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Register a new userAlternative Descriptions in Quaternionic Quantum Mechanics
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Giuseppe Scolarici dr
Publication date
2004
fields
Physics
and research's language is
English
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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.
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We analyze some features of alternative Hermitian and quasi-Hermitian quantum descriptions of simple and bipartite compound systems. We show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as tensor product of positive operators on component spaces. Some examples also show that such property could be strictly connected with symmetry properties of the non-Hermitian Hamiltonian.
In complete analogy with the classical situation (which is briefly reviewed) it is possible to define bi-Hamiltonian descriptions for Quantum systems. We also analyze compatible Hermitian structures in full analogy with compatible Poisson structures.
The ontological aspect of Bohmian mechanics, as a hidden-variable theory that provides us with an objective description of a quantum world without observers, is widely known. Yet its practicality is getting more and more acceptance and relevance, for it has proven to be an efficient and useful resource to tackle, explore, describe and explain such phenomena. This practical aspect emerges precisely when the pragmatic application of the formalism prevails over any other interpretational question, still a matter of debate and controversy. In this regard, the purpose here is to show and discuss how Bohmian mechanics emphasizes in a natural manner a series of dynamical features difficult to find out through other quantum approaches. This arises from the fact that Bohmian mechanics allows us to establish a direct link between the dynamics exhibited by quantum systems and the local variations of the quantum phase associated with their state. To illustrate these facts, simple models of two physically insightful quantum phenomena have been chosen, namely, the dispersion of a free Gaussian wave packet and Young-type two-slit interference. As it is shown, the outcomes from their analysis render a novel, alternative understanding of the dynamics displayed by these quantum phenomena in terms of the underlying local velocity field that connects the probability density with the quantum flux. This field, nothing but the so-called guidance condition in standard Bohmian mechanics, thus acquires a prominent role to understand quantum dynamics, as the mechanism responsible for such dynamics. This goes beyond the passive role typically assigned to this field in Bohmian mechanics, where traditionally trajectories and quantum potentials have received more attention instead.
We present the first experimental test that distinguishes between an event-based corpuscular model (EBCM) [H. De Raedt et al.: J. Comput. Theor. Nanosci. 8 (2011) 1052] of the interaction of photons with matter and quantum mechanics. The test looks at the interference that results as a single photon passes through a Mach-Zehnder interferometer [H. De Raedt et al.: J. Phys. Soc. Jpn. 74 (2005) 16]. The experimental results, obtained with a low-noise single-photon source [G. Brida et al.: Opt. Expr. 19 (2011) 1484], agree with the predictions of standard quantum mechanics with a reduced $chi^2$ of 0.98 and falsify the EBCM with a reduced $chi^2$ of greater than 20.
We study decoherence in a simple quantum mechanical model using two approaches. Firstly, we follow the conventional approach to decoherence where one is interested in solving the reduced density matrix from the perturbative master equation. Secondly, we consider our novel correlator approach to decoherence where entropy is generated by neglecting observationally inaccessible correlators. We show that both methods can accurately predict decoherence time scales. However, the perturbative master equation generically suffers from instabilities which prevents us to reliably calculate the systems total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.
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