### Loop Quantum Gravity

"The most successful programs of unification always came about when there was some urgent question to be answered. In most cases, the urgent question was that something seemed to be wrong in the present understanding, and the correct answer then turned out to be that the only way to get it right was to say that this effect and that effect cancel, and the only way to make them cancel was to put them into one big theory together and show that they were the same force."

-Gerard 't Hooft

"I think Physics is about escaping the prison of the received thoughts and searching for novel ways of thinking the world, about trying to clear a bit the misty lake of our insubstantial dreams, which reflect reality like the lake reflects the mountains."

-Carlo Rovelli

### The Domain

General Relativity and Quantum Mechanics are the two great fundamental theories of 20th Century physics the former governing the large-scale structure of spacetime and the latter the microcosm of the atomic and subatomic phenomena and quite a large domain of the meso and macrocosm as well. For, quantum phenomena is not confined to the microcosm alone and the border between the classical and quantum phenomena is not that sharply defined. General Relativity and Quantum Mechanics, therefore, serve to describe, explain and predict quite a large sector of the universe. Nevertheless, the attempt to apply both theories near the limits of their applicability force profound transformations in the theoretical, mathematical and structural aspects of the individual theories themselves and as a consequence, force a conceptual change, or rather more precisely, a revolution, as well. Of the varied attempts to arrive at a candidate theory of "quantum gravity," the term coined to describe the result of a possible unification of General Relativity and Quantum Mechanics, one of the most conservative is Loop Quantum Gravity inspired by Paul Dirac, Peter Bergman, Arnowitt-Deser-Misner, Bryce de Witt, John Archibald Wheeler, Chris Isham, Karel Kuchar, Roger Penrose and several others and formulated primarily by Abhay Ashtekar, Carlo Rovell and Lee Smolin. Another is Shape Dynamics pioneered and formulated by Julian Barbour and his collaborators.

At CFRCE, research in Quantum Gravity is driven primarily by **The Problem of Time**. This has inspired a series of investigations in Mathematical Physics and Mathematics. In Mathematical Physics, it has resulted in the Presymplectic formulation of the problem of time and in Mathematics, investigations into Loop Spaces, a special class of Lie Groups.

### The Field

"Space and time are hard to think about because they are the backdrop to all human experience. Everything that exists, exists somewhere, and nothing happens that does not happen at some time. So, just as one can live without questioning the assumptions in one's native culture, it is possible to live without asking about the nature of space and time. But there is at least a moment in every child's life when they wonder about time. Does it go on for ever? Was there a first moment? Will there be a last moment? If there was a first moment, then how was the universe created? And what happened just a moment before that? If there was no first moment, does that mean that everything has happened before? And the same for space: does it go on and on for ever? If there is an end to space, what is just on the other side of it? If there isn't an end, can one count the things in the universe?"

-Lee Smolin, *"Three Roads to Quantum Gravity"*

**Paul Dirac:** Dirac was the first to consider the Hamiltonian formulation of General Relativity as a prelude to the quantisation of gravity. Trying to do that led him to come to grips with singular constraints before he could construct the Legendra transform (Fibre Derivative in the modern Symplectic Geometry language). He soon realised that it was necessary to develop a new approach to dynamics. This made him formulate “Singular Constrained Dynamics,” a fine presentation of which is found in his Yeshiva University lectures as in his monograph, “Lectures on Quantum Mechanics.” This was the first time, "Singular Constrained Dynamics," as a discipline in its own right developed.

**Peter Bergmann:** Bergmann was a collaborator of Einstein at the “Institute For Advanced Study,” Princeton. He introduced the concepts of primary and secondary constraints. He and his school developed the theory of singular constraints further. This led to the well known Dirac-Bergman Algorithm in the theory.

**Julian Schwinger:** Schwinger attempted to quantise gravity from the point of view of his *Source Theory*. Schwinger inspired a generation of physicists and mathematicians towards Quantum Gravity. His students including, Bryce de Witt, Stanley Deser, Richard Arnowitt played a key role in the development of Canonical Quantum Gravity.

