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The Nature of Time Essay Contest (2008)
[back]

TOPIC: Time as a Universal Scaling Principle by Lawrence B. Crowell [refresh]

TOPIC: Time as a Universal Scaling Principle by Lawrence B. Crowell [refresh]

A correspondence between time and a scaling principle is examined here within the framework of nonassociative field theory. Imaginary time $t~=~hbar/kT$ determines a scaling principle for the universe where a scale invariance is given by the conformal structure of AdS, which further indicates the universe may results from a a quantum critical point.

I did my graduate work at Purdue University and have since worked in affiliation with the AIAS and industry. For the last several years I have been working to set up the problem outlined in this paper. Quantum cosmology is a quantum error correction process.

Dear Mr Crowell, I hope you don't mind if I ask a question on the surface of your essay. You mention "nonassociative field theory" and use AdS/CFT correspondence for some of the arguments in the essay. I could envision such a field theory to be scale invariant, but I don't see how special conformal translations (and therewith AdS correspondence) could be modeled on a nonassociative background. Am I overlooking something? Thanks, Jens

Hello Jens,

I of course spotted you paper, and have read a couple of Dzhunushaliev's papers on nonassociative geometry.

The argument comes down to tessellation. The 120-cell or octohedrachoron tessellates the AdS spacetime. The 120-cell is also a representation of the octonionic E_8, and the tessellation of the space is given by the roots of this group.

The noncommutative geometry of quantum spacetime is a quantum group which pertains locally, and the vacuum of the universe connects these quantum groups by associators (e_ie_j)e_k - e_i(e_je_k) = T_{ijk}^le_l. So the universe locally has some quantum group g and in another region another g' with A:g --- >g', or Ag = g' or that g^{-1}Ag = g^{-1}g'. This is the "overlap" of states predicted by associator transformed quantum groups. There is a set of 7 such quantum groups which form a basis of possible quantum groups. What one observer will measure as the vacuum state given by a quantum group is different from what another observer will probe. The associator preserves quantum information, but in a sense encrypts it. If we coarse grain over the associators we end up with the Bogoliubov transformation, which is the thermalized transformation involved with black hole radiation.

cheers L. C.

I of course spotted you paper, and have read a couple of Dzhunushaliev's papers on nonassociative geometry.

The argument comes down to tessellation. The 120-cell or octohedrachoron tessellates the AdS spacetime. The 120-cell is also a representation of the octonionic E_8, and the tessellation of the space is given by the roots of this group.

The noncommutative geometry of quantum spacetime is a quantum group which pertains locally, and the vacuum of the universe connects these quantum groups by associators (e_ie_j)e_k - e_i(e_je_k) = T_{ijk}^le_l. So the universe locally has some quantum group g and in another region another g' with A:g --- >g', or Ag = g' or that g^{-1}Ag = g^{-1}g'. This is the "overlap" of states predicted by associator transformed quantum groups. There is a set of 7 such quantum groups which form a basis of possible quantum groups. What one observer will measure as the vacuum state given by a quantum group is different from what another observer will probe. The associator preserves quantum information, but in a sense encrypts it. If we coarse grain over the associators we end up with the Bogoliubov transformation, which is the thermalized transformation involved with black hole radiation.

cheers L. C.

This essay is a bit of an overview of a part of research I have been doing for the last 2 to 2-1/2 years. The "big idea" is that quantum information is ultimately preserved. The bigger picture than this involves the Leech lattice /_{24}, which contains 3 E_8's in a Theta function construction.

The crux here is that the hyperbolic space is tiled by polytopes with pentagonal symmetry. This gives a "rule of fives" on quivers of vectors that is an associator. The 120-cell satisfies the essential symmtries required in four dimensions and is also the lattice representative of E_8, or the octonionic exceptional group.

Lawrence B. Crowell

Lawrence B. Crowell

The crux here is that the hyperbolic space is tiled by polytopes with pentagonal symmetry. This gives a "rule of fives" on quivers of vectors that is an associator. The 120-cell satisfies the essential symmtries required in four dimensions and is also the lattice representative of E_8, or the octonionic exceptional group.

