Narrow Results

The issues of holography and possible links with gauge theories in spacetime physics is discussed, in an approach quite distinct from the more restricted AdS-CFT correspondence. A particular notion of holography in the context of black hole thermodynamics is derived (rather than conjectured) from rather elementary considerations, which also leads to a criterion of thermal stability of radiant black holes, without resorting to specific classical metrics. For black holes that obey this criterion, the canonical entropy is expressed in terms of the microcanonical entropy of an Isolated Horizon which is essentially a local generalization of the very global event horizon and is a null inner boundary of spacetime, with marginal outer trapping. It is argued why degrees of freedom on this horizon must be described by a topological gauge theory. Quantizing this boundary theory leads to the microcanonical entropy of the horizon expressed in terms of an infinite series asymptotic in the cross-sectional area, with the leading 'area-law' term followed by finite, unambiguously calculable corrections arising from quantum spacetime fluctuations.

By making use of the grasping action of the area operator as an antisymmetrizer of the grasped strands in spin network and the Penrose binor identity, an equidistant area spectrum is deduced. Utilizing the spectrum to calculate the quantized area of black hole horizon, we recalculate the entropy of black hole H(A) = (8πℏG_N)^-1kA ln 2. By taking advantage of the smallest area quantum "½" excited by the spectrum via the Wilson loop in edge of spin network to approach the possible origin of qubit, the existences of entanglement of the area quanta in quantum space, as well as the nonlocal property of the entangled states are demonstrated.

We prove an upper bound for the evaluation of all classical SU₂ spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

An alternative framework underlying connection between tensor ${\mathrm{\text{sl}}}_2$-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of $\mathrm{\text{SL}}(2)$. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex values of chromatic evaluations.

We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.

Recent attempts to resolve the ambiguity in the loop quantum gravity description of the quantization of area has led to the idea that j=1 edges of spin-networks dominate in their contribution to black hole areas as opposed to j=1/2 which would naively be expected. This suggests that the true gauge group involved might be SO(3) rather than SU(2) with attendant difficulties. We argue that the assumption that a version of the Pauli principle is present in loop quantum gravity allows one to maintain SU(2) as the gauge group while still naturally achieving the desired suppression of spin-1/2 punctures. Areas come from j=1 punctures rather than j=1/2 punctures for much the same reason that photons lead to macroscopic classically observable fields while electrons do not.

The spin network simulator model represents a bridge between (generalized) circuit schemes for standard quantum computation and approaches based on notions from Topological Quantum Field Theories (TQFTs). The key tool is provided by the fiber space structure underlying the model which exhibits combinatorial properties closely related to SU(2) state sum models, widely employed in discretizing TQFTs and quantum gravity in low spacetime dimensions.

We analyze the effects of noise on quantum cloning based on the spin network approach. A noisy environment interacting with the spin network is modeled both in a classical scenario, with a classical fluctuating field, and in a fully quantum scenario, in which the spins are coupled with a bath of harmonic oscillators. We compare the realization of cloning with spin networks and with traditional quantum gates in the presence of noise, and show that spin network cloning is more robust.

We study quantum state transfer through a qubit network modeled by spins with XY interaction, when relying on a single excitation. We show that it is possible to achieve perfect transfer by shifting (adding) energy to specific vertices. This technique appears to be a potentially powerful tool for changing, and in some cases improving, the transfer capabilities of quantum networks. Analytical results are presented for all-to-all networks and for all-to-all networks with a missing link. Moreover, we evaluate the effect of random fluctuations on the transmission fidelity.

A spin-star system consisting of three peripheral two-state systems and a central one is considered, with the peripheral spins assumed to interact with each other, as well as with the central one. It is shown that such two couplings, each one being a thermal entanglement source, can significantly compete in the formation of quantum correlations in the thermal state to the point that they can destroy any thermal entanglement of the peripheral spins.

We show that quantum gravity states associated to open spin network graphs implicitly define maps from the bulk to the boundary of the corresponding region of quantum space. Employing random tensor network techniques, we then investigate under which conditions the flow of information from the bulk to the boundary is an isometric map, which is a necessary condition for holography.

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. The simplest case of these models is the Fibonacci model, itself universal for quantum computation. We here formulate these braid group representations in a form suitable for computation and algebraic work. In particular, we give quantum algorithms for computing colored Jones polynomials and the Witten-Reshetikhin-Turaev invariant of three-manifolds.

In this talk, we elaborate on the operation of graph contraction introduced by Gurau in his study of the Schwinger-Dyson equations. After a brief review of colored tensor models, we identify the Lie algebra appearing in the Schwinger-Dyson equations as a Lie algebra associated to a Hopf algebra of the Connes-Kreimer type. Then, we show how this operation also leads to an analogue of the Wilsonian flow for the effective action. Finally, we sketch how this formalism may be adapted to group field theories.