**Bryce de Witt:** After working with Julian Schwinger for his doctoral research, Bryce de Witt undertook the most extensive investigations into the quantisation of gravity. In a series of papers, entitled, “Quantum Theory of Gravity, Parts I, II, III,” he tackled major structural issues in its formulation. He elaborated this work further in his article in “Louis Witten edited, *Gravitation, An Introduction to Current Research*,” and in the Les Houches Summer School, article, *“Dynamical Theory of Groups and Fields,”* one of the most beautiful and insightful articles to read.

**Roger Penrose:** Penrose, one of the legendary figures of modern Theoretical and Mathematical Physics, is in a sense the father of the mathematical structure of Loop Quantum Gravity. As Lee Smolin writes in “Three Roads to Quantum Gravity,”

*“**One of the most beautiful results to have come from loop quantum gravity was the discovery that the loop states could be arranged in very beautiful pictures, which are called spin networks. These had actually been invented thirty years earlier by Roger Penrose. Penrose had also been inspired by the idea that space must be purely relational. Going directly to the heart of the matter, as is his nature, he had skipped the step of trying to derive a picture of relational space from some existing theory, as we had. Instead, having more courage, he had sought the simplest possible relational structure that might be the basis of a quantum theory of geometry. Spin networks were what he came up with."*

Penrose’s interest in Quantum Gravity perhaps had its origins in his and Stephen Hawking’s now famous Hawking-Penrose Singularity theorems. One of the ways out of the inevitability of singularities was for quantum effects to smooth them out. Penrose’s insight led him to attempt to figure out and place spacetime on a more fundamental basis. He realised that angular-momentum or spin like structures were the natural objects to do that. In "Ted Bastin edited, Quantum Theory and Beyond,” he outlined his *“Angular Momentum: An Approach to Combinatorial Space-Time,” * based on Cohomological considerations. At the start of that article he makes a prescient observation,

“The most obvious physical concept that one has to start with, where quantum mechanics says something is discrete, and which is connected with the structure of space-time in a very intimate way, is in *angular momentum*. The idea here, then, is to start with the concept of angular momentum -where one has a *discrete* spectrum -and use the rules for combining angular momenta together and see if in some sense one can construct the concept of *space* from this.”

Penrose himself soon developed these ideas into the framework of Twistor Theory in which, instead of beginning with spacetime as the fundamental concept, he used a four dimensional complex space. The inspiration for the concept of a twistor partly stemmed from the classification of spacetime into the Petrov types. In that context the concept of a nontrivial Killing Spinor also played a key role. This has been presented by him and Wolfgang Rindler in great detail in their beautiful Cambridge Monograph on Mathematical Physics, “Spinors and Spacetime Vols 1, 2.” As they were in the process of bringing spin networks fully into Quantum General Relativity, Carlo Rovelli and Lee Smolin realised that they needed Penrose’s help in learning to calculate with spin networks. As Lee Smolin writes,

"I had known for a long time that Penrose's spin networks should come into loop quantum gravity, but I had been afraid of working with them. When Penrose described them in his talks they always seemed so intricate that only he would be able to work with them without making mistakes. To do a calculation Penrose's way, one has to add up long series of numbers which are each either +1, 0 or 71. If you miss one sign, you're dead. Still, during a visit to Cambridge in 1994 I met Roger and asked him to tell me how to calculate with his spin networks. We did one calculation together, and I thought I had the hang of it. That was enough to convince me that spin networks would make it possible to calculate aspects of quantum geometry such as the smallest possible volume. I then showed what I had learned to Carlo, and we spent the rest of that summer translating our theory into the language of Penrose's spin networks."

This is only a very sketchy description of Roger Penrose’s many-sided contributions to Quantum Gravity, not to speak of his manifold contributions to Classical Gravity!

**Chris Isham:** The term Quantum Gravity is inextricably associated with the name of Chris Isham, one of the legendary figures in the field. Indeed, so great has been his influence on the field that it was almost customary for every conference proceedings on Quantum Gravity to carry a foreword by Isham. His 2011 Dirac Medal citation states,

“Chris Isham is a worldwide authority in the fields of quantum gravity and the foundations of quantum theory. Few corners of these subjects have escaped his penetrating mathematical investigations and few workers in these areas have escaped the influence of his fundamental contributions. Isham was one of the first to put quantum field theory on a curved background into a proper mathematical form and his work on anti-de Sitter space is now part of the subject’s standard toolkit.”