Lawrence B. Crowell

Lawrence B. Crowell

Lawrence,

Excellent paper, though it's obvious that it's been distilled from a much more extensive research program. That's not a bad thing; it makes one want to dig into more of your results.

I think your mathematical strategy is correct. If you get a chance to read my essay, "Time counts," you will find that we agree on many important points--e.g., scale invariance from first principles, a role for imaginary time, quantum information conservation.

Interestingly, we both have versions of sphere packing, though rather than Euclidean hyperbolic, I chose n-dimensional hyperspace, and focus on the n-dimension kissing number problem--with similar results in the higher dimensional model.

All best,

Tom

Excellent paper, though it's obvious that it's been distilled from a much more extensive research program. That's not a bad thing; it makes one want to dig into more of your results.

I think your mathematical strategy is correct. If you get a chance to read my essay, "Time counts," you will find that we agree on many important points--e.g., scale invariance from first principles, a role for imaginary time, quantum information conservation.

Interestingly, we both have versions of sphere packing, though rather than Euclidean hyperbolic, I chose n-dimensional hyperspace, and focus on the n-dimension kissing number problem--with similar results in the higher dimensional model.

All best,

Tom

Thanks. Yes this is one part of a "deep time" project of mine. I am drafting up a paper for the GRF essays which covers another part of this. The whole project is coming together.

The sphere packing I work with goes up to 24 dimensions, with the Leech lattice. This in Jacobi theta functions is @(L_{24}) ~ @(E_8)^3 which is a triality on E_8s. At lower energies the root space of one of these E_8s, the 120-cell actually two of them, is the tessellation of the emergent spacetime.

It is interesting that someone else has neural patterns similar to mine.

I have been picking my way through some of the papers in this blog. I have not voted on any yet, and I will have to read as many as possible before making that choice.

Cheers,

Lawrence B. Crowell

The sphere packing I work with goes up to 24 dimensions, with the Leech lattice. This in Jacobi theta functions is @(L_{24}) ~ @(E_8)^3 which is a triality on E_8s. At lower energies the root space of one of these E_8s, the 120-cell actually two of them, is the tessellation of the emergent spacetime.

It is interesting that someone else has neural patterns similar to mine.

I have been picking my way through some of the papers in this blog. I have not voted on any yet, and I will have to read as many as possible before making that choice.

Cheers,

Lawrence B. Crowell

One can deduce scaling from Physics, not Physics from scaling. This is nonsense. At the very beginning of your ideology, there was Nature and now there is only data computation. Why? You must prove that you created the world, L. Crowell.

LeRouge: I am a bit perpexed about what you wrote. Underlying this is the idea that the universe is similar to a quantum computer. I will admit that, and people such as Seth Lloyd have written much the same:

arXiv:quant-ph/0501135v8

I also do think that scaling or renormalization group physics is crucial. To be honest I think that numbers such as the fine structure constant may be determined in the way that Feigenbaum's number is --- or for that matter pi. The alternative is an ambiguity in how the physical vacuum is reached with lots of ideas about multiple universes. Of course these universes can't be observed, so this wrankles with my emperical sense of things.

L. Crowell

report post as inappropriate

arXiv:quant-ph/0501135v8

I also do think that scaling or renormalization group physics is crucial. To be honest I think that numbers such as the fine structure constant may be determined in the way that Feigenbaum's number is --- or for that matter pi. The alternative is an ambiguity in how the physical vacuum is reached with lots of ideas about multiple universes. Of course these universes can't be observed, so this wrankles with my emperical sense of things.

L. Crowell

report post as inappropriate

Dear Dr. Crowell,

I was unable to grasp the main idea of your essay; it sounds like Jack Sarfatti to me. Would you please elaborate in a more pedestrian fashion?

Regards,

Dimi Chakalov

post approved

I was unable to grasp the main idea of your essay; it sounds like Jack Sarfatti to me. Would you please elaborate in a more pedestrian fashion?

