Wieland's new action for 4d simplicial gravity can be significant

[marcus]

It is too new for me to know for sure but I think it probably is. The action has a new kind of momentum, consisting of tetrahedral volume flowing along an edge-network: a "one dimensional branched manifold" derived from a simplicial decomposition of the original 4d manifold.

Wieland calls the action the "spin-Regge" action and it is equation (50) on page 14 of this paper, which just appeared.

http://arxiv.org/abs/1407.0025
New action for simplicial gravity in four dimensions
Wolfgang M. Wieland
(Submitted on 30 Jun 2014)
We develop a proposal for a theory of simplicial gravity with spinors as the fundamental configuration variables. The underlying action describes a mechanical system with finitely many degrees of freedom, the system has a Hamiltonian and local gauge symmetries. We will close with some comments on the resulting quantum theory, and explain the relation to loop quantum gravity and twisted geometries. The paper appears in parallel with an article by Cortês and Smolin, who study the relevance of the model for energetic causal sets and various other approaches to quantum gravity.
26 pages, 2 figures.

 

[marcus]

Some interesting quotes:
==page 1==
... Here, we are, however, seeking for an answer to a subtler question: Is there a Hamiltonian formalism for discretized gravity in terms of first-order tetrad-connection variables available? This is a difficult question, because there is a conceptual tension: A Hamiltonian always generates a differential equation, it generates a Hamiltonian flow, while, on the other hand, discretized theories are typically governed by difference equations instead [16].
This article develops a proposal resolving the tension. Following the Plebański principle, we start with the topological BF action [17]. We introduce a simplicial decomposition of the four-dimensional spacetime manifold, and discretize the action. This leads us to a sum over the two-dimensional simplicial faces. Every face contributes a one-dimensional integral over its bounding edges, thus turning the topological action into an integral over the entire system of edges—into an action over a one-dimensional branched manifold.
==endquote==

==page 14==
Equation (50) is our final proposal, a proposal for an action for discretized gravity in first-order variables. At this point, neither do we know of any global solutions of the resulting equations of motion on an arbitrary two-complex, nor do we have a proof that they would correspond to any physical spacetime geometry. Yet, we do have positive evidence in favor of our proposal. First of all, we will see, that the constraint algebra closes, and that there are no secondary constraints. Then, the model has curvature. This curvature lies in the faces dual to the elementary triangles, and is given, just as in Regge calculus, by the sum over the boost-angles between the adjacent tetrahedra. Finally, and most importantly, the solutions of the equations of motion have a geometric interpretation and define a twisted geometry [24–26].

Twisted geometries are discrete geometries found in the semi-classical limit of loop quantum gravity [3, 28–30]. They are similar to Regge geometries insofar as they represent a collection of flat tetrahedra glued along their bounding triangles, but unlike Regge geometries there are no unique length variables: Every triangle has a unique area, and every tetrahedron has a unique volume, but the length of a triangle’s bounding side exists only locally.

III. DYNAMICS OF THE THEORY
In the last section, we gave a proposal for a gravitational action on a simplicial lattice. Now it is time to study the dynamics. The action (50) is local in t, and so are the resulting equations of motion, that tell us how the elementary configuration variables change as we move forward in t and go from one vertex to the next. This t-variable does however not have an immediate physical interpretation. It is no physical time, and does not measure duration as given by a clock.

A. Hamiltonian formulation
Symplectic structure: Let us first fix an arbitrary edge e in the discretization. Restricting our analysis to just a single edge is a matter of convenience. It allows us to use a condensed notation …
==endquote==

==pages 23 and 24==
IV. CONCLUSION
Summary We have split our presentation into two halves. The first half gave the derivation of the action. ... A simplicial decomposition of the four-dimensional manifold brought us to the discretized action, and we saw that this discretized action can be written as a one-dimensional integral over the simplicial edges—as an action over a one-dimensional branched manifold. ...We then studied the discretized simplicity constraints in the spinorial representation and added them to our one-dimensional action. This introduced an additional element to the theory: The volume-weighted time-normals of the elementary tetrahedra. We argued that a consistent theory is possible only if we also treat these time normals as dynamical variables in the action. At each simplicial vertex these volume-weighted normals sum up to zero—this representing the geometricity of the four-simplex itself. We argued that this closure constraint represents a momentum conservation law for a system of particles scattering in a locally flat auxiliary spacetime, and we made an explicit proposal for an action realizing this idea. Each of these particles corresponds to an elementary tetrahedron in the simplicial complex, with their mass representing the three-volume of the elementary tetrahedra, and each interaction vertex representing a four-simplex in the discretization...

