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Mathematics for Knots, Strings and Elementary Particles Page 1 of 63
Mathematics for Knots, Strings and Elementary Particles
Keith Murray stringfoams@gmail.com
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]”
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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.
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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.
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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.
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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/
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[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/
<<<<<<<<<<<<<<<<<<<<< ! >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
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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: khatsym@inp.nsk.su
How to get Feynman diagrams out of quantum area:
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
Max-Planck-Institut f•ur Gravitationsphysik
Schlaatzweg 1 D-14473 Potsdam
e-mail: loll@aei-potsdam.mpg.de
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
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Discrete Approaches to Quantum Gravity in Four Dimensions
Renate Loll
Max-Planck-Institut f•ur Gravitationsphysik
Schlaatzweg 1
D-14473 Potsdam e-mail: loll@aei-potsdam.mpg.de
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.
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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:elsner@math.uni.wroc.pl b Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland e-mail:pprzytyc@mimuw.edu.pl
Mathematics for Knots, Strings and Elementary Particles Page 36 of 63
And for those further interested, the discrete exterior calculus could be integrated fully with
Regge calculus.
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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.
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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.
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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
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Quark Confinement: The Hard Problem of Hadron Physics
https://arxiv.org/abs/hep-ph/0610365
R. Alkofer Institut f¨ur Physik, Karl-Franzens-Universit¨at, Universit¨atsplatz 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.
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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).
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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 F´isica e Matem´atica,
Dep. MTM, CEP 88.010-970, Florian´opolis, SC, Brazil
3Universidade de S˜ao Paulo, Instituto de F´isica-DFMA
Caixa Postal 66318, 05315-970, S˜ao Paulo, SP, Brazil
Mathematics for Knots, Strings and Elementary Particles Page 43 of 63
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: baez@math.ucr.edu, derek@math.ucr.edu
Alissa S. Crans Department of Mathematics
Loyola Marymount University/The Ohio State University
email: acrans@lmu.edu
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, Rim´anyi 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.
Mathematics for Knots, Strings and Elementary Particles Page 44 of 63
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.
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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.
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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”.
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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
Mathematics for Knots, Strings and Elementary Particles Page 51 of 63
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.
Mathematics for Knots, Strings and Elementary Particles Page 61 of 63
Ahttps://link.springer.com/article/10.1007/s10114-005-0659-5bstract