Regards,

Dimi Chakalov

post approved

At the core is that euclidean time t = hbar/kT determines the phase of the universe, a quantum critical point, for a huge range of temperatures. These temperatures or times, which ever point of view you want, scale great arcs in the 120-cell tessellated hyperbolic spacetime of AdS. These great arcs are similar to those on the Poincare 1/2-plane. The rest of it discusses this within the format of an E_8 tessellation of the spacetime and quantum error correction codes. So the quantum state of fields in the universe then obey a scale invariance, or renormalization group, for all energies 0 to infinity between the BTZ black hole and the AdS boundary.

Sorry that it might be a bit disjointed. I thought the deadline was the end of December or Jan 1. I then discovered a few days before Dec 1 that was due then. So wrote the paper up in a few days.

This is one segment of a general research program I have been working on with associators in lattices and the tessellation of space and spacetime with error correction codes. I also make no statement on whether time exists or not, for I don't think science really addresses these types of questions.

Cheers,

Lawrence B. Crowell

Sorry that it might be a bit disjointed. I thought the deadline was the end of December or Jan 1. I then discovered a few days before Dec 1 that was due then. So wrote the paper up in a few days.

This is one segment of a general research program I have been working on with associators in lattices and the tessellation of space and spacetime with error correction codes. I also make no statement on whether time exists or not, for I don't think science really addresses these types of questions.

Cheers,

Lawrence B. Crowell

Dear Lawrence,

It seems we disagree on what you wrote on Carlo Rovelli's thread - "I don't think that science can tell us about the existential status of geometric entities". I think, when analyzed rigorously, special relativity proves the existence of worldtubes and triangles (the twin paradox). If you disagree, try to provide an *explanation* (not just calculation) of the length contraction effect as briefly explained on Carlo Rovelli's thread (or in my essay).

Vesselin Petkov

It seems we disagree on what you wrote on Carlo Rovelli's thread - "I don't think that science can tell us about the existential status of geometric entities". I think, when analyzed rigorously, special relativity proves the existence of worldtubes and triangles (the twin paradox). If you disagree, try to provide an *explanation* (not just calculation) of the length contraction effect as briefly explained on Carlo Rovelli's thread (or in my essay).

Vesselin Petkov

Digging into the literature, I find it interesting that Mohammed Ansari and Lee Smolin at the Perimeter Institute have outlined requirements for a self organized critical (SOC) model of quantum gravity, using abstract spin network dynamics: http://arxiv.org/abs/hep-th/0412307

The idea is based conceptually on Per Bak's SOC avalanche model which necessitates scale invariance as a consequence of asymptotic functions as, e.g., found in phase transitions; in other words, self limiting functions that cohere in a self organized system.

Quoting the conclusion in Ansari & Smolin's paper: "We have proposed a propagation rule for colors to evolve on a 2d planar open spin network, which appears to exhibit self-organized critical behavior. It appears that with a special choice of evolution rule, the dynamics evolves the system to a dynamical equilibrium state, within which the behavior of the system appears to be scale invariant. This work is a step in the investigation of the hypothesis that the emergence of our classical world from a discrete quantum geometry is analogous to a self-organized critical process. Among the further steps are 1) the study of models in which the underlying graphs themselves evolve by local rules, analogous to those studied here, 2) the study of other correlation functions, including those that would be interpreted as propagation amplitudes for matter and gravitational degrees of freedom, and 3) an increase in the valence, from three to four valent graphs, which is expected to correspond to the dynamics of geometry in 3 + 1 dimensions, and 4) the demonstration that self-organized critical phenomena exists for quantum evolution and not just for ordinary statistical systems. These are considerable challenges, towards which the present results must be seen as just a first step."

Scale invariance is the fundamental feature that allows time-evolving geometric models to work in a background-independent context, because one can maintain the best features of a continuous field theory while recovering the discrete geometry of measured events at low energy.

Smolin's loop quantum gravity (LQG) contradicts smooth Lorentz invariant properties of space below the Planck limit--one may conjecture that within a scale invariant theory, though, this may turn out to be a technical problem rather than a fundamental obstacle to a mathematically complete theory of quantum gravity.

