Mathematics for Knots, Strings and Elementary Particles

Mathematics for Knots, Strings and Elementary Particles Page 1 of 63

Mathematics for Knots, Strings and Elementary Particles

Keith Murray [email protected]

October 2017

ABSTRACT

Survey of mathematics for modeling particles and fields based

on a knotted geometry and combine that logic with

representation theory.

Before reading the mathematics notes below (which are mainly a logical flow of Wikipedia

articles), please find the following (available on two websites):

Lorenz and modular flows: a visual introduction

A tangled tale linking lattices, knots, templates, and strange attractors.

http://www.ams.org/publicoutreach/feature-column/fcarc-lorenz

http://www.josleys.com/articles/ams_article/Lorenz3.htm

The last statement of this article is:

“It is known that the the family of curves ct tends to fill the space in a “uniform” way but

the quantitative estimate of the velocity of this phenomenon is equivalent to the famous

Riemann hypothesis, one of the most enticing open questions in mathematics! [15]”

http://www.ams.org/publicoutreach/feature-column/fcarc-lorenz

http://www.josleys.com/articles/ams_article/Lorenz3.htm

http://www.josleys.com/articles/ams_article/Lorenz3.htm#_edn15




Schematic of two strings colliding; each string composed of many

simplicial complexes, and the global curvature of each string causing an

SU(n) gauge interaction and gravitational interaction.


1. Introduction

If elementary particles are themselves some knotted space, or if particles bend space

into knots, or if particles are strings which are braided and closed at the ends (again,

knots), then we can take advantage of the rich mathematical analysis involving knots.

This includes concepts of quark confinement, Chern-Simons theory, knot polynomial

invariants, and categorical considerations. Immediately, the reader is referred to

some modern literature covering these topics in depth. See [1], [2], [3], [4], and [5].

[1] Knots and Physics. Louis H. Kauffman, Series on Knot and Everything – Vol 53,

ISBN 978-981-4383-00-4, ISBN 978-981-4383-01 (pbk).

[2] Gauge Fields, Knots, and Gravity. John Baez, Javier Muniain. Series on Knots

and Everything – Vol 4, 1994. ISBN 9810217293, 9810220340 (pbk).

[3] Functorial Knot Theory: Categories of Tangles, Coherence, Categorical

Deformations, and Topological Invariants. David N. Yetter, Series on Knots and

Everything, Vol. 26. ISBN 981-02-4443-6.

[4] Hopf Algebras. David E Radford, Series on Knots and Everything, Vol. 49. ISBN-

13: 978-981-4335-99-7. ISBN-10: 981-4335-99-1.

[5] Categorical Aspects of Topological Quantum Field Theories. Bruce Bartlett,

Master’s Thesis, Utrecht University, September 2005.

Let’s finish the introduction with Alexander’s Theorem, which provides a conceptual

bridge between knots and strings.

Let’s finish the introduction with Alexander’s Theorem, which provides a conceptual

bridge between knots and strings. We leave to the reader to investigate the proofs.

[yyyy] A Lemma on Systems of Knotted Curves. J. W. Alexander.

http://www.pnas.org/content/9/3/93, Proceedings of the National Academy of

Sciences, Vol. 9, 1923.

Theorem: Alexander’s Theorem. Every knot can be represented as a closed braid.

Corollary. The relationship between knots and braids is not one-to-one; a knot may

have many braid representations.

[zzz] Markov, A. A. "Über die freie Äquivalenz der geschlossenen Zöpfe." Recueil Math.

Moscou 1, 73-78, 1935.

Theorem (Markov’s Theorem). Equivalent braids expressing the same link are

mutually related by successive applications of two types of Markov moves.

http://www.pnas.org/content/9/3/93


Loop Quantum Gravity History

- 1986. Abhay Ashtekar, new variables.

- Lee Smolin and Ted Jacobson. New variables and Wheeler-DeWitt together imply loops.

- Jorge Pullin and Jerzy Lewandowski. The intersection of loops provide more structure for

canonical quantum gravity.

- 1994. Carlo Rovelli and Lee Smolin. LQG operators for area and volume have correspondences

to Roger Penrose’s spin networks.

- Poincaré groups applied to spin networks are analogous to Feynman diagrams.

- Thomas Thiemann. Canonical LQG (defined), which is anomaly-free, and where the Wheeler-

DeWitt equation implies there exists a background-free theory.

- 2008. Spin-foams can show finite ψexp(Q) amplitudes, the classical limit of these amplitudes contain

General Relativity, and it requires a positive cosmological constant which is in agreement with an accelerating

Universe.

