Search

We solve a very classical problem motivated by considerations in quantum gravity: providing a description of the geometry of a Euclidean tetrahedron from the initial data of the areas of the faces and the areas of the medial parallelograms of Yetter or equivalently of the pseudofaces of McConnell. In particular, we derive expressions for the dihedral angles, face angles and (an) edge length, the remaining parts being derivable by symmetry or by identities in the classic 1902 compendium of results on tetrahedral geometry of Richardson. We also provide an alternative proof using (bi)vectors of the result of Yetter that four times the sum of the squared areas of the medial parallelograms is equal to the sum of the squared areas of the faces. The updated version corrects several errors and provides better historical context for the result.

We investigate looking for black hole starships using very high energy gamma ray telescopes. We find they would be above the threshhold of visibility although there are many unknown factors., Comment: 6 pages contact crane@math.ksu.edu

We extend results of Pachner and Casali to give finite sets of moves relating triangulations of PL manifolds respecting filtrations by locally flat manifolds and stratifications in which a finite family of simple local models exists for neighborhoods of strata., Comment: 6 pages

Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group $E_6$, as explained by the quantum Mckay correspondence of Frenkel Jing and Wang. The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold., Comment: 15 pages, latex document proposal for unified field theory

In \cite{C1}, we proposed a new interpretation of the EPRL quantization of the BC model for quantum general relativity using a monoidal functor we call the time functor. In this preliminary draft we apply the theory of modules over monoidal functors \cite{Y1} to the time functor, to propose an extension of the EPRL model which would include the standard model. This is motivated by recent advances in neutrino Physics., Comment: 11 pages latex

In this research announcement, we propose a new interpretation of the EPR quantization of the BC model using a functor we call the time functor, which is the first example of a CLa-ren functor. Under the hypothesis that the universe is in the Kodama state, we construct a holographic version of the model. Generalisations to other CLa-ren functors and connections to model category theory are considered., Comment: research announcement. Latex file

We discuss a proposal to make small artificial black holes (ABH's) using a huge laser. Because of Hawking radiation, they would be extremely powerful energy sources. We investigate the technical problems of using them to make power plants and starships. The first suggestion is due to Hawking. Next, we consider what challenges the ABH proposal would pose for a future quantum theory of gravity. The form of a theory which would allow us to compute the necessary corrections to classical theory is considered. It is widely believed that every black hole produces a new baby universe on the other side of its singularity. If this is true, ABH technology will involve future humanity in the creation process of universes. Finally, we ponder the effects that the ABH proposal would have on the culture of a future society, particularly if the baby universe theory is correct. The changes in our economic life and understanding of our role in the cosmos would be so profound as to have a "spiritual" aspect., Comment: First prize winner FQXi contest "On the limits of Physics"

This document is the Special Issue of the First International Conference on the Evolution and Development of the Universe (EDU 2008). Please refer to the preface and introduction for more details on the contributions. Keywords: acceleration, artificial cosmogenesis, artificial life, Big Bang, Big History, biological evolution, biological universe, biology, causality, classical vacuum energy, complex systems, complexity, computational universe, conscious evolution, cosmological artificial selection, cosmological natural selection, cosmology, critique, cultural evolution, dark energy, dark matter, development of the universe, development, emergence, evolution of the universe evolution, exobiology, extinction, fine-tuning, fractal space-time, fractal, information, initial conditions, intentional evolution, linear expansion of the universe, log-periodic laws, macroevolution, materialism, meduso-anthropic principle, multiple worlds, natural sciences, Nature, ontology, order, origin of the universe, particle hierarchy, philosophy, physical constants, quantum darwinism, reduction, role of intelligent life, scale relativity, scientific evolution, self-organization, speciation, specification hierarchy, thermodynamics, time, universe, vagueness., Comment: 355 pages, Special Issue of the First International Conference on the Evolution and Development of the Universe (EDU 2008) (online at: http://evodevouniverse.com/wiki/Conference_2008) To be published in Foundations of Science. Includes peer-reviewed papers, commentaries and responses (metadata update)

We investigate whether it is physically possible to build starships or power sources using the Hawking radiation of an artificial black hole as a power source. The proposal seems to be at the edge of possibility, but quantum gravity effects could change the picture., Comment: Latex document 20 pages

