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Tripled Pauli Statistics

  1. Apr 7, 2007 #1


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    Back in 2002, Lubos Motl wrote a paper on

    Title: An analytical computation of asymptotic Schwarzschild quasinormal frequencies
    Lubos Motl, Adv.Theor.Math.Phys. 6 (2003) 1135-1162
    Recently it has been proposed that a strange logarithmic expression for the so-called Barbero-Immirzi parameter, which is one of the ingredients that are necessary for Loop Quantum Gravity (LQG) to predict the correct black hole entropy, is not another sign of the inconsistency of this approach to quantization of General Relativity, but is rather a meaningful number that can be independently justified in classical GR. The alternative justification involves the knowledge of the real part of the frequencies of black hole quasinormal states whose imaginary part blows up. In this paper we present an analytical derivation of the states with frequencies approaching a large imaginary number plus ln 3 / 8 pi M; this constant has been only known numerically so far. We discuss the structure of the quasinormal states for perturbations of various spin. Possible implications of these states for thermal physics of black holes and quantum gravity are mentioned and interpreted in a new way. A general conjecture about the asymptotic states is stated. Although our main result lends some credibility to LQG, we also review some of its claims in a critical fashion and speculate about its possible future relevance for Quantum Gravity.


    In the above paper, (equations 51-54) Lubos obtains a transmission amplitude for the spin-j energy emitted by a black hole. The spin-1/2 and spin-1 amplitudes follow the expected Fermi and Bose statistics:

    [tex]T(\omega) = 1/(e^{\beta_{\textrm{Hawking}}\;\omega} \; \mp 1)[/tex]

    where the + sign is the Fermi and the - sign is the Bose statistics, and

    [tex]\beta_{\textrm{Hawking}} \; = 1/T_{\textrm{Hawking} }\;.[/tex]

    What's strange is that the spin-2 amplitude follows the following rule:

    [tex]T(\omega) = 1/(e^{\beta_{\textrm{Hawking}}\;\omega} \;+ 3)[/tex]

    Lubos calls this "tripled Pauli statistics", writing:

    I would like to suggest that the above form can be put into the standard form by making two assumptions. First, there are degrees of freedom present in the system that are not apparent in the usual quantum mechanics. For each Pauli state, one finds three states at high temperature, which can be thought of as three colors. Second, the colored states are of extreme high energy (on the order of the Planck energy), and consequently are suppressed, leaving only the colorless state to contribute to the exponential.

    Any comments?
  2. jcsd
  3. Apr 9, 2007 #2


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    Goodness, Carl. Nobody seems interested in your remarks. Perhaps Lubos would care to comment on both this paper of his, and some of his more recent MHV work, which involves operad diagrams.
  4. Apr 9, 2007 #3
    As Lubos notes on page 7, the group SO(3) must replace SU(2) in order to recover the ln(3) numerical coincidence.

    In the 2004 paper Ln(3) and Black Hole Entropy, Dreyer discusses the implications of replacing SU(2) by SO(3). From the computational perspective, he remarks that Wheeler's “It from bit” is upgraded to “It from trit” as a result. This is another cool example of ternary logic making its appearance in black hole calculations.
  5. Apr 9, 2007 #4


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    I don't think I've supported the idea very well, Kea. Intuitively it makes sense to me, but it has big holes.

    Suppose you have two states, spin +z and spin -z, and you combine them into a statistical mixture. The resulting density matrix is:

    + 0.5\left(\begin{array}{cc}0&0\\0&1\end{array}\right)
    = \left(\begin{array}{cc}0.5&0\\0&0.5\end{array}\right)[/tex]

    This density matrix is the same one you get when you add two spin +x and spin -x:

    + 0.5\left(\begin{array}{cc}0.5&-0.5\\-0.5&0.5\end{array}\right)
    = \left(\begin{array}{cc}0.5&0\\0&0.5\end{array}\right)[/tex]

    So the information about the composition of the density matrix has been destroyed when the statistical mixture was produced. This gives the usual quantum statistics. Of course this has been well tested experimentally, but only at very low energies.

    The strange statistics appear with even spin objects. This would be spin-0 and spin-2. There are no fundamental spin-0 particles other than the Higgs, and its statistics have not been tested. The only fundamental spin-2 particle is the graviton, again with no experimentally tested statistics. For the moment, let us look at the spin-0 fundamental particles.

    My point of view, of course, is that the fundamental spin-1/2 (and spin-1) particles are not fundamental at all. But I think that the differences only appear at order Planck mass, and so a low energy approximation, like the one these gravitation theorists have been working on, will be unable to detect the difference. I don't think that the spin-0 and spin-2 particles are fundamental either.

    Tripled Pauli statistics has two problems for standard quantum mechanics, as far as being the statistics for the Higgs and graviton. The first problem is that they are Fermi statistics, rather than Bose statistics. The second problem is that damned factor of 3.

