Wieland's thesis foreword in rough German-to-English translation

[marcus]

Wieland's thesis was posted at tel.archives (the French open-archive website) on 3 March 2014.
http://tel.archives-ouvertes.fr/docs/00/95/24/98/PDF/diss.pdf
The title of the 160-page dissertation is The Chiral Structure of Loop Quantum Gravity.
It is a noteworthy thesis which could be valuable to learn from, at several levels.
It's almost all in English but there are brief passages at the beginning in German and French (German is Wieland's native language, and the work was done in Marseille). For the convenience of other readers, who might wish to have it, I'll post a rough translation of the Foreword. Here is the original, and a rough draft translation will follow in the next post.
==quote from Wieland "The Chiral Structure of LQG"==
VORWORT
Von allen Grundkräften der Physik passt nur die Schwerkraft nicht zur Quantentheorie. Als schwächste aller vier Wechselwirkungen (das sind die beiden Kernkräfte, die elektromagnetische Kraft und eben die Gravitation) spielt sie für die Physik des Mikrokosmos keine Rolle; die Gravitation beherrscht die Welt I am Großen. Die allgemeine Relativitätstheorie liefert den mathematischen Rahmen. Einstein erklärt die Schwerkraft aus der geometrischen Struktur von Raum und Zeit: Genauso wie die Krümmung der Erdoberfläche den Kurs eines Flugzeugs bestimmt (von Wien nach Washington folgt der Pilot der kürzesten Verbindung, einem Großkreis, keiner Geraden), genauso zwingt die Krümmung der Raumzeit die Erde auf ihre Bahn um die Sonne.
Die Quantenmechanik spielt für die Umlaufbahnen der Planeten keine Rolle. Sie beherrscht die Physik I am Kleinen. Ort und Impuls eines Teilchens lassen sich als kom- plementäre Variable nicht gleichzeitig scharf messen, sind als Zufallsgrößen unscharf verschmiert. Die Schrödingergleichung beschreibt diese Unschärfen als Wellenfeld in Raum und Zeit. „Das Elektron trifft in zehn Minuten am Ort x ein.“ So ein Satz ist der Quantentheorie ganz unbekannt, wir sagen stattdessen: „In zehn Minuten ist das Elektron mit p(x)-prozentiger Wahrscheinlichkeit am Orte x.“
Den alten Streit um die Frage, ob es kleinste Teilchen gebe, oder die Welt aus einem stofflichen Kontinuum bestehe, beendet die Quantenmechanik mit einem salomonischen Urteil. Beides ist gleichermaßen wahr, und hängt von der Fragestellung ab. In dem einen Experiment enthüllt sich die Quantennatur der Welt: Angeregte Atome senden Lichtteilchen nur ganz bestimmter Farbe aus – das charakteristische Orange der Straßenlaternen kommt vom Natrium. Ein anderer Versuch zeigt die Kontinuumseigenschaften der Materie: Bei aufmerksamem Blick in eine Straßenlaterne kann man Beugungsringe sehen, wenn I am Augapfel die Wellen des Natriumlichts an kleinen Hindernissen streuen.
Was hat das nun alles mit der Gravitation zu tun? Nach Einsteins allgemeiner Relativitätstheorie hat das Schwerefeld der Erde stets überall einen fest vorhersagbaren Wert. In der Quantenmechanik ist das nicht mehr so, hier gibt es nur mehr Wahrscheinlichkeitsaussagen. Es kann nicht beides stimmen; entweder es gibt echten Zufall, oder die Welt ist streng deterministisch. Außerdem folgt alle Materie der Quantentheorie, doch koppelt in immer gleicher Weise an die Gravitation. Damit müssen sich die Gesetze der Quantentheorie auch auf die Schwerkraft übertragen. Wie die Quantentheorie mit der Relativitätstheorie zu versöhnen sei, das weiß freilich niemand so genau. Trotz jahrzehntelanger, teils recht phantastischer Bemühungen, fehlt uns noch immer eine Theorie der Quantengravitation.
Was können wir von einer Quantentheorie der Gravitation erwarten? Zunächst müsste sie alle bisherigen experimentellen Tests bestehen. Sie muss uns aber auch Fragen beantworten, die über unser bisheriges Verständnis weit hinausgehen: Was geschah beim Urknall? Was sind die Quanten des Gravitationsfeldes? Ist vielleicht die Geometrie der Raumzeit selbst gequantelt, gibt es gleichsam kleinste Raumatome? Was geschieht I am Inneren eines schwarzen Lochs?
Meine Doktorarbeit beschäftigt sich mit nur einem Ansatz, diese Fragen zu beantworten, mit der loop quantum gravity wie die Theorie auf Englisch heißt. An erster Stelle steht die Frage: Gelingt der Übergang zur bekannten Physik? Dafür braucht es geeignetes mathematisches Handwerkszeug. Meine Doktorarbeit entwickelt solches Werkzeug, und untersucht den klassischen Grenzfall der Theorie. Ich kann zeigen, dass die Schleifentheorie I am klassischen Limes sich als Vielteilchentheorie deuten lässt. Die zugehörigen Punktteilchen bewegen sich allerdings nicht in Raum und Zeit, sondern leben in einem zweidimensionalen komplexen Vektorraum, I am Raume der Spinoren.
==endquote==

