Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basisComment: 30 pages, 10 figure

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of nonzero subdeterminants.Comment: 16 pages, 3 figure

Dynamically Triangulating Lorentzian Quantum Gravity

Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4. This includes a derivation of Lorentzian simplicial manifold constraints, the gravitational actions and their Wick rotation. We define a transfer matrix for the system and show that it leads to a well-defined self-adjoint Hamiltonian. In view of numerical simulations, we also suggest sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological phases found previously in Euclidean models of dynamical triangulations cannot be realized in the Lorentzian case.Comment: 41 pages, 14 figure

Symmetries, Sum Rules and Constraints on Effective Field Theories

Using unitarity, analyticity and crossing symmetry, we derive universal sum rules for scattering amplitudes in theories invariant under an arbitrary symmetry group. The sum rules relate the coefficients of the energy expansion of the scattering amplitudes in the IR to total cross sections integrated all the way up to the UV. Exploiting the group structure of the symmetry, we systematically determine all the independent sum rules and positivity conditions on the expansion coefficients. For effective field theories the amplitudes in the IR are calculable and hence the sum rules set constraints on the parameters of the effective Lagrangian. We clarify the impact of gauging on the sum rules for Goldstone bosons in spontaneously broken gauge theories. We discuss explicit examples that are relevant for WW-scattering, composite Higgs models, and chiral perturbation theory. Certain sum rules based on custodial symmetry and its extensions provide constraints on the Higgs boson coupling to the electroweak gauge bosons.Comment: 50 pages, 5 figures, 5 appendices; several typos fixed, discussions improved, references added; results unchange

The Ongoing Impact of Modular Localization on Particle Theory

Modular localization is the concise conceptual formulation of causal localization in the setting of local quantum physics. Unlike QM it does not refer to individual operators but rather to ensembles of observables which share the same localization region, as a result it explains the probabilistic aspects of QFT in terms of the impure KMS nature arising from the local restriction of the pure vacuum. Whereas it played no important role in the perturbation theory of low spin particles, it becomes indispensible for interactions which involve higher spin $s\geq1$ fields, where is leads to the replacement of the operator (BRST) gauge theory setting in Krein space by a new formulation in terms of stringlocal fields in Hilbert space. The main purpose of this paper is to present new results which lead to a rethinking of important issues of the Standard Model concerning massive gauge theories and the Higgs mechanism. We place these new findings into the broader context of ongoing conceptual changes within QFT which already led to new nonperturbative constructions of models of integrable QFTs. It is also pointed out that modular localization does not support ideas coming from string theory, as extra dimensions and Kaluza-Klein dimensional reductions outside quasiclassical approximations. Apart from hologarphic projections on null-surfaces, holograhic relations between QFT in different spacetime dimensions violate the causal completeness property, this includes in particular the Maldacena conjecture. Last not least, modular localization sheds light onto unsolved problems from QFT's distant past since it reveals that the Einstein-Jordan conundrum is really an early harbinger of the Unruh effect.Comment: a small text overlap with unpublished arXiv:1201.632

Maximal-entropy-production-rate nonlinear quantum dynamics compatible with second law, reciprocity, fluctuation-dissipation, and time-energy uncertainty relations

In view of the recent quest for well-behaved nonlinear extensions of the traditional Schroedinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, in this paper, together with a review of the general features of the nonlinear quantum (thermo)dynamics I proposed in a series of papers [see references in G.P. Beretta, Found.Phys. 17, 365 (1987)], I show its exact equivalence with the maximal-entropy-production variational-principle formulation recently derived in S. Gheorghiu-Svirschevski, Phys.Rev. A 63, 022105 (2001). In addition, based on the formalism of general interest I developed for the analysis of composite systems, I show how the variational derivation can be extended to the case of a composite system to obtain the general form of my equation of motion, that turns out to be consistent with the demanding requirements of strong separability. Moreover, I propose a new intriguing fundamental ansatz: that the time evolution along the direction of steepest entropy ascent unfolds at the fastest rate compatible with the time-energy Heisenberg uncertainty relation. This ansatz provides a possible well-behaved general closure of the nonlinear dynamics, compatible with the nontrivial requirements of strong separability, and with no need of new physical constants. In any case, the time-energy uncertainty relation provides lower bounds to the internal-relaxation-time functionals and, therefore, upper bounds to the rate of entropy production.Comment: RevTeX; 19 pages; submitted to Phys.Rev.A on Feb.9, 2001; revised version submitted on Sept.14, 2001 with slightly modified derivation in Section III, improved discussion on strong separability in Sections X and IX, added Eqs. 64b, 64c and 11

The Quest for Understanding in Relativistic Quantum Physics

We discuss the status and some perspectives of relativistic quantum physics.Comment: Invited contribution to the Special Issue 2000 of the Journal of Mathematical Physics, 38 pages, typos corrected and references added, as to appear in JM