To supplement my last post, I came across a 1979 paper by G. Galloway that reinforces the initial perception I had a couple of years ago that Joy's framework is powerful enough to solve the cosmological horizon problem. This follows from the framework's implied independence of time scale.
Tom
Hi Tom,
Thanks for the summary, but I am not sure how it is related to what Michael wants to prove. Let us wait and see what he has in mind, because so far I haven't understood what he wants to prove, or why it is not already proved in Chapter 7 of my book (or in the attached paper).
Best,
More recently from Professor Galloway
And seeing that this paper was communicated by no lesser light than S. T. Yau, I get the goosebumpy feeling that topology in physics is on the verge of a golden age.
Tom
Hi Joy (and Tom),
Yes, nothing quite stands in the way of clear communication like using the same language, but with different intended meanings. So on the point:
"You seem to be suggesting that what is responsible for entanglement---at least in the singlet state---is *light*"
This is not a suggestion, but the standard QFT view of the interacting particles in a spin singlet state. For most singlet cases in QFT, there is a virtual-particle connection between the particles in terms of the gauge bosons - photon, W/Z, gluons - depending upon the particle charges, and no fermionic particle is genuinely chargeless (unless you find a right-handed neutrino!). So in the QFT view, it is the spin of these bosons which is ultimately responsible for the strong quantum correlations between the spins of the particles in the spin singlet state. For the hidden variable framework to say that QT correlations come from somewhere else - i.e. not QT - it must successfully capture this QFT view in order to go beyond it.
The exception to this general QFT view is the neutrino correlations you pointed out, as the strong correlations in this case are between 2 left-handed neutrinos with different flavours. So the strong QT correlations are flavour correlations which aren't accounted for in QFT because they are not due to the spin of a photon, W/Z or gluon connection. This marks a weak point of the Standard Model - I'm suggesting this points to the non-associativity of the octonions, because my model with an octonion space reproduces particle flavours.
Then there is my suggestion, which comes from the fact that dimensional reduction of a pure geometric 11D GR yields the Standard Model Lagrangian (up to colour group only) with S7 gauge space - this can be read straight from the attached paper (sec 2.3-2.4 eqns 2.20 and 2.22 of attached paper give my eqns 14 and 18 in sec 4 of my paper) which is a review of the state of KK in 1987, before Witten falsely claimed KK couldn't give EW chirality - i.e. the dimensional reduction is standard stuff. As for all KK-theories, the 4D gauge connection is a *gravitational* connection in the full 11D - that is why KK can unify gravity with particle forces. So I have NO disagreement over the wording:
"I have been arguing, on the other hand, that what is responsible for entanglement---at least in the singlet state---is *the geometry and topology of the physical space*, or more precisely the gravitational interaction."
There is no "other hand" in my suggestion, the gauge connection of 4D field theory - i.e. QFT - IS about the *geometry and topology of the physical space*. What I'm looking for is a metaphysical proof in 3 parts (the highlighted words make it metaphysics):
1) Strong QT correlations *must* be due to the topology of physical space
2) That necessarily *requires* the existence of compactified dimensions - then the connection IS a form of gravitational interaction in its most general sense
3) The compactified dimensions *must* be S7
On first reading of your Ch 7 it seemed to me that you had already proven this, but you have been disputing this in our discussion - I still think that in the mathematical framework you have the core of the proof I'm seeking. In your proof, it is the expression of causation as a factorisation condition which gives the critical condition of mathematical closure that selects the fundamental number systems C, H, O - the C case being trivial.
In my case, I have causation expressed as GR. The topological condition for the existence of topological monopoles demands a mapping from S7 to S3, and then the causal dynamics of GR also gives S1. So the essential physics of our two approaches is the same - expressions of causation - and the topological conditions pick out the same fundamental number systems - C, H, O. In my case there is a very open issue between N and R, whereas I pointed out the same issue is implicitly present within the hidden variable framework. It seems to me that between us we have a round peg and round hole, and I can't see them not fitting. Because we have the same very basic underlying physics and maths, the claim that they are in contradiction strikes my as being similar to claiming that we have a proof that in physics 1=0. That is very unlikely!
Best,
Michael
Attachment failed as is 4.4MB. Reference was Rep. Prog. Phys. 50 (1987) 1087-1170. IOP journal behind a pay-wall, can't find where my pdf came from. Can e-mail if query my statement that it's standard stuff in a review article.
Michael
Thanks, Michael.
I actually know the report by Bailin and Love. I studied it some years ago. I have downloaded it again just in case, but I don't doubt your word for a second. I am more concerned about understanding the apparent incompatibilities in our views. I will reflect on your comments above and get back to you.