Isham’s “Lectures on Quantum Theory,” is a concise and precise treatment of Quantum Mechanics for those interested in Quantum Gravity. It treats structural, philosophical and mathematical issues in a pertinent manner.

Isham was also responsible for introducing Topos theory into Quantum Gravity. He was also one of the pioneering figures of *“The Problem of Time in Quantum Gravity,” * inspiring a generation of physicists and mathematicians towards the problem. His papers coauthored with Jeremy Butterfield, *“**Spacetime and the Philosophical Challenge of Quantum Gravity”* and *“On the Emergence of Time in Quantum Gravity**” *are two of the finest introductions to the problem of time.

**Richard Arnowitt, Stanley Deser, and Charles W. Misner,** the eminent trio of the ADM formalism, the 3+1 split of spacetime with a view to the Hamiltonian formation of General Relativity. Arnowitt and Deser were Schwinger’s students while Misner was John Wheeler’s. Their article in “Louis Witten ed, Gravitation, An Introduction to Current Research,” is a classic of Canonical Gravity and is still worth reading for the great insights it offers into the meaning of the 3+1 split. In the same article they use the Schwinger Action Principle and carry out a variety of constructions. Of course, a more complete treatment came a little later by Isenberg and Nester in “Held ed, General Relativity and Gravitation, Vol 1,” but the original ADM article is quite irresistible!

**Abhay Ashtekar: **With the introduction of the “New Variables,” Abhay Ashtekar is the originator of Loop Quantum Gravity. His was the greatest breakthrough since Dirac’s development of Singular Constrained Dynamics. Ashtekar made it completely possible to bring Canonical Gravity into the framework of modern Gauge Theories. Before Ashtekar, there had been several attempts to cast General Relativity as a gauge theory, notably the work of Kibble, Sciama, Walter Mayer and others.

But Ashtekar’s approach was the most powerful, novel and clear in that it departed from using the metric and its derivatives as the variables. Instead, the verbein and the connection were brought into play as the basic variables. Indeed, taking these in their pure form would not do. He had the great insight to choose, instead, a complex canonical formulation in which the tetrad and spin connection were the key variables. By adapting the Palatini formulation and with these as the variables, the ADM form of the Einstein-Hilbert Lagrangian took a simpler form. It was then possible to perform a Legendre transform and obtain the Hamiltonian in a polynomial form, a first requirement for quantisation. The Poisson brackets were then calculated and other details tackled. The entire work gave a completely new direction and impetus to physicists and mathematicians working on Classical and Quantum Gravity. The rest is the stuff of legend and history.

As Einstein was the originator and also the most extensive developer of Relativity, so also has been Ashtekar for Loop Quantum Gravity, with his further work on “Geometric Formulation of Quantum Mechanics,” “Isolated Horizons, Dynamical Horizons,” right up to the present times.

His “Lectures on Non-Perturbative Canonical Gravity,” is one of the most inspiring sources for the student to read. His and Anne Magnon’s translation of Elie Cartan’s classic, “Manifolds with an Affine Connection and the Theory of General Relativity,” is one of the most beautiful sources for the Einstein-Cartan theory.

**Lee Smolin: **Lee Smolin along with Abhay Ashtekar and Carlo Rovelli is one of the three major figures in the structural development of Loop Quantum Gravity. Lee Smolin and Carlo Rovelli formulated Canonical Gravity via spin-networks and converted it into its present form of Loop Quantum Gravity. Lee Smolin played a major role in the Problem of Time in Quantum Gravity and in delineating the role of background independence. He was also one of the first to explore possibilities of connections between Loop Quantum Gravity and String Theory and indirectly inspired the background independent formulation of String Theory. His books, “Life of the Cosmos,” and “Three Roads to Quantum Gravity,” are modern classics. “Three Roads to Quantum Gravity,” is more than a book. It is a road-map to Quantum Gravity proper and one of the most inspiring sources for any beginning student on Quantum Gravity and indeed in Theoretical Physics itself. His book, “The Trouble with Physics,” is a panacea for young physicists in the making who are about to enter into the troubled waters of the field of Theoretical Physics, especially fundamental physics and the whole scientific enterprise in general. It brings out what Einstein clearly held in his lecture “Principles of Research,” that is so significant especially to a beginning student of science that is well worth quoting in detail.

