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Jun 29, 2018
06/18

Jun 29, 2018
by
John C. Baez

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Sunada's work on topological crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie on the edges. We prove that for any connected graph $X$ without bridges, there is a canonical way to...

Topics: Combinatorics, Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1607.07748

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Jun 27, 2018
06/18

Jun 27, 2018
by
John C. Baez; Brendan Fong

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Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with marked input and output terminals. In this category, composition describes the process of attaching the outputs of one circuit to the inputs of another. We construct a functor, dubbed the "black box functor", that takes a circuit, forgets its...

Topics: Category Theory, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1504.05625

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Sep 24, 2013
09/13

Sep 24, 2013
by
John C. Baez; Emory F. Bunn

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This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of its consequences, and explain how the formulation given here is equivalent to the usual...

Source: http://arxiv.org/abs/gr-qc/0103044v5

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez; Danny Stevenson

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Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the first Cech cohomology of M with coefficients in G, or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by...

Source: http://arxiv.org/abs/0801.3843v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez; Mike Stay

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In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there...

Source: http://arxiv.org/abs/0903.0340v3

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez

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Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure'' on the space A of connections on P and for ``Haar measure'' on the group Ga of gauge transformations. More precisely, we define algebras of ``cylinder functions'' on the spaces A, Ga, and A/Ga, and define generalized measures on these spaces as continuous linear functionals on the corresponding algebras. Borrowing some ideas from...

Source: http://arxiv.org/abs/hep-th/9310201v1

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez; John Huerta

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The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify forces and particles by extending the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x SU(4). In this...

Source: http://arxiv.org/abs/0904.1556v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez

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For any subgroup G of O(n), define a "G-manifold" to be an n-dimensional Riemannian manifold whose holonomy group is contained in G. Then a G-manifold where G is the Standard Model gauge group is precisely a Calabi-Yau manifold of 10 real dimensions whose tangent spaces split into orthogonal 4- and 6-dimensional subspaces, each preserved by the complex structure and parallel transport. In particular, the product of Calabi-Yau manifolds of dimensions 4 and 6 gives such a G-manifold....

Source: http://arxiv.org/abs/hep-th/0511086v2

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Sep 23, 2013
09/13

Sep 23, 2013
by
John C. Baez; Urs Schreiber

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Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a manifold but a category with a suitable smooth structure. Where gauge theory uses Lie groups and Lie algebras, higher gauge theory uses their categorified analogues: Lie 2-groups...

Source: http://arxiv.org/abs/math/0511710v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Stephen Sawin

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We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a smooth principal G-bundle. A `cylinder function' on the space of smooth connections on P is a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves in M. We construct diffeomorphism-invariant functionals on the space of cylinder functions from `spin...

Source: http://arxiv.org/abs/q-alg/9708005v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Alissa S. Crans

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The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as morphisms and "linear natural transformations" as 2-morphisms. We define a "semistrict Lie 2-algebra" to be a 2-vector space L equipped with a skew-symmetric bilinear functor satisfying the Jacobi identity up to a completely...

Source: http://arxiv.org/abs/math/0307263v6

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Aaron D. Lauda

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A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse":...

Source: http://arxiv.org/abs/math/0307200v3

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; James Dolan

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`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category of finite sets serves as a categorified version of the set of natural numbers, with disjoint union and Cartesian product playing the role of addition and multiplication. We sketch how categorifying the integers leads naturally to the infinite loop space...

Source: http://arxiv.org/abs/math/0004133v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; S. Jay Olson

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Ng and van Dam have argued that quantum theory and general relativity give a lower bound of L^{1/3} L_P^{2/3} on the uncertainty of any distance, where L is the distance to be measured and L_P is the Planck length. Their idea is roughly that to minimize the position uncertainty of a freely falling measuring device one must increase its mass, but if its mass becomes too large it will collapse to form a black hole. Here we show that one can go below the Ng-van Dam bound by attaching the measuring...

Source: http://arxiv.org/abs/gr-qc/0201030v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez

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Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this...

Source: http://arxiv.org/abs/1101.5690v3

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Aristide Baratin; Laurent Freidel; Derek K. Wise

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A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional...

Source: http://arxiv.org/abs/0812.4969v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

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Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from...

Source: http://arxiv.org/abs/0812.4864v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Aaron Lauda

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This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest themselves in Feynman diagrams, spin networks, string theory, loop quantum gravity, and topological quantum field theory. Our chronology ends around 2000, with just a taste of later developments such as open-closed topological string theory, the categorification of quantum groups, Khovanov homology, and...

