# GAUGE FIELDS KNOTS AND GRAVITY PDF

This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a. GAUGE FIELDS,. KNOTS AND GRAVITY. John Baez. Department of Mathematics. University of California. Riverside. Javier P. Muniain. Department of Physics. Audiobook Gauge Fields, Knots and Gravity Kindle ready Download here: https:// resourceone.info?book=

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The loop representation of quantum gravity has many formal resemblances which the gauge field is concentrated, while an e citation of the. Javier P. Muniain, John C. Baez resourceone.info ISBN: , | pages | 6 Mb Download. Gauge Fields, Knots And Gravity by John Baez, , available at Book Depository with free delivery worldwide.

In section 2. I have divided results in two groups. At the cost of several repetitions, the structure of this review is very modular: Sections are to a large extent independent of each other, have different styles, and can be combined according to the interest of the reader.

A reader interested only in a very brief overview of the theory and its results can find this in section 9. Graduate students or persons of general culture may get a general idea of what goes on in this field and its main ideas from sections 2 and 7. If interested only in the technical aspects of the theory and its physical results, one can read sections 6 and 7 alone. Scientists working in this field can use sections 6 and 7 as a reference, and I hope they will find sections 2 , 3 , 5 and 8 stimulating.

I will not enter into technical details. However, I will point to the literature where these details are discussed. I have tried to be as complete as possible in indicating all relevant aspects and potential difficulties of the issues discussed. The literature in this field is vast, and I am sure that there are works whose existence or relevance I have failed to recognize.

I sincerely apologize to the authors whose contributions I have neglected or under-emphasized, and I strongly urge them to contact me to help me make this review more complete. Quantum Gravity: Where are We? This is a non-technical section in which I illustrate the problem of quantum gravity in general, its origin, its importance, and the present state of our knowledge in this regard.

The problem of describing the quantum regime of the gravitational field is still open. There are tentative theories, and competing research directions. For an overview, see [ ]. The two largest research programs are string theory and loop quantum gravity. But several other directions are being explored, such as twistor theory [ ], noncommutative geometry [ 68 ], simplicial quantum gravity [ 7 , 66 , 62 , 1 ], Euclidean quantum gravity [ , ], the Null Surface Formulation [ 85 , 86 , 87 ] and others.

String theory and loop quantum gravity differ not only because they explore distinct physical hypotheses, but also because they are expressions of two separate communities of scientists, scientists who have sharply distinct prejudices, and who view the problem of quantum gravity in surprisingly different manners.

What is the problem? Thanks to this success, now a few decades old, physics is in a position in which it has very rarely been: There are no experimental results that clearly challenge, or clearly escape, the present fundamental theory of the world. The theory we have encompasses virtually everything — except gravitational phenomena.

From the point of view of a particle physicist, gravity is then simply the last and weakest of the interactions. It is natural to try to understand its quantum properties using the strategy that has been so successful for the rest of microphysics, or variants of this strategy.

The search for a conventional quantum field theory capable of embracing gravity has spanned several decades and, through an adventurous sequence of twists, moments of excitement and disappointments, has lead to string theory. The foundations of string theory are not yet well understood; and it is not yet entirely clear how a supersymmetric theory in 10 or 11 dimensions can be concretely used for deriving comprehensive univocal predictions about our world.

In string theory, gravity is just one of the excitations of a string or other needed for the formulation and for the interpretation of the theory. This is the case not only in perturbative string theory, but, to my understanding, in the recent attempts at a non-perturbative definition of the theory, such as M theory.

Thus, for a physicist with a high energy background, the problem of quantum gravity is now reduced to an aspect of the problem of understanding: What is the mysterious nonperturbative theory that has perturbative string theory as its perturbation expansion? And how does one extract information on Planck scale physics from it?

The view of a relativist For a relativist, on the other hand, the idea of a fundamental description of gravity in terms of physical excitations over a background metric space sounds physically very wrong. The key lesson learned from general relativity is that there is no background metric over which physics happens unless, of course, in approximations. The world is more complicated than that.

Indeed, for a relativist, general relativity is much more than the field theory of a particular force. Rather, it is the discovery that certain classical notions about space and time are inadequate at the fundamental level; they require modifications which are possibly as basic as the ones that quantum mechanics introduced.

