It has been almost exactly a year since Hawking gave a talk in Dublin claiming to have found a resolution of the black hole information paradox. Tonight a preprint giving some details of his argument has appeared.
I’ll leave to the quantum gravity experts the evaluation of exactly how convincing Hawking’s argument is. It is based on using the Euclidean quantum gravity framework, which Hawking refers to as “the only sane way to do quantum gravity non-perturbatively”. I’ve always been fond of the idea that you have to think about QFTs using a Euclidean signature for the background, so I wouldn’t argue with him about this point, but I assume others will.
My comments about the article are available on my blog.
I say, so what? The Clifford algebra of spacetime is not H(2), because in the non-relativisitic limit the Dirac theory has to reduce to the Pauli 2-spinor theory. The H(2) theory would lead to Majorana fermions, and these are not seen. Dirac spinors are not quaternionic.
-drl
Peter Woit discusses “… Euclidean signature and with Minkowski signature, especially when there are spinors …”, saying
“… Mathematically the simplest way of saying what you have to do is to formulate the theory in Euclidean space and analytically continue. …”.
Danny Ross Lunsford then commented: “… Yes, but this introduces yet another complex structure that requires interpretation. …”.
John Baez at http://math.ucr.edu/home/baez/symplectic.html made an observation that may be relevant:
“… Fermions are quaternionic …”.
Therefore, if you want fermions/spinors in your (1+3)-dim spacetime, you want quaternionic structure.
Since you want fermions/spinors, look at the relevant Clifford algebras, which for (1+3)-dim spacetime can have signature either -+++ or +—.
Although their spin groups are isomorphic to each other (and to SL(2,C)), the entire Clifford algebras are distinct, one being M(4,R), the 4×4 real matrices, and the other being M(2,Q), the 2×2 quaternionic matrices.
If you follow John Baez and go to the quaternionic M(2,Q), you see that the vector spacetime is not a real 4-dim Minkowski space as to which you must add a complex structure to get analytic continuation,
but
is in fact a 1-dim quaternionic space which has its own built-in machinery for analytic continuation.
Therefore, if you follow John Baez’s observation to its logical conclusion, you get the benefits of Peter Woit’s analytic continuation without the introduction of Danny Ross Lunsford’s extraneous ad-hoc complex structure.
Tony Smith
http://www.valdostamuseum.org/hamsmith/
Peter,
Yes, but this introduces yet another complex structure that requires interpretation. The existing complex structures (e.g. electromagnetic duality, Dirac algebra and spinors, 2-spinors..) have tight geometric interpretations, much as the “circular points at infinity”, the solution to R^2=0, characterize Euclidean space. In fact if there is going to be any interpretation of this analytic continuation it will amount to mutating the light cone, which characterizes propagation, with the circular points at infinity, which charactize distance.
-drl
I’m certainly not sure what is the exact relation between QFTs in curved spaces with Euclidean signature and with Minkowski signature, especially when there are spinors, but something interesting is going on. Even in flat space, free field QFT, if you try and directly formulate it in Minkowski space you run into trouble (the propagator is given by an integration contour that goes through poles) and you have to do something. Mathematically the simplest way of saying what you have to do is to formulate the theory in Euclidean space and analytically continue.
Peter – it doesn’t bother you pretending that the signature of spacetime is definite?
-drl
I think that Hawking, as well as Penrose, are upset that Hollywood has overlooked them in their latest venture.
Has anything Hawking predicted been experimentally verified?
Has anything Penrose predicted been experimentally verified?
http://physicsmathforums.com/showthread.php?t=56
Tied Up & Strung Out: Hollywood String Theory Movie!!! Looking For Extras!!!
FOR IMMEDIATE RELEASE:
ALL TIED UP & STRUNG ALONG, a movie about String Theorists and their expansive theories which extend human ignorance, pomposity, and frailty into higher dimensions, is set to start filming this fall. Jessica Alba, John Cleese, Eugene Levie, Jackie Chan, and David Duchovney of X-files fame have all signed on to the $700 million Hollywood project, which is still cheaper than String Theory itself, and will likely displace less physicists from the academy.
“As contemporary physics is about money, hype, mythology, and chicks,” Ed Witten explained from his offices at the Princeton Institute for Advanced Study, “The next logical step was Hollywood, although I thought Burt Reynolds should play me instead of Eugene Levy.”
Brian Greene, the famous String Theorist who will be played by David “the truth is out there” Duchovney, explained the plot: “String theory’s muddled, contorted theories that lack postulates, laws, and experimentally-verified equations have Einstein spinning so fast in his grave that it creates a black hole. In order to save the world, we String Theorists have to stop reformulating String Theory faster than the speed of light. We are called upon to stop violating the conservation of energy by mining higher dimensions to publish more BS than can accounted for with the Big Bang alone.”
Greene continues: “At first my character is reluctant to stop theorizing, but when my love interest Jessica Alba is sucked into the black hole, I search my soul and find Paul Davies there, played by John Cleese. I ask him what he’s doing in my soul, and he explains that the answer is containined in the mind of God, which only he is privy too, but for a small fee, some tax and tuition dollars, a couple grants here and there, and an all-expense-paid book tour with stops in Zurich and Honolulu, he can let me in on it. But as soon as he points out that we can make more money in Hollywood than printing more coffee-table books that recycle Einstein, Bohr, Dirac, Feynman, and Wheeler, I am converted. I agree to turn my back on String Theory’s hoax and save Jessica Alba.”
But it’s not that easy, as standing in Greene’s way is Michio “king of pop-theory-hipster-irony-the-theory-of-everything-or-anything-made-
you-read-this” Kaku, played by Jackie Chan. Kaku beats the crap out of Greene for alomst blowing the “ironic” pretense his salary, benefits, and all-expense paid trips depend on. “I have better hair!” Kaku argues as he delivers a flying back-kick, “There can be ony ONE! I WILL be String Theory’s front man!!”
But Greene fights back, “Kaku! Kaku! (pronounced Ka-Kaw! Ka-Kaw! like Owen Wilson did in Bottle Rocket) It is theoretically impossible to build a coffee tables strong enough to support any more coffee-table physics books!!! And what the #&#%&$ does M stand for in M theory???”
How does it all end? Does physics go bankrupt funding theories that have expanded our ignorance from four dimensions into ten, twenty, and thirty dimensions? Do tax payers revolt? Do young physicists overthrow the hand-waving, contortionist bullies and revive physics with a classical renaissance favoring logic, reason, and Truth over meaningless mathematical abstractions? Does Moving Dimensions Theory (MDT) prevail with its simple postulate? We’ll all just have to wait!
But in the meantime, how do you think it will play out?
MDT’s postulate: THE FOURTH DIMENSION IS EXPANDING AT A RATE OF C RELATIVE TO THE THREE SPATIAL DIMENSIONS IN QUANTIZED UNITS OF THE PLANCK LENGTH, GIVING RISE TO TIME AND ALL QUANTUM MECHANICAL AND RELATIVISTIC PHENOMENA.
http://physicsmathforums.com/showthread.php?t=56