Sparring Sparling
Thanks to a commenter at Not Even Wrong we have this link to an article about the work of G. Sparling of twistor fame. The article is about 3 space and 3 time dimensions, and it contains this diagram 
which refers to three copies of twistor space. Here is the arxiv link to the paper. The article also mentions triality, Jordan algebras, category theory and condensed matter physics. It feels like Christmas!
which refers to three copies of twistor space. Here is the arxiv link to the paper. The article also mentions triality, Jordan algebras, category theory and condensed matter physics. It feels like Christmas!
posted by Kea | 3:08 PM
10 Comments:
Kea, this reminds me of a diagram on page 6 of this paper by Jay Yablon.
Dear Kea,
amusing accident that I wrote a commentary about this paper but in Finnish. I am too tired to translate it.
As you know the 6-D space CP_3 of twistors can be regarded as a U(1) bundle (corresponding to spin) on space of light-like geodesics (direction and position at which the geodesic goes through time = constant hyper-surface making 5 dimensions), 6 altogether.
With this interpretation 3+3 signature is to my opinion not natural. The kinematics associated with this signature is definitely non-physical unless one invents some trick to eliminate 3-D continuum of energies. I would not regard the 6-D space not as the fundamental space-time.
The "world of classical worlds" consisting of lightlike partonic 3-surfaces reduces at the point-like limit and full determinism (light curves replaced with light-like geodesics) to twistor space apart from the phase which could be interpreted as a remnant of the spinor bundle of "world of classical worlds". Light-like curves still allow a conformal symmetry with respect to curve parameter: Virasoro algeba emerges from light-likeness conditions. Therefore generalization of ordinary conformal invariance by lightlike randomness gives stringy conformal invariance.
I think that the starting point, octonionic trinity, is more fundamental than the outcome. TGD counterpart for the holy trinity is trinity of 8-D imbedding space and two 8-D complex(!) spinors corresponding to two chiralities of imbedding space spinors identified as quarks and leptons (quark color is angular momentum like in TGD).
Matti
Thank you very much indeed for this news. On 3 space plus 3 time like dimensions, I'd like to mention D. R. Lunsford's unification of electrodynamics and gravitation, “Gravitation and Electrodynamics over SO(3,3)”, International Journal of Theoretical Physics, Volume 43, Number 1 / January, 2004, Pages 161-177, as summarized here.
Lunsford discusses why, despite being peer reviewed and published, arXiv blacklisted it, in his comment here. Lunsford's full paper is available for download, however, here.
Lunsford succeeds in getting a unification which actually makes checkable predictions, unlike the Kaluza-Klein unification and other stuff: for instance it predicts that the cosmological constant is zero, just as observed!
The idea is to have three orthagonal time dimensions as well as three of the usual spatial dimensions. This gets around difficulties in other unification schemes, and although the result is fairly mathematically abstract it does dispense with the cosmological constant. This is helpful if you (1) require three orthagonal time dimensions as well as three orthagonal spatial dimensions (treating the dimensions of the expanding universe as time dimensions rather than space dimensions makes the Hubble parameter v/t instead of v/x, and thus it becomes an acceleration which allows you to predict the strength of gravity from a simple mechanism, since outward force of the big bang is simply f=ma where m is the mass of the universe, and newton's 3rd law then tell's you that there is equal inward reaction force, which - from the possibilities known - must be gravity causing gauge boson radiation of some sort, and you can numerically predict gravity's strength as well as the radial gravitational contraction mechanism of general relativity in this way), and (2) it require no cosmological constant:
(1) The universe is expanding and time can be related to that global (universal) expansion, which is entirely different from local contractions in spacetime caused by motion and gravitation (mass-energy etc.). Hence it is reasonable, if trying to rebuild the foundations, to have two distinct but related sets of three dimensions; three expanding dimensions to describe the cosmos, and three contractable dimensions to describe matter and fields locally.
(2) All known real quantum field theories are Yang-Mills exchange radiation theories (ie, QED, weak and QCD theories). It is expected that quantum gravity will similarly be an exchange radiation theory. Because distant galaxies which are supposed to be slowing down due to gravity (according to Friedmann-Robertson-Walker solutions to GR) are very redshifted, you would expect that any exchange radiation will similarly be “redshifted”. The GR solutions which slow slowing should occur are the “evidence” for a small positive constant and hence dark energy (which provides the outward acceleration to offset the presumed inward directed gravitational acceleration).
