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Worldline lengths in loop quantum gravity? 
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#1
Mar608, 10:46 AM

P: 7

Hello everyone,
I'm currently occupying myself with Loop Quantum Gravity and wonder about some question. In LQG, space is built up from a spin network. However, how is movement of material particles realized in this network? One could tend to the idea, particles are simply hopping from one network node to the next one, like IP packets transmitted from router to router in Internet communication. Assuming this, one could think that a particle's wordline length  which is in theory of relativity identical to the time passing for the particle  is determined by the relation between the number of Planck time intervals where the particle stays on a node and the number of intervals where it hops to next node: every interval without hopping contributes one Planck time for the passing time, every interval with hopping delivers a zero time contribution. So, for a static particle without any movement, that always stays at same network node, passing time is maximal, a particle with sublight speed, that sometimes hops and sometimes rests, feels some time dilation because of a finite portion of zero time intervals, and finally a photon, travelling with light speed, always hops in every time interval, has no time passing and therefore woldline length of zero. In other words: Minkowsky length of worldline element ds^2 = (cdt)^2  dl^2 becomes S = N_rest/N_total * t_Planck where N_rest is the count of intervals where the particles remains at a node and N_total the total count of time intervals. However, I do not really believe that LQG really delivers such an easy picture. Googling for "particles movement in loop quantum gravity", I found some hints that particles movement is closely related to fluctuations of space geometry, where nodes unify, split up, or reconfigure connections to neighbour nodes. And I guess, it might be incompatible with LQG's philosophy, to differ between particles movement and changes in geometry that strict. So, particles movement rather should be realized by changes in network configuration than by hopping through a static network. Therefore I assume a particle's worldline length  or its passing time  is not calculated in that easy scheme I mentioned above. That's why I ask if there's someone here who can tell me how to calculate those things? 


#2
Mar608, 01:31 PM

Astronomy
Sci Advisor
PF Gold
P: 23,228

Hi Bareil, welcome to the forum!
I thought worldline length was a classical notion. In a quantum picture perhaps one cannot say what path a particle took to get from here to there. Interesting question. I hope some others here respond to it. Personally, in my reading I have not encountered a calculation of worldline length or proper time in the Loop or the Spinfoam context. The closest thing, in the spirit of what you are asking, is the work of Laurent Freidel and Etera Livine. Look up Freidel in arxiv. Spinfoams= Feynman diagrams. 


#3
Mar608, 01:43 PM

P: 587

Is any of this incorrect? Is there more to the present understanding of LQG than spacetime? 


#4
Mar608, 04:17 PM

P: 7

Worldline lengths in loop quantum gravity?
And as far as I know, LQG is able to predict different speeds of light for different EM radiation frequencies, this indicates LQG must ship a method to consider matter  or at least nongravity fields like EM. 


#5
Mar608, 07:01 PM

Astronomy
Sci Advisor
PF Gold
P: 23,228

About energy dependent speed of light (or equivalently frequency dependent) I have tried to follow this and I have not been able to find any definite prediction. There was hope of this, and a result was derived in a lowerdimensional case, by Freidel and others. There was some talk about it mostly back in 2006, but the bottom line was NO prediction as far as I could see. However you may have found some definite explicit hard prediction in some published paper that I don't know about. If so, please give us a link to it. The gammaray instrument GLAST is scheduled to go into orbit in a couple of months and it would be the perfect thing to test such a prediction, if one could be made. 