Joy
Excellent paper Joy.
I keep imagining variations on the theme Michael was unfolding above, with nested or coupled gyroscopes, as being perhaps a little less messy than exploding balls, but Joy Christian's exploding balls conjecture does have a certain ring to it. Maybe some combination of gyroscopes, torsion balances, and interferometry, would give a more precise read out of the effect intended to measure.
But I like the concept.
Regards,
Jonathan
Intriguing developments.
This thread has been white hot with interesting ideas to compare. I can barely keep up with reading the parts I like, but this twist in the compatible or incompatible debate is a bit surprising - and it gives me a lot of food for some very intense thought. Whether to consider your two theories as possibilities that might be real or realities that might be possible is part of the issue, I think. I still have the gut feeling that your ideas can be blended, or that nature's reality incorporates aspects of both your models, but digesting all of the comments above will take time.
I keep returning to the notion that size is relative, and so is interiority/exteriority among forms, when non-commutativity and non-associativity enter the picture, so that the physical interpretation of geometric realities may differ from their intrinsic arrangement. We do not possess a God's eye view, as inhabitants in a physical reality. So there are still legitimate questions about what constitutes a privileged viewpoint, and what an ordinary observer would see in a given setting.
All the Best,
Jonathan
Hi Jonathan,
Full appreciation of the depth of Relativity is critical, where it can be viewed as having third orders of meaning:
1) First order meaning is the relative measurement of rotations - i.e. fermionic spinors - and speeds - as in the speed of light is the same for all local observers at the same cosmological time. It is one of the themes of this year's essay contest, that even this level of Relativity isn't being properly grasped in the mainstream. It is quite trivial in SR that there is a unique reference frame for every event in the universe, namely the co-moving reference frame. Apply this fact to a closed expanding universe, and it has a unique co-moving reference frame - a global cosmological reference frame with a uniquely defined cosmological time, and all our local measurements are relative to this. As the stars are being dragged along by the expanding fabric of reality, we have a realisation of Mach's principle. So it's fairly trivial to show in GR that many of this year's essays, and the 'process physics' paper you mentioned in your thread below (plus more) are correct about the mainstream presentation of Relativity being screwy. Other essays have objected to the mainstream cosmological constant, which is an oxymoron in Relativity (the clue is in the word). That the cosmological term is defined to be 'constant' relative to the metric is even clearly evident on Wikipedia - the term has to be 'constant' within the surface defined by the metric. But for a closed expanding universe where the metric is parameterised by radial scale factor, because it is an extra-dimensional parameter, the cosmological term can also be parameterised by the same factor and still be 'constant' as demanded - as could the speed of light and the gravitational coupling constant. Again, it is trivial to show that the mainstream presentation of GR is screwy, as many of us saying.
2) Second order meaning is what you referred to, which is physically realised in the case of all space-time measurements being relative to the physical scale of compactified dimensions. The simple relativism of this solves the apparent mystery of compactified dimensions being of 'constant' size - ALL physical measurements are relative to the size of the compactified dimensions, and measuring their scale in terms of themselves gives a constant relative size. In an absolute sense, they're not of a constant size. When the compactified dimensions are octonion in character, then as you imply, this relativism gets really interesting. As I mentioned in your thread below, when the scale measurements used in the construction of space-time differentials are relative to octonion compactified dimensions, it is not unreasonable to expect the resulting differential structure for the 4D space-time manifold to be 'exotic'. My KK-theory indicates that the non-commutivity gives a gauge structure, whereas the non-associativity gives a family structure to the 'emergent' 4D space-time. The left/right split in the octonion representation could well be associated with the different chiralities of the EW vacuum - this appears as though it would be the case from my work. The various ways of expressing the octonions after this left/right split does seem to raise legitimate questions about the existence of privileged viewpoints.
3) Third order meaning of Relativity is that the definitions of some physical quantities are defined relative to the number of dimensions of the space. This will be irrelevant, except when there is the compacification of dimensions, which seems to be the case. Just assuming that a black hole has a compactified surface is all that is needed to derive Hawking's expression for black hole entropy in basic thermodynamics. But the derivation reveals that the definition of entropy for radiative modes involving compactified dimensions depends upon the number of spatial dimensions left uncompactified. This changes from 3 in normal space, to 2 in the compactified surface of a black hole, which implies that the black hole 'information paradox' is simply due to an entropy anomaly of not comparing like definitions of entropy. The classical thermodynamics derivation also gives black hole radiation - with the correct inverse relationship between temperature and radius - which implies that radiation in a compactified surface exerts an outwards radiation pressure. This agrees with the cosmological 'constant' not being constant, and solves the mystery of where the motive force for spatial inflation and dimensional compactification comes from.