Lawrence Crowell provides the geometry that preserves and encrypts the quantum information (think of Bohm & Hiley's metaphor of the ink dot in a rotating vat of gelatin)in a way that claims to guarantee recovery at all energy scales, and therefore at all time scales. Personally, I like the mathematical strategy, very much.

Tom

The idea is based conceptually on Per Bak's SOC avalanche model which necessitates scale invariance as a consequence of asymptotic functions as, e.g., found in phase transitions; in other words, self limiting functions that cohere in a self organized system.

Quoting the conclusion in Ansari & Smolin's paper: "We have proposed a propagation rule for colors to evolve on a 2d planar open spin network, which appears to exhibit self-organized critical behavior. It appears that with a special choice of evolution rule, the dynamics evolves the system to a dynamical equilibrium state, within which the behavior of the system appears to be scale invariant. This work is a step in the investigation of the hypothesis that the emergence of our classical world from a discrete quantum geometry is analogous to a self-organized critical process. Among the further steps are 1) the study of models in which the underlying graphs themselves evolve by local rules, analogous to those studied here, 2) the study of other correlation functions, including those that would be interpreted as propagation amplitudes for matter and gravitational degrees of freedom, and 3) an increase in the valence, from three to four valent graphs, which is expected to correspond to the dynamics of geometry in 3 + 1 dimensions, and 4) the demonstration that self-organized critical phenomena exists for quantum evolution and not just for ordinary statistical systems. These are considerable challenges, towards which the present results must be seen as just a first step."

Scale invariance is the fundamental feature that allows time-evolving geometric models to work in a background-independent context, because one can maintain the best features of a continuous field theory while recovering the discrete geometry of measured events at low energy.

Smolin's loop quantum gravity (LQG) contradicts smooth Lorentz invariant properties of space below the Planck limit--one may conjecture that within a scale invariant theory, though, this may turn out to be a technical problem rather than a fundamental obstacle to a mathematically complete theory of quantum gravity.

Lawrence Crowell provides the geometry that preserves and encrypts the quantum information (think of Bohm & Hiley's metaphor of the ink dot in a rotating vat of gelatin)in a way that claims to guarantee recovery at all energy scales, and therefore at all time scales. Personally, I like the mathematical strategy, very much.

Tom

Invariant intervals are what an observer measures on that world line. So ds = sqrt{g_{ij}dx^idx^j} is the time that general relativity defines. The coordinate time "t" is a metric dependent element, more of a bookkeeping device than anything physical. Quantum field theory requires the employment of "t" in a dynamical wave equation, which puts the two theories on different footings with respect to how they treat time.

So is ds real? Maybe, for we can multiply it by mc^2 and define an action

S = mc^2 int ds,

which appears real in some sense. It has units of action, or angular momentum, which is a measurable quantity. Yet there is something a bit troublesome about all of this. How does the observer on this world line actually measure this interval? A clock is employed which must have some system of oscillations, such as a spring. Yet this is measuring the invarant interval according to something carried on that world line that deviates from the world line. Hence some sort of nongeodesic motion is being used to define or measure an interval along a geodesic path. Of course I am thinking primarily of a mechanical clock, but an atomic one still appears to hold for an EM field must be applied to knock electrons in the Ce atoms.

So our Lagrangian is being defined according to something which is not intrinsic to mc^2. So we might then consider that action as dS = pdq - Hdt. Now we have some Hamiltonian, which might include a part for the dynamics of the clock. Hamiltonians must be specified on some Cauchy surface of data with a coordinate time direction. Yet this has gotten us into some funny issue, for to define an invariant interval it appears that we need a coordinate defined clock.

Of course the invariant interval is defined on a hyperbolic metric space which leads to the Lorentz group. All of that is pretty standard stuff.

The issue of time is a bit slippery. I am not out to deny the existence of time, but it is something which appears to be geometrical and as such "relational." It relates kinematic entities to dynamical ones. As I see it the important question is not whether time exists, but as a relational quantity "what does it tell us?"

