Gluon Phenomenology and a Linear Topos by Marni Dee Sheppeard A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at the University of Canterbury 2007 Disclaimer I hereby declare that this thesis is entirely my own creation, based on work done with my collaborator Michael Rios and under the supervision of Dr William Joyce (chapters 5 and 6). No part of the thesis will be used towards a qualifi- cation at any other institution. Acknowledgements There are many people who have helped me through the years of this work. I would like to thank them all, but I can only mention a few. First and fore- most, I must thank my new collaborator Michael Rios, whom I am very much looking forward to meeting one day. Prof. Ross Street suggested I talk about M-theory and certainly helped me steer things in the right direction. I am also indebted to my secondary supervisors, Dr David Wiltshire for his long term support, and Prof. Louis Crane for giving me the great opportunity to work in an exciting field and for his immense generosity. There were many discussions and seminars with other University of Canter- bury category theorists and constructivists, including Philip Catton, Douglas Bridges, Jeremy Martin, Ewan Orr and Sonja Rendell. Sonja’s courage and good nature on a certain occasion will not be forgotten. A big thank you to Ewan for proof–reading, and a special thank you to my long–suffering office mates, Benedict Carter, Ben Leith and Alex Nielsen, along with Kahae Han. Belinda Mellish has been a great friend, as has Prof. Peter Jarvis. Undying thanks to those on the rescue teams that helped pull Sonja and me off Mt Iso- bel in 2003, and to the guys who hauled me out of a crevasse on the Grosser Aletsch glacier earlier that year. For their wonderful hospitality and kindness I must mention Gill Evans, Rob Goldblatt, Matt Visser, Ross Street, John Barrett, Louise Gauld, Caterine Crane, the Perimeter Institute in Waterloo, the CNRS in Montpellier, Tao Mo, the Weerasekara family of Christchurch, and my own family. Last, but not least, I want to thank my friends Kerie and Allan for believing in me. But when I said that nothing had been done I erred in one important matter. We had definitely committed ourselves and were half-way out of our ruts. We had put down our passage money - booked a sailing to Bombay. This may sound too simple, but is great in consequence. Until one is committed, there is hesitancy, the chance to draw back, always ineffectiveness. Concerning all acts of initiative (and creation), there is one elementary truth, the ignorance of which kills countless ideas and splendid plans: that the moment one definitely commits oneself, then Providence moves too. All sorts of things occur to help one that would never otherwise have occurred. A whole stream of events issues from the decision, raising in one’s favour all manner of unforeseen incidents and meetings and material assistance, which no man could have dreamt would have come his way. W. H. Murray, The Scottish Himalayan Expedition, 1951 Contents 1 Introduction 4 2 Basic Categories and Toposes 11 2.1 Whatisacategory?......................... 11 2.2 PullbackLemmasandSheaves. 15 2.3 TwoDimensionalStructures . 18 2.4 Internalisation............................ 22 2.5 Whatisatopos? .......................... 27 2.6 TheCombinatoricsofOperads. 28 3 MHV Amplitudes and Ribbon Graphs 34 3.1 ClassicalGravity .......................... 35 3.2 MHVAmplitudes.......................... 40 3.3 RibbonGraphM-Theory. 41 4 Towards Gluon Amplitudes 45 4.1 RP1 ModuliSpaces ......................... 46 4.2 PhysicalAmplitudes . 51 5 Characterisation and Toposes 55 5.1 MonicCharacterisation. 56 5.2 QuantumStates........................... 58 5.3 Monic Cospan Characterisation . 59 5.4 DefinitionofaQuantumTopos . 66 5.5 MonoidalStructure . 68 6 The Logic of a Linear Quantum Topos 73 6.1 Lattices ............................... 74 6.2 SubobjectsandMeets . 75 6.3 Conjunction and Monoidal Conjunction . 76 6.4 Implication ............................. 81 6.5 Universal Quantification and Bra States . 83 6.6 InternalSemantics . 85 6.7 TheFundamentalTheorem . 87 6.8 Truth and Falsity for Vector Spaces . 95 6.9 Properties of Complementation . 96 6.10 QuantumLattices. 98 6.11Disjunction .............................102 6.12 Example: Vector Spaces . 104 6.13 TheInternalLanguage . .107 7 Higher Categorical Directions 111 7.1 Strict Cubical Tricategories . 111 7.2 Tricategories in Physics . 