Lessons from Quantum Field Theory - Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurences of this or closely related Hopf algebras in other mathematical domains, such as foliations, Runge Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.Comment: Survey paper, 12 pages, epsf for figures, dedicated to Mosh\'e Flato, minor corrections, to appear in Lett.Math.Phys.4

Kinetic theory and classical limit for real scalar quantum field in curved space-time

Starting from a real scalar quantum field theory with quartic self-interactions and non-minimal coupling to classical gravity, we define four equal-time, spatially covariant phase-space operators through a Wigner transformation of spatially translated canonical operators within a 3+1 decomposition. A subset of these operators can be interpreted as fluctuating particle densities in phase-space whenever the quantum state of the system allows for a classical limit. We come to this conclusion by expressing hydrodynamic variables through the expectation values of these operators and moreover, by deriving the dynamics of the expectation values within a spatial gradient expansion and a 1-loop approximation which subsequently yields the Vlasov equation with a self-mass correction as a limit. We keep an arbitrary classical metric in the 3+1 decomposition which is assumed to be determined semi-classically. Our formalism allows to systematically study the transition from quantum field theory in curved space-time to classical particle physics for this minimal model of self-interacting, gravitating matter. As an application we show how to include relativistic and self-interaction corrections to existing dark matter models in a kinetic description by taking into account the gravitational slip, vector perturbations and tensor perturbations.Comment: This version matches the one accepted for publication in Physical Review

Robustness of delocalization to the inclusion of soft constraints in long-range random models

Motivated by the constrained many-body dynamics, the stability of the localization-delocalization properties to the inclusion of soft constraints is addressed in random matrix models. These constraints are modeled by correlations in long-ranged hopping with the Pearson correlation coefficient different from zero or unity. Counterintuitive robustness of delocalized phases, both ergodic and (multi)fractal, in these models, is numerically observed and confirmed by the analytical calculations. First, the matrix inversion trick is used to uncover the origin of such robustness. Next, to characterize delocalized phases, a method of eigenstate calculation, sensitive to correlations in long-ranged hopping terms, is developed for random matrix models and approved by numerical calculations and the previous analytical ansatz. The effect of the robustness of states in the bulk of the spectrum to the inclusion of soft constraints is generally discussed for single-particle and many-body systems

Localization & Exact Holography

We consider the AdS_2/CFT_1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT_1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS_2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving `exact holography' in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes

No-signaling, perfect bipartite dichotomic correlations and local randomness

The no-signaling constraint on bi-partite correlations is reviewed. It is shown that in order to obtain non-trivial Bell-type inequalities that discern no-signaling correlations from more general ones, one must go beyond considering expectation values of products of observables only. A new set of nontrivial no-signaling inequalities is derived which have a remarkably close resemblance to the CHSH inequality, yet are fundamentally different. A set of inequalities by Roy and Singh and Avis et al., which is claimed to be useful for discerning no-signaling correlations, is shown to be trivially satisfied by any correlation whatsoever. Finally, using the set of newly derived no-signaling inequalities a result with potential cryptographic consequences is proven: if different parties use identical devices, then, once they have perfect correlations at spacelike separation between dichotomic observables, they know that because of no-signaling the local marginals cannot but be completely random.Comment: Published in 'Proceedings of the International Conference Advances in Quantum Theory', AIP Conference Proceedings, vol. 1327, 2011. pp. 36-5

Lexicons of Key Terms in Scholarly Texts and Their Disciplinary Differences: From Quantum Semantics Construction to Relative-Entropy-Based Comparisons

Complex networks are often used to analyze written text and reports by rendering texts in the form of a semantic network, forming a lexicon of words or key terms. Many existing methods to construct lexicons are based on counting word co-occurrences, having the advantage of simplicity and ease of applicability. Here, we use a quantum semantics approach to generalize such methods, allowing us to model the entanglement of terms and words. We show how quantum semantics can be applied to reveal disciplinary differences in the use of key terms by analyzing 12 scholarly texts that represent the different positions of various disciplinary schools (of conceptual change research) on the same topic (conceptual change). In addition, attention is paid to how closely the lexicons corresponding to different positions can be brought into agreement by suitable tuning of the entanglement factors. In comparing the lexicons, we invoke complex network-based analysis based on exponential matrix transformation and use information theoretic relative entropy (Jensen–Shannon divergence) as the operationalization of differences between lexicons. The results suggest that quantum semantics is a viable way to model the disciplinary differences of lexicons and how they can be tuned for a better agreement

Lexicons of Key Terms in Scholarly Texts and Their Disciplinary Differences: From Quantum Semantics Construction to Relative-Entropy-Based Comparisons

Complex networks are often used to analyze written text and reports by rendering texts in the form of a semantic network, forming a lexicon of words or key terms. Many existing methods to construct lexicons are based on counting word co-occurrences, having the advantage of simplicity and ease of applicability. Here, we use a quantum semantics approach to generalize such methods, allowing us to model the entanglement of terms and words. We show how quantum semantics can be applied to reveal disciplinary differences in the use of key terms by analyzing 12 scholarly texts that represent the different positions of various disciplinary schools (of conceptual change research) on the same topic (conceptual change). In addition, attention is paid to how closely the lexicons corresponding to different positions can be brought into agreement by suitable tuning of the entanglement factors. In comparing the lexicons, we invoke complex network-based analysis based on exponential matrix transformation and use information theoretic relative entropy (Jensen–Shannon divergence) as the operationalization of differences between lexicons. The results suggest that quantum semantics is a viable way to model the disciplinary differences of lexicons and how they can be tuned for a better agreement