Review of Elementary Loop Quantum Gravity

- General Covariance - Invariance of physics laws in any frame of reference. Frequently referred

to as “diffeomorphism invariance”.

- Diffeomorphism – isomorphism in category of smooth manifolds.

- Einstein. Physical entities are located with respect to one another, and not with respect to a

spacetime manifold. (Quantum mechanics on a classical spacetime manifold can be removed by

assuming the classical manifold is interacting quantum mechanically with Feynman diagrams;

no need for Copenhagen Interpretation).

- Ashtekar. (1) An SU(2) Yang-Mills with diffeomorphism invariance, and a vanishing

Hamiltonian.

(2) Ashtekar variables: G(x ) = ∫ G j (x ) λ j (x ) d3x

(3) Gauge field Aia(x) is the configuration variable analogous to qi in ordinary

mechanics; Eai is the conjugate “momentum” (e.g., electric field).

(4) Kia (extrinsic curvature) = Kab Eai / √det a

(5) Recover classical GR if Aia ia – iKia, an SU(2) chiral spin connection, with

covariant derivative Da = ∂a + Aia.

[ insert some formula for spin connection]

Figure 1. Figure 2.


Figure 3. Figure 4.
























2. Background

We suppose first that elementary particles and the gauge forces are artefacts of the

Big Bang (see for example [zzzz paper about 3-d knots and the big bang]). Then, we

are less concerned with the source of energy and geometry, and we can focus on

conceptual models for particles and gauges.

Beginning with Feynman diagrams, we consider spin networks. Let’s note that spin

networks can be related to Feynman diagrams by saying the spin network group is

Poincaré.

[xxxx] See “Spin Networks, Spin Foams, and

Quantum Gravity.” Lecture at the

Minnowbrook Symposium on Space-Time

Structure. John Baez, 1999.

http://math.ucr.edu/home/baez/foam/


[xxxx] See “Spin Networks,

Spin Foams, and Quantum

Gravity.” Lecture at the

Minnowbrook Symposium

on Space-Time Structure.

John Baez, 1999.

http://xxx.lanl.gov/pdf/gr-

qc/9709052v3 or arXiv:gr-

qc/9709052v3 23 Apr 1998

Using all of the above

available knowledge, like

Rovelli and Reisenberger

(see image below), we will

try to demonstrate that

spacetime arises from an ℝ

x S configuration.

<<<<<<< place the following inside the cellular automata article >>>>>>>>>

Strong force strength, and comparing to the other forces:

https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-known-forces-

of-nature/the-strength-of-the-known-forces/

<<<<<<<<<<<<<<<<<<<<< ! >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

http://xxx.lanl.gov/pdf/gr-qc/9709052v3

http://xxx.lanl.gov/pdf/gr-qc/9709052v3

https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-known-forces-of-nature/the-strength-of-the-known-forces/

https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-known-forces-of-nature/the-strength-of-the-known-forces/


3. Quantum Areas

Area expectation values in quantum area Regge calculus V.M.Khatsymovsky, Budker Institute of Nuclear Physics, Novosibirsk, 630090, Russia

E-mail address: [email protected]

How to get Feynman diagrams out of quantum area:


Discrete Approaches to Quantum Gravity in Four Dimensions

Renate Loll

Max-Planck-Institut f•ur Gravitationsphysik

Schlaatzweg 1 D-14473 Potsdam

e-mail: [email protected]

Published 15 Dec 1998

www.livingreviews.org/Articles/Volume1/1998-13loll

Living Reviews in Relativity Published by the Max-Planck-Institute for Gravitational Physics Albert Einstein Institute,

Potsdam, Germany


Discrete Approaches to Quantum Gravity in Four Dimensions

Renate Loll

Max-Planck-Institut f•ur Gravitationsphysik

Schlaatzweg 1

D-14473 Potsdam e-mail: [email protected]

Published 15 Dec 1998

www.livingreviews.org/Articles/Volume1/1998-13loll

Living Reviews in Relativity

Published by the Max-Planck-Institute for Gravitational Physics Albert Einstein Institute, Potsdam, Germany

Just to give the reader an idea of the current sophistication for those not actively researching

in the field, the following authors have expositions which extend simplexes into categories.


Square complexes and simplicial nonpositive curvature Tomasz Elsnera_ and Piotr

Przytyckiby a Mathematical Institute, University of Wrocław, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland e-mail:[email protected] b Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland e-mail:[email protected]


And for those further interested, the discrete exterior calculus could be integrated fully with

Regge calculus.