*“In the temple of science are many mansions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior intellectual power; science is their own special sport to which they look for vivid experience and the satisfaction of ambition; many others are to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside…”*

Smolin calls the physicists who are left inside as the “seers,” and the others as “craftsmen,” and as he points out it is not at all that one type of physicists are superior to the other. It is rather, that both types are supremely necessary and significant for the cause of science, for the balance of ecology in the scientific discipline. It is just that in the present times, the balance is in danger of being offset exclusively in favour of the craftsmen to the near extinction of the seers. For, the seers by their very nature are often “*somewhat odd, uncommunicative, solitary fellows*,” and certainly not likely to possess the “survival skills” that to the craftsmen are quite natural given their more adaptable and communicable character. The fact of the matter is that both are absolutely necessary for the scientific enterprise. This recalls to us the statement that Thomas Thiemann makes in his “Lectures on Loop Quantum Gravity.”

*“To be sure, it is a shame that one has to justify fundamental research at all, a situation unheard of in the beginning of the 20’th century which probably was part of the reason for why so many breakthroughs especially in fundamental physics have happened in that time. Fundamental research can work only in absence of any pressure to produce (mainstream) results, otherwise new, radical and independent thoughts are no longer produced.”*

It is the seers who despite or rather because of insurmountable difficulties, persist in fundamental research in the spirit of the “religious worshipper or the lover, the daily effort coming from no deliberate intention or program, but straight from the heart.”

Presenting these sentiments with great care, Smolin appears to echo and illustrate Einstein’s words,

*“I am quite aware that we have just now lightheartedly expelled in imagination many excellent men who are largely, perhaps chiefly, responsible for the buildings of the temple of science; and in many cases our angel would find it a pretty ticklish job to decide. But of one thing I feel sure: if the types we have just expelled were the only types there were, the temple would never have come to be, any more than a forest can grow which consists of nothing but creepers. For these people any sphere of human activity will do, if it comes to a point; whether they become engineers, officers, tradesmen, or scientists depends on circumstances. Now let us have another look at those who have found favor with the angel. Most of them are somewhat odd, uncommunicative, solitary fellows, really less like each other, in spite of these common characteristics, than the hosts of the rejected. What has brought them to the temple? That is a difficult question and no single answer will cover it. To begin with, I believe with Schopenhauer that one of the strongest motives that leads men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one's own ever shifting desires. A finely tempered nature longs to escape from personal life into the world of objective perception and thought; this desire may be compared with the townsman's irresistible longing to escape from his noisy, cramped surroundings into the silence of high mountains, where the eye ranges freely through the still, pure air and fondly traces out the restful contours apparently built for eternity.”*

And captures the feeling with which the seers engage in their quest,

*“The state of mind which enables a man to do work of this kind is akin to that of the religious worshiper or the lover; the daily effort comes from no deliberate intention or program, but straight from the heart.”*

As Wolfgang Pauli was known as the *“Conscience Keeper of Physics,”* in the 20^{th} Century, perhaps Lee Smolin, by his work and profound writings, especially *“The Trouble with Physics,”* would be called by posterity as the *“Conscience Keeper of 21*^{st}* Century Theoretical Physics!” *

**Carlo Rovelli:** Carlo Rovelli, along with Abhay Ashtekar and Lee Smolin is one of the three major figures in the structural development of Loop Quantum Gravity. Carlo Rovelli and Lee Smolin formulated Canonical Gravity via spin-networks and converted it into its present form of Loop Quantum Gravity. Rovelli has been one of the most insightful conceptual thinkers of the era with a philosophical grasp rare in modern times. Rovelli played a leading role in the Problem of Time in Quantum Gravity. Right from the beginning he proceeded with a deep insight, that time is not a fundamental concept in physics. The seeds of this are to be seen in his Ph. D thesis and in his paper, “On Quantum Mechanics.”