The second half of the paper studied the dynamics of the theory as derived from the action. Let us say it clearly: We have not shown that the equations of motion for the discretized theory would correspond to some version of the Einstein equations on a simplicial lattice. Nevertheless, we do have some definite results: First of all, there is a Hamiltonian formulation for the discretized theory, there is a phase space, constraints and a Hamiltonian. The Hamiltonian generates the t-evolution along the elementary edges of the discretization, and preserves both the first- and second class constraints. There are no secondary constraints. Next, we showed that the solutions of the equations of motion have a geometric interpretation in terms of twisted geometries.

Twisted geometries are piecewise flat geometries generalizing Regge geometries: In Regge calculus the edge lengths are the fundamental configuration variables, in twisted geometries there is no unique notion of length: Every tetrahedron has a unique volume, and every triangle has a unique area, but the length of a segment depends on whether we compute it from the flat metric in one simplex or the other.

Finally we gave an argument why the model has curvature. Going around a triangle we pick up a deficit angle, which is a measure for the curvature in the dual plane. We showed that this deficit angle will generically not vanish, in fact it is given by the integral of the Lagrange multiplier λ …

The relevance of the model The action (50) describes a system of finitely many degrees of freedom propagating and interacting along the simplicial edges. The system has a phase space, local gauge symmetries and a Hamiltonian. What happens if we quantize this model? Do we get yet another proposal for a theory of quantum gravity? Recent results [23, 27, 36, 58] point into a more promising direction and suggest a convergence of ideas: The finite-dimensional phase space can be trivially quantized. The constraints of the theory glue the quantum states over the individual edges so as to form a Hilbert space over the entire boundary of the underlying simplical manifold. The boundary states represent projected spin-network functions [59, 60] in the kinematical Hilbert space of loop quantum gravity. It is clear what should be done next: For any fixed boundary data we should define a path integral over the field configurations along the edges in the bulk. At this point, many details remain open, and we have only finished this construction for the corresponding model in three-dimensions [58], yet we do know, that whatever the mathematical details of the resulting amplitudes will be, they will define a version of spinfoam gravity [61].

Finally, there is the motion of the volume-weighted time normals, which endow the entire simplicial complex with a flow of conserved energy-momentum. As shown by Cortês and Smolin in a related paper [27], these momentum-variables introduce a causal structure, and allow us to view the simplicial complex as an energetic causal set [51, 52]—a generalization of causal sets carrying a local flow of energy-momentum between causally related events.
==endquote==

I'll quote the abstract and give a link to the Cortês and Smolin paper in the next post.

 

[David Horgan]

Thanks Marcus this pair of papers are very valuable. Wieland' s work is outstanding isn't it. Definitely time to review again the Cambridge Monographs on Mathematical physics on sponsors and twistors.

 

[marcus]

Thanks Marcus this pair of papers are very valuable. Wieland' s work is outstanding isn't it. Definitely time to review again the Cambridge Monographs on Mathematical physics on spinnors and twistors.

I agree! Really outstanding! And you are probably right at least as applies to me about reviewing the math.

You remind me that I mustn't forget to post the link and abstract to the companion paper by Marina Cortês and Lee Smolin.

http://arxiv.org/abs/1407.0032
Spin foam models as energetic causal sets
Marina Cortês, Lee Smolin
(Submitted on 30 Jun 2014)
Energetic causal sets are causal sets endowed by a flow of energy-momentum between causally related events. These incorporate a novel mechanism for the emergence of space-time from causal relations. Here we construct a spin foam model which is also an energetic causal set model. This model is closely related to the model introduced by Wieland, and this construction makes use of results used there. What makes a spin foam model also an energetic causal set is Wieland's identification of new momenta, conserved at events (or four-simplices), whose norms are not mass, but the volume of tetrahedra. This realizes the torsion constraints, which are missing in previous spin foam models, and are needed to relate the connection dynamics to those of the metric, as in general relativity. This identification makes it possible to apply the new mechanism for the emergence of space-time to a spin foam model.
16 pages, 4 figures.

 

[marcus]

Listening to Wieland's ILQGS talk and the accompanying discussion helps to understand the July paper.
Slides:
http://relativity.phys.lsu.edu/ilqgs/wieland091614.pdf
Audio:
http://relativity.phys.lsu.edu/ilqgs/wieland091614.wav

The talk generated active discussion (Ashtekar, Smolin, Speziale, Bianchi, Rovelli, ...) so the audio extends for more than an hour and 15 minutes.

To get the July paper (if you don't recall the arxiv number 1407.0025) sometimes it's swifter to google it instead of having to go through an arxiv search.
I just google "wieland new action".

https://www.physicsforums.com/threa...ssical-action-wielands-talk-is-online.771379/

 

[marcus]

Wieland's action lives on the one-dimensional branched manifold consisting of the edges of the spin foam.

