Loops, braids and Knots approach is forming a dynamical existence model where the "big bang" is just the initial setup while the existential evolution is still going on ...
I recent weeks I had started to read "Approaches to Quantum Gravity", edited by Daniele Oriti. A quite expensive (166 €) but very good book, worth every single Euro.
Especially remarkable is the introduction written by Carlo Rovelli, title: "Unfinished Revolution". And indeed, this is true in a doubled sense: it is a revolution and it is unfinished.
Rovelli tends quite well to a more fundamental approach where the physical heuristics is not just based on the evolution of understanding of our math based tools.
The second article I want the refer to is the one of Olaf Dreyer called "Emergent relativity".
Olaf Dreyer refers to macroscopic systems as internal relative in the sense that they establish the framework of space-time just by the relation of events to each other, so space-time is not existent for the single particle but is emergent with bigger quantum systems.
Olaf Dreyer start's with the only prerequisite of "coherent degrees of freedom", a setting that constitutes an external framework where the "internal" system is embedded in, so that it can't view it's embedding in an exterior system.
So, there is an isolation between the embedded / created system and the embedding / creating system.
This is quite fundamental and I had developed a similar approach where I coined these degrees of freedom "unoccupied potentials" or "UP".
Those unoccupied potentials are isolated in a way that we can not define an universal associativity between those potentials, but possibly they define a topology.
The "UP" belong to a system (maybe a multiverse or an embedding reality of unknown nature) that does not "use" functionally those potentials in the way that they are stable enough to organize towards a new, functional decoupled system, which is our universe or other universes.
Let's denote the set of the components of our world, created by that process "N".
Let us call the set of the unoccupied potentials "D", let us call the probability space of the evolution of the parts of N "A".
We can call A the "quantum level".
Let us now define a function set F = H + G, where H maps N->A and G maps N->D.
D->A->N is something like the "big bang", a fundamental cosmological event, where N is created. It marks the advent of universal associativity.
As you already know (I wrote it here for some times):
N=F{A,D} =H{A{D}} + G{D}, (equation 1)
where N_h = H{A{D}} (equation 1.1)
... and:
N_g = G{D} (equation 1.2)
"Relativity" in the sense of relativistic behavior is encoded within such a setting in the part G{D}.
It comes up always if the "delta N" affects D, the set of the unoccupied potentials (v->c, mass and energy concentration) insofar dependent from N.
Or if the D acts back on N in the "rare" case that one or more unoccupied potential of the infinite "UP" will become occupied (by the embedding reality / multiverse again) and the d_i "d element D" will fall out of the equation 1.
This can be seen as infinitesimal dynamics on the one hand (mathematically), but it is a sort of entropy of the existence giving system "UP" (physically) by shrinking or expanding D by some more or some less d_i (element of D).
The average setting of the "UP" with regard to creations and removals of those infinitesimal elements + / - d_i is in my model responsible for the specific setting of the cosmological constant.
And now, quite suddenly, time:
t = N_h / N_g = H{A{D}} / G{D} (equation 2)
The functions N_h and N_g must have some "topological support" -that can be make more dense further to algebraic structures - on the unoccupied potential "UP" in order to get functionally connected via sub-algebras.
This means:
Although we have no general associativity between the elements of the "UP", there are "rare" / "finite" cases where elements or sets of elements can fit together anyway. These cases could be algebraic in their nature or just mapped that way.
"Finite" could mean something like the possible baryon count for our universe of 10 exp 80.
The back-reaction of the created system composed of {N}, basically isolated from the creating system based on {D}, towards the creating system via another mechanism / an alternative topology, that is the essence of models like this.
The map:
X = D->[f]->N->A
... is different from:
Y = N->A->[g]->D.
Both maps, X and Y are maps that are creating relativistic effects.
Y by movement -> c and acceleration, X by mass concentration locally as gravity and globally by the change of Kappa, the cosmological constant.
We can possibly define another map Z, that looks like:
Z= N->[m]->A->D
..., which describes a case, where a "macroscopic configuration change" effects coherently the particle / quantum level and by this induces a process / map "A->D", which is a non-local process with many a_i "element A" involved.
And if any configuration in N realizes the map Z, then the function "m" can modify D, say: we are acting back onto the level or creation or existence. As we know nothing about this system, such a back reaction would probably create just a sort entropy, that is gravity.
An example for this could be NASA's QThruster or EMDrive or my proposal for a Gyropeatron.
The bases for such processes is non-locally "coherent" quantum activity, such that the underlying virtual quantum processes (VQP) are not arbitrary but in a certain correlation, thus inducing a chain of virtual quantum activity ("VQA") that acts "coherent" back on D. For the QThruster this result in a setting, that produces micro-gravity.