Lawrence B. Crowell

So is ds real? Maybe, for we can multiply it by mc^2 and define an action

S = mc^2 int ds,

which appears real in some sense. It has units of action, or angular momentum, which is a measurable quantity. Yet there is something a bit troublesome about all of this. How does the observer on this world line actually measure this interval? A clock is employed which must have some system of oscillations, such as a spring. Yet this is measuring the invarant interval according to something carried on that world line that deviates from the world line. Hence some sort of nongeodesic motion is being used to define or measure an interval along a geodesic path. Of course I am thinking primarily of a mechanical clock, but an atomic one still appears to hold for an EM field must be applied to knock electrons in the Ce atoms.

So our Lagrangian is being defined according to something which is not intrinsic to mc^2. So we might then consider that action as dS = pdq - Hdt. Now we have some Hamiltonian, which might include a part for the dynamics of the clock. Hamiltonians must be specified on some Cauchy surface of data with a coordinate time direction. Yet this has gotten us into some funny issue, for to define an invariant interval it appears that we need a coordinate defined clock.

Of course the invariant interval is defined on a hyperbolic metric space which leads to the Lorentz group. All of that is pretty standard stuff.

The issue of time is a bit slippery. I am not out to deny the existence of time, but it is something which appears to be geometrical and as such "relational." It relates kinematic entities to dynamical ones. As I see it the important question is not whether time exists, but as a relational quantity "what does it tell us?"

Lawrence B. Crowell

Larry: Thank you for your efforts. If you wish to comment on my efforts to *think* of gravitational energy as being both localizable and non-localizable (cf. my postings above), please do it at your thread and I'll jump there, with utmost pleasure. I believe all this pertains to the nature of time in GR, since nobody has managed to separate time from energy. Surely in textbook GR there isn't such animal like the one I propose, perhaps because I address this puzzle in GR after proposing a solution to the measurement problem in QM.

Dimi

----------------

Interestingly quantum mechanics and general relativity have according to their Galois field representation, or their Coxeter-Weyl group representation. There are a few algebraic similarities between quantum mechanics and gravitation which are suggestive. By comparison, it is not difficult to show that a Schild's ladder construction of parallel translation of vectors in general relativity is equivalent to GF(4). The Galois field is GF(4) = (0, 1, z, z^2) with z = {1/2}(i sqrt{3} - 1) with z^2 = z^*. GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8). The properties of the basis elements that produce an algebra of commutators are

z^2 = z + I, z^3 = I, z^* = z^2,

and defines the hexacode system C_6. This system is an elementary example of a Golay code. This system is also the code for the quaternionic system of 1/2hbar spins. So there is a relationship between gravitation and quantum mechanics on this elementary algebraic level.

So there is this sort of functorial equivalency, modulo the hyperbolic nature of relativity. I think this may have some bearing on the relationship between quantum nonlocality and the nonlocality of energy in GR, but I don't understand this currently.

In the case of cosmology, where a conformal structure gives as a special case synchronized coordinates, the identification between proper time and coordinate time may on some global (eg cosmological equivalency principle) remove some of the problems with nonlocalizability of energy. Maybe least up to some conformal Killing time make some global identification of energy which is perfectly localizable.

Lawrence B. Crowell

Dimi

----------------

Interestingly quantum mechanics and general relativity have according to their Galois field representation, or their Coxeter-Weyl group representation. There are a few algebraic similarities between quantum mechanics and gravitation which are suggestive. By comparison, it is not difficult to show that a Schild's ladder construction of parallel translation of vectors in general relativity is equivalent to GF(4). The Galois field is GF(4) = (0, 1, z, z^2) with z = {1/2}(i sqrt{3} - 1) with z^2 = z^*. GF(4) is the Dynkin diagram for the Lie Algebra D_4 = spin(8). The properties of the basis elements that produce an algebra of commutators are

z^2 = z + I, z^3 = I, z^* = z^2,

and defines the hexacode system C_6. This system is an elementary example of a Golay code. This system is also the code for the quaternionic system of 1/2hbar spins. So there is a relationship between gravitation and quantum mechanics on this elementary algebraic level.

So there is this sort of functorial equivalency, modulo the hyperbolic nature of relativity. I think this may have some bearing on the relationship between quantum nonlocality and the nonlocality of energy in GR, but I don't understand this currently.