Source: http://arxiv.org/abs/0908.2469v1

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

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Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally...

Source: http://arxiv.org/abs/0908.4305v3

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; Derek K. Wise; Alissa S. Crans

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After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a...

Source: http://arxiv.org/abs/gr-qc/0603085v2

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Sep 22, 2013
09/13

Sep 22, 2013
by
John C. Baez; James Dolan

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The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of n-category suited for this purpose, and outline a program in which n-dimensional TQFTs are to be described as n-category representations. First we describe a "suspension" operation on n-categories, and hypothesize that the k-fold suspension of a weak n-category stabilizes for k >= n+2....

Source: http://arxiv.org/abs/q-alg/9503002v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez; Christopher L. Rogers

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Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2 is relevant to string theory: we call this 2-plectic geometry. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2-plectic manifold form a "Lie 2-algebra", which is a categorified...

Source: http://arxiv.org/abs/0901.4721v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez

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An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of n-category, with an emphasis on `weak' n-categories, in which all rules governing the composition of j-morphisms hold only up to equivalence. (An n-morphism is an equivalence if it is invertible, while a j-morphism for j < n is an equivalence if it is...

Source: http://arxiv.org/abs/q-alg/9705009v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez; Derek K. Wise

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We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincare 2-group as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Conversely, every flat strict 2-connection on this 2-bundle arises in this way...

Source: http://arxiv.org/abs/1204.4339v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez; Tobias Fritz; Tom Leinster

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There are numerous characterizations of Shannon entropy and Tsallis entropy as measures of information obeying certain properties. Using work by Faddeev and Furuichi, we derive a very simple characterization. Instead of focusing on the entropy of a probability measure on a finite set, this characterization focuses on the `information loss', or change in entropy, associated with a measure-preserving function. Information loss is a special case of conditional entropy: namely, it is the entropy of...

Source: http://arxiv.org/abs/1106.1791v3

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez

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The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical physics and topology. At the formal level, an example of such a measure is given by the Chern-Simons path integral. Certain measures of this sort also play the role of states in quantum gravity in Ashtekar's formalism. These measures define link invariants, or more generally multiloop invariants; as...

Source: http://arxiv.org/abs/hep-th/9305045v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez

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In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links in R^3. The physically correct inner product on this space of states is not yet known, or in other words, the *-algebra structure of the algebra of observables has not been determined. In order to approach this problem, we consider a larger space H of solutions of the Hamiltonian constraint, which...

Source: http://arxiv.org/abs/hep-th/9205007v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez

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In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining...

Source: http://arxiv.org/abs/gr-qc/9905087v1

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez

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A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a *-structure, conjugate-linear on the hom-sets, satisfying = = . We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H*-algebras, braided 2-H*-algebras, and symmetric 2-H*-algebras, and we describe the relation between these and tangles in 2,...

Source: http://arxiv.org/abs/q-alg/9609018v2

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Sep 21, 2013
09/13

Sep 21, 2013
by
John C. Baez; Alexander E. Hoffnung; Christopher L. Rogers

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A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic...

Source: http://arxiv.org/abs/0808.0246v1

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Sep 20, 2013
09/13

Sep 20, 2013
by
John C. Baez

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Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G = GL(4,R) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented...

Source: http://arxiv.org/abs/q-alg/9507006v1

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Sep 20, 2013
09/13

Sep 20, 2013
by
John C. Baez; Michael Shulman

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The goal of these talks was to explain how cohomology and other tools of algebraic topology are seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov towers, the theory of "n-stuff", and n-categories for n = -1 and -2. The talks were very informal, and so are these notes. A lengthy appendix clarifies certain puzzles and ventures into deeper waters such as higher topos theory. For readers who want more details, we include an annotated...

Source: http://arxiv.org/abs/math/0608420v2

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Sep 20, 2013
09/13

Sep 20, 2013
by
John C. Baez; James Dolan

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Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called `coherence laws'. Iterating this process requires a theory of `n-categories', algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms....

Source: http://arxiv.org/abs/math/9802029v1

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Sep 20, 2013
09/13

Sep 20, 2013
by
John C. Baez

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General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n-1)-dimensional manifolds representing "space" and whose morphisms are n-dimensional cobordisms representing "spacetime". Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe "states", and whose morphisms...