With a full appreciation of Relativity, a lot of the apparent mysteries surrounding the mainstream view of Relativity just disappear. It is probably worth noting that many of the mysteries disappear if we assume that the universe is closed. I show that this assumption also solves the mystery of what the fermionic particles are, and why EW theory is chiral. The relativism of space-time to an octonion structure seems to be a significant feature that remains to be explored.
Best,
Michael
Hello again,
I'm going to jump in here, and go way out on a limb at risk of appearing foolish, because I've been morphing spheres in my head, trying to conceptualize the differences in Joy's and Michael's constructions - imagining that visualizations will reveal what symbolic Math cannot. In the case of nested spheres, one must consider that being nearer to a center or surface makes it appear larger than structure which is not in proximity. The fact that octonions are the natural embedding space defining perspective (projective geometry) for all object-observer relations, however, gives us more degrees of freedom creating a relational definition of extendedness we can use in higher dimensional spaces.
In Michael's construction; two 3-spheres that share a topological boundary have a growing-shrinking relationship, where the shrinking of one allows the other to expand. I particularly like this notion that an action carries across (or is carried by) the topological boundary separating the two spheres, in a way that swings the gate allowing the energy or space to pass from one to the other. While I feel that nature must adhere to the laws of geometry; I am also a firm believer that the physical dynamism which drives the evolution of form is a powerful and necessary element of any physically realistic description.
It is largely my observation that these two must fit together seamlessly which drives my investigation into this subject. That is; it is wise to assume that geometry is a determiner of the ways form can evolve, but to realize that it is not the only set of rules to which nature must conform. I think time has the fewest rules, needing to adhere only to the nature of process. Space needs to follow rules of geometrical evolution. Energy drives evolutionary dynamism, but must follow rules of propagation and circulation. And matter must follow a few more rules, or is the most constrained. This is the hierarchy I spelled out in my essay for the 2nd contest (my first).
I don't think one can do realistic Physics in higher dimensional spaces without considering the realities of geometry and topology in those spaces. However; I think one must consider also the rules of procedural evolution or process dynamism to properly describe spacetime, given my understanding of time. So just plotting the decomposition of higher-d spheres into component entities is not enough to fully describe the Physics IMO, as it is in some ways a static picture. There is a need to figure in the physical dynamism. But with 28 possible smooth/differential structures for S7, a figure which plays a strong role in both your theories, I think there are some variations on a theme yet to explore.
I'll leave off there for now,
Jonathan
Thinking further on this,
There may be some relation to Dixon's construction based on T = C x H x O, where he posits a time-reversed antimatter universe, as a companion to our observable cosmos. This has some appeal for me, as the same notion arises in my Cosmology derived from the Mandelbrot Set. But it would imply that the two copies of S3 in your theory, Michael, can be viewed as interchangeable if time-reversal is admitted. Which sphere is compact is relative to the time arrow, but this may be flexibly defined.
I once wrote a paper on the brain hemispheres, suggesting that their respective views of process are reversed in time - and they see the same reality both ways at once - which may be relevant. I'll forward a copy at some point. But now I have other work to do.
More later,
Jonathan
Michael,
Nice to hear acceptance of the fact there is a right and left character for octonion algebra. You would not know this from most of the available literature. Curious about your actual position on the difference between the 8 algebraic structures of like chirality you brought up. Do you think it is reasonable to think one may be preferable to another? Clearly I do not think so, for it is the cornerstone of my work on applying octonion algebra to physics. The ramifications of acceptance that NONE of the 16 ways to roll out octonion algebra is preferable to another is the voice behind the algebra telling us precisely how nature must look. It is not a coincidence that all of these old school things like observable currents, force, work, energy density, energy flux, momentum, and conservation of energy and momentum, etc. all fall in line with my law of algebraic invariance when properly represented in an octonion analytic framework.
I would be wary of any mathematical construction that requires a subset of octonion algebraic constructions to work out.
I do not know if Tevian Dray still favors one over another, but you might look here as a starting point if you are unfamiliar with the work.
Rick
Thank you so much Michael!
I especially appreciate your treatment of the 2nd order meaning of relativity, and I second Rick's appreciation for your grasp of the subtlety that both the left and right hand algebras come into play. In a way; it's a matter of where you are coming from and where you are going to - procedurally speaking. But your explication is elegant.
