Embedding the Bilson-Thompson Model in a LQG-like framework by Deepak Vaid

This week I have been thinking and reading about Xiao-Gang Wen‘ s work on quantum Boson systems and his work on condensed matter and the origins of fermions and photons. I would see it going something like Quantum Tetrahedron/spin network  used via Group field theory to give an  emergent spacetime consisting of a Bose-Einstein Condensate, then using the string net interpretation of spin networks,  bosons and fermions could arise as theorised by Xiao-Gang Wen in his work.

It is known that for string nets labelled by the positive integers, string-nets are the spin networks studied in loop quantum gravity. This has led to the proposal by Levin and Wen,and Smolin, Markopoulou and Konopka[ that loop quantum gravity’s spin networks can give rise to the standard model of particle physics through this mechanism, along with fermi statistics and gauge interactions. To date, a rigorous derivation from LQG‘s spin networks to Levin and Wen’s spin lattice has yet to be done and one idea for doing  so is quantum graphity and in a recent paper, Tomasz Konopka, Fotini Markopoulou and Simone Severini argued that there are some similarities to spin networks that gives rise to U(1) gauge charge and electrons in the string net mechanism.

This is really encouraging because others are working on this idea, also quantum graphity is a fun theory to work with so now I have more excuse to work on that area some more.I’ll be writting about this over the coming weeks,

In the meantime I have been looking at other attempts to add matter to LQG.  One popular model that fits my interests is Sundance Bilson-Thompson‘s work on braids. Here I review an interesting paper by Deepvak Vaid about this.

Introduction

LQG  remains a few steps away from giving a coherent description of quantum gravity which naturally incorporates the particles of the SM. Any notion of particles as topological structures should find a natural home in LQG and should have a realization as particles of a QFT obtained by coarse-graining over the microscopic degrees of freedom. Sundance Bilson-Thompson proposed proposed such a model  for the particles of the Standard Model or at least those in the first generation: the leptons consisting of the electron, electron-neutrino and the up and down quarks and the gauge bosons could be given a unified representation in terms of the irreducible elements of the first non-trivial braid group (B3). He showed that the irreducible elements of  B3 can be put into one-to-one correspondence with the first generation of the SM particles in a very natural manner.

Topological considerations

LQG is derived from a 3+1 decomposition of the Einstein action when formulated in connection and tetrad variables yielding the Hamiltonian H of General Relativity  without matter. This Hamiltonian is found to be the sum of constraints as would be expected to be the case for a diffeomorphism invariant theory called the Gauss Diffeomorphism and Hamiltonian constraints. These constraints are functions of the connection Aia and triad eai living on a 3-manifold which is a slice of the complete
four dimensional spacetime on which the action was originally defined. In the general case can have internal boundaries, topologically two spheres S² and/or external ones corresponding to a cosmological horizon.

This will be our arena: A 3-dimensional manifold with arbitrary number of internal boundaries which are topologically S² and an outer boundary which would correspond to a cosmological, deSitter, horizon.

In LQG as it stands today, there is no natural way to incorporate the particles of the Standard Model. Bilson-Thompson’s  of model may be a way of overcoming this obstacle. In this paper it is shown how this can be accomplished. It is here that topology comes into play. The internal boundaries used in this paper replace the usual vertices of spin networks. The edges of graphs are replaced by framed ribbons which fit neatly onto the surfaces of the internal boundaries.

It also turns out that the braiding which is essential to describe the SM particles, emerges naturally on consideration of the action of discrete symmetries of the internal boundaries.

We start with showing how the gravitational action on manifolds with boundaries decomposes into a bulk and boundary term, the latter being identical to Chern-Simons theory.

Consider a 4-manifold M of signature +1 with holes. The boundaries of these holes are topologically S² R. On M, there lives a gravitational connection AIμ a one-form which takes
values in the sl(2,C) lie-algebra and a tetrad eμI. In terms of these variables the action for GR can be written as:

bilequ1

The curvature is given by:

bilequ2

where g is the gauge coupling. Then we have:

bilequ3

this gives

bilequ4

Doing an integration by parts in the above
equation yields:

bilequ5

The boundary term looks similar to the
action for Chern-Simons theory. This similarity
can be made more apparent by the inclusion of
the cosmological term in the action

bilequ6

The last term must be evaluated in the
bulk and also on the boundary. Consequently
SGRΛ takes the form:

bilequ7

where the last term is corresponds to the Chern-Simons theory of spinors living on the boundary.

The physical picture of matter we have is that of a sea of defects living on many internal boundaries which are floating in the bulk manifold. In this case we have multiple disconnected
boundaries and the boundary action is more appropriately
written as:

bilequ10

That is, the sum of the Chern-Simons action on all the boundaries in M.

This  leads to the following picture of pre-geometry: an abstract 3 dimensional topological space with a fermionic sea, which undergoes spontaneous symmetry breaking leading to the formation of a condensate, where the natural interpretation of the resulting quasi-particles is that of quanta of geometry. The resulting situation is depicted below:

bilfig1We are left with a gas-like state of geometry. Such a state is unlikely to yield an approximate continuum geometry because we have no natural measure on this space of tetrahedra. What we require is a graphlike picture as in the spin-network construction.

For this purpose the tetrahedra must somehow be connected to each other. In the process of trying to discover the right way to glue these tetrahedra together we are naturally led to the emergence of
the Bilson-Thompson model.

Braids on inner boundaries

It has been rigorously shown in that in canonical LQG, the state
space of the punctures on the surface of an isolated horizon ,i.e. inner boundary, corresponds to that of Chern-Simons theory. The punctures living on the isolated horizon can be considered to be fermionic degrees of freedom which undergo condensation to form quanta of geometry. The interpretation of topological punctures, characterized by a deficit angle θ, as particles is consistent with
the results of and is also in line with various proposals for realizing matter as topological structures in QG.

Most viable schemes for constructing a theory of quantum geometry exploit the partition function approach. This involves replacing the bulk continuous manifold by a discrete one constructed from simplices whose faces, edges and vertices are labelled by various spins and operators whose values determine the quantum state of geometry for each simplex.

Focussing  on a hole in a three dimensional manifold, whose surface S² is punctured by lines carrying gravitational flux. The smallest non-trivial triangulation of a 2-sphere requires four triangles – giving us a tetrahedral approximation to the surface.

bilfig2

We imagine each face our tetrahedron is pierced by one flux line, endowing that face with an area. The enclosed volume represents the smallest irreducible atom of geometry

 

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