Knotted String Foams and Mathematical Physics Keith Murray 08 September 2017 ABSTRACT We attempt to show that there is sufficient mathematical evidence to consider knots as “foundational” particles in mathematical physics, by presenting relationships to the mathematics of General Relativity and Yang-Mills. 1. Introduction Shortly after the onset of the millennium, I was wrestling with a direct quantization of the Einstein and stress-energy tensors which would produce square-integrable wave functions. While I was able to find some results, the interpretation of the wave functions were ambiguous and in limiting cases the equations resembled the Klein-Gordon equations. This caused me to think that I had rediscovered nothing more than an alternative expression for a freely moving photon traveling in curved spacetime. Furthermore, were I to embark on the arduous journey of generalizing these expressions, I may arrive merely at an analog of the Dirac equation on curved spacetime. Rather than pursuing that path further, I decided it was probably a better idea to investigate what generalizations could be hypothesized for modern objects in mathematical physics. During my survey of objects in mathematical physics, I frequently encountered a generic and recurring theme for which I invoked symbolic logic out of necessity. This was simply a symbolic string: 퐓 = ∅ + 퐇 + 퐀 (1) Here, T can be a “theory” in the logical or model theoretical sense. The ∅ empty set symbol may literally represent the empty set, the concept of zero, an arbitrarily small epsilon in mathematical analysis, or any other null character which the theory may require. H may be a vector space over a field, module over a ring, or any tensored product thereof; by logical extension, H could be any object we may find in any category. And A is a connection (gauge) or some other object required to satisfy the axioms of the theory with respect to H. Finally, “+” is the familiar plus sign, or Abelian operator; but more generally, it may represent any commutative binary operation, e.g., triples (+, H, A) = (+, A, H). While this was all fine, in terms of mathematical physics my theoretical boat was slipping further from the safety of the coastline and into the white-capped seas of mathematical logic. To escape this, I then decided to focus on constructing a general framework for elementary particles that would be in alignment with Eqn. 1. The solution my mind produced: a school of fish could be an excellent toy model for elementary particles. A school of fish is driven by swarm behavior, so immediately we have some nice concepts summarized in a few words: discrete, stochastic, optimization, Markov chains, hitting times. Perhaps spacetime or simply space if it is discrete – and the Loop Quantum Gravity folks will remind us that there is no a priori reason to believe space is discrete, but what the heck – is distinguishable to a test particle because of the way swarms interact. For example, a flock of birds meets a swarm of insects, or a shoal of fish meets a swarm of bottle-nosed dolphins. These different types of “particles” will interact. However, then I thought that although swarms (like a bait ball of fish) can have a very recognizable global geometry (a sphere), in the end swarm equations are only behavioral functions. Swarm models of behavior follow very precise codes, and can be simulated using computer programming languages. So while these emulate the sort of behavior one may like to think about for a quantum geometric model of particles, straight away we know that the Lie groups are not required to model swarm behavior. But Lie groups are required to model the interactions of elementary particles. How to reconcile these thoughts? We do not know what Standard Model point particles are, but the math works in the laboratory. We do not know what strings are, but there is some ancestral relationship with point particle laboratory physics. We do not know what spin foams are, but they can be used to recover General Relativity. We do not know what particles are, really, but we know that particles cannot be modelled by swarm behavior alone. Yet swarms are nice because they have dynamics, they are discrete, and the global geometry is spherical. If only there was a way to connect swarms with Lie algebras, like an embedding! But that yet appears insufficient, because although we can embed swarms in tensored vector spaces or in GL(n)-valued spaces, we still have only sort-of spherical geometry with a swarm, and spherical geometry is only useful for a limited number of physical scenarios – both classical and quantum. What we really need is an idea that includes more general geometric structure, and it would be convenient if there were some pre-established mathematical concepts relating more general geometric structures