#6
Mar708, 07:29 PM

P: 7

May be it's interesting what my original intention has been
My idea was (or still is) to extend LQG, in a way that it enables particles to travel faster than the light. This requires two steps: first, we need a prefered frame of reference, to have an absolute simultanity to prevent causality loops. In the second step, we need to keep an FTL particle's worldline length timelike, so that real time is passing for the particle. The first step, LQG ships naturally: the spin network forms a spacelike hypersurface and by this defines a frame of reference. Concerning second step, two solutions are delivered by General Relativity: the wormhole solution and the Alcubiere solution, where in the latter, space is contracting in front of the particle and expanding behind it. However, both approaches are not practicable. The trouble with wormholes is that one cannot determine the place where the wormhole's exit occurs. The exit appears at a completely random position in space, no chance to cause it building up where one needs it. Alcubiere solution, on the other hand, lacks on the need for a matter distribution that itself can be created only with sublight speed. In other words: you need to build up a railway, before you can ride over it with FTL speed. This is where some LQG extension could introduce a better opportunity. In discrete spacetime, ony may assume a type of spin network configuration transitions where a spin not only changes its connections to neighbour spins, but changes its neighbourhood: the spin "cuts" the connections to its previous neighbours, and builds up new connections to a couple of spins far away, resulting in a relocation of the spin in the network. This is similar to the wormhole solution in GR, but with the advantage, that discrete spacetime discloses the possibility of some dynamics that enables one to control direction and range of such a relocation, where range is measured in number of network nodes between initial and relocated position. For example, one could introduce a set of relocation operators, with each of them annihilating initial state with the spin connected to a multiplet of spins and creating new state with the spin connected to new multiplet of spins, in some way similar to climbup and climbdown operators for quantum harmonic oscillator. Inserting those relocation operators into Hamiltonian in a appropriate way, one could derive transition rates for the relocations. For example, a relocation where new position is 1000 nodes away from initial position could cost more energy than a relocation with only 10 nodes distance. A particle housed by the relocated spin would have been transported with FTL speed then, in the 1000 nodes case with 1000 times lightspeed, as relocation process takes place in one Plancktime time step. However this concept lives from the idea, that the housed particle does not "feel" the repositioning of the spin as movement in space. Therefore my interest for worldline length. 


#7
Mar808, 03:17 AM

Sci Advisor
PF Gold
P: 9,445

I think the energy dependant speed of light thing has already had a funeral.



#8
Mar808, 10:02 AM

P: 7

http://www.edge.org/3rd_culture/smol...n03_print.html 


#9
Mar2208, 07:32 PM

P: 7

some news about my idea of node relocation process:
such relocation is problematic when one regards the background independence of LQG. If spin network had a fixed, predefined structure, one could consider it like a chess board, and node relocations like the move of a Queen from one field to another one with several fields distance. Relocation would simply mean movement by n hops in some direction. However, in backgroundindependent LQG, spin network does not have such a fixed structure. From the point of view of a node, is not automatically clear what n hops in direction Omega means. The solution is, to assume every node has some information about the network's current configuration available. By this, a node that is going to relocate has a pool of information present, what types of relocations are available in current network configuration. This implies, that there is an instantaneous information transfer from whole network to each node, i.e. with infinite speed, that informs each node about every change in configuration, no matter how far away. That does not require signals running through the network  LQG tells that on basic layer of reality, there's no spatiality, all nodes are in some way direct neighbours, this enables instanteneous information transfer. As a modification of this concept, one could introduce some "mesoscopic" field, that is defined on a high number of nodes, i.e. a big portion of network, big enough to handle the field as classic. This fields defines directions, by its wave modes. Instead of directly refering to the network's configuration, a relocating node could refer to the field  by doing a relocation into a direction with respect to a field mode and a range measured in number of hops in this direction, or in wave lengths of the field mode. In the latter, one had to assume a fixed wave length for the field, e.g. one hop, or one Planck length. Concerning the details of relocation process, I have three different ideas: 1) simple relocation, the node starts with connections to a set of nodes (initial neighbourhood) and ends with connections to a new set of nodes (final neighbourhood). 2) "burst" of next neighbour transitions: a node only does a next neighbour transition, which are part of conventional LQG, but there's a whole "line" of nodes doing the same in same time step, resulting in the initial node being relocated to the end of the line after the time step. There should be a "trace" of the transitions viewable along the line. 3) topological loop: the relocating loop establishes new connections to a set of nodes far away (like in 1)), but with at first keeping the connections to old set, resulting in a wormhole topology with a loop, consisting of the long path to the new neighbours and one of the new connections. After another time step, the old connections collapse, and the loop with them. 


#10
Mar2408, 10:32 AM

P: 7

Concerning nongravitational matter fields in LQG, I found this resource:
http://www.aei.mpg.de/pdf/doctoral/HSahlmann_02.pdf In chapter 5.1, as far as I can see, it is derived for KleinGordon field that the field is simply defined on the vertices of the spin network and the dynamics of the field is happening on a fixed network configuration, without influencing or changing the configuration. This would lead to the picture, that KleinGordon particles are simply hopping around on the spin network, from vertex to vertex. Or do I misunterstand the article? 


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