11
11

Jul 26, 2016
07/16

Jul 26, 2016
by
John C. Baez

texts

#
eye 11

#
favorite 0

#
comment 0

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

15
15

Apr 21, 2015
04/15

Apr 21, 2015
by
John C. Baez; Brendan Fong

texts

#
eye 15

#
favorite 0

#
comment 0

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

129
129

Feb 25, 2013
02/13

Feb 25, 2013
by
John C. Baez; Mike Stay

texts

#
eye 129

#
favorite 0

#
comment 0

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

169
169

Sep 17, 2012
09/12

Sep 17, 2012
by
John C. Baez; Jacob Biamonte

texts

#
eye 169

#
favorite 0

#
comment 0

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

60
60

Aug 5, 2012
08/12

Aug 5, 2012
by
John C. Baez; John Huerta

texts

#
eye 60

#
favorite 0

#
comment 0

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

45
45

Jun 8, 2012
06/12

Jun 8, 2012
by
John C. Baez; Derek K. Wise

texts

#
eye 45

#
favorite 0

#
comment 0

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

113
113

Mar 9, 2012
03/12

Mar 9, 2012
by
John C. Baez; Brendan Fong

texts

#
eye 113

#
favorite 0

#
comment 0

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

54
54

Nov 18, 2011
11/11

Nov 18, 2011
by
John C. Baez; Tobias Fritz; Tom Leinster

texts

#
eye 54

#
favorite 0

#
comment 0

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

93
93

Jun 6, 2011
06/11

Jun 6, 2011
by
John C. Baez

texts

#
eye 93

#
favorite 0

#
comment 0

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

70
70

Apr 18, 2011
04/11

Apr 18, 2011
by
John C. Baez

texts

#
eye 70

#
favorite 0

#
comment 0

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

65
65

Feb 9, 2011
02/11

Feb 9, 2011
by
John C. Baez; Aristide Baratin; Laurent Freidel; Derek K. Wise

texts

#
eye 65

#
favorite 0

#
comment 0

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

59
59

Oct 21, 2010
10/10

Oct 21, 2010
by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

texts

#
eye 59

#
favorite 0

#
comment 0

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

85
85

Jul 24, 2010
07/10

Jul 24, 2010
by
John C. Baez

texts

#
eye 85

#
favorite 0

#
comment 0

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

158
158

Jul 24, 2010
07/10

Jul 24, 2010
by
John C. Baez; John Huerta

texts

#
eye 158

#
favorite 0

#
comment 0

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

65
65

May 1, 2010
05/10

May 1, 2010
by
John C. Baez; John Huerta

texts

#
eye 65

#
favorite 0

#
comment 0

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

104
104

Mar 23, 2010
03/10

Mar 23, 2010
by
John C. Baez; John Huerta

texts

#
eye 104

#
favorite 0

#
comment 0

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

61
61

Jan 25, 2010
01/10

Jan 25, 2010
by
John C. Baez; Alissa S. Crans

texts

#
eye 61

#
favorite 0

#
comment 0

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

54
54

Dec 29, 2009
12/09

Dec 29, 2009
by
John C. Baez; John Huerta

texts

#
eye 54

#
favorite 0

#
comment 0

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

94
94

Oct 8, 2009
10/09

Oct 8, 2009
by
John C. Baez; Alexander E. Hoffnung

texts

#
eye 94

#
favorite 0

#
comment 0

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

917
917

Aug 18, 2009
08/09

Aug 18, 2009
by
John C. Baez; Aaron Lauda

texts

#
eye 917

#
favorite 0

#
comment 0

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

57
57

Jul 27, 2009
07/09

Jul 27, 2009
by
John C. Baez; Danny Stevenson

texts

#
eye 57

#
favorite 0

#
comment 0

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

101
101

Jun 6, 2009
06/09

Jun 6, 2009
by
John C. Baez; Mike Stay

texts

#
eye 101

#
favorite 1

#
comment 0

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

47
47

Jan 29, 2009
01/09

Jan 29, 2009
by
John C. Baez; Christopher L. Rogers

texts

#
eye 47

#
favorite 0

#
comment 0

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

53
53

Dec 30, 2008
12/08

Dec 30, 2008
by
John C. Baez; Alexander E. Hoffnung; Christopher D. Walker

texts

#
eye 53

#
favorite 0

#
comment 0

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

46
46

Aug 2, 2008
08/08

Aug 2, 2008
by
John C. Baez; Alexander E. Hoffnung; Christopher L. Rogers

texts

#
eye 46

#
favorite 0

#
comment 0

