Sundance Bilson Thompson- Braided Topology and the Emergence of Matter

8/3/2019 Sundance Bilson Thompson- Braided Topology and the Emergence of Matter

http://slidepdf.com/reader/full/sundance-bilson-thompson-braided-topology-and-the-emergence-of-matter 1/31

Introduction

Interactions

Braided Networks

Extracting the standard model

Braids and Twists

Braided topology and the emergence of matter

Sundance Bilson-Thompson

School of Chemistry and PhysicsUniversity of Adelaide

Adelaide, Australia

Sundance Bilson-Thompson Braided topology and the emergence of matter

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

INTRODUCTION

Ideas developed in collaboration/discussion with

Jonathan Hackett

Lou Kauffman

Lee Smolin

Fotini Markopoulou-Kalamara

Isabeau Premont-Schwarz

Yidun Wan


http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

FERMIONS FROM HELONS

Based on the Shupe-Harari models (1979).

Assume three basic components called helons: H +, H −, H 0

Combine into triplets. H + and H − together not allowed.

Possible combinations are;

H + H + H + (e+) H + H + H 0 (qu) H + H 0 H + (qu) H 0 H + H + (qu)

H 0 H 0 H 0 (ν e) H 0 H 0 H + (qd ) H 0 H + H 0 (qd ) H + H 0 H 0 (qd )

H − H − H − (e−) H − H − H 0 (qu) H − H 0 H − (qu) H 0 H − H − (qu)

H 0 H 0 H − (qd ) H 0 H − H 0 (qd ) H − H 0 H 0 (qd )

NB: No anti-neutrino

Permutations define colour.


http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists













http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists













B id d N k

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

RELEVANCE TO LQG

Represent spacetime structure by networks (framed).

Nodes dual to volumes, connections dual to areas.

Twisting and braiding allowed, but these DoFs don’t affectarea and volume operators.

View helons as extended ribbon-like structures

Electric charge of helons is twist of ribbons

Interpret topology of connections between nodes using

helon model


B id d N t k

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

RELEVANCE TO LQG

Represent spacetime structure by networks (framed).

Nodes dual to volumes, connections dual to areas.

Twisting and braiding allowed, but these DoFs don’t affectarea and volume operators.

View helons as extended ribbon-like structures

Electric charge of helons is twist of ribbons

Interpret topology of connections between nodes using

helon model


Braided Networks

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

RELEVANCE TO LQG


Braided Networks

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

FIRST GENERATION FERMIONS

Construct half the 1st generation fermions from +ve andnull twists on a braid


Braided Networks

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists



Construct the anti-particles as mirror images


Braided Networks

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists



Construct the anti-particles as mirror images


I t d tiBraided Networks

http://find/

http://goback/



Introduction

Interactions

Braided Networks


Braids and Twists

BRAID/TWIST EQUIVALENCE

Fermions are defined by braiding (crossings) and twists.

We can flip a node over to exchange twist ←→ crossing.

This induces one specific triple of twists; [+, +,−]


I t d tiBraided Networks

http://find/

http://goback/



Introduction

InteractionsExtracting the standard model

Braids and Twists


We can imagine the node at the "top", and the legs bent"downwards"

Let σ i be the crossing of leg i over leg i+1

Let σ −1

i be the crossing of leg i under leg i+1;

[+, +,−]↔ σ 1 [−,−, +]↔ σ −

11 (1)

[−, +, +]↔ σ 2 [+,−,−]↔ σ −1

2(2)

σ 1, . . . ,σ N −1 are generators of the braid group on N strands.


IntroductionBraided Networks

http://find/

http://goback/



Introduction


Braids and Twists


We can imagine the node at the "top", and the legs bent"downwards"

Let σ i be the crossing of leg i over leg i+1

Let σ −1

i be the crossing of leg i under leg i+1;

[+, +,−]↔ σ 1 [−,−, +]↔ σ −

11 (1)

[−, +, +]↔ σ 2 [+,−,−]↔ σ −1

2(2)

σ 1, . . . ,σ N −1 are generators of the braid group on N strands.



http://find/

http://goback/



Introduction


Braids and Twists

PURE TWIST NUMBERS

We can unravel a braid to obtain its "pure twist form"

The twists define a triplet of numbers [a, b, c]

Braids with the same twist numbers are topologicallyequivalent

Yields the linking numbers for an equivalent link/knot

Only works in 3-valent case



http://find/

http://goback/



Introduction


Braids and Twists

KEEPING IT SIMPLE

Can construct arbitrary braids, in general

This “node flipping” trick combines them into equivalenceclasses

(Hopefully) limits the number and type of fermions

If more complex crossings give higher generations, does

this limit the number of generations?



