L. Solombrino

Quaternionic eigenvalue problem

We discuss the (right) eigenvalue equation for H, C and R linear quaternionic operators. The possibility to introduce an isomorphism between these operators and real/complex matrices allows us to translate the quaternionic problem into an equivalent real or complex counterpart. Interesting applications are found in solving differential equations within quaternionic formulations of quantum mechanics.

The theoretical apparatus of semantic realism: A new language for classical and quantum physics

The standard interpretation of quantum physics (QP) and some recent generalizations of this theory rest on the adoption of a rerificationist theory of truth and meaning, while most proposals for modifying and interpreting QP in a “realistic” way attribute an ontological status to theoretical physical entities (ontological realism). Both terms of this dichotomy are criticizable, and many quantum paradoxes can be attributed to it. We discuss a new viewpoint in this paper (semantic realism, or briefly SR), which applies both to classical physics (CP) and to QP. and is characterized by the attempt of giving up verificationism without adopting ontological realism. As a first step, we construct a formalized observative language L endowed with a correspondence truth theory. Then, we state a set of axioms by means of L which hold both in CP and in QP. and construct a further language Lv which can express bothtestable andtheoretical properties of a given physical system. The concepts ofmeaning andtestability do not collapse in L and Le hence we can distinguish between semantic and pragmatic compatibility of physical properties and define the concepts of testability and conjoint testability of statements of L and Le. In this context a new metatheoretical principle (MGP) is stated, which limits the validity of empirical physical laws. By applying SR (in particular. MGP) to QP, one can interpret quantum logic as a theory of testability in QP, show that QP is semantically incomplete, and invalidate the widespread claim that contextuality is unavoidable in QP. Furthermore. SR introduces some changes in the conventional interpretation of ideal measurements and Heisenberg’s uncertainty principle.

Semantic realism versus EPR-Like paradoxes: The Furry, Bohm-Aharonov, and Bell paradoxes

We prove that the general scheme for physical theories that we have called semantic realism(SR) in some previous papers copes successfully with a number of EPR-like paradoxes when applied to quantum physics (QP). In particular, we consider the old arguments by Furry and Bohm- Aharonov and show that they are not valid within a SR framework. Moreover, we consider the Bell-Kochen-Specker und the Bell theorems that should prove that QP is inherently contextual and nonlocal, respectively, and show that they can be invalidated in the SR approach. This removes the seeming contradiction between the basic assumptions of SR and QP, and proves that some problematic features that are usually attributed to QP, us contextuality and nonlocality, occur because of the adoption of a verificationist position, from one side, and from an insufficient adherence to the operational principles that have inspired QP itself, from the other side.

Some remarks on assignment maps

We study the properties of general linear assignment maps, showing that positivity axiom can be suitably relaxed, and propose a new class of dynamical maps (generalized dynamics). A puzzling result, arising in such a context in quantum information theory, is also discussed.

On the Painlevé property of nonlinear field equations in 2+1 dimensions: The Davey–Stewartson system

With the purpose of clarifying some aspects of the complete integrability of nonlinear field equations, a singular‐point analysis is performed of the Davey–Stewartson system, which can be considered as an extension in 2+1 dimensions of the nonlinear Schrodinger equation. It is found that the system under consideration possesses the Painleve property and allows a set of Backlund transformations obtained by truncating the series expansions of the solutions about the singularity manifold.

Symmetry properties and exact patterns in birefringent optical fibers.

We carry out a group-theoretical study of the pair of nonlinear Schrödinger equations describing the propagation of waves in nonlinear birefringent optical fibers. We exploit the symmetry algebra associated with these equations to provide examples of specific exact solutions. Among them, we obtain the soliton profile, which is related to the coordinate translations and the constant change of phase.

Nonlinear evolution equations and nonabelian prolongations

A systematic analysis of the class of nonlinear evolution equations ut +uxxx +φ(u, ux)=0 is carried out within the Estabrook–Wahlquist prolongation scheme.

Quaternionic symmetry groups and particle multiplets

We consider the quaternionic complete symmetry group of a massive physical system, obtained extending the connected Poincare group and the internal symmetry group by means of the CPT and the generalized parity operators. We classify the irreducible Q-representations of this group crossing the generalized Wigner and Frobenius–Schur classifications, and obtain 14 different cases. Some novelties arise in this context, such as the failure of the statement that only irreducible representations must be associated with particle multiplets, and a suggestion on the possible forms of a parity-violating Hamiltonian.

Irreducible linear–antilinear representations and internal symmetries

We reconsider a well‐known classification, due to Wigner, of the unitary–antiunitary finite‐dimensional irreducible group representations within a somewhat generalized mathematical framework, where, in particular, any algebraically closed field K with an involutory automorphism j is considered in place of the complex field C. We show that each case of the classification can be characterized by the set U′l of the linear mappings that commute with the given set U, which is now assumed to be an irreducible semigroup of linear and antilinear (i.e., j‐semilinear) mappings, and explicitly exhibit U′l. Then, this classification is crossed with the classification of the sets of linear and antilinear mappings that has been obtained in some previous work and that generalizes the old classification of the unitary representations introduced by Frobenius and Schur. We obtain a new classification in which every case can be characterized by the group Uc of the invertible linear and antilinear mappings which commute with...

Real Versus Complex Representations and Linear-Antilinear Commutant

A proposition of Autonne , Frobenius and Schur concerning the equivalence of any unitary finite-dimensional complex group representation to a real representation and a related classification of such complex representations are both generalized to every family U of arbitrary mappings in any vector space over a (commutative or non-commutative) field endowed with a conjugation (non-identical involutory automorphism) j. The subject is connected with the group of the linear and “antilinear” (that is semilinear with respect to j) invertible mappings that commute with U.