One of these inadequate notions is precisely the notion of a background metric space, flat or curved , over which physics happens. This profound conceptual shift has led to the understanding of relativistic gravity, to the discovery of black holes, to relativistic astrophysics and to modern cosmology.

From Newton to the beginning of this century, physics has had a solid foundation in a small number of key notions such as space, time, causality and matter. In spite of substantial evolution, these notions remained rather stable and self-consistent.

## The Fiber Bundle at the Gates of Metaphysics. Challenging Tim Maudlin’s Proposal

In the first quarter of this century, quantum theory and general relativity have deeply modified this foundation. The two theories have obtained solid success and vast experimental corroboration, and can now be considered established knowledge. Each of the two theories modifies the conceptual foundation of classical physics in an more or less internally consistent manner, but we do not have a novel conceptual foundation capable of supporting both theories.

This is why we do not yet have a theory capable of predicting what happens in the physical regime in which both theories are relevant, the regime of Planck scale phenomena, cm. General relativity has taught us not only that space and time share the property of being dynamical with the rest of the physical entities, but also —more crucially— that spacetime location is relational only see section 5.

Therefore, we need a relational notion of a quantum spacetime in order to understand Planck scale physics. Thus, for a relativist, the problem of quantum gravity is the problem of bringing a vast conceptual revolution, begun with quantum mechanics and with general relativity, to a conclusion and to a new synthesis. Unlike perturbative or nonperturbative string theory, loop quantum gravity is formulated without a background spacetime.

Loop quantum gravity is thus a genuine attempt to grasp what quantum spacetime is at the fundamental level. Accordingly, the notion of spacetime that emerges from the theory is profoundly different from the one on which conventional quantum field theory or string theory is based.

## Loops, knots, gauge theories and quantum gravity

Strings or loops? Above I have emphasized the radically distinct cultural paths leading to string theory and loop quantum gravity. Here I attempt to compare the actual achievements the two theories have obtained so far regarding the description of Planck scale physics. Once more, however, I want to emphasize that, whatever prejudices this or that physicist may have, both theories are tentative: As far as we really know, either, or both, theories could very well turn out to be physically entirely wrong.

And I do not mean that they could be superseded: I mean that all their specific predictions could be disproved by experiments. Nature does not always share our aesthetic judgments, and the history of theoretical physics is full of enthusiasm for strange theories turned into disappointment.

The arbiters in science are experiments, and not a single experimental result supports, not even very indirectly, any of the current theories that go beyond the Standard Model and general relativity.

To the contrary, all the predictions made so far by theories that go beyond the Standard Model and general relativity proton decay, supersymmetric particles, exotic particles, solar system dynamics have for the moment been punctually falsified by experiments. Comparing this situation with the astonishing experimental success of the Standard Model and classical general relativity should make us very cautious, I believe.

Lacking experiments, theories can only be compared on completeness and aesthetic criteria — criteria, one should not forget, that according to many favored Ptolemy over Copernicus at some point. The main merits of string theory are that it provides a superbly elegant unification of known fundamental physics, and that it has a well defined perturbation expansion, finite order by order. Its main incompletenesses are that its non-perturbative regime is poorly understood, and that we do not have a background-independent formulation of the theory.

In a sense, we do not really know what the theory we are talking about is. Because of this poor understanding of the non perturbative regime of the theory, Planck scale physics and genuine quantum gravitational phenomena are not easily controlled: Except for a few computations, there has not been much Planck scale physics derived from string theory so far. There are, however, two sets of remarkable physical results.

The first is given by some very high energy scattering amplitudes that have been computed see for instance [ 3 , 4 , 5 , 6 , , ]. An intriguing aspect of these results is that they indirectly suggest that geometry below the Planck scale cannot be probed —and thus in a sense does not exist— in string theory. The second physical achievement of string theory which followed the d-branes revolution is the recent derivation of the Bekenstein-Hawking black hole entropy formula for certain kinds of black holes [ , , , ].

The main merit of loop quantum gravity, on the other hand, is that it provides a well-defined and mathematically rigorous formulation of a background-independent, non-perturbative generally covariant quantum field theory. The theory provides a physical picture and quantitative predictions of the world at the Planck scale. The main incompleteness of the theory is regarding the dynamics, formulated in several variants.