E i h d d d l

http://find/

http://goback/



Introduction


Braids and Twists

KEEPING IT SIMPLE








E t ti th t d d d l

http://find/

http://goback/




Braids and Twists

KEEPING IT SIMPLE









http://find/

http://goback/




Braids and Twists

KEEPING IT SIMPLE


This “node flipping” trick combines them into equivalence

classes





Introduction Weak Interactions

http://find/

http://goback/



Interactions Bosons

WEAK INTERACTIONS

Braid product links braids top-to-bottom(σ i . . .σ j)∗ (σ k . . .σ l) = σ i . . .σ jσ k . . .σ l

Twists can spread up and down the strands

Hence charges can be exchanged, turning up quarks into

down quarks, electrons into neutrinos, and so on



http://find/

http://goback/



Interactions Bosons

WEAK INTERACTIONS

Braid product links braids top-to-bottom(σ i . . .σ j)∗ (σ k . . .σ l) = σ i . . .σ jσ k . . .σ l

Twists can spread up and down the strands

Hence charges can be exchanged, turning up quarks into

down quarks, electrons into neutrinos, and so on



http://find/

http://goback/



Interactions Bosons

BOSONS

Weak interactions suggest bosons are braids which inducetrivial permutations

Simplest case;

Other braids which induce trivial permutations are

possible, in principle


Introduction

I i

Weak Interactions

B

http://find/

http://goback/



Interactions Bosons

INTERACTIONS IN NETWORKS

Interactions are quite constrained - must not undo all

network structure. Need to create twists in opposing pairs

Braid product requires that braids join ”base to base”

Nodes act like (composite) 4-valent nodesNeed a move that allows opposing twists to form over

4-valent nodes


Introduction

I t ti

Weak Interactions

B

http://find/

http://goback/



Interactions Bosons

INTERACTIONS IN NETWORKS

Interactions are quite constrained - must not undo all

network structure. Need to create twists in opposing pairs

Braid product requires that braids join ”base to base”

Nodes act like (composite) 4-valent nodesNeed a move that allows opposing twists to form over

4-valent nodes


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



Interactions Bosons

3-VALENT OR 4-VALENT?

Yidun Wan developed ideas of braids on 4-valent

networks, using dual Pachner moves (1-4 and 2-3)


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



Interactions Bosons


Braids can be made to interact

Wan’s braids seemed to naturally fall into two categories(fermions and bosons?)

“Node flipping” only reduces braids to pure twist form in the

3-valent case


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



Interactions Bosons


Braids can be made to interact

Wan’s braids seemed to naturally fall into two categories(fermions and bosons?)

“Node flipping” only reduces braids to pure twist form in the

3-valent case


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



Interactions Bosons

COMBINING THE BEST BITS

Can convert twist to crossings and vice-versa

Take four 3-valent nodes in helon model, to make two

4-valent nodes

Twist in one strand becomes crossing in others before we

shrink them down

Obtain a restricted version of Wan’s braids (one strand is

twist-free)


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



te act o s oso s

PROBLEMS THAT KEEP ME AWAKE AT NIGHT

Do we have exotic particle species, or processes?

How do we make interactions occur?

Can we describe Cabbibo mixing, neutrino oscillations?

Origin of inertial mass?

Is there a limit to the number of generations?

Why is the weak interaction so weird?


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



PROBLEMS THAT KEEP ME AWAKE AT NIGHT

Do we have exotic particle species, or processes?

How do we make interactions occur?

Can we describe Cabbibo mixing, neutrino oscillations?

Origin of inertial mass?

Is there a limit to the number of generations?

Why is the weak interaction so weird?


Introduction

Interactions

Weak Interactions

Bosons

http://find/

http://goback/



REFERENCES AND FURTHER READING

Related papers

hep-ph/0503213 A topological model of composite preons

(S. Bilson-Thompson)

hep-th/0603022 QG and the standard model (S.

Bilson-Thompson, F. Markopolou, L. Smolin)arXiv:0804.0037 Particle identifications from symmetries of

braided ribbon network invariants (S. Bilson-Thompson, J.

Hackett, L. Kauffman, L. Smolin)

arXiv:0903.1376 Particle topology, braids, and braided belts (S. Bilson-Thompson, J. Hackett, L. Kauffman)

arXiv:0710.1548 Propogation and interaction of chiral

states in Quantum Gravity (Y. Wan, L. Smolin)


http://find/

http://goback/

Documents

Sundance Bilson Thompson- Braided Topology and the Emergence of Matter