Topological Quantum Gravity

[Bogie]

Good post Jeltz. I'll read your link with interest. Thanks for elaborating.

[austintorn@aol.com]

I am making a good-hearted attempt to also explain everything using just space and geometry,, along the lines of this thread.

First of all, space is real, for its existential quantity is volume. Space is physical, but not material, per sey; however, its warps, deflections, and curvature (all are the same thing) give rise to what we call "material", or mass-energy.

So, what about time? It is the difference of space (the motion), and, conversely, space is the difference of time (the distance), if you want to look at it that way, but space is the more primary of the two.

More next time, perhaps about "points" and geometry.

[AntonioLao]

I do not believe that there remain any numerical constants in the standard model or outstanding cosmological observations not explained at least in principle above.

After going through your mathematically sparse details, I failed to see how you resolve the negative dimensional coupling constant of general relativity of which resolution is needed for the renormalization of the field equations.

[Jeltz]

After going through your mathematically sparse details, I failed to see how you resolve the negative dimensional coupling constant of general relativity of which resolution is needed for the renormalization of the field equations.

I agree the mathematical details are sparse, and I don't claim a proved theory but a conjecture. One of the difficulties is that GR and quantum theory are presented in differential calculus notation. But the topological constraint I present relates how neighbouring points in the manifold may move relative to each other.

Comparing the two is not easy. One way is to numerically simulate the manifold and constraint and to see if the local point to point derivatives of manifold co-ordinate dimensions relative to each other match that of QR/GM calculus. My initial attempts at this are at least demonstrating the constraint is not incompatible with QM/GR theory and it is this which has made me somewhat confident in my approach. Making a formal mathematical proof is going to be difficult, and this is what I am working on at present.

Because the constraint equation explicitly allows manifold point co-ordinates to become congruent with each other, the local derivatives of one dimension co-ordinate relative to another can go to infinity. It is this fact I believe that drives the need for re-normalisation in differential calculus theories (re-normalisation meaning that the parametric co-ordinate system overlaid on the manifold has to be rotated so that these infinites do not occur). But in fact this infinity is only a mathematical artefact of trying to formulate differential calculus equations to describe the manifold. Re-normalisation removes these artefacts but does not alter the underlying equations which is why the whole concept of re-normalisation has finally been accepted as a valid thing to do by physicists.

The manifold itself however carries along just fine, it does not see the infinity at all! The infinity is only an indication that from one perspective a point in the manifold has moved directly behind another (become congruent in some subset of co-ordinates). In a black hole for example, all the points in the manifold just pile up on each other (in four of the dimensions anyway) From the manifolds perspective it is just a bunch of points that are congruent in four dimensions...no big deal, but try to calculate points per unit volume and you get infinity which we try rather stupidly to interpret as infinite mass density and therefore makes us nervous because putting an infinite stress energy tensor into GR equations not surprisingly does not work very well.

Perhaps if you can define what you mean by the negative coupling constant in GR a little bit more fully (coupling what to what!) Through tensor notation GR is notionally a 10 dimensional theory and couples all sorts of things to each other.

[AntonioLao]

Topological constraint is the same as the genus of a given topology. Now, if your delta-psi squared is equivalent to the genus then your are simply implying that your topology is equivalent to a Poincare sphere (regardless of how many folding is applied to the manifold or better yet primafold) which was already proven by Perelman into a theorem and not a conjecture anymore.

Negative dimensional coupling constant is the same as Newton's constant of his law of universal gravitation. Since gravity is always attractive, the direction of the centralized force is toward a fictitious center equivalent to the center of a sphere and for ellipsoid, there must be two fictitious centers but only one is effective as suggested by the elliptical orbits of planets. mathematically, all directions toward an arbitrary center is considered as preceded by a negative sign but for a quantized theory of the gravitational field this coupling constant must be expanded in its power series expansion. Unfortunately, only even powers can make the coupling constant to give positive results while odd powers give negative answers, these make the infinite series diverge and insoluble infinities cannot be eliminated hence not renormalizable. A sensible theory must always gives finite answers. On the other hand, QED has the coupling constant of 1/137. Its power series expansion is convergent giving finite answers hence QED is renormalizable.

[Jeltz]

austintorn@aol.com, yes time is a very interesting thing to ponder. The manifold has to be viewed as what reality is since it contains the mathematical points that make up the wave functions that we see as matter around us. It is very hard to argue against the observation that the manifold is constantly changing shape and configuration around us (yes that bird really did just fly past my window). In the 4C manifold we can track these changes in the point co-ordinates of the manifold with respect to something we might call universal time. But this is just a mathematical convenience, a parametric variable designed purely to make sense of the order in which things change in the manifold. And it turns out this time is not linearly related to our everyday sense of time.