## Search form

So far, the theory has lead to two main sets of physical results. The first is the derivation of the Planck scale eigenvalues of geometrical quantities such as areas and volumes. Finally, strings and loop gravity may not necessarily be competing theories: There might be a sort of complementarity, at least methodological, between the two.

This is due to the fact that the open problems of string theory are with respect to its background-independent formulation, and loop quantum gravity is precisely a set of techniques for dealing non-perturbatively with background independent theories. Perhaps the two approaches might even, to some extent, converge. Undoubtedly, there are similarities between the two theories: first of all the obvious fact that both theories start with the idea that the relevant excitations at the Planck scale are one dimensional objects — call them loops or strings.

I understand that in another living review to be published in this journal Lee Smolin explores the possible relations between string theory and loop gravity [ ].

History of Loop Quantum Gravity, Main Steps The following chronology does not exhaust the literature on loop quantum gravity. It only indicates the key steps in the construction of the theory, and the first derivation of the main results.

For more complete references, see the following sections. Due to the attempt to group similar results, some things may appear a bit out of the chronological order. In this formulation, the field variable is a self-dual connection, instead of the metric, and the canonical constraints are simpler than in the old metric formulation. The idea of using a self-dual connection as field variable and the simple constraints it yields were discovered by Amitaba Sen [ ].

Abhay Ashtekar realized that in the SU 2 extended phase space a self-dual connection and a densitized triad field form a canonical pair [ 8 , 9 ] and set up the canonical formalism based on such pair, which is the Ashtekar formalism. Recent works on the loop representation are not based on the original Sen-Ashtekar connection, but on a real variant of it, whose use has been introduced into Lorentzian general relativity by Barbero [ 40 , 41 , 42 , 43 ].

Soon after the introduction of the classical Ashtekar variables, Ted Jacobson and Lee Smolin realized in [ ] that the Wheeler-DeWitt equation, reformulated in terms of the new variables, admits a simple class of exact solutions: the traces of the holonomies of the Ashtekar connection around smooth non-self-intersecting loops. In other words: The Wilson loops of the Ashtekar connection solve the Wheeler-DeWitt equation if the loops are smooth and non self-intersecting.

## Origin Of Gauge Symmetry

Quantum states can be represented in terms of their expansion on the loop basis, namely as functions on a space of loops. The difficulties of the loop representation in the context of Yang-Mills theory are cured by the diffeomorphism invariance of GR see section 6.

The relation to the connection representation was originally derived in the form of an integral transform an infinite dimensional analog of a Fourier transform from functionals of the connection to loop functionals. Several years later, this loop transform was shown to be mathematically rigorously defined [ 13 ]. The immediate results of the loop representation were two: The diffeomorphism constraint was completely solved by knot states loop functionals that depend only on the knotting of the loops , making earlier suggestions by Smolin on the role of knot theory in quantum gravity [ ] concrete; and suitable [ , ] extensions of the knot states with support on non-selfintersecting loops were proven to be solutions of all quantum constraints, namely exact physical states of quantum gravity.

The investigation of exact solutions of the quantum constraint equations, and their relation to knot theory in particular to the Jones polynomial and other knot invariants started soon after the formulation of the theory and has continued since [ , 57 , 58 , 59 , 60 , , 92 , 94 , , 89 ].

The first indication that the theory predicts that Planck scale discreteness came from studying the states that approximate geometries flat on large scale [ 24 ]. In [ ] Junichi Iwasaki and Rovelli studied the representation of gravitons in loop quantum gravity.

These appear as topological modifications of the fabric of the spacetime weave. Matter coupling was beginning to be explored in [ , ]. In [ 15 , 16 , 17 ] Ashtekar and Lewandowski set the basis of the differential formulation of loop quantum gravity by constructing its two key ingredients: a diffeomorphism invariant measure on the space of generalized connections, and the projective family of Hilbert spaces associated to graphs. Using these techniques, they were able to give a mathematically rigorous construction of the state space of the theory, solving long standing problems deriving from the lack of a basis the insufficient control on the algebraic identities between loop states.

Using this, they defined a consistent scalar product and proved that the quantum operators in the theory were consistent with all identities.