We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime disappears, replacing it with a discrete approximation to the causal path space described as an object in a model category. One of the versions of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration., Comment: 19 pages latex

In this paper, we outline a new approach to quantum gravity; describing states for a bounded region of spacetime as eigenstates for two classes of physically plausible gedanken experiments. We end up with two complementary descriptions in which the point set continuum disappears. The first replaces the continuum of events with a handlebody decomposition of loop space. We conjecture that techniques fron algebraic topology will allow us to extend state sum models on spacetime to loop space. The second picture replaces the continuum with a nondistributive lattice; the classical limit seems more tractible in this picture., Comment: latex document. Submitted to GRF

We survey indications from different branches of Physics that the fine scale structure of spacetime is not adequately described by a manifold. Based on the hints we accumulate, we propose a new structure, which we call a quantum topos. In the process of constructing a quantum topos for quantum gravity, we propose a new, operational approach to the problem of the obervables in quantum gravity, which leads to a new mathematical point of view on the state sum models., Comment: 25 pages, latex file

We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites., Comment: Contribution to the Cambridge University Press volume on Quantum Gravity

We propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural "tangent 2-bundle," analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well., Comment: 21 pages, 3 eps figures; v2: added references; to appear in JMP

We give the construction modulo normalization of a new state sum model for lorentzian quantum general relativity, using the construction of Dirac's expansors to include quantum operators corresponding to edge lengths as well as the quantum bivectors of the Barrett-Crane model, and discuss the problem of its normalization. The new model gives rise to a new picture of quantum geometry in which lengths come in a discrete spectrum, while areas have a continuum of values., Comment: 11 pages latex

We show that the sum over geometries in the Lorentzian 4-D state sum model for quantum GR in [1] includes terms which correspond to geometries on manifolds with conical singularities. Natural approximations suggest that they can be interpreted as gauge bosond for the standard model, plus fermions plus matter., Comment: 9 pages, latex no figs

We show that the normalized Lorentzian state sum is finite on any triangulation. It thus provides a candidate for a perturbatively finite quantum theory of general relativity in four dimensions with Lorentzian signature., Comment: 11 pages, 3 pictures

We propose a new line of attack to create a finite quantum theory which includes general relativity and (perhaps) the standard model in its low energy limit. The theory would emerge from the categorical approach. A structure is observed on the category of unitary representations of the lorentz group which we call hypergravity. This, combined with a study of the relationship between Feynman diagrams and words in tensor categories, leads to the proposal., Comment: 16 pages, latex

We give a relativistic spin network model for quantum gravity based on the Lorentz group and its q-deformation, the Quantum Lorentz Algebra. We propose a combinatorial model for the path integral given by an integral over suitable representations of this algebra. This generalises the state sum models for the case of the four-dimensional rotation group previously studied in gr-qc/9709028. As a technical tool, formulae for the evaluation of relativistic spin networks for the Lorentz group are developed, with some simple examples which show that the evaluation is finite in interesting cases. We conjecture that the `10J' symbol needed in our model has a finite value., Comment: 22 pages, latex, amsfonts, Xypic. Version 3: improved presentation. Version 2 is a major revision with explicit formulae included for the evaluation of relativistic spin networks and the computation of examples which have finite values

We propose a new 4D state sum model, related to the balanced model, which is constructed using the octonions, or equivalently, triality. An effective continuum physical theory constructed from this model coupled to the balanced model would have a non-vanishing cosmological constant, chiral asymmetry, and a gauge group related to the octonions., Comment: 9 pages, Latex

We discuss the interpretation of the state sum of Barrett and Crane., Comment: 4 pages latex

Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) times SU(2). Relativistic quantum spins are related to the geometry of the 2-dimensional faces of a 4-simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3-simplex. This leads us to suggest that there may be a 4-dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3-dimensional quantum gravity from ordinary spin networks., Comment: 10 pages, amstex, some errors corrected, more references

We propose a model for the quantum theory of gravity. the model has diffeomorphism invariance, a natural length scale, and (plausibly) propagating modes. in the new addendum, we alter the model in a way which makes the propagating modes much easier to approach., Comment: 11 pages, latex, important addendum

We define the cohomology and formal deformation theories for algebra and bialgebra categories. We suggest some approaches to finding nontrivial deformations of the categories associated to the quantum groups by the work of Lusztig., Comment: 12 pages latex

