This thesis studies matter emergent as topological excitations of
quantum geometry in quantum gravity models. In these models, states
are framed four-valent spin networks embedded in a topological three
manifold, and the local evolution moves are dual Pachner moves.
We first formulate our theory of embedded framed four-valent spin
networks by proposing a new graphic calculus of these networks. With
this graphic calculus, we study the equivalence classes and the
evolution of these networks, and find what we call 3-strand braids,
as topological excitations of embedded four-valent spin networks.
Each 3-strand braid consists of two nodes that share three edges
that may or may not be braided and twisted. The twists happen to be
in units of 1/3. Under certain stability condition, some 3-strand
braids are stable.
Stable braids have rich dynamics encoded in our theory by dual
Pachner moves. Firstly, all stable braids can propagate as induced
by the expansion and contraction of other regions of their host spin
network under evolution. Some braids can also propagate actively, in
the sense that they can exchange places with substructures adjacent
to them in the graph under the local evolution moves. Secondly, two
adjacent braids may have a direct interaction: they merge under the
evolution moves to form a new braid if one of them falls into a
class called actively interacting braids. The reverse of a direct
interaction may happen too, through which a braid decays to another
braid by emitting an actively interacting braid. Thirdly, two
neighboring braids may exchange a virtual actively interacting braid
and become two different braids, in what is called an exchange
interaction. Braid dynamics implies an analogue between actively
interacting braids and bosons.
We also invent a novel algebraic formalism for stable braids. With
this new tool, we derive conservation laws from interactions of the
braid excitations of spin networks. We show that actively
interacting braids form a noncommutative algebra under direction
interaction. Each actively interacting braid also behaves like a
morphism on non-actively interacting braids. These findings
reinforce the analogue between actively interacting braids and
bosons.
Another important discovery is that stable braids admit seven, and
only seven, discrete transformations that uniquely correspond to
analogues of C, P, T, and their products. Along with this
finding, a braid's electric charge appears to be a function of a
conserved quantity, effective twist, of the braids, and thus is
quantized in units of 1/3. In addition, each $CPT$-multiplet of
actively interacting braids has a unique, characteristic
non-negative integer. Braid interactions turn out to be invariant
under C, P, and T.
Finally, we present an effective description, based on Feynman
diagrams, of braid dynamics. This language manifests the analogue
between actively interacting braids and bosons, as the topological
conservation laws permit them to be singly created and destroyed and
as exchanges of these excitations give rise to interactions between
braids that are charged under the topological conservation rules.
Additionally, we find a constraint on probability amplitudes of
braid interactions.
We discuss some subtleties, open issues, future directions, and work
in progress at the end.
| Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/4550 |
| Date | 01 August 2009 |
| Creators | Wan, Yidun |
| Source Sets | University of Waterloo Electronic Theses Repository |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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