This 1D manifolds could be thought of as the web of world-lines of a swarm of tetrahedra (like a cloud of gnats or flock of birds).
As the tetrahedra move around (dividing and coalescing) along the edgework manifold, the geometry evolves.

It's interesting that there is no global time. An evolution parameter "t" is introduced to define the action. t runs along the edges, but it does not correspond to any physical time that could be read from clocks. At least not in general. In the ILQGS discussion Lee Smolin offered the example of a special case where "t" actually was a time parameter.
We're familiar with this sort of thing: in GR the "coordinate time" is not, in general, something that can be observed. It is not the time of any physical clock. But in special cases one can arrange for it to correspond to physical time in some particular solution, as in Friedmann model cosmology.

Wieland emphasized (as he also does in the July "new action" paper) that the evolution parameter "t" is not any sort of physical time. However I think that each individual tetrahedron (like each individual gnat or bee in a swarm) can have its own time, along its own path in the 1D branched manifold.

 

[marcus]

Listening to Wieland's ILQGS talk and the accompanying discussion helps to understand the July paper.
Slides:
http://relativity.phys.lsu.edu/ilqgs/wieland091614.pdf
Audio:
http://relativity.phys.lsu.edu/ilqgs/wieland091614.wav

The talk generated active discussion (Ashtekar, Smolin, Speziale, Bianchi, Rovelli, ...) so the audio extends for more than an hour and 15 minutes.

To get the July paper (if you don't recall the arxiv number 1407.0025) sometimes it's swifter to google it instead of having to go through an arxiv search.
I just google "wieland new action".

https://www.physicsforums.com/threa...ssical-action-wielands-talk-is-online.771379/

I want to emphasize that the September talk helps understand the July "New Action" paper, for example here is the CONCLUSIONS slide of the seminar talk:
==quote "Key features of the spin foam action" ==
Branched-continuity: The action is an integral over the entire system of edges, an action for a one-dimensional branched manifold (c.f. relative locality).

Causal spinfoams: The edges are oriented, we can distinguish future-pointing from past-pointing edges.

Twisted geometries: The Hamiltonian generates twisted geometries.

Particle picture: Every tetrahedron represents a massive particle, with internal SU(2)_n little-group color DOFs. The particles’ momenta p are the volume-weighted four-normals p = Vol n. Mass turns into quantized volume.
==endquote==
So it appears we are dealing with a new type of spin foam. The edges are oriented. So the vertices (corresponding to 4-simplices, i.e. to space-time elements, or events) constitute a partially ordered set.
So this type of spin foam isd akin to what is studied in the "causal sets" approach to QG (pursued by Rafael Sorkin, Fay Dowker, David Rideout, and others)
and in his conclusions Wieland refers to what he is studying as "causal spin foams"

 

[marcus]

David Horgan (post #3) correctly observed that what we are dealing with here is a PAIR of papers, and the emergence of the idea of the causal spin foam. You can get the Cortes-Smolin paper by googling "cortes causal spin foam" and it provides, I think, the easiest to understand explanation of how the whole thing works and the role Wieland's "new action" paper plays. I'll try to given an impressionistic sketch. First let's recall the Cortes-Smolin paper which is essentially the source of what I have to say.

I agree! Really outstanding! And you are probably right at least as applies to me about reviewing the math.

You remind me that I mustn't forget to post the link and abstract to the companion paper by Marina Cortês and Lee Smolin.

http://arxiv.org/abs/1407.0032
Spin foam models as energetic causal sets
Marina Cortês, Lee Smolin
(Submitted on 30 Jun 2014)
Energetic causal sets are causal sets endowed by a flow of energy-momentum between causally related events. These incorporate a novel mechanism for the emergence of space-time from causal relations. Here we construct a spin foam model which is also an energetic causal set model. This model is closely related to the model introduced by Wieland, and this construction makes use of results used there. What makes a spin foam model also an energetic causal set is Wieland's identification of new momenta, conserved at events (or four-simplices), whose norms are not mass, but the volume of tetrahedra. This realizes the torsion constraints, which are missing in previous spin foam models, and are needed to relate the connection dynamics to those of the metric, as in general relativity. This identification makes it possible to apply the new mechanism for the emergence of space-time to a spin foam model.
16 pages, 4 figures.

The gist is that the space-time past is being built up out of Pachner move interactions of tetrahedra . Every 4-simplex added to the body of the past represents a Pachner interaction involving 5 tets . there are 4 possible interactions 1↔4, or 2↔3
The reason we can say this is that in the causal spin foam the 4-simplices are dual to vertices and the tets are dual to edges and the edges are TIME-ORIENTED. So at each vertex we can identify the incoming and outgoing edges . So the vertex is an interaction.