I also liked the fact you elucidated the 3rd order meaning entirely. My point was not wasted on you! In fact, you nicely showed the hierarchy of intended meanings, or a progression within the theoretical basis. The problem is, most physicists don't even know that the subtleties of 2nd and 3rd order relativity exist, much less that they are important to consider.
And Thank You Rick,
As to your question about the 8 algebraic structures, and their usage - I think it is exactly like the 5 different string theories and supergravity each having classes of problems for which they are soluble formulas - where other areas of the parameter space find use for a different flavor of string theory, and they morph into each other through dualities.
Likely some of the 8 possible structures for octonion algebra will be easier to solve for a given class of problem. But as Michael says, this aspect of things still needs to be explored. A very interesting problem to consider, though.
All the Best,
Jonathan
Hi Rick,
My awareness of the distinction between the right and left-handed octonions came from you, so thank you. Currently the left/right word usage for chirality and octonions is two different meanings. But the left/right chirality for EW is a clear binary split where the internal particle symmetries are exactly the same in both cases, and the left/right octonion algebras is a clear binary split with the same structure in both cases. For the octonion structure to be that of the internal particle symmetries, the two would have to match up on left/right meaning, which seems entirely reasonable and feasible. But before investigating that, I think you and Jonathan have a valid point about the 8 different algebraic structures having some role to play in physics.
I am aware that the identification of S7 with the internal particle symmetries gives a colour group problem. My theory has it, and if Joy's framework does have the implications that I think it does, then it will have it as well. The S4 base-space of S7 can readily be identified with the EW symmetry space, complete with giving a closed geometric formula for the Weinberg angle - sin^2theta_W=sqrt(5/21)=0.238 - that is within the experimental range. But this leaves the colour symmetry space as being the S3 fibre of S7, which naturally gives the colour symmetry group as being Spin(3) and not SU(3). However, the 8-fold character of the octonions allows SU(3) to be found within them, which is what Geoffrey Dixon does in his SM extension. But as you say, there is a another 8-fold invariance in the 8 algebraic representations of the octonions.
In the old paradigm of QT being fundamental, the ethos was that fermionic fields could just be added by hand to a field theory Lagrangian in any group representation that happened to take your fancy, and the expectation was that it would be physics. But we have 2 independent proofs that QT is not fundamental, and this really does give a new paradigm where the old arbitrariness can no longer be expected to yield meaningful physics. If Bell's change in meaning of his hidden variable has the significance I expect it to have from my work, then QT is conceptually the result of describing a discrete particle world in a continuous analogue way. So there could be a pure representational issue beyond the underlying physical reality, and this is where the 8-fold nature of octonion algebra representation could come in. Such a feature seems to me to be the only way of squaring a S3 colour space from the topology of physical space with SU(3) in QFT within the new paradigm of QT not being fundamental. Stating this view as a proposition would give something along the lines of:
The 8-fold invariance of octonion algebraic representation turns the Spin(3) colour symmetry group, of the physical S3 colour fibre of the compactified S7 dimensions, into a SU(3) representational invariance in the dimensionally reduced Lagrangian, after the representational shift from the natural-numbers of physical fermions to real-number valued fermionic fields that gives QFT.
If this proposition is true, then my pure geometric KK-theory succeeds in deriving the SM Langrangian of QFT as it stands, and so unifies the 3 particle forces as given by the SM with the force of gravity given by GR. If the proposition isn't true, then I think the metaphysical character of my and Joy's work is saying that the colour group of SM isn't SU(3), but Spin(3), the symmetry group with group space corresponding to the physical S3 fibre of S7. So, is the proposition true?
Best
Michael
Hi Michael,
You wrote: "With a full appreciation of Relativity, a lot of the apparent mysteries surrounding the mainstream view of Relativity just disappear."
I am compelled to agree, without qualification.
"It is probably worth noting that many of the mysteries disappear if we assume that the universe is closed."
Problem is, the observational evidence won't let us make this assumption. The value of Omega still appears to be very close to 1, on the border of open and closed. In the moments of my wooliest thinking (pp 32 -- 35 & esp. fig. S2.2 I expect that the external closure of correlated quanta of S^2 point properties on the S^3 manifold map perfectly to an open, continuous and perfectly flat internal plane on S^7. This continuous mapping would therefore compel the propagation of point particle dynamics on S^3 X S^7, as you allow. What I found is that the middle value -- the central point from where observation originates -- adds one point to *every* axis of rotation. For years, I had no idea what this means -- Joy Christian's work made me realize that the answer was already built into principles of topology that I understood mathematically though not physically; these are orientability and complete measurement functions. Once initial condition was introduced and followed through 4pi rotation, I grokked how Hopf fibration and torsion applies to the physical model.