DISCRETE EXTERIOR CALCULUS MATHIEU DESBRUN, ANIL N. HIRANI, MELVIN LEOK, AND JERROLD E. MARSDEN Abstract. We present a theory and applications of discrete exterior calculus on simplicial

complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space.

Our theory includes not only discrete differential forms but also discrete vector fields and the

operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous

attempts at discrete exterior calculus have addressed only differential forms. We also introduce

the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in

this field has been well understood, but previous researchers have used barycentric

subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete exterior calculus that admits both vector fields and forms.


4. Categories, TQFT, and Knots

The file I have is named “monoidal cats and Feynman diagrams”, but it is the following

paper by Bruce Bartlett.

[X] Categorical Aspects of Topological Quantum Field Theories, Bruce Bartlett,

Masters Thesis, Utrecht University, September 2005.


Let’s examine the following regarding the “flux tube” profile for a three quark

interaction in QCD ([7] H. Suganuma et al. arXiv:hep-lat (2003) 0312031.)

This statement was originally found from a paper entitled “Lattice Gauge Theory””

PHY 509 Final Presentation: Lattice Gauge Theory, C.M. Melby-Thompson, A.E. Miller, W.W.

Wong, January 2004


Quark Confinement: The Hard Problem of Hadron Physics

https://arxiv.org/abs/hep-ph/0610365

R. Alkofer Institut für Physik, Karl-Franzens-Universität, Universitätsplatz 5, A-8010

Graz, Austria

J. Greensite Physics and Astronomy Department, San Francisco State University, San

Francisco, CA 94132, USA

Abstract. We give a brief overview of the problem of quark confinement in hadronic

physics, and outline a few of the suggested explanations of the confining force.


Hopf Algebras, Series on Knots and Everything — Vol. 49, David E Radford, ISBN-13: 978-981-

4335-99-7 (hardcover : alk. paper), ISBN-10: 981-4335-99-1 (hardcover : alk. paper).


Quantum Topology Change in (2 + 1)d

A.P. Balachandran1, E. Batista2, I.P. Costa e Silva3 and P. Teotonio-Sobrinho3

1Department of Physics,Syracuse University Syracuse, NY 13244-1130, USA

2Universidade Federal de Santa Catarina, Centro de Física e Matemática,

Dep. MTM, CEP 88.010-970, Florianópolis, SC, Brazil

3Universidade de São Paulo, Instituto de Física-DFMA

Caixa Postal 66318, 05315-970, São Paulo, SP, Brazil


1. Strings and Knots

Exotic Statistics for Strings in 4d BF Theory

John C. Baez and Derek K. Wise Department of Mathematics

University of California

Riverside, CA 92521, USA

email: [email protected], [email protected]

Alissa S. Crans Department of Mathematics

Loyola Marymount University/The Ohio State University

email: [email protected]

April 20, 2006

Abstract

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 copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can give a pre-

sentation of the whole loop braid group, which turns out to be isomorphic to the ‘braid

permutation group’ of Fenn, Rimányi and Rourke. In the context 4d BF theory this group

naturally acts on the moduli space of flat G-bundles on the complement of a collection of

unlinked unknotted circles in R3. When G is unimodular, this gives a unitary representation of the loop braid group. We also discuss ‘quandle field theory’, in which the gauge group G is

replaced by a quandle.

Building (mathematics) From Wikipedia, the free encyclopedia

(Redirected from Building theory)

In mathematics, a building (also Tits building, Bruhat–Tits building, named after François Bruhat and Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type, the theory has also been used to study the geometry and topology of homogeneous spaces of p-adic Lie groups and their discrete subgroups of symmetries, in the same way that trees have been used to study free groups.

https://en.wikipedia.org/w/index.php?title=Building_theory&redirect=no

https://en.wikipedia.org/wiki/Mathematics

https://en.wikipedia.org/wiki/Fran%C3%A7ois_Bruhat

https://en.wikipedia.org/wiki/Fran%C3%A7ois_Bruhat

https://en.wikipedia.org/wiki/Jacques_Tits

https://en.wikipedia.org/wiki/Flag_manifold

https://en.wikipedia.org/wiki/Projective_plane

https://en.wikipedia.org/wiki/Riemannian_symmetric_space

https://en.wikipedia.org/wiki/Riemannian_symmetric_space

https://en.wikipedia.org/wiki/Group_of_Lie_type

https://en.wikipedia.org/wiki/Group_of_Lie_type

https://en.wikipedia.org/wiki/Homogeneous_space

https://en.wikipedia.org/wiki/P-adic_Lie_group

https://en.wikipedia.org/wiki/Tree_(mathematics)

https://en.wikipedia.org/wiki/Free_group


Lie bialgebra From Wikipedia, the free encyclopedia

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.