Professor Philip Anderson argues against Professor Sean Carroll here that: “the flat universe is just not decelerating, it isn’t really accelerating ... there’s a bit of the “phlogiston fallacy” here, one thinks if one can name Dark Energy or the Inflaton one knows something about it. And yes, inflation predicts flatness, and I even conditionally accept inflation, but how does the crucial piece Dark Energy follow from inflation?–don’t kid me, you have no idea.”
My arguments in favour of lambda = 0 and 6 dimensions (3 time like global expansion, and 3 contractable local spacetime which describes the coordinates of matter) are at places like this and other sites.
Hey, nice diagram... I wish I understood category theory. You seem to be having some fun.
I did enjoy group theory when I was a student... I remember the "aha!" feeling when I started to master the Young diagrams. But what you teach in your posts is usually above my head :( I keep reading them tho.
Cheers,
T.
Thanks, everyone.
I think that the starting point, octonionic trinity, is more fundamental than the outcome.
Yes, Matti, I quite agree, but it is nice to see something so closely related to our thinking in the news!
Hi Kea
Two trinities, eight trinities, maybe more?
Triplets of many types appear throughout the mathematical and physical spectrum.
Are they related?
John Baez at TWF week 250 has 3 trinities [spatial, rotational, transformational] and one time for 10 dimensions.
With one more dimension [time?] this may be M-theory, with two more [both time?] then F-theory?
http://math.ucr.edu/home/baez/week250.html
I found this paper, bringing complex numbers into Max-Plus or by some called Tropical Algebra attempting "ultradiscretization".
'Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation'
Authors: Tetsu Yajima, Keisuke Nakajima, Naruyoshi Asano
(Submitted on 26 May 2005)
Abstract: A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number of complex variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Stretched coordinates are introduced to obtain the max-plus transformation whose imaginary part coinsides with a phase of the discrete Fourier transformation.
http://arxiv.org/abs/nlin/0505056
Since Max-Plus uses the term NODE rather than Vertex and now may be complex, then this algebra might be called Node Operator Algebra [NOA] - perhaps related to VOA?
Nature has an article RE: von Neumann relationship from two to three dimensions [and more?].
Nature: Editor’s Summary 26 April 2007
'Added Dimensions'
Cellular structures or tessellations are ubiquitous in nature: examples include foams and crystalline grains in metals and ceramics. In many situations, the cell/grain/bubble walls move under the influence of surface tension (capillarity), with a velocity proportional to their mean curvature. As a result, the cells evolve and the structure coarsens. Over 50 years ago, the Hungarian-born mathematician John von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structure. Now the much-sought extension of this result into three (or more) dimensions has been found. The formula should lead to predictive models for various industrial and commercial processes, from the heat treatment of metals to controlling the head on a glass of beer.
News and Views: 'Mathematical physics: Added dimensions to grain growth'
A long-standing mathematical model for the growth of grains in two dimensions has been generalized to three and higher dimensions. This will aid our practical understanding of certain crucial properties of materials.
doi:10.1038/446995a
Full Text | PDF (200K)
Article: 'The von Neumann relation generalized to coarsening of three-dimensional microstructures'
Robert D. MacPherson & David J. Srolovitz
doi:10.1038/nature05745
Abstract | Full Text | PDF (188K) | Supplementary information
http://www.nature.com/nature/journal/v446/n7139/edsumm/e070426-09.html
Doug, unfortunately I cannot access the Nature article without a subscription, but cheers.
Still a comment about problems of 3+3 signature. I checked Lunsford's article but he said nothing about the severe problems raised by the new kinematics in particle physics unless the new time dimensions are compactified to small enough radius.
For instance, for a given four-momentum one should have 2-D continuum values of energies in two additional time directions. This would increase dramatically the phase space for final states in particle reactions. Also thermodynamics would be affected profoundly since one woul have three temperatures.
Second implication is existence of full spectrum of tachyons in the usual sense of the word: if the "first" energy is the standard one then one can have states for which two other energies are non-vanishing and first 3-momentum is non-vanishing.