 

[marcus]

Anyone who wants can try Google translator or other online translators on this. That may get some parts better than my attempt here.
==rough translation from Wieland "The Chiral Structure of LQG"==
FOREWORD
Of all fundamental forces of physics only gravity doesn't fit with quantum theory. As the weakest of all four interactions (the two nuclear forces, the electromagnetic force, and gravity itself) it plays no role in the micro-world--gravity rules the world at large. General Relativity provides mathematical Space.
Einstein explained gravity as the geometric structure of space and time: just as the curvature of the Earth's surface determines the course of an airplane (from Vienna to Washington, a pilot follows the shortest connection: a great circle, not a straight line), in the same way the curvature of space-time guides the Earth on its way around the Sun.
Quantum mechanics plays no role in the orbits of the planets. It rules physics in the small. As complementary variables, the position and momentum of a particle do not let themselves be precisely measured, but are smeared out as random variables. The Schrödinger equation describes these indefinites as wave-functions in space and time. "The electron arrives at position x in ten minutes." Such a statement is unknown to Quantum Theory--instead we say
"In ten minutes the electron will be with probability p(x)-percent at position x."
The old argument over the question whether there are smallest particles or the world consists of a material continuum was settled by Quantum mechanics with a Solomon-like judgement: both are equally true and it depends on how the question is posed. The Quantum nature of the world is contained in a single experiment: excited atoms emit only photons of definite colors--the characteristic orange of streetlights comes from sodium. Another test shows the continuity of matter: by looking attentively at a streetlight one can see diffraction fringes, scattered by minute obstacles in the eyeball.
What does all this have to do with gravity? According to Einstein's theory of General Relativity, the gravitational field of the Earth has a definite predictable value everywhere. In Quantum Mechanics that is no longer so, here there are only more probability expressions. Both cannot be true; either there is genuine randomness or the world is strictly deterministic. Furthermore all matter follows Quantum Theory, but likewise always couples to gravity. Therefore the laws of Quantum Theory must also be applied to gravity. Frankly, no one knows exactly how to reconcile Quantum Theory with General Relativity. Despite decades of some really fantastic effort we still lack a theory of Quantum Gravity.
What could we expect from a quantum theory of gravitation? Firstly it must survive all previous experimental tests. But also it must answer questions for us which until now are far beyond our understanding: what happened at the Big Bang? What are the quanta of the gravitational field? Could the geometry of space-time itself be quantized, so that there are smallest atoms of space? What happens inside a black hole?
My doctoral thesis is concerned with just one approach to answering these questions, with loop quantum gravity, as the theory is known in English. The first thing to ask is: does the bridge connect with known physics? For that one needs the proper mathematical tools. My thesis develops such tools and investigates the Classical Limit of the theory. I am able to show that in the classical limit the loop theory can be seen as a many-particle theory. However the point particles belonging to it do not move in space and time, but live in a vector space of two complex dimensions: in spinor space.
==endquote==

 

[marcus]

Loop gravity is reaching (or has attained already) a new level of maturity in establishing that canonical LQG and Spinfoam QG are equivalent and describe the same theory---LQG ≈SF.
It looks like Wieland has achieved this using a spinor formulation of canonical LQG.
But his is not the only research aimed in this directions--the current (1st quarter 2014) MIP poll has 3 papers by members of Bianca Dittrich's group that present ways to close the gap by reformulating LQG closer to SF, and also to address the issues of refining, coarse-graining, and the continuum limit.
The Introduction of Wieland's thesis has, I think, an especially clear statement of the situation and what is in progress. I will only quote a brief excerpt and urge anyone interested to look at the Introduction pages 1-7
http://tel.archives-ouvertes.fr/docs/00/95/24/98/PDF/diss.pdf