In the case of cosmology, where a conformal structure gives as a special case synchronized coordinates, the identification between proper time and coordinate time may on some global (eg cosmological equivalency principle) remove some of the problems with nonlocalizability of energy. Maybe least up to some conformal Killing time make some global identification of energy which is perfectly localizable.

Lawrence B. Crowell

You desired my comments on your post. I tend to agree with many of the postings on this essay,e.g. scaling can not describe Physics but Physics can use scaling to understand some phenomenon. One may use a geometrical tool to comprehend a phemenon in Physics but one can't say that Geometry provides the base for Physics.

I don't know what else you desire me to post on this site, as per your post on my essay!!

I don't know what else you desire me to post on this site, as per your post on my essay!!

I am slightly perplexed as I did not post on your essay site. I am putting this post on both mine and your area. I looked at your essay some time back, particularly since you appear to be grabbing up public votes like a champion fisherman. I'd have to look again at your essay, but as I recall I did not understand it very well, and it appears you are using different intellectual modalities from what I am familiar with.

My essay discusses one aspect of a general problem. The AdS spacetime is a mathematical representation for a spacetime with conformal structure. Maldacena illustrated that for N large in SU(4) that the AdS is dual to conformal fields. The scaling comes about with the "energy" of the geodesics that arc through the spacetime. These may start and end at the conformal boundary of the AdS, or with an event horizon of a BTZ black hole. How this energy scales is determined by the tessellation of the geometry, which provides a basis (or quivers) of quaternion valued fields. These fields exist in a general E_8 type of grassmannian framining, which in the 120-cell tessellation of the AdS are the roots for the E_8. That is one nice thing about E_8, the root space for the group defines the group. In a more formal setting this involves a functor which converts the geometric picture into an algebraic one.

I did not discuss the physical cosmology we actually live in. That would have taken more space, yet the basic structure in the AdS version carries over to the deSitter spacetime. The AdS has features which make it more convenient to work in.

Lawrence B. Crowell

My essay discusses one aspect of a general problem. The AdS spacetime is a mathematical representation for a spacetime with conformal structure. Maldacena illustrated that for N large in SU(4) that the AdS is dual to conformal fields. The scaling comes about with the "energy" of the geodesics that arc through the spacetime. These may start and end at the conformal boundary of the AdS, or with an event horizon of a BTZ black hole. How this energy scales is determined by the tessellation of the geometry, which provides a basis (or quivers) of quaternion valued fields. These fields exist in a general E_8 type of grassmannian framining, which in the 120-cell tessellation of the AdS are the roots for the E_8. That is one nice thing about E_8, the root space for the group defines the group. In a more formal setting this involves a functor which converts the geometric picture into an algebraic one.

I did not discuss the physical cosmology we actually live in. That would have taken more space, yet the basic structure in the AdS version carries over to the deSitter spacetime. The AdS has features which make it more convenient to work in.

Lawrence B. Crowell

Hi Lawrence, Perplexing progression of posts. I think Plato said it best, geometry is the basis of physics. And he had something to say about the Geometer too.

Thanks for mentioning the items to consider on my essay page.

Happy New Year to all!

Thanks for mentioning the items to consider on my essay page.

Happy New Year to all!

Geometry is extremely useful because you have visual pictures in your mind. The equivalence with algebra is of course important, but I put a big value on imagination and that big TV we have in our heads. It really is better than the ones you buy --- even the flat screen HD BlueRay etc ones.

L. C.

L. C.

The A5 icosahdral symmetry is the permutathdral group for sporadic groups and E_8. Also the underlying homotopy group describes a nonassociative algebra.

The 24-cell is the minimal sphere packing for four dimensions. The 24-cell (tetrahedachoron) has 24 vertices and 96 edgelinks, where the 24-cell is self dual. So consider the 48-cell given by this cell plus its dual. A reassignment of...

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The 24-cell is the minimal sphere packing for four dimensions. The 24-cell (tetrahedachoron) has 24 vertices and 96 edgelinks, where the 24-cell is self dual. So consider the 48-cell given by this cell plus its dual. A reassignment of...