113 8 Conclusions 118 A Bicategories and Tricategories 120 A.1 Bicategories .............................120 A.2 Tricategories.............................124 Abstract In thinking about quantum causality one would like to approach rigorous QFT from outside the perspective of QFT, which one expects to recover only in a specific physical domain of quantum gravity. This thesis considers issues in causality using Category Theory, and their application to field theoretic ob- servables. It appears that an abstract categorical Machian principle of duality for a ribbon graph calculus has the potential to incorporate the recent calcula- tion of particle rest masses by Brannen [Bra], as well as the Bilson-Thompson characterisation of the particles of the Standard Model [BT]. This thesis shows how Veneziano n-point functions may be recovered in such a framework, using cohomological techniques inspired by twistor theory and recent MHV techniques introduced in [Wit04a]. This distinct approach fits into a rich framework of higher operads [Bata], leaving room for a gener- alisation to other physical amplitudes. The utility of operads raises the question of a categorical description for the underlying physical logic. We need to consider quantum analogues of a topos. Grothendieck’s concept of a topos [Gro] is a genuine extension of the notion of a space that incorporates a logic internal to itself. Conventional quantum logic has yet to be put into a form of equal utility, although its logic has been formulated in category theoretic terms [Coe]. Axioms for a quantum topos are given in this thesis, in terms of braided monoidal categories. The associated logic is analysed and, in particular, ele- ments of linear vector space logic are shown to be recovered. The usefulness of doing so for ordinary quantum computation was made apparent recently by Coecke et al [Pava]. Vector spaces underly every notion of algebra, and a new perspective on it is therefore useful. The concept of state vector is also readdressed in the language of tricategories. 3 1 Introduction “It is surprising to be told that a trivial system suffers from in- tractable infinities.” E. Witten [Wit88] Physical theory is currently undergoing a revolution of thought on a par with that of the Copernican revolution. For a long time the incompatibility of the successful Standard Model and the theory of General Relativity has gen- erated countless attempts at modifying the principles of one or the other, or both. Now experimental anomalies in the standard picture have forced us to search even further for completely new fundamental principles. On the math- ematical side, real progress has been made on formulating a rigorous language for quantum field theory. But how do such techniques arise in a quantum gravitational context? Without a clearer understanding of new physics, it is uncertain how even the most advanced techniques could be used to predict quantitative results for upcoming experiments such as the LHC. Heisenberg pointed out that particles were not fundamental because every particle in some sense contained all others [ed.73]. In viewing particles as building block systems this is no doubt true. However, the particles of QFT do represent fundamental kinds of proposition. Thus logic and algebra, rather than algebra alone, are necessary for a clear description of both the Standard Model and the new theory of quantum gravity. In mathematics, only category theory can combine these two disciplines. Category theory is also about geometry. The concept of point, or spacetime event, is greatly abstracted from the idea of a point as an element of the set Rn, the model for manifolds. In quantum gravity one tends to view the classical spacetime as an emergent phenomenon arising from the combination of large numbers of basic gravitational states. Unfortunately, this loose viewpoint is borrowed from the meaning of observable in particle physics and, by itself, adds nothing to our understanding of mass. On the face of it, rest mass is a classical concept, even as it characterises the inertia of a charged quantum particle as measured in a mass spectrometer, because the mass depends on the curvature of the particle’s path as it travels at a known small velocity in a magnetic field. This requires a background 4 template of rods and clocks to define a laboratory frame. However, mass can potentially be viewed as a quantum observable not unlike spin. Recall that the Stern-Gerlach experiment takes a beam of electrons and observes its twofold splitting through a magnetic field, defining spin. This beam conveniently con- tains only electrons, and not muons or tau particles. But we could imagine a beam of (spin up) electrons, muons and tau particles which is split threefold by the particle masses. In his Clifford algebra density matrix approach to QFT, Brannen [Bra] has constructed mass operators whose eigenvalues yield the charged lepton masses to within experimental precision. These operators rely at present on a small splitting parameter whose theoretical origin is not yet understood, but they clearly indicate the potential for a first principles derivation of particle masses within that framework, which is very distinct from string theory or LQG or other popular approaches to gravity. A test of any new approach is its ability to explain the recent successes of the MHV diagram technique and its spinoffs [Wit04a][Svr]. The physical relevance is unquestionable. It may be pointed out [Svr], for instance, that the multijet production at the LHC will be dominated by tree level QCD scattering, and at tree level the supersymmetric Yang-Mills theory really does behave like QCD.