Like you, I think that we're all on the same path. After all, how many ways are there to prove the Pythagoras theorem?
Best,
Tom
Previous in this thread is mine.
Hi Tom,
Problem is, observational evidence in cosmology is not direct, but interpreted through a model of GR where it is unphysical for certain "constants" to actually be constant. The cosmological "constant" is the number one unphysical "constant" of the mainstream presentation of GR, but legitimate suspicions can also be raised about the speed of light and gravitational constant not varying with the radial scale factor of the universe. I'm not the only one in the essay contest smelling a rat on these points. For GR to be physics these corrections to unphysical constancy must be made prior to feeding the observational data into GR for a closed universe, and this isn't happening. Until these corrections are made, the claims for what the observational data means in GR is not credible physics.
With a closed universe - and my representational propostion I gave in my response to Rick being true - then the particle physics of the SM can be derived in a theory that unifies with gravity - the theory itself is just GR in 11D, and nothing else. The experimental basis for particle physics is much more reliable than that of cosmology, where we also know that the although the assumption of QT being fundamental is wrong, we also know that its features which give experimental results must be reproduced nonetheless. We are on a path where particle physics is going to say that the standard cosmology is wrong, and it is fairly straightforward to identify the "constants" in the unphysical interpretation of mainstream GR as being the problem.
Best,
Michael
Michael,
You wrote: "The experimental basis for particle physics is much more reliable than that of cosmology, where we also know that the although the assumption of QT being fundamental is wrong, we also know that its features which give experimental results must be reproduced nonetheless."
Sure, I agree.
"We are on a path where particle physics is going to say that the standard cosmology is wrong, and it is fairly straightforward to identify the 'constants' in the unphysical interpretation of mainstream GR as being the problem."
I don't agree that there is a boundary between Einstein's cosmology and whatever quantum gravity turns out to be. What I mean is (and I think Vesselin Petkov did an outstanding job of explaining this in his essay) the absence of inertia is a telling sign that the universe is flat (I interpret this as the S^7 physical space, following Joy) without being necessarily open. That is, as Petkov notes, "The equal status of geodesics in flat and curved spacetimes is encoded in the fall of different masses with the same acceleration. By the geodesic hypothesis, their fall is inertial and indeed the motion of falling particles is unsurprisingly similar to motion by inertia in the absence of gravity -- particles that move by inertia do so irrespective of their masses."
Why would we think that mass effects in the standard model of particle physics differ from those in relativity? A relativistic quantum cosmology would eliminate the need for quantized spacetime without obviating particle quanta as manifest in high energy experiments and correlated properties -- these are where the locally unmeasured results that you call unphysical are corrected by topology, because the topology frees us from imagined distance boundaries that turn out to be irrelevant to the fundamental physics. Indeed, "all physics is local," just as Einstein prescribed.
Best,
Tom
Hello all,
I have some thoughts I'd like to summarize here, relating to what was said a few items up the stack. Non-associative geometry explores the ways in which aspects of interiority/exteriority become relational in higher-dimensional spaces. The dynamics are explored in Michael's description of 2nd and 3rd order relativity above - which I think was brilliantly well-stated.
Briefly; in complex figures like S7, the fact there are multiple (28) smooth/differential structures possible is in some way of a reflection that the figure has multiple throats, or radii, where various portions of the same figure can be within or outside others, topologically speaking. It can be effectively round, toroidal, or pretzel-like, and encode a twist or torsion when parallelized.
The associative property or grouping calculations within an equation has a structural equivalent in geometry, in this property where forms can exist inside or outside of another figure. To some extent; this allows the higher-d exotic spheres to be explained. But this insight is similar in form to the observation that commutativity is connected with notions of size and distance, and non-commutativity requires us to redefine those concepts.
More later,
Jonathan
Hi Tom,
If you don't mind me asking, what are your reasons for a belief in quantum gravity? The main justification has always been the ethos of the paradigm of QT being fundamental that everything had to be quantised, including gravity. The reason for believing that QT had to be fundamental was Bell's supposed "theorem" saying that there could be no other option. But that's not true, and so ends the paradigm of QT being fundamental - with it goes the condition that gravity *has* to be quantised. There has never been any experimental evidence for quantum gravity, and without the paradigm saying that it *has* to exist, it seems to become an unsupported speculation. Is there some other basis for thinking there is quantum gravity?
Best,
Michael