    In standard quantum mechanics, statistics are tied up with spin and it is impossible for spin-0 particles, even fundamental ones that have not yet been observed, to have any sort of Fermi statistics. This was an issue that I lost my virginity on 2 years ago. Let me try and explain it.

    Given a spin-1/2 particle, one can write it as a sort of linear combination of two chiral halves, say right handed and left handed. If one were unaware of the weak force and all that, one could justify treating these two halves as parts of an undividable whole makes complete sense. But the forces (other than mass interaction) preserve handedness, and this, together with the clean way that the left and right handed particles interact in the mass term, implies that the fundamental particles are not the complicated spin-1/2 things, but instead the simpler handed particles.

    Thus the spin-1/2 particles are built up from two chiral handed particles.

    Now you know that the spin-1/2 particles do Fermi statistics. But what about the handed particles? To the extent that they are the limit obtained by accelerating a spin-1/2 particle parallel or antiparallel to their direction of spin their statistics must be the same as that of the spin-1/2 particles, and they therefore must take Pauli statistics.

    However, the chiral halves are not spin-1/2 particles in that they have only one complex degree of freedom. The situation is similar to how massless spin-1 particles lose a degree of freedom and can take only spin+1 and spin -1. All this sort of suggests that a fundamental scalar particle should take Fermi statistics, but this is in violation to the spin statistics theorem.

    The spin statistics theorem relies on the Poincare group, (for a short description see Baez.) It says that spin-s must have Fermi or Bose statistics according as s is even or odd. To violate this, one must give up on the Poincare group, and therefore, on relativity. Uh, I lost my virginity on relativity four years ago when I first started working on the proper time geometry.

    Now if you ignore the Poincare group and the spin statistics theorem, you can assign the incorrect statistics to spin-0 particles if you like. But to get here, you have to drop relativity.

    The other problem is the factor of three. To make a spin-0 particle from a spin-1/2, one can combine two opposite spins together. As mentioned above, this is normally treated as a single state, but the density matrix can be built three different ways.

    The three ways you can combine spin-1/2 become three different velocity directions for handed particles that travel at speed c. Intuitively, these are a lot more difficult to imagine as not contributing to three different statistical mixtures, and this gives the correct statistics for the spin-0 particles.

    Physics is like a cloth that is woven from very long threads. The problem with tripled Pauli statistics is that it implies that a very large number of threads that have been accepted for a very long time are not parts of reality, but instead are just mathematical coincidences. For this reason, it is a lot easier to just ignore tripled Pauli statistics and I expect that this is what theorists will mostly do.
  6. Apr 9, 2007 #5


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    Indeed, Carl. Methinks this may indeed be the case. And don't worry about the ln 3. It follows perfectly naturally from the meta-operad entropy invariant acting on the prime 3.

    Convincing the powers that remain may prove to be difficult.

  7. Apr 9, 2007 #6


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    The big problem with SO(3) is that it is the classical rotation group. This is the rotation group one expects for classical velocity instead of a quantum object. The implication is that quantum mechanics should have a classical foundation.

    I've seen the Wheeler "it from bit" speculation, and I think it is the pointless guessing of a guy who doesn't have the slightest clue what the foundations of physics look like. He was proposing that reality was not physical. This is just yet another example of the sophomoric philosophical speculation that Mach started with his claiming that atoms were not physical, and that got us into the difficulty that physics is currently mired in.

    The best quote from your link is this one, which exactly expresses my disdain for Wheeler's thinking: Wheeler has condensed these ideas into a phrase that resembles a Zen koan ...

    Zen koans are beautiful, but they are not physics and never will be. They are one of the two last desperate attempts that a physicist who has given up trying to understand the foundations makes. The other is to begin justifying the equations of physics by recourse to the anthropic principle.
  8. Apr 9, 2007 #7


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    Thank you, Carl.
  9. Apr 9, 2007 #8


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    Well, of course we think differently on this point. Perhaps people should have thought a bit more about why Mach was so against atoms.

  10. Apr 10, 2007 #9
    What's nice about SO(3) is that it exhibits 'triality', in the sense of three inequivalent subgroup embeddings. However, so do SU(3), Sp(6) and F4. In each case, we're still talking about swapping idempotents. You can see where I'm going with this. :wink:
    Last edited: Apr 10, 2007
  11. Apr 12, 2007 #10


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    I just invited Lubos to join our discussion (in all seriousness) by posting on both his blog and Woit's blog. Unfortunately, since both Motl and Woit tend to delete my posts, I am not sure whether he will consider the invitation seriously.
  12. Apr 12, 2007 #11


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    Good luck. I'm thinking maybe Woit, probably not Motl, but hope to be wrong on the second count.
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