A Rejoinder on Quaternionic Projective Representations

In a series of papers published in this Journal (J. Math. Phys.), a discussion was started on the significance of a new definition of projective representations in quaternionic Hilbert spaces. The present paper gives what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues.Comment: AMStex, 6 Page

On the mathematical Structure of Quantum Measurement Theory

We show that the key problems of quantum measurement theory, namely the reduction of the wave packet of a microsystem and the specification of its quantum state by a macroscopic measuring instrument, may be rigorously resolved within the traditional framework of the quantum mechanics of finite conservative systems. The argument is centred on the generic model of a microsystem, S, coupled to a finite macroscopic measuring instrument, I, which itself is an N-particle quantum system. The pointer positions of I correspond to the macrostates of this instrument, as represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces, or 'phase cells', are the simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables, M, and, crucially, are of astronomically large dimensionalities, which incease exponentially with N. We formulate conditions on the conservative dynamics of the composite (S+I) under which it yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant state of S; and we show that these conditions are fulfilled by the finite version of the Coleman-Hepp model.Comment: 20 pages, minor correstions installed, to appear in Rep. Math. Phy

Alternative Descriptions in Quaternionic Quantum Mechanics

We characterize the quasianti-Hermitian quaternionic operators in QQM 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.Comment: 16 page

Unbounded normal operators in octonion Hilbert spaces and their spectra

Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.Comment: 50 page

Projective Group Representations in Quaternionic Hilbert Space

We extend the discussion of projective group representations in quaternionic Hilbert space which was given in our recent book. The associativity condition for quaternionic projective representations is formulated in terms of unitary operators and then analyzed in terms of their generator structure. The multi--centrality and centrality assumptions are also analyzed in generator terms, and implications of this analysis are discussed.Comment: 16 pages, no figures, plain Te

New classical properties of quantum coherent states

A noncommutative version of the Cramer theorem is used to show that if two quantum systems are prepared independently, and if their center of mass is found to be in a coherent state, then each of the component systems is also in a coherent state, centered around the position in phase space predicted by the classical theory. Thermal coherent states are also shown to possess properties similar to classical ones

Quasi-permutable normal operators in octonion Hilbert spaces and spectra

Families of quasi-permutable normal operators in octonion Hilbert spaces are investigated. Their spectra are studied. Multiparameter semigroups of such operators are considered. A non-associative analog of Stone's theorem is proved.Comment: 20 page

BUT WHAT IS IT \u3ci\u3eSAYING\u3c/i\u3e? TRANSLATING THE MUSICAL LANGUAGE OF STRAVINSKY’S \u3ci\u3eTHREE PIECES FOR CLARINET SOLO\u3c/i\u3e

In response to questions of interpretation of his music, Igor Stravinsky has said simply to let the notes speak for themselves. In this paper I will translate the language of Stravinsky’s music in his Three Pieces for Clarinet Solo. I will demonstrate the following: how Stravinsky was able to derive a harmonic structure out of melodic content, thereby creating a two-dimensional space; the formal structure of each of the three Pieces; and relationships between Three Pieces and another of Stravinsky’s works, L’Histoire du Soldat. This analysis will serve as my translation of Stravinsky’s musical language, which will then be compared to scholarly research conducted regarding the Three Pieces