It is a bialgebra where the comultiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.

They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson-Lie group.

Lie bialgebras occur naturally in the study of the Yang-Baxter equations.


https://en.wikipedia.org/wiki/Bialgebra

https://en.wikipedia.org/wiki/Lie_algebra


https://en.wikipedia.org/wiki/Lie_coalgebra

https://en.wikipedia.org/wiki/Bialgebra

https://en.wikipedia.org/wiki/Comultiplication

https://en.wikipedia.org/wiki/Self-complementary_graph

https://en.wikipedia.org/wiki/Jacobi_identity


https://en.wikipedia.org/w/index.php?title=Cocycle_(algebraic_topology)&action=edit&redlink=1


https://en.wikipedia.org/wiki/Poisson-Lie_group

https://en.wikipedia.org/wiki/Yang-Baxter_equation



Modular group (see Hecke group section

of Modular Group article and golden ratio

article, below)

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.


https://en.wikipedia.org/wiki/Projective_linear_group

https://en.wikipedia.org/wiki/Fractional_linear_transformation

https://en.wikipedia.org/wiki/Moduli_space

https://en.wikipedia.org/wiki/Modular_arithmetic




Note the Golden Ratio is present in Hecke

groups, above. Then see the articles in the

pages following below, for Braid Group,

Alexander’s Theorem, and the “Interactions

of Braid Groups”.


See Hecke Group from Modular Group article

Braid group In mathematics, the braid group on n strands (denoted Bn), also known as the Artin braid

group,[1] is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's


https://en.wikipedia.org/wiki/Braid_group#cite_note-1

https://en.wikipedia.org/wiki/Braid_theory

https://en.wikipedia.org/wiki/Ambient_isotopy

https://en.wikipedia.org/wiki/Ambient_isotopy

https://en.wikipedia.org/wiki/Group_operation

https://en.wikipedia.org/wiki/Braid_group#Introduction

https://en.wikipedia.org/wiki/Knot_theory

https://en.wikipedia.org/wiki/Alexander%27s_theorem

https://en.wikipedia.org/wiki/Mathematical_physics


canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.[2]

If particles are knots, or if particles bend space into knots, or if particles are strings which are

braided and tied together at the ends, then we can take advantage of the rich mathematical

analysis involving knots and quark confinement, Chern-Simons theory, and knot invariants.

Immediately, the reader is referred to some modern literature covering these topics in depth.

[1] Knots and Physics. Louis H. Kauffman, Series on Knot and Everything – Vol 53, ISBN 978-

981-4383-00-4, ISBN 978-981-4383-01 (pbk).

[2] Gauge Fields, Knots, and Gravity. John Baez, Javier Muniain. Series on Knots and

Everything – Vol 4, 1994. ISBN 9810217293, 9810220340 (pbk).

Alexander's theorem From Wikipedia, the free encyclopedia

In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid. The theorem is named after James Waddell Alexander II, who published its proof in 1923.

Braids were first considered as a tool of knot theory by Alexander. His theorem gives a positive answer to the question Is it always possible to transform a given knot into a closed braid? A good construction example is found on page 130 of Adams's The Knot book (see ref. below). However, the correspondence between knots and braids is clearly not one-to-one: a knot may have many braid representations. For example, conjugate braids yield equivalent knots. This leads to a second fundamental question: which closed braids represent the same knot type?

That question is addressed in Markov's theorem, which gives ‘moves’ relating any two closed braids

that represent the same knot.

https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation

https://en.wikipedia.org/wiki/Braid_group#Basic_properties

https://en.wikipedia.org/wiki/Braid_group#Basic_properties

https://en.wikipedia.org/wiki/Monodromy

https://en.wikipedia.org/wiki/Braid_group#cite_note-2

https://en.wikipedia.org/wiki/Knot_(mathematics)

https://en.wikipedia.org/wiki/Link_(knot_theory)

https://en.wikipedia.org/wiki/Braid_theory

https://en.wikipedia.org/wiki/James_Waddell_Alexander_II

https://en.wikipedia.org/wiki/Knot_theory

https://en.wikipedia.org/wiki/Bijection











Ahttps://link.springer.com/article/10.1007/s10114-005-0659-5bstract



https://aip.scitation.org/doi/10.1063/1.2209771

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Mathematics for Knots, Strings and Elementary Particles