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Hi Lawrence,

You wrote in part: `"The running through energy scales then defines a renormalization group, which as yet I honestly don't understand that well. This is connected with Ricci flow equations, such as the Hamilton-Perelman theory used to prove the Poincare conjecture."

Donal O'Shea in his excellent book, The Poincare′ Conjecture: in search of the shape of the universe (Walker & Co, NY, 2007), said "...(Perelman) alluded to a connection between the Ricci flow and a very different flow in physics that connects space at different resolutions. Here, the parameter is not time, but scale—and our space is modeled not by a manifold with a metric, but by a hierarchy of manifolds and metrics connected to the Ricci flow equation. This sort of fundamental shift in point of view was reminiscent of Riemann’s probationary lecture. The mathematics belongs squarely to the new century and the new millennium, but the notion of a hierarchy of metrics would have pleased Riemann.

“Perelman wrote `Note that we have a paradox here: the regions that appear to be far from each other at a large distance scale may become close at a smaller distance scale; moreover if we allow Ricci flow through singularities, the regions that are in different connected components at a larger distance scale may become neighboring…’ This is the stuff of science fiction. Then, back to Earth. He wrote, ‘Anyway, this connection between the Ricci flow and the RG [renormalization group] flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation.’ Well, almost back to earth. Gradient flows are relatively well understood, but to say that the Ricci flow can be regarded as a gradient flow represented another fundamental insight.”

As you know, Lawrence, I am with you all the way on your mathematical strategy of scale invariant functions. One can find application of the above principles in my ICCS 2006 paper “Self organization in real and complex analysis”: I wrote, “… a discrete step in time is defined by an exchange of continuous curves for discrete points. Therefore, if every point of n-dimensional space is homeomorphic to a 3-sphere, and time is a physical quantity (of zero measure) on an n-dimensional self-avoiding random walk, a move in time of any magnitude bridges an infinite gulf…by avoiding infinity—‘counting’ is over the manifold of a 3-sphere embedded in the Hilbert space; continuous curves are exchanged for discrete points in a complex network of randomly oriented self-similar metrics, in a self-limiting system projected between S^3 and S^1.”

All best,

Tom

You wrote in part: `"The running through energy scales then defines a renormalization group, which as yet I honestly don't understand that well. This is connected with Ricci flow equations, such as the Hamilton-Perelman theory used to prove the Poincare conjecture."

Donal O'Shea in his excellent book, The Poincare′ Conjecture: in search of the shape of the universe (Walker & Co, NY, 2007), said "...(Perelman) alluded to a connection between the Ricci flow and a very different flow in physics that connects space at different resolutions. Here, the parameter is not time, but scale—and our space is modeled not by a manifold with a metric, but by a hierarchy of manifolds and metrics connected to the Ricci flow equation. This sort of fundamental shift in point of view was reminiscent of Riemann’s probationary lecture. The mathematics belongs squarely to the new century and the new millennium, but the notion of a hierarchy of metrics would have pleased Riemann.

“Perelman wrote `Note that we have a paradox here: the regions that appear to be far from each other at a large distance scale may become close at a smaller distance scale; moreover if we allow Ricci flow through singularities, the regions that are in different connected components at a larger distance scale may become neighboring…’ This is the stuff of science fiction. Then, back to Earth. He wrote, ‘Anyway, this connection between the Ricci flow and the RG [renormalization group] flow suggests that Ricci flow must be gradient-like; the present work confirms this expectation.’ Well, almost back to earth. Gradient flows are relatively well understood, but to say that the Ricci flow can be regarded as a gradient flow represented another fundamental insight.”

As you know, Lawrence, I am with you all the way on your mathematical strategy of scale invariant functions. One can find application of the above principles in my ICCS 2006 paper “Self organization in real and complex analysis”: I wrote, “… a discrete step in time is defined by an exchange of continuous curves for discrete points. Therefore, if every point of n-dimensional space is homeomorphic to a 3-sphere, and time is a physical quantity (of zero measure) on an n-dimensional self-avoiding random walk, a move in time of any magnitude bridges an infinite gulf…by avoiding infinity—‘counting’ is over the manifold of a 3-sphere embedded in the Hilbert space; continuous curves are exchanged for discrete points in a complex network of randomly oriented self-similar metrics, in a self-limiting system projected between S^3 and S^1.”

