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

 

Polyhedral quantum geometry

This week I’ve been reading Spinfoams: Simplicity Constraints and Correlation Functions PhD thesis by Ding. I’m reviewing the  section on Polyhedral quantum geometry  which is relevant to my work on the quantum tetrahedron.

Consider the truncation of the LQG Hilbert space HLQG and restrict ourself to a single graph Hilbert space H(Γ) and decompose it in terms of SU(2)-invariant spaces Hn associated to each node n. Here I’ll briefly review the state that this node space Hn is the quantization of the space of shapes of the geometry of solids figures tetrahedra, or more general polyhedra . See the posts:

Let’s start with the classical phase space of shapes of a flat polyhedron in Rwith fixed area. A classically flat three-dimensional polyhedron can be described by a set of L vectors Al, l = 1…L, satisfying the following closure constraint:

ding131

Here the L vectors Al can be interpreted as the vectorial areas of the L triangles in the boundary of the polyhedron, in the sense that the norm al = |Al| is the area of the polygon l and normalized vector nl = Al/|Al| is the normal when embedded in to a R3 Euclidean space.
To introduce a symplectic structure, one can associate to each normal Aia generator of the algebra of SO(3).

ding132 A quantum representation of this Poisson algebra is precisely defined by the generators of SU(2) on the space Hn for a 4-valent node n. The operator corresponding to the area al = |Al| is the Casimir of the representation jl, therefore the space quantizes
the space of the shapes of the tetrahedron with areas jl(jl +1). Furthermore, the Hamiltonian flow of G, generates the rotations of the tetrahedron in R3.

By imposing

ding131

and factoring out the orbits of this flow, one obtains the intertwiner space Kn.

In this way, one gives an intertwiner a geometrical interpretation in terms of quantum polyhedron.

There is a  relation among spinfoam formalism, kinematical Hilbert space and polyhedral quantum geometry. For example the boundary space of the simplicial EPRL spinfoam model can be obtained from simplicity constraints, which is the simplicial truncation of LQG kinematical Hilbert space and the boundary state has a geometrical interpretation in terms of quantum tetrahedron geometry. This consistent picture can be generalized into an arbitrary-valence spinfoam formalism. It is also possible to compute the two-point correlation function of Lorentian EPRL spinfoam model and show it matches the one from Regge geometry.

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Group field theory as the 2nd quantization of Loop Quantum Gravity by Daniele Oriti

This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference  paper – Group field theory: A quantum field theory for the atoms of space  and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:

Introduction

We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,
there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus
define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be  the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical
dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators  and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is  obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization  and Effective quantum cosmology from GFT condensates.

Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise

gfteqtfig1

The group elements Gij are assigned to each link of the graph, with Gij=Gij-1. Assume  gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

Spin networks in 2nd quantization

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum –  the natural background independent, diffeo-invariant vacuum state.

The  2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible
with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field  quantum is the spin network vertex or tetrahedron – the so called building block of space.

gftfig4

A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

gftfig5The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties  from results in quantum simplicial geometry.

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