Indeed, one following the evolution of his ideas can clearly glimpse his insights on time concretising into his reformulation of quantum mechanics into “Relational Quantum Mechanics,” on one and his timeless, presymplectic hamiltonian formulation of General Relativity on one hand. Thus, after eliminating time from both Quantum Mechanics and General Relativity, Rovelli brings them together in his papers, “Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance,” “The Century of the Incomplete Revolution,” “Partial Observables,” “A Note on the Foundation of Relativistic Mechanics, Part 1 and Part 2” and culminating in “Forget Time,” that won the ‘First Community Prize' of the FQXi 'The Nature of Time' Essay Contest. The full wealth of insights and results he has been gathering are to be found in his beautiful Cambridge Monograph on Mathematical Physics, “Quantum Gravity.”

In Loop Quantum Gravity itself, Carlo Rovelli and Reisenberger laid down the framework of Temperley Lieb Algebras and calculations leading to the quanta of space and Black Hole entropy. Rovelli’s papers are a treat to read and contain insights not to be easily had elsewhere. He is perhaps an exception to the division that Lee Smolin speaks of, of seers and craftsmen and has the rare balance between both groups. He is a seer when putting forth the timeless view and asking to forget time. He is a craftsman when building up the formalism corresponding to that view.

**Thomas Thiemann:** Thomas Thiemann is perhaps the most brilliant Mathematical Physicist in Loop Quantum Gravity deploying powerful mathematical tools with great insight and dexterity. His “Introduction to Modern Canonical Quantum General Relativity” is perhaps the most rigorous and complete mathematical treatment of Canonical Quantum Gravity. Besides, it is of great aesthetic appeal and reminds one inevitably of Hermann Weyls’s *“Group Theory and Quantum Mechanics”*, John Von Neumann’s *“Mathematical Foundations of Quantum Mechanics”*, and Streater and Whitman’s *“PCT, Spin, Statistics and all That.”*

Chris Isham writes in the foreword to the same book,

"I can still remember the shock I felt when I first read the papers he put onto the web dealing with the Hamiltonian constraint. Suddenly, someone with a top-rate mathematical knowledge had addressed this critical question anew, and with considerable success. Indeed, Thiemann succeeded with loop quantum gravity where I had failed with the old Wheeler–DeWitt equation, and he has gone on since that time to become one of the internationally acknowledged experts in loop quantum gravity…

…My graduate students not infrequently ask me what I think of the current status of canonical quantum gravity and, in particular, what I think the chances are of ever making proper mathematical sense of the constraints that define the theory. For some years now I have replied to the effect that, if anybody can do it, it will be Thomas Thiemann and, if he cannot do it, then probably nobody will."

**John Baez: **John Baez is one of the most brilliant Mathematicians and Mathematical Physicists in Quantum Gravity and one of the most illuminating expositors of Theoretical Physics and Mathematics. Since its inception, his column, “This Week’s Finds in Mathematical Physics,” was a great source of inspiration, insight and guidance for the new generation of physicists. His beautiful book with Muniain, *“Gauge Fields, Knots and Gravity,” *is the finest introduction not only to Quantum Gravity but also the most captivating mathematical structures in Topology, Differential Geometry and Analysis. His books are proof of Dirac’s view that an improvement in notation precedes an improvement in understanding of a theoretical framework. All his papers are elegant and in neat and concise notation.

**Domnico Giulini: **Domenico Giulini’s article, “Ashtekar Variables in Classical General Relativity” in Ehlers ed, “Canonical Gravity: From Classical to Quantum,” is perhaps the clearest introduction to Ashtekar Variables. It is one of the most elegantly presented articles and quickly takes a beginning student into the heart of the matter. As it uses modern differential geometric methods it is also easy to read. Several other papers of Giulini are equally insightful. His, “That Strange Procedure Called Quantization,” is an insightful article on the obstructions to quantisation as captured by the No-Go theorem of Groenewold and van Howe, which states that a naive transcription of Dirac’s quantisation rules cannot work. Giulini’s insight into the Thin-Sandwich conjecture and his early anticipation that Julian Barbour’s work would perhaps be most significant for Quantum Gravity have been validated by the development of Barbour’s Shape Dynamics.

**Gambini and Pullin:**Developed the loop space formulation of Loop Quantum Gravity further and cast it into the framework of an elegant calculus that mirrored the connection and curvaturte structures of Differential Geometry. In their beautiful Cambridge Monograph, "Loops, Knots, Gauge Theories and Quantum Gravity," they presented their formalism in a very clear manner. This monograph is still worth reading for its fine treatment of the of LQG just around the time spin networks were being introduced.