All best,

Tom

Dear Lawrence,

it is my shortcoming that you are not able to comprehend my essay. The fact of the matter is that i have a label of Low energy Experimental Nuclear Physicist. i just dabbled into Cosmology as a layman and found something worth providing some propects as such. It was done with the feeling that i may indicate a few unbiased ideas that may have future persuance. But i can't be sure of its usefulness.You may well be right in your assessment!

it is my shortcoming that you are not able to comprehend my essay. The fact of the matter is that i have a label of Low energy Experimental Nuclear Physicist. i just dabbled into Cosmology as a layman and found something worth providing some propects as such. It was done with the feeling that i may indicate a few unbiased ideas that may have future persuance. But i can't be sure of its usefulness.You may well be right in your assessment!

Ricci flow equations in a general relativistic perspective assume some sort of 4-metric of the form

ds^2 = -dt^2 + F(dx^2 + dy^2 + dz^2)

and this can be reduced to the Ricci flow as due to a gradient flow

nabla^2X + &X/&t = 0,

for F = e^{-2X}. The hitch comes when there are "defects" in this metric, such as some local region where

ds^2 = -Adt^2 + Bdr^2 + r^2domega^2

Of course one can by multiplying by A^{-1} through the whole equation get synchronous coordinates, but they may not be conformally equivalent to the above ones. This may be due to the presence of a singularity in the spacetime, such as a black hole.

Hence in general the spacetime of the universe may only exhibit nice Ricci flows in various local regions (star shaped regions etc), according to saddle point integrations. These different regions "patch together" in a general theory of Morse indices. Transitions between these regions in a quantum mechanical sense are similar to tunnelling states, or they don't occur according to a global time. The correspondence with the Wheeler DeWitt equation is that by imposing some scalar oscillatory field in the spacetime local killing vectors might be defined. These might extend into some conformal time, but where there are still regions of the universe where this type of "flow" does not occur.

Physically these are associated with inhomeogenieties in the universe. As the universe expands these should "even out" or in a Ricci flow type of manner become less important. For black holes this will of course involve the quantum decay of these black holes. So a universal Ricci flow appears to involve something beyond diffeomorphisms of three dimensional spaces.

Lawrence B. Crowell

ds^2 = -dt^2 + F(dx^2 + dy^2 + dz^2)

and this can be reduced to the Ricci flow as due to a gradient flow

nabla^2X + &X/&t = 0,

for F = e^{-2X}. The hitch comes when there are "defects" in this metric, such as some local region where

ds^2 = -Adt^2 + Bdr^2 + r^2domega^2

Of course one can by multiplying by A^{-1} through the whole equation get synchronous coordinates, but they may not be conformally equivalent to the above ones. This may be due to the presence of a singularity in the spacetime, such as a black hole.

Hence in general the spacetime of the universe may only exhibit nice Ricci flows in various local regions (star shaped regions etc), according to saddle point integrations. These different regions "patch together" in a general theory of Morse indices. Transitions between these regions in a quantum mechanical sense are similar to tunnelling states, or they don't occur according to a global time. The correspondence with the Wheeler DeWitt equation is that by imposing some scalar oscillatory field in the spacetime local killing vectors might be defined. These might extend into some conformal time, but where there are still regions of the universe where this type of "flow" does not occur.

Physically these are associated with inhomeogenieties in the universe. As the universe expands these should "even out" or in a Ricci flow type of manner become less important. For black holes this will of course involve the quantum decay of these black holes. So a universal Ricci flow appears to involve something beyond diffeomorphisms of three dimensional spaces.

Lawrence B. Crowell

We do not live in a AdS universe. We asymptotically evolve to a dS universe with an asymptotically constant entropy-area of ~ 10^123 BITs. Therefore, how is Crowell's model relevant to the actual facts of precision cosmology?

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