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Jul 20, 2013
07/13

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John C. Baez

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Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional BF theory on any framed surface Sigma embedded in M. In these solutions the cotetrad field e (and thus the metric) vanishes outside a neighborhood of Sigma, while inside this neighborhood the connection A and the field E = e ^ e satisfy the equations of 4-dimensional BF theory. Moreover, there is a correspondence between...

Source: http://arxiv.org/abs/gr-qc/9702051v1

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Sep 23, 2013
09/13

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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 18, 2013
09/13

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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 21, 2013
09/13

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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 20, 2013
09/13

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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 22, 2013
09/13

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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 21, 2013
09/13

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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 20, 2013
09/13

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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 21, 2013
09/13

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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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Jul 20, 2013
07/13

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John C. Baez

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This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a `background-free quantum theory with local degrees of freedom propagating causally'. We describe the insights provided by work on topological quantum field theories such as quantum gravity in 3-dimensional spacetime. These are background-free quantum theories lacking local degrees of freedom, so they...

Source: http://arxiv.org/abs/gr-qc/9902017v1

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Sep 19, 2013
09/13

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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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Jun 29, 2018
06/18

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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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Sep 19, 2013
09/13

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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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Jul 20, 2013
07/13

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John C. Baez

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Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P -> M, there is a canonical `generalized measure' on the space A/G of smooth connections on P modulo gauge transformations. This allows one to define a Hilbert space L^2(A/G). Here we construct a set of vectors spanning L^2(A/G). These vectors are described in terms of `spin networks': graphs phi embedded in M, with oriented edges labelled by irreducible unitary representations of G, and with vertices...

Source: http://arxiv.org/abs/gr-qc/9411007v1

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Sep 18, 2013
09/13

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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

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John C. Baez

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A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3-dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity,...

Source: http://arxiv.org/abs/gr-qc/9504036v2

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Sep 21, 2013
09/13

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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 19, 2013
09/13

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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
09/13

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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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Jul 20, 2013
07/13

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John C. Baez

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We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the evolution operators of the perturbed model as convergent power series in the coupling constant governing the perturbation. The terms in the power series can be efficiently computed when the unperturbed model is a topological quantum field theory. Moreover, in this case we can explicitly sum the whole...

Source: http://arxiv.org/abs/gr-qc/9910050v1

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Sep 21, 2013
09/13

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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 19, 2013
09/13

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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 23, 2013
09/13

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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 20, 2013
09/13

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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 23, 2013
09/13

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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

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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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Jul 20, 2013
07/13

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John C. Baez; Brendan Fong

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Noether's theorem links the symmetries of a quantum system with its conserved quantities, and is a cornerstone of quantum mechanics. Here we prove a version of Noether's theorem for Markov processes. In quantum mechanics, an observable commutes with the Hamiltonian if and only if its expected value remains constant in time for every state. For Markov processes that no longer holds, but an observable commutes with the Hamiltonian if and only if both its expected value and standard deviation are...

Source: http://arxiv.org/abs/1203.2035v1

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Sep 18, 2013
09/13

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John C. Baez; Jacob Biamonte

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Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: "stochastic Petri nets". But if we look at these theories from the perspective of quantum theory, they turn out to...

Source: http://arxiv.org/abs/1209.3632v1

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Sep 17, 2013
09/13

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John C. Baez; John Huerta

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Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n+1)-cocycle on a Lie superalgebra yields a "Lie n-superalgebra": that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the...

Source: http://arxiv.org/abs/1003.3436v2

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Sep 19, 2013
09/13

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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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Sep 18, 2013
09/13

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John C. Baez; Alejandro Perez

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BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in three dimensions generalizes in a direct way to BF theory in d dimensions coupled to (d-3)-branes. In the resulting model, the connection is flat except along the membrane world-sheet, where it has a conical singularity whose strength is proportional to the membrane tension. As a step towards...

Source: http://arxiv.org/abs/gr-qc/0605087v1

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Sep 19, 2013
09/13

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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 17, 2013
09/13

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John C. Baez; John Huerta

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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie...

Source: http://arxiv.org/abs/1003.4485v1

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Sep 20, 2013
09/13

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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 23, 2013
09/13

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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

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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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Jul 20, 2013
07/13

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John C. Baez; James Dolan

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We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O+ whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I(0) = I, I(i+1) = I(i)+, we call the operations of I(n-1) the `n-dimensional opetopes'. Opetopes form a...

Source: http://arxiv.org/abs/q-alg/9702014v1

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Sep 20, 2013
09/13

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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 23, 2013
09/13

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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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Jun 27, 2018
06/18

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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 22, 2013
09/13

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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

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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 19, 2013
09/13

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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 22, 2013
09/13

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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 18, 2013
09/13

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John C. Baez; Laurel Langford

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Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. Using the work of Carter, Rieger and Saito, we prove that this 2-category is the `free semistrict braided monoidal 2-category with duals on one unframed self-dual object'. By this universal property, any unframed self-dual object in a braided...

Source: http://arxiv.org/abs/math/9811139v3

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Sep 24, 2013
09/13

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

45
45

Sep 21, 2013
09/13

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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67

Sep 22, 2013
09/13

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

46
46

Sep 22, 2013
09/13

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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94

Jul 22, 2013
07/13

by
John C. Baez; Alexander E. Hoffnung

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A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological...

Source: http://arxiv.org/abs/0807.1704v4