to physics, in order to avoid reinventing lots of math. So, very close to the swarm idea is the knot! Knots themselves may be a great way to think about particles, because they have holonomy, inherent relationships to algebra (Jones polynomial, for example), and if they are framed – recall framed knots are more like tubes or rubber bands instead of 1-dimensional strings – then knots could still have the “school of fish” swimming inside the tubes or along frames with a certain periodicity as these “fish” swim around the race track. Also, since knots have holonomy there is a notion of connection, referring back to our simple statement in Eqn. 1, of a logical mathematical physics “theory”. Admittedly, since we do not know much about treating knots themselves as elementary particles, let us rather say we shall treat knots as “foundational” particles, whatever that means, and proceed to lay some groundwork. 2. Knots as Foundational Particles <<under construction>> 3. Categorical Physics To keep our discussion in order, let us review some basic properties of analysis and establish some of the techniques we will be using. [For example, the fundamental group is a functor pi1 : Top -> Grp:, A History of n-Categorical Physics, DRAFT VERSION!, John C. Baez] Example: the fundamental group, with F, G, H other functors. Categorical String Theory [see Categorical Aspects of Topological QFT, Bruce Bartlett: 3.4.2 TopString, 4.1.3 Triangulated spaces, 4.1.2 The connection as a functor] Relations to Topos, Sheaves, Sites [see Baez, Gauge Fields, p 333 – 334] Relationships to Model Theory and Categorical Logic From Model Theory: Finitary Models can be [First-Order, Categorical Theories]. First order theories cannot be categorical, i.e., they cannot be described up to isomorphism, unless that model is finite. Reworded: given a finite model, a first order theory may be categorical. Two famous model theoretic theorems deal with the weaker notion of K-categoricity for a cardinal K. A theory T is called K-categorical if any two models of T that are of cardinality K are isomorphic. For a complete first-order theory T in a finite or countable signature the following conditions are equivalent (Ryll-Nardzewski Theorem): - T is Aleph0-categorical - For every natural number n, the Stone Space Sn(T) is finite - For every natural number n, the number of formulas f(x1, x2, x3, … , xn) in n free variables, up to equivalence modulo T, is finite. ℎ(푥) (5) 4. TOE Unification Model Let’s start with a note from Wikipedia’s GUT article: Let’s compare [Quantum Theory from Quantum Gravity, Fotini Markopoulou∗ and Lee Smolin†] the abstract from Lee Smolin to Mark Hadley. Winding numbers, Jones Polynomial, Witten, and Standard Model Spacetime Algebras and Lie Algebras as distinct but related entities. SU(3)SU(2)U(1) can be found. Spacetime algebra: [CHERN-SIMONS GAUGE THEORY AS A STRING THEORY, Edward Witten, ⋆School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540] [Discrete Approaches to Quantum Gravity in Four Dimensions, Renate Loll, Max-Planck-Institut fur Gravitationsphysik, 3 Quantum Regge Calculus, and 4. Dynamical triangulations] [4 Dynamical triangulations . .23 4.1 Introduction . .. 23 4.2 Path integral for dynamical triangulations . 23 4.3 Existence of an exponential bound? . 24 4.4 Performing the state sum . .. 24 4.5 The phase structure . 25 4.6 Evidence for a second-order transition? . 26 4.7 Influence of the measure . .. 26 4.8 Higher-derivative terms . 27 4.9 Coupling to matter fields . .. 27 4.10 Non-spherical lattices . 28 4.11 Singular congurations . 28 4.12 Renormalization group . .. 28 4.13 Exploring geometric properties . 28 4.14 Two-point functions . 29 4.15 Summary . .. 30] [QUASIALGEBRA STRUCTURE OF THE OCTONIONS, Helena Albuquerque, Shahn Majid, December, 1997 – February, 1998; Abstract We show that the octonions are a twisting of the group algebra of Z2 × Z2 × Z2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle.] Octonion X-product and Octonion E8 Lattices, Geoffrey Dixon Baez, Octonions and relationship to E8. [A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold Sarita Rosenstock1, a) and James Owen Weatherall1, b)] Spacetime Algebras and Lie Algebras as distinct but related entities. SU(3)SU(2)U(1) can be found. Brief Review: Spacetime algebra: Clifford Algebra and Discrete Manifolds, and Clifford Algebra Approach to Curvature Extending Finite Element Exterior Calculus to Finite Element Clifford Algebra. The objective of including this in this survey is to think of using discrete forms as manifolds, so the two may be used interchangeably. From http://math.arizona.edu/~agillette/research/pd11talk.pdf. Some Clifford/Spacetime calculations arriving at curvature over continuous manifolds are given in: In: International Journal of Theoretical Physics Vol.