Basically we are talking about a new "ontology", a way of thinking the world. there is no instantaneous 3d space, but there is (past) spacetime. And (past) spacetime is made of interactions (that have happened).

There's a swarm of tets and they are constantly interacting and becoming different ones, 4 might collide and 4→1 become one, or one might explode 1→4 and become 4. the AMPLITUDES for these moves haven't been thoroughly worked out. But Wieland has treated the swarm of tets as a classical system and proposed a Hamiltonian. That is what the "new action" is about.

Once some tets have interacted, they are DEAD and become part of the fossilized or crystallized PAST. The past is something you can't affect any more, you can't interact with it, and it can't cause anything more than it already has and is already in progress. So it is immaterial in a sense. You can only see and feel and touch what you can INTERACT with.

Our experience is in the present, in a kind of confusion of interacting tetrahedra which have not finished interacting yet. Since we equate interactions to 4-simplex "cells" or "atoms" of space-time, these 4d cells are still forming. Or in the language of Sorkin and Dowker, the atoms are being born.
The order in which the past is covered over with a new layer of interactions is, as Dowker says, "pure gauge". It is not physically meaningful in which order it happens because you can't interact with the order. There isn't time. There isn't time to discriminate between different orders of interactions freezing up. This is the experience of the present moment, and the PASSAGE OF TIME.

The reason Cortes-Smolin "causal spin foam" and Wieland "new action" papers are significant is basically because they find a way to represent our experience of the passage of time in the context of Covariant Lqg (CLQG) aka Spin Foam QG

 

[marcus]

Since there is potentially a kind of partial merger going on between Spin Foam QG and Causal Sets, I want to understand Causal Sets better. Fay Dowker's May 2014 paper about how the specialness of the present moment and the experience of PASSAGE is realized in Causal Sets has been very helpful. She argues effectively that a basic world theory (improving on GR) should have intrinsic features representing the present moment and the passage of time. And Causal Sets does. You can POINT TO where it is, in the theory.
So that's good. I've mentioned this before. If you want to look at Dowker"s May article google "Dowker passage arxiv".This stuff carries over to Wieland's work because his version of spin foams have ORIENTED EDGES which are dual to tetrahedra of the triangulation or the simplicial complex. At any node or vertex of the spin foam you can say which edges are INCOMING and which are OUTGOING. Each "pent" or space-time cell is an interaction of tets. It is a "geometric interaction" of the most basic sort. Some tets come in, collide, react, and some other tets go off. These interaction events COMPRISE the spacetime because each interaction is a pentachoron, that is the technical name for a 4-simplex.

I had some thoughts about Dowker's May paper, in some cases just questions I was wondering about. They're somewhat foggy and unresolved, but I'll share them, as part of trying to understand Causal Sets and this trend towards a synthesis with SF type of LQG.

Operationally, time is a COUNT of some periodic process. You need an oscillator and a counter to measure time. If it's so brief that even in principle you cannot measure it, does it have any meaning? If below some range of duration, or above some range of frequency, you cannot measure, if it is not measurable even in principle, does it mean anything?

Causation is more basic than time. It is not quantified except by itself. A sequence of events that follow each other immediately according to the discreteness of the world. Each causing the next. Is there a smallest separation, a shortest duration? But if those amounts cannot even be measured, all there is is the sequence. Number. Below some critical scale, which might be (say) Planck scale, how would you build a clock? Even in principle.

The proliferation of space-time cells constituting a causal set does not happen IN time. It is causation itself. It provides a basis on which things like clocks and concepts like time can be constructed.

I am thinking of the orderless commotion at the frontier of the past, where new interactions are occurring, new events happening, new cells of space-time (which wieland identifies with basic geometric interactions) are accumulating and accruing to the past. The vital frontier of causation itself.

Let's think of building up a partial order by adjoining nodes (interaction events) in a certain sequential order. Now the same partial order (think of a tree diagram, as one possible example) may be able to be built up in several different sequential orders. If, of two events, it is impossible (even in principle) to say which happened first, then the sequential order is "pure gauge" physically meaningless, merely a matter of mathematical convenience. This is why I'm describing the frontier as an orderless commotion. An arbitrary choice of sequential order may be useful for description, but permuting the sequential order of construction (as long as the causal set itself is unchanged) makes no physical difference.

Dowker points out the analogy with GR, which has "diffeomorphism invariance" aka "general covariance". In both cases, if you are careful you can mess around with the description (whether it is a building sequence or a system of coordinates) and it does not change the physics.