThus, we find that the distortion of the measured particle by the gravitational field of the particle used for measurement prevents the resolution of arbitrarily small structures. Resolution is bounded by the Planck length, which is about 10-33cm. The Planck length thus plays the role of a minimal length.

(You might criticize this argument because it makes use of Newtonian gravity rather than general relativity, so let me add that, in his paper, Mead goes on to show that the estimate remains valid also in general relativity.)

As anticipated, this minimal length is far too small to be of relevance for actual microscopes; its relevance is conceptual. Given that Heisenberg's uncertainty turned out to be a fundamental property of quantum mechanics, encoded in the commutation relations, we have to ask then if not this modified uncertainty too should be promoted to fundamental relevance. In fact, in the last 5 decades this simple argument has inspired a great many works that attempted exactly this. But that is a different story and shall be told another time.

To finish this story, let me instead quote from a letter that Mead, the author of the above argument, wrote to Physics Today in 2001. In it, he recalls how little attention his argument originally received:
"[In the 1960s], I read many referee reports on my papers and discussed the matter with every theoretical physicist who was willing to listen; nobody that I contacted recognized the connection with the Planck proposal, and few took seriously the idea of [the Planck length] as a possible fundamental length. The view was nearly unanimous, not just that I had failed to prove my result, but that the Planck length could never play a fundamental role in physics. A minority held that there could be no fundamental length at all, but most were then convinced that a [different] fundamental length..., of the order of the proton Compton wavelength, was the wave of the future. Moreover, the people I contacted seemed to treat this much longer fundamental length as established fact, not speculation, despite the lack of actual evidence for it."

1. While extraordinary claims demand extraordinary evidence, I am always amazed when claims which are obvious in hindsight, even mathematical theorems, are treated sceptically. Maybe it is because people don't want to admit that they are ashamed that they didn't think of it first.

2. http://www.nature.com/scitable/topicpage/human-chromosome-number-294

Feynman's Nobel Prize pivoted on a V−A (left-handed) Lagrangian for weak interactions. Everybody knew it was S-T. It had to be S-T. Sudarshan was correct much earlier on. He was told to piss off, as were Yang and Lee. Yang and Lee had Madame Wu. Feynman had Feynman (and Gell-Mann). Sudarshan got nothing.

Gravitation is not a fashion statement. The plural of "anecdote" is not "data," and data are not information. Why must the vacuum be fundamentally continuous and isotropic toward fermionic mass?

3. I dunno, Al. Why not? You tell us.

4. Also, thanks Bee, for this wonderful blogpost.

I can't help thinking though, you've only told us half a story, which is cool, because it builds interest therefore anticipation.

This is the sort of stuff that makes Science fun!

Can't wait to hear how this pans out. Good on ya.

5. Hi Bee,

I am just trying to orientate from your perspective.:)At that Planck length of course one runs into trouble with some geometrical description so how indeed would some quasi-description ever be satisfied as to defining the shape of things in a matter orientated world?

LISA will be sensitive to waves in the frequency band between 0.03 milliHertz to 100 milliHertz, including signals from massive black holes that merge at the center of galaxies, or that consume smaller compact objects; from binaries of compact stars in our Galaxy; and possibly from other sources of cosmological origin, such as the very early phase of the Big Bang, and speculative astrophysical objects like cosmic strings and domain boundaries

Sir Roger Penrose of course has his own ideas too. What is the basis of his experimental views?

Accepting that wavefunctions are physically real, Penrose believes that things can exist in more than one place at one time. In his view, a macroscopic system, like a human being, cannot exist in more than one position because it has a significant gravitational field. A microscopic system, like an electron, has an insignificant gravitational field, and can exist in more than one location almost indefinitely. See:The Penrose interpretation

Hmmm.......

Best,

6. Bee,

Your choice of font makes it difficult to perceive italicized elements of quoted sources as they are demonstrated in comment section. Has comment section been given an italicized choice? This is new I think?

If no desire to change will adapt to the way quotes are demonstrated according to that selection. No problem for the future.

Best,

7. Shouldn't that be generalized to a Planck volume instead?

Which you could combine the the Heisenberg uncertainty principle to ensure that singularities are not observable?

Namely:
$\Delta volume \times \Delta mass \geq \frac{\hbar \times length_planck}{c}$

8. Hi Bee,

The derivation of the Heisenberg principle, interestingly enough, does not involve the electric charge, i.e., how the photon couples to the particle is not relevant.

For the gravity case, G enters, however, we are only after Δx. I suppose a careful argument using EM analogous to the Mead argument would yield

Δ x ≥ fine structure constant * compton wavelength (or some such).

Since the gravity bound is much more stringent than this, I can see why people were skeptical of Mead's argument.

Thanks for the post!
-Arun

9. Hi Plato,

It's not so much my choice of font as the default that came with the template. You are right, the italics are difficult to see. I'll try to change the font, I'm not a big fan of arial anyway. Best,

B.

10. Hi Bee,

The Compton wavelength limit normally would have a priority. But I guess he considers non relativistic physics only.

11. Hi Giotis,

No, he does it fully relativistic also, I just haven't added the more complete argument here. If you turn up the energy of the photon you can get down the wavelength. If you don't take into account gravity, you can do this arbitrarily. The point is here that when you reach Planckian energies, you start perturbing the particle you are trying to measure in such a way that going to even higher frequencies doesn't help. Best,

B.