John Baez showed how the measure can be used in the context of conventional connections, extended it to the non-gauge invariant states allowing the E operator to be defined and developed the use of the graph techniques [ 29 , 30 , 28 ]. In my opinion, the most significant result of loop quantum gravity is the discovery that certain geometrical quantities, in particular area and volume, are represented by operators that have discrete eigenvalues. This was found by Rovelli and Smolin in [ ], where the first set of these eigenvalues was computed.

Shortly after, this result was confirmed and extended by a number of authors, using very diverse techniques. In particular, Renate Loll [ , ] used lattice techniques to analyze the volume operator and corrected a numerical error in [ ]. Ashtekar and Lewandowski [ , 18 ] recovered and completed the computation of the spectrum of the area using the connection representation, and new regularization techniques.

Frittelli, Lehner and Rovelli [ 84 ] recovered the Ashtekar-Lewandowski terms of the spectrum of the area, using the loop representation. DePietri and Rovelli [ 77 ] computed general eigenvalues of the volume. Complete understanding of the precise relation between different versions of the volume operator came from the work of Lewandowski [ ].

A long standing problem with the loop basis was its overcompleteness. A technical, but crucial step in understanding the theory has been the discovery of the spin-network basis, which solves this overcompleteness. This step was taken by Rovelli and Smolin in [ ] and was motivated by the work of Roger Penrose [ , ], by analogous bases used in lattice gauge theory and by ideas of Lewandowski [ ].

Shortly after, the spin network formalism was cleaned up and clarified by John Baez [ 34 , 35 ]. After the introduction of the spin network basis, all problems deriving from the incompleteness of the loop basis are trivially solved, and the scalar product could be defined also algebraically [ 77 ]. In [ 76 ], Roberto DePietri proved the equivalence of the two formalisms, using ideas from Thiemann [ ] and Lewandowski [ ]. The first version of the loop hamiltonian constraint is in [ , ]. An important step was made by Rovelli and Smolin in [ ] with the realization that certain regularized loop operators have finite limits on knot states see [ ].

The search culminated with the work of Thomas Thiemann, who was able to construct a rather well-defined hamiltonian operator whose constraint algebra closes [ , , ]. Variants of this constraint have been suggested in [ , ] and elsewhere. A derivation of the Bekenstein-Hawking formula for the entropy of a black hole from loop quantum gravity was obtained in [ ], on the basis of the ideas of Kirill Krasnov [ , ].

Recently, Ashtekar, Baez, Corichi and Krasnov have announced an alternative derivation [ 12 ]. The resulting covariant theory turns out to be a sum over topologically inequivalent surfaces, realizing earlier suggestions by Baez [ 31 , 28 , 34 , 25 ], Reisenberger [ , ] and Iwasaki [ ] that a covariant version of loop gravity should look like a theory of surfaces.

Baez has studied the general structure of theories defined in this manner [ 26 ]. Smolin and Markoupolou have explored the extension of the construction to the Lorentzian case, and the possibility of altering the spin network evolution rules [ ].

Presently, it is being kept updated by Christopher Beetle and Alejandro Corichi. The latest version can be found on the net in [ 45 ].

More detailed but less up to date presentations are listed below. The part of the book on the loop representation is essentially an authorized reprint of parts of the original Rovelli Smolin article [ ].

For this quantum part, I recommend looking at the article, rather than the book, because the article is more complete. This is more oriented to a reader with a physics background. The other book by Baez, and Muniain [ 37 ], is a simple and pleasant introduction to several ideas and techniques in the field.

Hopefully, this will become a recurrent meeting. This may be the right place to go for learning what is going on in the field. For an informal account of the last of these meetings August , see [ ]. Jean-marc Ginoux. Jan Awrejcewicz. Bernadette Bouchon-Meunier. Andrew Adamatzky.

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Popular Features. New in Gauge Fields, Knots And Gravity. Description This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes.

The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations.One could conventionally split the spacetime metric into two terms: one to be considered a background, which gives a metric structure to spacetime; the other to be treated as a fluctuating quantum field.

The metaphysics within physics. The book is about some of the recent approaches to find a models, known as the theory of everything. Journal of Philosophy — Leon O. Studies in History and Philosophy of Modern Physics —

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