The 4C manifold though is a very strange thing. Our local sense of time comes from local clocks and local changes in the manifold around us. Local time is fundamentally related to the rate at which electromagnetic processes happen at the atomic level. General relativity (and countless empirical observations) demonstrate beyond any doubt that the rate at which these atomic processes happen is not constant but depends on where we are in the manifold with respect to gravitational potential and how quickly we move with respect to other objects in the manifold (lorentz transformations of special relativity etc). This means that in general when you go on a journey and return to the starting point your clock will no longer agree with a clock that was left behind. Your clock won't even agree with that of another traveller unless you have undertaken an identical journey. And so our sense of local time is distorted from the universal time by the very presence of the manifold around us and how we move through it, and the objects around us. When you add this distortion of time into our concept of motion and velocity (being the derivative of space with respect to time) it is hardly any wonder that some counter-intuitive observations result.

I am nor entirely happy with this explanation but it is the best I have at the moment.

[Jeltz]

Antonio,
Ok I see what you are on about now. You must remember this constraint acts in 8 dimensions (4 complex co-ordinates). Any action to fold up an otherwise uniform manifold to form a distinctive localised shape (the quantum wave function) in one of the dimensions, can only do so by stealing space from other dimensions. This is what the constraint means in practice. Any folding of any sort in one set of dimensions will always act to reduce the amount of space available in the other dimensions. The curvature this induces in the surrounding manifold to accommodate the folding then will always be positive towards the source of the folding. therefore gravity will always be attractive.

A rough but incomplete analogy in one dimension is a bicycle chain. In this analogy the links take on the role of the topological constraint (in this case requiring neighbouring points to maintain a fixed distance from each other). If you take a straight chain and fold it in the middle to form a wave function shape, it does not matter how you choose to fold it, the ends of the chain must move towards each other to accommodate the folding. i.e. gravity is always attractive.

[AntonioLao]

Poincare topology works in any dimension. The hardest to prove was the 3 dimensions but Perelman proved it using Ricci flow, a concept of multidimensional differential manifolds. Sadly, strict rules of algebraic topology do not allow the definition of length between points implying infinite density. Pure topology is not measureable. For a sphere, inward direction signifies attraction while outward direction repulsion. In a complete theory, antigravity can exist.

[austintorn@aol.com]

…

Existence:


I meant to say that the quantity of space is volume (not quality). What exists has quantity. No quantity; no existence.


The Point:

Points are physical, not abstractions. They have no properties except of position relative to other points. (The simplest things are the most important things.)

A point is shapeless, a consequence of infinite smallness (just as the universe of infinite largeness has no boundary).

The cosmic size scale has points at one end and infinite volume at the other.

In a way, the large may be the same as the small, for there are only two directions away from finiteness, infinite largeness and infinite smallness, each perhaps leading to nothing, one by dispersion to zero and the other by compactness to zero, although the universe truly has neither a floor nor a ceiling. At any rate, we are perched in between in our finiteness.

Zero and infinity cause many of the same problems in algebra, but are kind of two different viewpoints of the same thing. Particle physicists are ever trying to reconcile the macro and the micro.


Big Bang:

The concept of space-time rushing outward in a rapid initial expansion, called the Big Bang, has some problems: Expanding into what exactly? A region already present by definition?


Space:

Space is infinite. This is not math, but a property of the universe (the one and only). It has closure. All action remain within the unbounded universe. This is also called completeness.

Space’s three dimensions are equivalent, due to symmetry; thus, space is shapeless, indicating a flat space-time geometry, not open (saddle shaped) or closed (spherical).

The totality of the universe may be cubic, although not a cube as such, for it has no boundary.


(More next time.)

[AntonioLao]

A universe constructed from a Mobius topology implies the reality of infinite number of zero length trivectors. Pointing in one lateral direction indicates expansion, pointing in the other opposite direction indicates contraction, and pointing in the cross section indicates equilibrium. The one dimensional analog is a Hopf link and clearly only two distinct Hopf links exist, the H-plus link and the H-minus link. 2 links create a neutrino, 4 links create a down quark, 6 links create an up quark, 8 links create either a photon or an electron, 10 links create a W boson, 16 links create a Z0 boson. Nevertheless, there are 8 unique trivectors and each represents a gluon.