Geon Eon
Carl Brannen's guest post at Tommaso's blog contains a link to the papers of Mark Hadley. In particular, there is a 1997 paper which considers particles as non-trivial topologies in spacetimes with closed timelike curves (4-geons).
It takes seriously the idea of obtaining non-classical logic from General Relativity. That is, propositions determining states do not necessarily obey the distributive law. Typical 2-valued propositions ask whether or not a certain region contains a particle. This is an argument for the requirement of higher dimensional toposes in gravity, because the logic of a 1-topos is always distributive, whether or not it is Boolean. Recall that distributivity in general is naturally expressed by a map $ST \Rightarrow TS$ for two monads $S$ and $T$.
The basic idea bears a little resemblance to Louis Crane's geometrization proposal for four dimensional spin foam models, which allows for a relaxation of the manifold condition at a point. Hadley only discusses manifolds, for ordinary GR, as if quantization of gravity were unnecessary, but no analysis of solutions to Einstein's equations is actually given. His conclusion is that gravitons do not exist, because as 3-geons they would lack the topological structure needed to localise them.
Note that Hadley's later papers have even more grandiose titles, without much accompanying mathematical analysis.
It takes seriously the idea of obtaining non-classical logic from General Relativity. That is, propositions determining states do not necessarily obey the distributive law. Typical 2-valued propositions ask whether or not a certain region contains a particle. This is an argument for the requirement of higher dimensional toposes in gravity, because the logic of a 1-topos is always distributive, whether or not it is Boolean. Recall that distributivity in general is naturally expressed by a map $ST \Rightarrow TS$ for two monads $S$ and $T$.
The basic idea bears a little resemblance to Louis Crane's geometrization proposal for four dimensional spin foam models, which allows for a relaxation of the manifold condition at a point. Hadley only discusses manifolds, for ordinary GR, as if quantization of gravity were unnecessary, but no analysis of solutions to Einstein's equations is actually given. His conclusion is that gravitons do not exist, because as 3-geons they would lack the topological structure needed to localise them.
Note that Hadley's later papers have even more grandiose titles, without much accompanying mathematical analysis.
posted by Kea | 1:11 PM
7 Comments:
Thanks to Carl. Hs nice post shows again that science is really advanciung somewhere.
Speed typing, Louise? Lol.
The manifold structure for space-time is often regarded as something God given. There are exceptions to this thinking (intersecting branes and orbifolds in M-theory).
In the length scale resolution of our sensory perception we however find non-manifolds everywhere. Think only book like structures with pages glued along the back of the book.
In my own Universe non-manifolds emerge naturally. The lines of Feynman diagrams are replaced with light-like 3-surfaces meeting along commong partonic 2-surface. Vertices become manifolds unlike in string models. An interesting implication is that number theoretical braids replicate in particle creation vertex. Keeping in mind topological quantum computation this brings strongly in mind copying of classical information and non-faithful copying of quantum information and quantum communication by parton exchange.
Also generalized imbedding space is a non-manifold forming a book like structure. In two senses: real imbedding space and its p-adic variants are glued along common rationals and algebraics and imbedding spaces corresponding to different values of Planck constant are glued together along 4-D M^2xS^2.
Carl talked in Tommaso's blog about how to achieve a variable light-velocity and this means preferred frame. The choise of quantization axes coded by M^2xS^2 at the level of geometry defines a preferred frame, not the presence of "aether". Poincare and color invariance are not broken at the level of world of classical worlds since WCW is union of sectors corresponding to all choices of M^2xS^2.
Perhaps Kea is refering to his paper on the various choices of parity when she complains that Hadley's later papers are grandiose without a lot of math. I believe he was driven to write this paper because GR didn't have parity. I find his arguments persuasive, but I haven't looked at this in a while. But my recollection was that this adjustment of E&M was probably necessary in a unified treatment that is based on the vibrations of space-time and stuff like that (as opposed to a unified treatment that doesn't claim to describe space-time).
The second thing I should mention is that in my point of view, Hadley's work is about quantum PARTICLES, and not at all about quantum waves. The stuff I work on is all about waves, not particles so much. This means that in a certain way, the two approaches are duals to one another, they fill in each other's missing ingredients.
And the third thing is that I don't believe much in GR, and I certainly don't believe in the "closed time-like curves" that is, paths that fail to move forward in time.
Because of this, I'd like to see Hadley's work recast in Euclidean relativity form, with a hidden circular dimension. The CTCs would become paths in the hidden circular dimension that are "time-like" in the proper time form, that is, they would actually be moving forward in the usual coordinate time, but would be stationary in proper time.
Another way of describing the kind of CTCs I'd like to see Hadley's work redone with is that they are the path integrals of stationary particles. They move forward in coordinate time by cycling through proper time. The "closed" part of the path comes from the fact that topologically, in my favorite flavor of Euclidean relativity, proper time is a circle.
Finally, one might compare these sorts of ideas (CTCs) with the string theorist's closed and open strings. See what Alejandro Rivero discusses on Tommaso's blog.
And as a PS, I should probably guess that due to Hadley's divergence from the mainstream it will be tough for him to get tenure, or much notice. I don't mean to imply that approval from me or Louise is always the kiss of death. Uh, I'll let the reader decide that one for themselves.
Dear Kea and All,
I made quite a big step of progress in understanding how DNA could act as a topological quantum computer. As a matter fact, in the recent formulation of quantum TGD entire Universe acts like a topological quantum computer.