12. Hi Aaron,

Yes, without spherical symmetry one may expect that volumes are the relevant quantity to talk about. This argument has been made eg here (page 5/6), and though plausible it is not particularly bloggable if you see what I mean. Best,

B.

13. and what is the relation with the Compton wavelength?

14. I am probably misunderstanding the question. The Compton wavelength of what?

15. The Compton wavelength as a limit for which positions measurements become ill defined due to creation of particles. I mean if the energy of the photon exceeds some limit according to QFT particles would be created.

16. The photon's energy always exceeds the Planck energy in some restframe. But yes, if that is what you mean, if it interacts with the particle at very high energies, a microscope isn't anymore a very good analogy since, as you say, you'd have a very inelastic scattering and you'd have to figure out what was going on from the outgoing particles rather than watching photons on a screen.

17. Thanks Bee for taking the time to answer these questions.

18. Where could I get a free copy of the paper? I am very interested into study it!
Best wishes!

19. Hi Juan,

Unfortunately, I cannot be of help. I only have a printed version of the paper, and presently no journal access, so cannot download a PDF. Maybe somebody else can send one? Best,

B.

20. This comment has been removed by the author.

21. Hi All,

This link expires in 10 days.

Best wishes

22. Given frequency, wavelength inversely varies with refractive index. Anomalous dispersion reduces local speed of light to 340 m/s (ruby), to 17 meters/sec (BEC), and to zero. Electromagnetic Planck length may suffer interaction wildly altering scale. Gravitation is interaction. Equivalence Principle (EP) composition, field, and photon tests are inert to 5x10^(-14) difference/average. Physics cannot derive fermionic mass parity asymmetries. They are manually inserted. If gravitation is geometry, the test is massed geometry. Opposite shoes violate the EP.

Chemically and macroscopically identical, enantiomorphic crystal lattices violate the EP. 1) Left- versus right-handed alpha-quartz or gamma-glycine single crystals, Eotvos experiment. 2) North-south aligned enantiomorphic crystal lattices and changing enthalpies of fusion to identical achiral melts over 24 hours, benzil.

Physics, like Euclid, contains no errors. Euclid fails on a sphere. Falsification occurs external to founding postulates.

23. Hi Bee,

It’s indeed interesting to wonder what if anything can be defined as the minimum of length and yet as J.S. Bell would point out quite another thing to consider what exactly it is we are attempting to have measured as to be so defined.

“The concept of 'measurement' becomes so fuzzy on reflection that it is quite surprising to have it appearing in physical theory at the most fundamental level. Less surprising is perhaps that mathematicians, who need only simple axioms about otherwise undefined objects, have been able to write extensive works on quantum measurement theory - which experimental physicists do not find it necessary to read. Mathematics has been well called 'the subject in which we never know what we are talking about’ [ Bell quoting Bertrand Russell]. Physicists confronted with such questions, are soon making measurement a matter of degree, talking of ‘good’ measurements and ‘bad’ ones. But the postulates quoted no nothing of ‘good’ and ‘bad. And does not any analysis of measurement require concepts more fundamental than measurement? And should not the fundamental theory be about these more fundamental concepts?”

-J.S. Bell “Quantum Mechanics for Cosmologists” Speakable and Unspeakable in Quantum Mechanics (First Edition) p.117-118

Best,

Phil

24. Hi Genorb,

Thanks! That is very useful. Best,

B.

25. Does lightphotons and neutrinos interact at all? Neutrinos are still leptons?

At c does the photon behave relativistic also then? A photon is massless. c is measured by em-force as Uncle Al points out. You say Planck energy is always higher.

26. In dense aether model the observable reality appears like fractal landscape under the fog (or like the undulating water surface being observed via its own ripples). The density fluctuations of dark matter replicate the foamy structure of space-time at short scales (Higgs field). After then two AdS/CFT dual approaches could be applied here:

1) Nothing smaller than these density fluctuations can be observed there in similar way, like the objects outside of visibility scope of landscape under the fog.

2) These Universe at such small scales doesn't differ from our Universe at the human observer scale, we just cannot observe it clearly because of omnipresent quantum noise.

27. http://physicsforme.wordpress.com/2012/01/22/are-opera-neutrinos-faster-than-light-because-of-non-inertial-reference-frames/

28. L.H. Thomas used something Frame like to analyze the precession of the electron, parallel transported around an orbit, so something is not quite a geometrical point. Perhaps one might argue that quanta are more fundamental than idealized spacetime geometry, but merely consistent with it.

29. Just as exceeding the speed of light is "impossible" because it would turn time "inside out," so exceeding the Plank length limit is "impossible" because it would turn space "inside out." So maybe this is where we need to look when we look for those tiny, curled up "extra dimensions."

30. I guess patience is a virtue.

31. About the speed of light limit. I would like to say that in multitemporal relativities in which other speeds of light could appear (I know, it is a crazy idea, there is no evidence of that although DM/DE are puzzling too, and people usually argue hardly against multitime theories and other geometries beyond the riemannian one), and other extensions of relativistic symmetries, the limit is no longer a limit. The speed of light limit is closely related to the Lorentz invariance of our 3+1 world. Either if you change the (pseudo)riemannian structure of the theory, or you include new "degrees of freedom" like extra-times (curiously it doesn't happen with spatial-like coordinates) or some other multivector structures related to Finsler-like objects, there is no problem with speeds greater than c. If vacuum is some kind of medium, it is also reasonable that could exist something faster than light. I found myself puzzled when Bee published her paper about quantum superpositions of speeds of light. Indeed, there is something there to be understood better.