We make a direct connection between the construction of three dimensional topological state sums from tensor categories and three dimensional quantum gravity by noting that the discrete version of the Wheeler-DeWitt equation is exactly the pentagon for the associator of the tensor category, the Biedenharn-Elliott identity. A crucial role is played by an asymptotic formula relating 6j-symbols to rotation matrices given by Edmonds., Comment: 10 pages, amstex, uses epsf.tex. New version has improved presentation

We construct tensor and bitensor categories with given Grothedieck rig (fusion algebra) in simple cases. The results provide examples on which to test the conjectural construction of 4-D TQFT's proposed by Crane and Frenkel and shed light on several other constructions of TQFT's., Comment: LaTeX Sun Release 4.1, uses amssym.def and amsssym

We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing topological quantum field theories.The algebraic tools are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context., Comment: To appear in special issue of JMP. Latex document

We provide, with proofs, a complete description of the authors' construction of state-sum invariants announced in [CY], and its generalization to an arbitrary (artinian) semisimple tortile category. We also discuss the relationship of these invariants to generalizations of Broda's surgery invariants [Br1,Br2] using techniques developed in the case of the semi-simple sub-quotient of $Rep(U_q(sl_2))$ ($q$ a principal $4r^{th}$ root of unity) by Roberts [Ro1]. We briefly discuss the generalizations to invariants of 4-manifolds equipped with 2-dimensional (co)homology classes introduced by Yetter [Y6] and Roberts [Ro2], which are the subject of the sequel. (citations refer to bibliography in the paper), Comment: 46 pages

We propose a new mwthod of constructing 4D-TQFTs. The method uses a new type of algebraic structure called a Hopf Category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases., Comment: 38 pages

We consider the implications of some simple assumptions about the nature of the quantum theory of gravity which are plausible for a class of possible theories I have been attempting to construct. The simple assumptions turn out to have surprizingly wide implications of a type one might call philosophical. The paper is short and nontechnical. ( This builds on recent unpublished work of Lee Smolin, but comes to opposite conclusions)., Comment: 5 pages, latex document

Recent work of Roberts has shown that the first surgical 4-manifold invariant of Broda and (up to an unspecified normalization factor) the state-sum invariant arising from the TQFT of Crane-Yetter are equivalent to the signature of the 4-manifold. Subsequently Broda defined another surgical invariant in which the 1- and 2-handles are treated differently. We use a refinement of Roberts' techniques developped by the authors in hep-th/9309063 to show that the "improved" surgical invariant of Broda also depends only on the signature and Euler character, Comment: 4 pages (revised to state correctly Lemma 1--the correct lemma was proved, and used throughout, but the original statement jumbled some phrases--and add an explanatory footnote on an abuse of language)

We provide an explicit formula for the invariant of 4-manifolds introduced by Crane and Yetter (in hep-th 9301062). A consequence of our result is the existence of a combinatorial formula for the signature of a 4-manifold in terms of local data from a triangulation. Potential physical applications of our result exist in light of the fact that the Crane-Yetter invariant is a rigorous version of ideas of Ooguri on B wedge F theory., Comment: 4 pages

Motivated by the similarity between CSW theory and the Chern Simons state for General Relativity in the Ashtekar variables, we explore what the universe would look like if it were in a state corresponding to a 3D TQFT. We end up with a construction of propagating ststes for parts of the universe and a Hilbert space corresponding to a certain approximation. The construction avoids path integrals, using instead recombination diagrams in a certain tensor category., Comment: 10 pages to appear in Knots and Quantum Gravity ed John Baez, Oxford University Press Latex document

We show that the construction of Ocneanu, which yields 1 for any 4D manifold, is not identical to our construction, which gives different numbers for different manifolds., Comment: 2 pages

We construct a four dimensional topological Quantum Field Theory from a modular tensor category. We complete the proof in the case of SU(2)q at a root of unity. Our construction may be important in the physical interpretation of the Chern Simons state in the Ashtekar variables., Comment: 11 pages

I propose a new mathematical form for the quantum theory of gravity coupled to matter. The motivation is from the connection between CSW TQFT and the Ashtekar variables. I also connect the algebraic structure of TQFT with some thoughts about the interpretation of quantum mechanics in a general relativistic setting., Comment: 9 pages amstex