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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 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 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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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 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 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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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 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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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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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 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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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 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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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 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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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 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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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 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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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 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 22, 2013
09/13

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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

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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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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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Jul 22, 2013
07/13

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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

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Sep 22, 2013
09/13

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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

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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 21, 2013
09/13

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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 22, 2013
09/13

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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 18, 2013
09/13

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John C. Baez; Alissa S. Crans; Danny Stevenson; Urs Schreiber

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We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a categorified version of a Lie group. If G is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras g_k each having Lie(G) as its Lie algebra...

Source: http://arxiv.org/abs/math/0504123v2

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Sep 22, 2013
09/13

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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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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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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 21, 2013
09/13

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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 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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Sep 21, 2013
09/13

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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 22, 2013
09/13

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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 24, 2013
09/13

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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 18, 2013
09/13

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John C. Baez; J. Daniel Christensen

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The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis...

Source: http://arxiv.org/abs/gr-qc/0110044v4

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Sep 19, 2013
09/13

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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 18, 2013
09/13

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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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Sep 18, 2013
09/13

by
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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Jul 20, 2013
07/13

by
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 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 20, 2013
09/13

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 22, 2013
09/13

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 19, 2013
09/13

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 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