Sparling covered octonions, Jordan algebras and Freudenthal triple systems in cond-mat/0401015. His definition for a point of the octonionic projective plane coincides with the form of the eigenmatrices of the Jordan eigenvalue problem, which I thought was pretty cool. I referenced Sparling in my recent paper.
"I checked Lunsford's article but he said nothing about the severe problems raised by the new kinematics in particle physics unless the new time dimensions are compactified to small enough radius." - Matti Pitkanen
Thanks for responding, Matti. But the time dimensions aren't extra spatial dimensions, and they don't require compactification. Lunsford does make it clear, at least in the comment on Woit's blog, that he mixes up time and space.
The time dimensions describe the expanding vacuum (big bang, Hubble recession of matter), the 3 spatial dimensions describe contractable matter.
There's no overlap possible because the spatial dimensions of matter are contracted due to gravity, while the vacuum time dimensions expand.
It's a physical effect. Particles are bound against expansion by nuclear, electromagnetic and gravitational (for big masses like planets, stars, galaxies, etc.) force.
Such matter doesn't expand and so it needs a different coordinate system to describe it from the vacuum in the expanding universe inbetween lumps of bound matter (galaxies, stars, etc.).
Gravitation in general relativity causes a contraction of spatial distance, the amount of radial contraction of mass M being approximately (1/3)MG/c^2. This is 1.5 mm for earth's radius.
The problem is that this local spatial contraction of matter is quite distinct from the global expansion of the universe as a whole. Attractive forces over short ranges, such as gravity, prevent matter from expanding and indeed cause contraction of spatial dimensions.
So you need one coordinate system to describe matter's coordinates. You need a separate coordinate system to describe the non-contractable vacuum which is expanding.
The best way to do this is to treat the distant universe which is receding as receding in time. Distance is an illusion for receding objects, because by the time the light gets to you, the object is further away. This is the effect of spacetime.
At one level, you can say that a receding star which appears to you to be at distance R and receding at velocity v, will be at distance = R + vt = R + v(R/c) = R(1 + v/c) by the time the light gets to you.
However, you then have an ambiguity in measuring the spatial distance to the star. You can say that it appears to be at distance R in spacetime where you are at your time of t = 10^17 seconds after the big bang (or whatever the age of the universe is) and the star is measured at a time of t - (R/c) seconds after the big bang (because you are looking back in time with increasing distance).
The problem here is that the distance you are giving relates to different values of time after the big bang: you are observing at time t after the big bang, while the thing you are observing at apparent distance R is actually at time t - (R/c) after the big bang.
Alternatively you get a problem if you specify the distance of a receding star as being R(1 + v/c), which allows for the continued recession of the star or galaxy while its light is in transit to us. The problem here is that we don't can't directly observe how the value of v varies over the time interval that the light is coming to us. We only observationally know the value of recession velocity v for the star at a time in the past. There is no guarantee that it has continued receding at the same speed while the light has been in transit to us.
So all possible attempts to describe the recession of matter in the big bang as a function of distance are subjective. This shows that to achieve an unequivocal, unambiguous statement about what the recession means quantitatively, we must always use time dimensions, not distance dimensions to describe the recession phenomena observed. Hubble should have realized this and written his empirical recession velocity law not as v/R = constant = H (units reciprocal seconds), but as a recession velocity increasing in direct proportion to time past v/T = v/(R/c) = vc/R = (RH)c/R = Hc.
This has units of acceleration, which leads directly to a prediction of gravitation, because that outward acceleration of receding matter means there's outward force F = m.dv/dt ~ 104^3 N. Newton's 3rd law implies an inward reaction, carried by exchange radiation, predicting forces, curvature, cosmology. Non-receding masses obviously don't cause a reaction force, so they cause asymmetry => gravity. This "shadowing" is totally different from LeSage's mechanism of gravity, which predicts nothing and involves all sorts of crackpot speculations. LeSage has a false idea that a gas pressure causes gravity. It's really exchange radiation in QFT. LeSage thinks that there is shielding which stops pressure. Actually, what really happens is that you get a reaction force from receding masses by known empirically verified laws (Newton's 2nd and 3rd), but no inward reaction force from a non-receding mass like the planet earth below you (it's not receding from you because you're gravitationally bound to it). Therefore, because local, non-receding masses don't send a gauge boson force your way, they act as a shield for a simple physical reason based entirely on facts, such as the laws of motion, which are not speculation but are based on observations.