So, the narrative that Quantum Gravity is only an issue at or near the Planck scale, do possibly not hold in those special cases of "coherent VQA" where it QG could come around already at the micro scale and subsequently is in our technological reach if we can transform quantum processes to UP-coherence.
In order to get this done, we have to listen to that process first, we have to "listen" at quantum processes first.
Of course, this back-reaction to D is to define in greater detail and clarity. Possibly we can coin Olaf Dreyer' s "coherent degrees of freedom" this way more detailed using this conceptual route.
Some thoughts about Dark matter in the "UP" / internal relativity context:
If we change our equation one as follows to a multiverse setting:
N_n=F_j{A_m,D_i} =H{A_m{D_i}} + G_j{D_i}, (equation 3),
... and the X_h, Y_k and Z_p are describing the respective maps in all universes of the multiverse.
We want to suppose that the UP's of those universes are separated, but not completely, so there is possibly an intersection I_d off all D_i which is subject to all maps X_h, Y_k and Z_p.
According to our above described relativistic model all X->I_d, Y->I_d and Z_I_d are creating in this case relativistic effects in all universes, contributing that way the missing amount of mass that can contribute to the gravitational settings of the galaxy's.
Knots and Loops Model:
One can imagine this construction by a geometric-algebraic - where the geometric parts supports our intuition and the algebraic parts make it mathematically concrete - model of Loops, Braids and Knots while the knots are arising from the loops and then feeds back partially to the loop (G{D}) and moreover creating secondary loops, forming braids that are transporting and exchanging energy.
The loop is represented by D / UP and the knot is representing N, the general n-fold. The knot (which represents essentially matter) is not purely local, it is connected by other loops and braids with other knots.
The knot is represented by a sub-algebra over D, which is picking certain elements k_j out of D into N.
The whole mechanism is about Entropy and Information, which are forming a system in an "active state of existence".
This model is describing "existence" as a dynamical process which could have initial bursts ("big bang") but then evolves further in interaction with the base of existence as well as with the created function set in the shape of a "universe" over D.
So:
E{D} = K{D} + B{D} - I{D} (equation 4)
G{D} = K{D} + B{D} (equation 5)
and:
E{D} = G{D} - I{D} (equation 6)
where:
+ E{D} is the overall entropy going back to the loop / D,
+ G{D} is positive entropy expressed withing N as gravity,
+ K{D} is the knot, representing matter components,
+ B{D} is the braid and
+ I {D} represents information.
Thus E{D} represents the overall entropy brought back on D / UP, where is G{D} is the source of gravity, subtracted by the information I{D}, coming from the knots.
If E{D} works "coherent" back to D, (which is the case for E{D} < 0, so I{D} > G{D}, the d_i count will increase and Kappa, the cosmological constant, will be negative, the universe will expand.
If G{D} > 0, which implies G{D} > I{D}, the d_i count will decrease, kappa will grow and the universe will shrink.
Parts of the outgoing loops from the knots are branching into other loops, forming ("energy-like") braids.
Physical interactions in general (except gravity) are going also via braids in the loops.
Loops between some knots are closed while loops over many knots are producing themselves knots, then representing an inertial system.
Knots in general are increasing the entropy of the loop because they extracting "information" from the loops ("D").
But if the knots are feeding the extracted energy into other loops which are creating themselves knots, then they could act back coherently to D, which is I{D}.
I{D} is always coming from equipping loops with a knot, which have to be for that purpose in a coherent state, which means that it is stable and represents a function towards D, so D = f(I) or I{D} = - E{D}
This energy is coming back to the Loop / to "D" as entropy, as some of the d_i of the D are branching out into the knot-forming sub-algebra.
to be continued...
Some personal remarks:
I know that this is pretty heavy stuff for some people, maybe especially for whom the world ends with Standard Model, Planck Scale and General Relativity.
What we can think it in principle, what we can describe is the measure for our mind, the outer boundary for what is possible in principle, for what could be possible in principle.
So, we can model completely new aspect of reality but we have to detail it also much more.
This is not only about physics though. It is about deep and so far hidden connections between mathematics and physics. Especially important is the supposed fact, that mathematics is still able to describe the "UP" - the unoccupied potential.
Another thing is the fundamental importance of virtual quantum processes (domain "A") as a layer of systematic "action" (not: "interaction"!) between the UP and our physical world, a layer of reality, where general commutative and associative behavior is not given from the onset.
Equation Z= N->[m]->A->D offers in principle the possibility to address gravity technologically by designing chains of micro-(quantum-) processes, implementing matter-energy-based technological solutions far before the Planck scale, like QThruster / EMDrive is one, opening completely new and hitherto unthinkable technologies.
So, the road map I describe above, is to be seen as a practical way for designing technological systems which are implementing Quantum Gravity practically.
Best regards from Berlin,
Robert Gallinat