Parity and the origin of neutrino mass

In the LHC era the issue of the origin and nature of neutrino mass has attained a new meaning and a renewed importance. The growing success of the Higgs–Weinberg mechanism behind the charged fermion masses paves the way for answering the question of neutrino mass. We have shown recently how the spontaneous breaking of parity in the context of the minimal left–right symmetric model allows to probe the origin of neutrino mass in complete analogy with the charged fermions masses in the Standard Model. We revisit here this issue and fill in the gaps left in our previous work. In particular we discuss a number of different mathematical approaches to the problem of disentangling the seesaw mechanism and show how a unique analytical solution emerges. Most important, we give all the possible expressions for the neutrino Dirac mass matrix for general values of light and heavy neutrino mass matrices. In practical terms what is achieved is an untangling of the seesaw mechanism with clear and precise predictions testable at hadron colliders such as LHC.

Solving q-Virasoro constraints

We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for $$\beta $$ β -ensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint $$T_{-1} \mathcal {Z} = 0$$ T - 1 Z = 0 . We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on $$D^2 \times S^1$$ D 2 × S 1 and $$S^3$$ S 3 .

Phase-conjugate mirror for water waves driven by the Faraday instability

The Faraday instability appears on liquid baths submitted to vertical oscillations above a critical value. The pattern of standing ripples at half the vibrating frequency that results from this parametric forcing is usually shaped by the boundary conditions imposed by the enclosing receptacle. Here, we show that the time modulation of the medium involved in the Faraday instability can act as a phase-conjugate mirror––a fact which is hidden in the extensively studied case of the boundary-driven regime. We first demonstrate the complete analogy with the equations governing its optical counterpart. We then use water baths combining shallow and deep areas of arbitrary shapes to spatially localize the Faraday instability. We give experimental evidence of the ability of the Faraday instability to generate counterpropagating phase-conjugated waves for any propagating signal wave. The canonical geometries of a point and plane source are implemented. We also verify that Faraday-based phase-conjugate mirrors hold the genuine property of being shape independent. These results show that a periodic modulation of the effective gravity can perform time-reversal operations on monochromatic propagating water waves, with a remarkable efficiency compared with wave manipulation in other fields of physics.

A global perspective to connections on principal 2-bundles

For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define connections on principal 2-bundles as Lie 2-algebra-valued 1-forms on the total space Lie groupoid of the 2-bundle, satisfying a condition in complete analogy to connections on ordinary principal bundles. We carefully treat various notions of curvature, and prove a classification result by the non-abelian differential cohomology of Breen–Messing. This provides a consistent, global perspective to higher gauge theory.

Hêlēl ben-Šaḥar and the Chthonic Sun: A New Suggestion for the Mythological Background of Isa 14:12-15

This paper will explore the oft investigated problem of the mythological referents which inform Isa 14:12-15. Crucial to this will be a reinterpretation of the mysterious hêlēl ben-šaḥar of v. 12, almost universally understood by commentators and translators alike to refer to the ‘Day Star, son of the Dawn’, and thus taken to refer to the ‘morning star’, the planet Venus. Much of the scholarship has approached the ancient Near Eastern material with this meaning in mind, yet no myth hitherto proposed has provided a complete analogy to Isa 14:12-15. Thus I will begin by exploring the problems with these previous analogies, before reconsidering the meaning of hêlēl ben-šaḥar. Understanding the phrase to metonymically remind of the sun itself, the Ugaritic conception of the chthonic sun will be proposed to provide a much more satisfying parallel with our Isaiah passage.

ON THE ORDER STRUCTURE OF REPRESENTABLE FUNCTIONALS

The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on *-algebras. We describe the extreme points of order intervals, and give a non-trivial sufficient condition to decide whether or not the infimum of two representable functionals exists. To this aim, we offer a suitable approach to the Lebesgue decomposition theory, which is in complete analogy with the one developed by Ando in the context of positive operators. This tight analogy allows to invoke Ando's results to characterize uniqueness of the decomposition, and solve the infimum problem over certain operator algebras.

Is left–right symmetry the key?

In collaboration with Jogesh Pati, Abdus Salam challenged the chiral gauge nature of the Standard Model by paving the road towards the left-right symmetric electro-weak theory. I describe here the logical and historical construction of this theory, by emphasising the pioneering and key role it played for neutrino mass. I show that it is a self-contained and predictive model with the Higgs origin of Majorana neutrino mass, in complete analogy with the SM situation regarding charged fermions.

Linear Irreversible Phenomenological Thermodynamics of Polarization Processes in Rigid Unmagnetic Insulators

We show how classical irreversible thermodynamics is used to derive relaxation equations for dielectric polarization processes in insulators. We calculate susceptibilities for multiple polarization processes and show how coupling arises thermodynamically. Furthermore, we derive evolution equations for electromagnetic fields by combining the dielectric relaxation equations with Maxwell’s equations. Analytical solutions for various frequency regimes will be briefly discussed. A complete analogy exists between the dielectric problem, the Kelvin–Voigt viscoelasticity of solid media, and the non-equilibrium (reactive, vibrational) gasdynamic flow. Also, numerical solutions, using the method of characteristics, are given for a generic signal problem in half-space.

Constraints on the reciprocal propagation of a quantum particle through a one-dimensional localized complex potential

In the propagation of an electron through a one-dimensional asymmetric complex potential, it is known that while the conventional Green function reciprocity symmetry will ensure transmission to be symmetric between a “left-incident” and a “right-incident” beam, no such symmetry exists for the case of reflection. Here we derive generalized reciprocity relations for both the amplitude and phase of the reflected waves as constraints on the left- and right-incident beams, in complete analogy to what was established in optics. We further provide illustrations of these relations via direct analytical calculations in the case of a real potential, and via numerical studies in the case of a complex potential.