The 1.5 mm contraction of Earth's radius according to general relativity causes the problem that Pi would change because circumference (perpendicular to radial field lines) isn't contracted. Hence the usual explanation of curved spacetime invoking an extra dimension, with the 3 known spatial dimensions a curved brane on 4 dimensional spacetime. However, that's too simplistic, as explained, because there are 6 dimensions with a 3:3 correspondence between the expanding time dimensions and non-expanding contractable dimensions describing matter. The entire curvature basis of general relativity corresponds to the mathematics for a physical contraction of spacetime!
The contraction is a physical effect. In 1949 some kind of crystal-like Dirac sea was shown to mimic the SR contraction and mass-energy variation, see C.F. Frank, ‘On the equations of motion of crystal dislocations’, Proceedings of the Physical Society of London, A62, pp 131-4: ‘It is shown that when a Burgers screw dislocation [in a crystal] moves with velocity v it suffers a longitudinal contraction by the factor (1 - v^2 /c^2)^1/2, where c is the velocity of transverse sound. The total energy of the moving dislocation is given by the formula E = E(o)/(1 - v^2 / c^2)^1/2, where E(o) is the potential energy of the dislocation at rest.’
Because constant c = distance/time, a contraction of distance implies a time dilation. (This is the kind of simple argument FitzGerald-Lorentz used to get time dilation from length contraction due to motion in the spacetime fabric vacuum. However, the physical basis of the contraction is due to motion with respect to the exchange radiation in the vacuum which constitutes the gravitational field, so it is a radiation pressure effect, instead of being caused directly by the Dirac sea.)
You get the general relativity contraction because a velocity v, in the expression (1 - v^2 /c^2)^1/2, is equivalent to the velocity gravity gives to mass M when it falls from an infinite distance away from M to distance R from M: v = (2GM/R)^{1/2}. This is just the escape velocity formula. By energy conservation, there is a symmetry: the velocity a body gains from falling from an infinite distance to radius R from mass M is identical to the velocity needed to escape from mass M beginning at radius R.
Physically, every body which has gained gravitational potential energy, has undergone contraction and time dilation, just as an accelerating body does. This is the equivalence principle of general relativity. SR doesn't specify how the time dilation rate of change varies as a function of acceleration, it merely gives the time flow rate once a given steady velocity v has been attained. Still, the result is useful.
The fact that quantum field theory can be used to solve problems in condensed matter physics, shows that the vacuum structure has some analogies to matter. At very low temperatures, you get atoms characterized by outer electrons (fermions) pairing up to form integer spin (boson like) molecules, which obey Bose-Einstein statistics instead of Fermi-Dirac statistics. As temperatures rise, increasing random, thermal motion of atoms breaks this symmetry down, so there is a phase transition and weird effects like superconductivity disappear.
At higher temperatures, further phase transitions will occur, with pair production occurring in the vacuum at the IR cutoff energy, whereby the collision energy is equal to the rest mass energy of the vacuum particle pairs. Below that threshold, there's no pair production, because there can be no vacuum polarization in arbitrarily weak electric fields or else renormalization wouldn't work (the long range shielded charge and mass of any fermion would be zero, instead of the finite values observed).
The spacetime of general relativity is approximately classical because all tested predictions of general relativity relate to energies below the IR cutoff of QFT, where the vacuum doesn't have any pair production.
So the physical substance of the general relativity "spacetime fabric" isn't a chaotic fermion gas or "Dirac sea" of pair production. On the contrary, because there is no pair production in space where the steady electric field strength is below 10^18 v/m, general relativity successfully describes a spacetime fabric or vacuum where there are no annihilation-creation loops; it's merely exchange radiation which doesn't undergo pair production.
This is why field theories are classical for most purposes at low energy. It's only at high energy that you get within a femto metre from a fermion, so QFT loop effects like pair production begin to affect the field due to vacuum polarization of the virtual fermions and chaotic effects.
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