Giuseppe Scolarici

Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n -dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.

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.

proper versus improper mixtures: Toward a quaternionic quantum mechanics

The density operators obtained by taking partial traces represent improper mixtures of subsystems of a compound physical system because the coefficients in the convex sums expressing them never bear the ignorance interpretation. Assigning states to these subsystems is consequently problematic in standard quantum mechanics (subentity problem). In the semantic realism interpretation of quantum mechanics, it is instead proposed to consider improper mixtures true nonpure states conceptually distinct from proper mixtures. Based on this proposal, we show that proper and improper mixtures can be represented by different density operators in the quaternionic formulation of quantum mechanics and can hence be distinguished even from a mathematical standpoint. We provide a simple example related to the quantum theory of measurement.

Pseudo-Hermitian Hamiltonians, indefinite inner product spaces and their symmetries

We extend the definition of generalized parity P, charge-conjugation C and time-reversal T operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular, we show that T P and CT P are the generators of the P-antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian H to admit a P-reflecting symmetry which generates the P-pseudounitary and the P-pseudoantiunitary symmetries. Finally, a physical example is considered and some hints on the P-unitary evolution of a physical system are also given.

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.

Alternative descriptions in quaternionic quantum mechanics

We characterize the quasianti-Hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences between them.

Complex positive maps and quaternionic unitary evolution

The complex projection of any n-dimensional quaternionic unitary dynamics defines a one-parameter positive semigroup dynamics. We show that the converse is also true, i.e. that any one-parameter positive semigroup dynamics of complex density matrices with maximal rank can be obtained as the complex projection of suitable quaternionic unitary dynamics.

Pseudoanti-Hermitian operators in quaternionic quantum mechanics

We introduce the concept of pseudoanti-Hermitian operators in quaternionic quantum mechanics and give a complete characterization of their spectra. We highlight some physical properties related to time-reversal symmetry of the pseudoanti-Hermitian quaternionic Hamiltonians.

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.

Quasianti-Hermiticity and Quaternionic Quantum Systems

A necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in an infinite-dimensional (right) quaternionic Hilbert space is proven. According to such results we obtain (and briefly discuss) two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics.