Reformulation of Loop gravity in progress, comment?

marcus

Since we just turned a page, I'll copy post #19 as a reminder of what the reformulation topic-of-the-thread is about. It's interesting that things are in flux because we are now effectively in the runup to Loops 2013 which will be held at Perimeter Institute in about one year's time.

====quote post #19==
The reformulation of Loop now being explored is complex, and some parts seem still tentative.
I see three main initiatives:

A. Immirzi-less BH entropy.
Bianchi and others find S = A/4. The coefficient of area no longer depends on Immirzi parameter γ. So gamma is unclamped. arxiv:1204.5122 arxiv:1205.5325

B. un-Diracly quantizing GR.
Jacobson proposed a new goal. Find the correct quantum "molecules" of spacetime geometry for which Einstein's GR equation is the thermodynamic equation of state.
It could turn out that the Spinfoam description of geometric evolution already provides the correct degrees of freedom, and GR is simply the equation of state of spinfoam.
So that instead of quantizing GR Diracly, one has quantized it un-Diracly.
arxiv:1204.6349 arxiv:1205.5529

C. The sign of the tetrad--could one detect a region of "antispacetime"?
One possible crude picture of spacetime geometry is that of a partially coherent swarm of tetrads. Like flocking birds or shoals of fish, these tetrads tend to be oriented coherently with their neighbors. But in principle, divisions might occur: there could appear patches with opposite orientation. The set-up described in the May paper "Discrete Symmetries in Covariant LQG" arxiv:1205.0733 allows for this to happen. The usual Holst action is modified in a significant way---by introducing the sign of the tetrad, a symbol s which can be +1, 0, or -1 depending on the sign of the determinant of the tetrad.
Since fermions couple to the tetrad, phase can evolve in either of two senses and a double slit experiment can in principle detect reversed geometry by a shift of the interference pattern.
==endquote==

atyy

Rovelli's programme has not yet been shown to succeed. My point, and Tom's, I think, is that if it succeeds, then the covariant quantization will be equivalent to a canonical quantization.

marcus

That sounds reasonable, there is a Loop research community and a Loop program which involves a number of DIFFERENT approaches and versions. Bianchi has laid out several different ones. Etera Livine has some great ideas. Engle has too. Rovelli's view is obviously in flux. Lewandowski and Ashtekar are clearly major players in the program. That's only the beginning of a list

So there is a Loop program. That is something real and it may or may not succeed. And if it succeeds it MAY OR MAY NOT contain a background independent QFT that was derived by some preconceived "quantization" method which you have in mind. So that is all real enough and makes sense.

But sometimes you sound as if you actually believe there is a definite permanently fixed "Rovellian" approach to Loop QG. And you go on about how this conflicts with what Engle says or what Jacobson says etc etc. This sounds peculiar to me. When you talk about the "Rovellian" this or that as if you knew of some permanent definite approach it does not seem based on reality.

As far as I can see, Loop is rapidly evolving and advancing on several fronts and seems to change every two or three years. So far it has been up to Ashtekar and Rovelli to present a coherent in-a-manner-of-speaking "OFFICIAL" version every 2 or 3 years. In any given year they are the ones normally asked to supply the principle review paper and give the overview conference talk. With Ashtekar concentrating on the cosmology side.

However the lineup could change. Younger people could be invited to start filling these roles. And this year Jerzy Lewandowski is doing a great job reviewing organizing representing the program.
Also Jorge Pullin. A program has to have leaders and if there is rapid progress then every 2 or 3 years you need an official redefinition or reformulation. But who and how it's done can change. We will see how it shapes up at Loops 2013.

I don't known enough to even begin to give you a complete accurate portrait, of course.
But obviously back in 2011 a defining role was played by Rovelli's Zakopane lectures
arxiv 1102.3660 and May presentation at the Madrid Loops conference. Now we can expect something new and we can wonder what shape will it take this time? What will take the place of arxiv 1102.3660 when people gather for Loops 2013 at Perimeter. Who will give the main overview? What new work will stand out? It could be several peoples' work.

atyy

Oh, I usually mean very specific statements of X are in conflict with very specific statements of Y. I never mean all statements of X are in conflict with all statements of Y, which would be absurd. In this case, it has to do with the possibility of canonically quantizing GR, even in the UV. Rovelli does seem to alternate between two views. Sometimes he does seem to indicate that one could have a successful spin foam quantization which does not meet up with the canonical formalism. But I think the he mostly approaches spin foams as a way to meet up with canonical quantization. You can trace this line of thinking quite consistently over a period of more than 10 years, including the latest paper about anti-spacetime:

http://arxiv.org/abs/gr-qc/9806121 (bottom of p1)
"Here, we complete the translation of canonical loop quantum gravity into covariant spacetime form initiated in [6]. The “quantum gravity Feynman graphs” are two-dimensional colored branched surfaces, and the theory takes the form of a “spin foam model” ..."

http://arxiv.org/abs/0708.1236 (abstract)
"... providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. ...."

http://arxiv.org/abs/0711.0146 (abstract)
"These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions."

http://arxiv.org/abs/1205.0733
p2: "In canonical loop gravity one works in the time gauge and chooses a linear combination of the connection and its Hodge dual as a canonical variable. The corresponding conjugate momentum is the Ashtekar electric field Eai, but (confusingly) one finds two different expressions for this field in the literature ... The two expressions differ by the sign s and can be derived from S' and S�", respectively."

footnote 5: "... we know from canonical loop quantum gravity that links with j = 0 can be erased from the spin-network. ..."

tom.stoer

I think this is what the objective of the whole community - find a mathematical consistent and physically reasonable quantization. They use different approaches - canonical, covariant canonical, spin foams, group field, ... - not b/c these different approaches are mutually exclusive but complementary views, just like in ordinary quantum mechanics.

marcus

Good point! Different approaches can indeed complement each other and help to deepen and fill out the understanding. I remember Eugenio Bianchi saying this same thing--he has developed/worked on several alternate formulations of Loop gravity--they can improve or supplement each other. I don't recall his exact words.

BTW this just came out today. It has to do with the topic I called "The Sign of the Tetrad" (the possibility of having regions of spacetime geometry where the phase of a fermion rotates in reverse).

http://arxiv.org/abs/1206.3903
How to detect an anti-spacetime
Marios Christodoulou, Aldo Riello, Carlo Rovelli
(Submitted on 18 Jun 2012)
Is it possible, in principle, to measure the sign of the Lapse? We show that fermion dynamics distinguishes spacetimes having the same metric but different tetrads, for instance a Lapse with opposite sign. This sign might be a physical quantity not captured by the metric. We discuss its possible role in quantum gravity.
6 pages, 8 figures. Article awarded with an "Honorable Mention" from the 2012 Gravity Foundation Award.

marcus

==quote post #52==
B. un-Diracly quantizing GR.
Jacobson proposed a new goal. Find the correct quantum "molecules" of spacetime geometry for which Einstein's GR equation is the thermodynamic equation of state.
It could turn out that the Spinfoam description of geometric evolution already provides the correct degrees of freedom, and GR is simply the equation of state of spinfoam.
So that instead of quantizing GR Diracly, one has quantized it un-Diracly.
arxiv:1204.6349 arxiv:1205.5529
==endquote==

Regarding this general theme I should mention recent work by Thomas Thiemann and the group at Erlangen. Abstracts of several paper are given here
http://physicsforums.com/showthread....12#post3964712
with some comment. They seem to be exploring paths to a kinda-sorta Hamiltonian-style quantization without being constrained to a strictly Dirac format. If someone has a different interpretation of what's happening in those 3 new papers, please share it. I'd be interested to know how you see it. I like Derek Wise and Steffen Gielen's paper that uses the concept of a field of observers (straight out of standard cosmology).

In this same connection we should also look at a paper by a German PhD student David Schroeren, now at Marseille. He makes what seems to me creative and effective use of some ideas of Gell-Mann, Hartle, and others. See Hartle's 1993 Les Houches account http://arxiv.org/abs/gr-qc/9304006 .
As described there by Hartle an important motivation was to restructure Quantum Mechanics so that it would be more suitable for Cosmology (where there is no separate Observer, since the System is the whole universe.) Obviously Quantum Theory must be reformulated if it is going to be applied to the whole universe, and when reformulated it might in fact be GENERALLY BETTER and turn out to be useful for other applications besides Cosmology.

So we get proposals with names like "decoherent histories" QM and "consistent histories" QM with some slightly different formalism. Now Schroeren has tried applying these heretical ideas about Quantumtheory to Spinfoams.
It leads to a different kind of quantization of General Relativity, so I list this paper too.
http://arxiv.org/abs/1206.4553
Decoherent Histories of Spin Networks
David P.B. Schroeren
(Submitted on 20 Jun 2012)
The decoherent histories formalism, developed by Griffiths, Gell-Mann, and Hartle is a general framework in which to formulate a timeless, 'generalised' quantum theory and extract predictions from it. Recent advances in spin foam models allow for loop gravity to be cast in this framework. In this paper, I propose a decoherence functional for loop gravity and interpret existing results as showing that coarse grained histories follow quasiclassical trajectories in the appropriate limit.
13 pages

marcus

Decoherent Histories (DH) quantum mechanics looks interesting. I think the most active proponent is James Hartle (UC Santa Barbara)

Other authors are Murray Gell-Mann and Robert Griffiths, but I think of it primarily as "Hartle-QM"

It is a definition of QM that depends less heavily on the Observer making Measurements with a classical instrument. There is no essential split of the universe into a quantum system and a classical observer.

It is a "path integral" or Histories approach. The basic mathematical objects are PARTITIONS of all possible histories.

A partition is a collection of disjoint subsets whose union is the whole. Generally a partition involves many subsets, but a simple example could be a partition into just two:
"the ball went into the hole" versus "the ball did not go into the hole"

Partitions of all possible histories can represent things that we might care about, which matter to us, or which we might want to risk betting on, like whether the flight will land safely in Seattle or a certain flip will flop or a bridge not break. We may want to know which set of histories the world is in whether or not we are classical creatures and whether or not we are making measurements at the moment.

"the moon is there" versus "the moon is not there" has an approximate welldefined probability even when no one is looking. The set of histories in which it is there has high probability.

So Hartle-QM frees quantum mechanics from a kind of ontological dependence. One can invoke approximate probabilities of the subsets in a partition when the partition is sufficiently decoherent
(almost by definition) and a key part of Hartle-QM is formalizing when partitions are sufficiently unambiguous in this sense.

I'd like to see Hartle-QM applied to Spinfoam QG. I'll be interested to see the outcome.
I'll bring over some links.
Hartle Gell-Mann 2011 paper: http://arxiv.org/abs/1106.0767
Hartle 2008: http://arxiv.org/abs/0801.0688 (appendix A especially helpful)
Hartle 2006: http://arxiv.org/abs/gr-qc/0602013 (generalizing QM for quantum spacetime)

marcus

In just a week from tomorrow, on Tuesday 10 July, Fay Dowker is going to talk about something which I think is important to the development of Loop gravity. It will go into the PIRSA online video archive. I for one am certainly going to watch the talk.

PIRSA:12070001
Title: The Path Integral Interpretation of Quantum Mechanics
Speaker(s): Fay Dowker - Imperial College
Abstract: In 1932 Dirac wrote that the lagrangian approach to classical mechanics was probably more fundamental than the hamiltonian approach because the former is relativistically invariant whereas the latter is "essentially nonrelativistic". In quantum theory the hamiltonian approach leads to canonincal quantisation, Hilbert space, operators and the textbook rules for state vector "collapse", which are all indeed more or less divorced from the spacetime nature of the physical world as revealed by relativity. The "essentially relativistic" lagrangian approach on the other hand leads to the path integral, as shown by Dirac in 1932 and developed by Feynman. I will show how the interpretation of quantum mechanics in a path integral framework is based directly on events in spacetime and show that it leads to a second "fork in the road" depending on whether it is necessary for probabilities to play a fundamental role in the theory.
Date: 10/07/2012 - 3:30 pm
Series: Quantum Foundations
Location: Time Rm
URL: http://pirsa.org/12070001/
================

Basically I think this goes back to Jim Hartle's talk to the 2005 Solvay Conference (on the "Quantum Structure of Space and Time"). The talk was written up and posted in early 2006. I'll get the abstract:
http://arxiv.org/abs/gr-qc/0602013
Generalizing Quantum Mechanics for Quantum Spacetime
James B. Hartle (University of California, Santa Barbara)
(Submitted on 2 Feb 2006)
Familiar textbook quantum mechanics assumes a fixed background spacetime to define states on spacelike surfaces and their unitary evolution between them. Quantum theory has changed as our conceptions of space and time have evolved. But quantum mechanics needs to be generalized further for quantum gravity where spacetime geometry is fluctuating and without definite value. This paper reviews a fully four-dimensional, sum-over-histories, generalized quantum mechanics of cosmological spacetime geometry. This generalization is constructed within the framework of generalized quantum theory. This is a minimal set of principles for quantum theory abstracted from the modern quantum mechanics of closed systems, most generally the universe. In this generalization, states of fields on spacelike surfaces and their unitary evolution are emergent properties appropriate when spacetime geometry behaves approximately classically. The principles of generalized quantum theory allow for the further generalization that would be necessary were spacetime not fundamental...
31 pages. 4 figures.

To paraphrase, states and evolution of fields defined on spacelike surfaces are ONLY appropriate as math idealizations when geometry behaves APPROXIMATELY CLASSICALLY. In more general situations such idealizations are NOT appropriate.
They are, as Dowker put it, "more or less divorced from the spacetime nature of the physical world".

marcus

Ultimately if you think of Loop gravity as based on a fixed set of discrete points or a smooth manifold "continuum" then you aren't likely to understand the line of future progress I'm talking about.

The Dirac canonical quantization applied to GR leads to a lot of paraphernalia which it is NOT appropriate to assume (Hartle suggests) unless spacetime geometry is behaving in approximately classical manner. It's a picture that only "emerges" under specific tame circumstances.
That goes for approaches using EMBEDDED spin networks as well. They need a manifold--i.e. extra baggage.

Conversely the spinfoam dynamics approach, by now familiar to everybody, does not involve extra baggage--in particular, no manifold. It is based on what Dowker's abstract seems to be talking about: events. Related combinatorially. No infinite sets, just a finite web of facts/predictions, depending entirely on the history!

So we'll see. We'll watch the video of Dowker's talk and see if it fits with and extends what Hartle had to say to the 2005 Solvay Conference.

Dowker may steer the talk in the Causal Sets direction but that's all right. Loop and Causal Sets share foundation roots--to some extent a common rationale. Eventually "Quantum Foundations" considerations are going to influence the development of Loop gravity--indeed they may already have influenced it to a considerable extent.
================================
It may help clarify the issues if I paste in a short summary of how I see Hartle's "Decoherent Histories" (DH) version of quantum mechanics. This was originally a post in the "Loop future" thread:


Hartle and friends propose a reformulation of Quantum theory we can call "Histories" QM which basically says that the machinery of Dirac quantization does not exist--it is merely emergent at low energies, a convenient workable approximation to reality over a limited range. The spacelike 3D manifold does not exist in reality. To formulate QM, you need three things:
A. Histories
B. Partitions of histories (grouping, classifying, "coarsegraining" them)
C. a Decoherence functional that tells you when a given partition is bettable.

Sets in a partition represent things you might like to know or to predict. A given partition is bettable when you can assign fair odds (approximate conventional probabilities) to it, make predictions, settle bets, in other words make honest book on it.
The Decoherence functional tells you when a partition of the histories is sufficiently uncorrelated that the probabilities will be additive---interference is small enough to be considered negligible.

Hartle Histories QM is, I believe gaining acceptance. So it makes sense to me, in that light, that the Erlangen group should be moving away from a strict Dirac quantization and in the direction of DUST.
....[That gets you to a nice effective halfway station. Since it's not fundamental, why not make life easy and assume some dust? Going further down that road brings you to Histories=Foams]

tom.stoer

I think this is misleading.

I agree that spin networks are constructed from a manifold and that one get's rid of the manifold during quantization, constraint fixing an "projecting" to phys. d.o.f. = spin networks. Therefore in some sense spin networks have (or had) this extra baggage (historically).

But I do not agree that SFs do not have this extra baggage. They are constructed using the same ideas as spin networks; the only difference is that one switched from networks to foames rather late. There is no conceptual difference between spin networks and spins. It's a matter of taste whether you postulate a kinematical Hilbert space and a Hamiltonian or whether you postulate vertex amplitudes and PI measures.

Spin foams and spin networks share the same weakness; historically they are rooted in a picture using a manifold - and their derivation is by no means complete. Not deriving but postulating them has a different weak point, namely guessing ;-)

Nevertheless I agree that the main weak points could be that one is simply quantizing the wrong degrees of freedom (just like QFT applied to Navier-Stokes equations). These are essentially two weak points
1) wrong d.o.f.
2) quantization (which is never unique)

marcus

Sorry, you misunderstood. In the modern treatment spin networks are NOT constructed from a manifold. It used to be the case that spin networks were EMBEDDED.
When I say "embedded spin network" I mean to older object.

When I simply say "spin network" it is a combinatorial object, as per the standard Loop source paper. It does not have the extra baggage.

As you say, SF are constructed using the same ideas. Therefore they do NOT have manifolds or other extra baggage.

It seems you understood completely opposite from what I intended.
I must try to write more clearly.

tom.stoer

marcus, it's not fair to say that non-embeded networks are not constructed from a manifold; yes, they are combinatorial objects, but nevertheless they share many features with the embedded one; they are not completely bagge-free, even uif this baggage may be deeply hidden.

The first baggage I see is SU(2) or SO(3); why not SU(7) ???

marcus

I am just talking about the facts. The standard LQG formulation is http://arxiv.org/abs/1102.3660 ("Zakopane lectures") and in that paper the theory is developed using non-embedded networks and foams. No manifold representing spacetime continuum.

We both recognize this.

HISTORICALLY much of this grew out of work with similar structures EMBEDDED in a manifold.

So let's make a clean break. We recognize that the theory is now defined with combinatorial objects that represent geometric information. Measurements, predictions, hypothetical measurements, events of one sort or another.

There is no continuum in the theory, all we have is relationships among geometric info.

Now you ask "What about the Lie groups? What about SU(2)?"

Well I'm no authority--I can only tell you how I personally understand it. The choice of Lie group, for me, says something about the kinds of measurements that are being made at various points in the network.

We are trying to DESCRIBE Nature and how she responds to geometric measurement and how her geometry evolves. We think manifolds are unrealistic so we throw them out. Now we have a web of measurements (areas volumes angles...). We pick the best Lie group that describes the symmetries of measurement as we experience them. We pick the group that works best.

That's just how I personally understand it. So then the graph Hilbert space automatically comes out to be the square integrable ("L₂") functions on a product of as many copies of the group G as there are links in the graph( G^L)
Some redundancy has to be factored out but basically that's the graph Hilbert space.

You're surely familiar with this--I'm sure you've read the Zako lectures paper.

I suppose using SU(2) is a way of noting that our world has 3D rotations. We don't go so far as to assign it a differential manifold structure, that would be adding a lot of extra. Unnecessary extra. But we do observe that a local observer can turn and tip things.
So we put in that detail about Nature---rotation.

We are painting a portrait, and SU(2) is the color of her eyes.

So the Hilbert space turns out to be L₂[SU(2)^L]

tom.stoer

SU(2) is one critical relict of 3+1 dim. spacetime; you can't explain why to use SU(2) w/o referring to 3+1 dim. spacetime.

It is not clear what happens if you start with SU(7) - as an example; it is not clear to which manifold this reduces in the semiclassical limit - or if there is convergence to a Riemann or Riemann-Cartan manifold at all - classically there is no Riemann-Cartan manifold with SU(7) structure group.

I agree that the algebraic structures of non-embedded spin networks do not contain any directly visible relict of the manifold, but besides the structure group there are others: In the canonical formulation there is the operator algebra G^a, V^a and H; at least H survives! in addition when using H there is the requirement for a global foliation like R³*T (with T being the time direction). In the SF framework there are the simplicity constraints which are understandable only when referring to a manifold structure from which the theory has been created, and of course there are vertex amplitudes which are related to some - unknown - hamiltonian H.

So yes, the relicts are deeply hidden, but they are present even for non-embedded spin networks.

marcus

Nice comment! I think it would be interesting in a theoretical/mathematical sense for someone to explore what happens when you use some different Lie groups in the Loop setup.

I think there HAS to be some way of telling the theory about the dimensionality we live in and giving it SU(2) is a kind of minimal way.

Youi don't give it a whole differential manifold with all that extra machinery, you just tell it the rotational symmetry that belongs to our world.

For me that's very satisfying. It is a minimal way of telling the theory what dimensionality we live in. I don't expect the theory to tell me why there MUST be 3+1 dimensions to the world (although perhaps some day a theory WILL tell us that--it would be exciting, to be sure!)

==================

I've been thinking about the *embedded* issue and I wonder if we couldn't find a recent paper analogous to 1102.3660 that presents the embedded approach--so then we could have a DUAL standard. Would you like this? Then there need be no tension. At the beginning of my post I could say I am talking about purely combinatorial networks+foams as in 1102.3660 and at the beginning of your post you could say you are talking about embedded ones as in 11xx.yyyy. Maybe some recent paper by Lewandowski?

marcus

Talking about things that might figure in a reformulation of Loop gravity (showing up next year at GR20 and Loops 2013), one thing we seem to have completely overlooked is the new, hard, and potentially very important OSN line of development by Lewandowski's group.
http://arxiv.org/abs/1107.5185
This is the systematic way to do spinfoams without spinfoams. But you have to learn it like a new language. Jerzy is a mathematician's mathematician. Check it out.
http://arxiv.org/abs/1107.5185
Feynman diagrammatic approach to spin foams
Marcin Kisielowski, Jerzy Lewandowski, Jacek Puchta
(Submitted on 26 Jul 2011)
"The Spin Foams for People Without the 3d/4d Imagination" could be an alternative title of our work. We derive spin foams from operator spin network diagrams} we introduce. Our diagrams are the spin network analogy of the Feynman diagrams. Their framework is compatible with the framework of Loop Quantum Gravity. For every operator spin network diagram we construct a corresponding operator spin foam. Admitting all the spin networks of LQG and all possible diagrams leads to a clearly defined large class of operator spin foams. In this way our framework provides a proposal for a class of 2-cell complexes that should be used in the spin foam theories of LQG. Within this class, our diagrams are just equivalent to the spin foams. The advantage, however, in the diagram framework is, that it is self contained, all the amplitudes can be calculated directly from the diagrams without explicit visualization of the corresponding spin foams. The spin network diagram operators and amplitudes are consistently defined on their own. Each diagram encodes all the combinatorial information. We illustrate applications of our diagrams: we introduce a diagram definition of Rovelli's surface amplitudes as well as of the canonical transition amplitudes. Importantly, our operator spin network diagrams are defined in a sufficiently general way to accommodate all the versions of the EPRL or the FK model, as well as other possible models. The diagrams are also compatible with the structure of the LQG Hamiltonian operators, what is an additional advantage. Finally, a scheme for a complete definition of a spin foam theory by declaring a set of interaction vertices emerges from the examples presented at the end of the paper.
36 pages, 23 figures

And then just recently there was the followup on this, which (of course) is included in the 2nd quarter MIP poll!

http://arxiv.org/abs/1203.1530
One vertex spin-foams with the Dipole Cosmology boundary
Marcin Kisielowski, Jerzy Lewandowski, Jacek Puchta
(Submitted on 7 Mar 2012)
We find all the spin-foams contributing in the first order of the vertex expansion to the transition amplitude of the Bianchi-Rovelli-Vidotto Dipole Cosmology model. Our algorithm is general and provides spin-foams of arbitrarily given, fixed: boundary and, respectively, a number of internal vertices. We use the recently introduced Operator Spin-Network Diagrams framework.
23 pages, 30 figures

Note that Jerzy is bilingual---he can talk and think non-embedded Loop (purely combinatorial structures) and also the physical (non-Dirac, non-constraint) Hamiltonian approach with DUST--using (if he chooses) the older embedded structures in a manifold. Here is the talk he gave today at Stockholm MG13:

Lewandowski, Jerzy
Quantizable canonical LQG
Abstract :The canonical quantization scheme can be completed with the framework of Loop Quantum Gravity for several examples of the gravitational field coupled to matter fields. Explicitly, that has been accomplished for the generic dust, non-rotating dust, and massless scalar field. Those results will be presented and recent progress will be discussed...
================================

EDIT TO RESPOND TO NEXT POST.
Hi Tom, since I can still edit I will reply to your post here. You've got an interesting perspective that I want to think about. I can't say much at the moment. I'm intrigued. I need to think about it some more. I'd also like to understand Lewandowski OSN diagrams better---is his approach really useful? I think it is but am not completely sure yet.

Alesci, who is currently Thiemann's postdoc, has chosen to go to Warsaw this autumn, for his next postdoc fellowship. It is ridiculous for me to imagine myself young, and wonder what I would do in his place. But I think, ridiculous as it is, that I would do the same as Alesci, at this point.