Zero energy ontology and understanding of M-matrix in terms of Connes tensor product, the new understanding of number theoretical braids involving geometrization of Higgs, the hierarchy of Planck constants and of dark matter, the notion of magnetic body understood as controller of biological body through genome (gene expression), and the hierarchy of genomes (super-genomes for coherent gene expression for organs, etc..): these are the pieces.
These pieces combined together lead to a very beautiful vision about biological evolution as evolution of topological quantum cpmputation.
For instance, introns ("junk DNA"!!) can be understood as the part of genome devoted to topological quantum computation not therefore expressing itself chemically. The minimum number of three strands for braids (non-commutative braid group) relates naturally to the number three of nucleotides in codon). A,T,C,G could be interpreted as strand color so that strands can connect only identical nucleotides: this means restriction to subgroup of braid group.
DNA double strand defines the minimal braid and the appearance of copies of genome in each cell make more more complex computations possible. Hierarchy of genomes defines a hierarchy of programs using programs of earlier level as submodules.
Spacelike Connes tensor product defines memory storage of functions as entanglement coefficients. Time-like Connes tensor product between positive and negative energy states defines functions and topological quantum computation. Decay of braid strands defines a replication of classical information and exchange of 2-D parton surface defines quantum communication.
See my blog.
Generally I don't think people appreciate just how simple Hadley's idea is: Classical general relativity is the fundamental theory, and quantum mechanics is a phenomenological consequence of CTCs in the space-time manifold. It would mean that all attempts to see quantum mechanics, nonclassical logic, noncommutative probabilities, etc, as fundamental are misguided. Instead, they would arise solely from the "context dependence" of questions asked in the vicinity of an unknown non-simply-connected topology.
I am certainly in favor of the idea that classical logic is ultimately correct, and Bohmian mechanics shows you can get the predictions of quantum theory without having to believe in objectively indeterminate properties, so it would be nice to know whether Hadley's idea can work or not. Yet it doesn't seem to have advanced in ten years. We still only have the same basic qualitative argument. So how might one move forward?
I have read a number of times that quantum field theory in n space dimensions is "the same as" classical statistical mechanics in n+1 space dimensions. And since Hadley's idea, as I understand it, is to derive quantum probabilities from a classical four-dimensional perspective, one might imagine that there is a connection. In fact there have been many people who have tried to interpret quantum theory in terms of temporally bidirectional conditional probabilities, but I have never ever seen a derivation. How hard can it be to show whether the idea makes sense? So let me just think out loud for a moment.
QFT in n dimensions gives you transition probabilities between an initial state and a final state (which may be asymptotic). Classical stat mech attaches a classical probability to each of the possible spatial configurations of a field (for example). A "classical stat mech derivation of quantum field theory", if such a thing is possible, would associate classical probabilities with space-time configurations (over the space-time volume of interest), and would then re-derive the quantum transition probabilities as simple conditional probabilities (probability of the future boundary of the space-time volume being in a particular state, given that the past boundary was in some other state).
Right away one faces the problem of relating quantum states to the boundary states of the classical space-time volume. Presumably they must be associated with classical probability distributions over the possible classical boundary states; but which ones? One might assume that a coherent state (in the quantum-field sense) is just the corresponding classical state, with 100% probability; but what about "momentum eigenstates" - which classical distributions would they correspond with? (This might even be where a disproof of this idea becomes possible, perhaps along Kochen-Specker lines.)
Anyway, if one assumes the existence of a dictionary relating states of the quantum theory to classical probability distributions over the classical theory, how can we understand the emergence of the global conditional probabilities?
The intuition of "temporal zigzag" theorists is that spacelike correlations exist because you have correlations forward in time, to the point where the future light-cones intersect, and then backward in time down the other light-cone. Now certainly, in a classical stat-mech state with local correlations, you can get distant correlations by way of zigzag paths (i.e. by zigzag chains of local correlations), because every direction is just a spatial direction. So if one thinks about a classical probability distribution over space-time configurations, the potential for space-like correlations equally appears to be there.
In the case of classical stat mech, there's no dynamics yet in an instantaneous probability distribution. Mathematically you can set up whatever correlation structure you like, and then the role of the dynamics is to tell you how the distribution changes. Since here we are talking about a probability distribution over space-time histories, I suppose the first question is: Can one set up such a distribution at all, over the set of solutions to a classical theory, such that, given the right dictionary for the quantum states, one can recover the quantum transition probabilities? And the second question: If it is possible, is it just a contrivance, or is there some meaning to the set of global probabilities one uses? And finally, can those probabilities be justified by considerations like those of Hadley's? Because another way to pose Hadley's idea is to say that space-time is populated with small non-time-orientable patches (those are the CTCs), and they are what can cause a timelike chain of local correlations to end up globally in a hairpin shape, a temporal zigzag. It would be more than interesting if one could quantitatively derive, say, the quantum S-matrix from the classical field theory plus an assumption about the frequency of CTCs in a space-time volume; it would be revolutionary. And it would not be revolutionary, but it would certainly be progress to prove that one can not derive QM from the classical theory in this way. Unfortunately I don't seem to be equal to either task tonight. :-)
On the topic of deriving everything from classical general relativity, there is a weird old paper from Physica Scripta, 1989, "Nothing matters" by R.A. McCorkle, which purports to do just that. I must look it up in the morning...
"I have read a number of times that quantum field theory in n space dimensions is "the same as" classical statistical mechanics in n+1 space dimensions."
Yes, that's why I'd like to see Hadley's stuff redone in 5+1 dimensions. The extra spatial dimension being a circle that can be thought of as "proper time". This is the Euclidean relativity version of relativity.
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