==excerpt page 6==

Penrose’s spin network functions [50] form the most common orthonormal basis in the resulting Hilbert space. Excitations of geometry can neatly be visualised as fuzzy polyhedra glued among their facets [51–55]. Area, angle, length and volume turn into self-adjoint operators with a discrete Planckian spectrum [38,39,56–59]. The Hilbert space of a single tetrahedron may serve as a minimal example to illustrate the resulting quantum geometry. A classical tetrahedron is characterised by six numbers, e.g. the lengths of the six bones bounding the triangles. These numbers depend on the metric tensor, hence turn into operators once gravity is quantised. Yet, we cannot diagonalise all of them, simply because they do not commute among another [41]. Therefore, a quantised tetrahedron lacks a true shape. If we make some of its geometrical properties sharp, others become fuzzy. The most common choice for a complete set of commuting operators consists of the four areas and one dihedral angle, or four areas and the volume of the tetrahedron.

For the dynamics the situation is different, no such clean physical picture is available. There are two ideas of how to define the dynamics of the theory. The first idea [19,40] follows Dirac’s program of canonical quantisation [60]. This uses the Hamiltonian formulation of the theory, which rests upon a spatio-temporal decomposition of the spacetime manifold. Picking a time-coordinate breaks general covariance, only spatial diffeomorphisms remain manifest. Four-dimensional coordinate invariance is restored only dynamically by the Hamiltonian constraint. Its quantisation yields the Wheeler– DeWitt equation [61]. The second idea looks for a covariant path integral formulation. This comes under the name of spinfoam gravity [20,62], which is the main focus of this thesis.

However these two approaches will ever manifest themselves, they should just be two ways to define the very same physical theory, and indeed, at least at a formal level, this it what happens [63] in the Wheeler–DeWitt theory: The path integral gives transition amplitudes that formally solve the Wheeler–DeWitt equation. Whether this is true also for loop gravity is one of the most important consistency checks for the theory. I cannot give a conclusive answer to this question, but I can show that spin foam gravity comes from the canonical quantisation of a classical theory. This is a version of first-order Regge calculus [64], with spinors as the fundamental configuration variables. I will present this result in chapters 3 and 4. It should be a convincing evidence that spinors provide a universal language to bring the two sides of the theory together.

==endquote==

 

[tom.stoer]

hi marcus; here it comes, the long expected summary of Wieland's work, which could very well be a guideline for the next couple years; unfortunately - besides some success - major problems remain:

Equivalence discrete - continuum
Completeness of constraints
Role of Torsion (seems to be clarified to a large extent)
Recovery of continuum only when all e.o.ms are fulfilled
Fixed Graph!

I think that the discretization (of the classical theory) is already very problematic (refer to Wieland's summary)

 

[marcus]

hi marcus; here it comes, the long expected summary of Wieland's work, which could very well be a guideline for the next couple years; unfortunately - besides some success - major problems remain:

Equivalence discrete - continuum
Completeness of constraints
Role of Torsion (seems to be clarified to a large extent)
Recovery of continuum only when all e.o.ms are fulfilled
Fixed Graph!

I think that the discretization (of the classical theory) is already very problematic (refer to Wieland's summary)

Hi Tom! I appreciate your commenting. For me continuous vs. discrete is, in a way, analogous to wave vs. particle or to unobserved vs. observed. The occurrence of a geometry is necessarily discrete.

An observer can only make a finite number of measurements. Same for an instrument. So discretizing suggests a way of acknowledging that there is a USER of the theory.

So I am prepared to tolerate quite a bit of discreteness Indeed, to me a spacetime continuum looks like the continuous trajectory of a particle (and I do not believe in such a thing, there are I think no continuous trajectories of particles).

I think you are right to point out that the role of Torsion has been clarified to large extent.

You give us a useful list of things that are to some extent being worked on currently and do definitely need further work!

 

[tom.stoer]

For me continuous vs. discrete is, in a way, analogous to wave vs. particle or to unobserved vs. observed. The occurrence of a geometry is necessarily discrete.[/QUOTE
With continuous vs. discrete I simply mean that discretization causes technical problems and may introduce artefacts which are not "real".