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

70
70

Oct 27, 2007
10/07

Oct 27, 2007
by
John C. Baez; Michael Shulman

texts

#
eye 70

#
favorite 0

#
comment 0

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

89
89

Jun 28, 2006
06/06

Jun 28, 2006
by
John C. Baez; Urs Schreiber

texts

#
eye 89

#
favorite 0

#
comment 0

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

48
48

May 15, 2006
05/06

May 15, 2006
by
John C. Baez; Alejandro Perez

texts

#
eye 48

#
favorite 0

#
comment 0

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

64
64

May 9, 2006
05/06

May 9, 2006
by
John C. Baez; Derek K. Wise; Alissa S. Crans

texts

#
eye 64

#
favorite 0

#
comment 0

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

116
116

Jan 5, 2006
01/06

Jan 5, 2006
by
John C. Baez; Emory F. Bunn

texts

#
eye 116

#
favorite 0

#
comment 0

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

79
79

Nov 9, 2005
11/05

Nov 9, 2005
by
John C. Baez

texts

#
eye 79

#
favorite 0

#
comment 0

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

72
72

Apr 19, 2005
04/05

Apr 19, 2005
by
John C. Baez; Alissa S. Crans; Danny Stevenson; Urs Schreiber

texts

#
eye 72

#
favorite 0

#
comment 0

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

67
67

Oct 27, 2004
10/04

Oct 27, 2004
by
John C. Baez; Aaron D. Lauda

texts

#
eye 67

#
favorite 0

#
comment 0

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

48
48

Apr 14, 2004
04/04

Apr 14, 2004
by
John C. Baez

texts

#
eye 48

#
favorite 0

#
comment 0

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

51
51

Mar 6, 2004
03/04

Mar 6, 2004
by
John C. Baez; James Dolan

texts

#
eye 51

#
favorite 0

#
comment 0

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

58
58

Nov 4, 2002
11/02

Nov 4, 2002
by
John C. Baez; J. Daniel Christensen; Greg Egan

texts

#
eye 58

#
favorite 0

#
comment 0

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

67
67

Jul 21, 2002
07/02

Jul 21, 2002
by
John C. Baez; J. Daniel Christensen; Thomas R. Halford; David C. Tsang

texts

#
eye 67

#
favorite 0

#
comment 0

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

78
78

Jun 17, 2002
06/02

Jun 17, 2002
by
John C. Baez

texts

#
eye 78

#
favorite 0

#
comment 0

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

86
86

Jun 12, 2002
06/02

Jun 12, 2002
by
John C. Baez; J. Daniel Christensen

texts

#
eye 86

#
favorite 0

#
comment 0

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

107
107

Apr 23, 2002
04/02

Apr 23, 2002
by
John C. Baez

texts

#
eye 107

#
favorite 0

#
comment 0

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

46
46

Jan 9, 2002
01/02

Jan 9, 2002
by
John C. Baez; S. Jay Olson

texts

#
eye 46

#
favorite 0

#
comment 0

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

63
63

Aug 10, 2001
08/01

Aug 10, 2001
by
John C. Baez; John W. Barrett

texts

#
eye 63

#
favorite 0

#
comment 0

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

43
43

Oct 14, 2000
10/00

Oct 14, 2000
by
John C. Baez; Laurel Langford

texts

#
eye 43

#
favorite 0

#
comment 0

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

72
72

Apr 20, 2000
04/00

Apr 20, 2000
by
John C. Baez; James Dolan

texts

#
eye 72

#
favorite 0

#
comment 0

`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

96
96

Oct 14, 1999
10/99

Oct 14, 1999
by
John C. Baez

texts

#
eye 96

#
favorite 0

#
comment 0

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

48
48

May 21, 1999
05/99

May 21, 1999
by
John C. Baez

texts

#
eye 48

#
favorite 0

#
comment 0

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

44
44

Mar 16, 1999
03/99

Mar 16, 1999
by
John C. Baez; John W. Barrett

texts

#
eye 44

#
favorite 0

#
comment 0

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

65
65

Feb 4, 1999
02/99

Feb 4, 1999
by
John C. Baez

texts

#
eye 65

#
favorite 0

#
comment 0

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

55
55

May 31, 1998
05/98

May 31, 1998
by
John C. Baez; Laurel Langford

texts

#
eye 55

#
favorite 0

#
comment 0

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

60
60

Apr 23, 1998
04/98

Apr 23, 1998
by
John C. Baez

texts

#
eye 60

#
favorite 0

#
comment 0

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