Source: http://arxiv.org/abs/quant-ph/0404040v2

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Sep 20, 2013
09/13

Sep 20, 2013
by
John C. Baez; John Huerta

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Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again...

Source: http://arxiv.org/abs/1205.2447v3

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez; John Huerta

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Supersymmetry is deeply related to division algebras. Nonabelian Yang--Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green--Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain trilinear expression involving a spinor field. The reason for this, in turn, is the existence of normed division algebras in dimensions 1, 2, 4 and 8: the real numbers, complex...

Source: http://arxiv.org/abs/0909.0551v2

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez

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The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an approximation to QCD, in which the strings represent flux tubes of the gauge field. A heuristic path-integral approach indicates a duality between background-free string theories and generally covariant gauge theories, with the loop transform relating the two. We review progress towards making this duality...

Source: http://arxiv.org/abs/hep-th/9309067v1

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Sep 19, 2013
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Sep 19, 2013
by
John C. Baez

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The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V_infinity containing elements g_i satisfying the usual braid group relations and elements a_i satisfying g_i - g_i^{-1} = epsilon a_i, where epsilon is a formal variable that may be regarded as measuring the failure of g_i^2 to equal 1. Topologically, the elements a_i...

Source: http://arxiv.org/abs/hep-th/9207041v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez

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While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining...

Source: http://arxiv.org/abs/gr-qc/9709052v3

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez; J. Daniel Christensen; Greg Egan

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The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a...

Source: http://arxiv.org/abs/gr-qc/0208010v3

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Sep 19, 2013
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Sep 19, 2013
by
John C. Baez

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Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of "higher-dimensional Yang-Mills theory". It turns out that to do this, one should replace the Lie group by a "Lie 2-group", which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps...

Source: http://arxiv.org/abs/hep-th/0206130v2

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez; Kirill V. Krasnov

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We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A x F, where A is the space of...

Source: http://arxiv.org/abs/hep-th/9703112v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez; Laurel Langford

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Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R^4 may be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. In this announcement we give a purely algebraic characterization of the 2-category of unframed unoriented 2-tangles in 4 dimensions as the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. A forthcoming paper will contain a proof of...

Source: http://arxiv.org/abs/q-alg/9703033v4

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez

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Given a principal G-bundle over a smooth manifold M, with G a compact Lie group, and given a finite-dimensional unitary representation of G, one may define an algebra of functions on the space of connections modulo gauge transformations, the ``holonomy Banach algebra'' H_b, by completing an algebra generated by regularized Wilson loops. Elements of the dual H_b* may be regarded as a substitute for measures on the space of connections modulo gauge transformations. There is a natural linear map...

Source: http://arxiv.org/abs/hep-th/9301063v1

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Sep 19, 2013
09/13

Sep 19, 2013
by
John C. Baez; Mike Stay

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Algorithmic entropy can be seen as a special case of entropy as studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of 'microstates', and treat any function on X as an 'observable'. For any collection of observables, we...

Source: http://arxiv.org/abs/1010.2067v2

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63

Sep 18, 2013
09/13

Sep 18, 2013
by
John C. Baez; John W. Barrett

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The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L^2 functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is...

Source: http://arxiv.org/abs/gr-qc/0101107v2

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Sep 18, 2013
09/13

Sep 18, 2013
by
John C. Baez

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The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

Source: http://arxiv.org/abs/math/0105155v4

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Sep 18, 2013
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Sep 18, 2013
by
John C. Baez; J. Daniel Christensen; Thomas R. Halford; David C. Tsang

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Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where...

Source: http://arxiv.org/abs/gr-qc/0202017v4

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93

Sep 18, 2013
09/13

Sep 18, 2013
by
John C. Baez

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The Renyi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Renyi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can do as it moves to equilibrium at the new temperature, divided by the change in temperature, equals the system's Renyi entropy in its original state. This result applies to both classical and...

Source: http://arxiv.org/abs/1102.2098v3

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Sep 18, 2013
09/13

Sep 18, 2013
by
John C. Baez; John W. Barrett

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Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries of the tetrahedron in R^3, we use geometric quantization to obtain a Hilbert space of states. This Hilbert space has a basis of states labeled by the areas of the faces of the tetrahedron together with one more quantum number, e.g. the area of one of the parallelograms formed by midpoints of the...

Source: http://arxiv.org/abs/gr-qc/9903060v1