Matrix Algebras in Non-Hermitian Quantum Mechanics

In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schrödinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian) commutators.
These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.     

Remarks on Mohrhoff's Interpretation of Quantum Mechanics

In a recently proposed interpretation of quantum mechanics, U. Mohrhoff advocates original and thought-provoking views on space and time, the definition of macroscopic objects, and the meaning of probability statements. The interpretation also addresses a number of questions about factual events and the nature of reality. The purpose of this note is to examine several issues raised by Mohrhoff's interpretation, and to assess whether it helps providing solutions to the long-standing problems of quantum mechanics.     

Why Should We Interpret Quantum Mechanics?

The development of quantum information theory has renewed interest in the idea that the state vector does not represent the state of a quantum system, but rather the knowledge or information that we may have on the system. I argue that this epistemic view of states appears to solve foundational problems of quantum mechanics only at the price of being essentially incomplete.     

Towards a Neo-Copenhagen Interpretation of Quantum Mechanics

The Copenhagen interpretation is critically considered. A number of ambiguities, inconsistencies and confusions are discussed. It is argued that it is possible to purge the interpretation so as to obtain a consistent and reasonable way to interpret the mathematical formalism of quantum mechanics, which is in agreement with the way this theory is dealt with in experimental practice. In particular, the essential role attributed by the Copenhagen interpretation to measurement is acknowledged. For this reason it is proposed to refer to it as a neo-Copenhagen interpretation.     

N=4超对称量子力学

利用Clifford代数构造了N=4超对称量子力学的一般形式,并讨论了它的实现.     

Relativistic Quantum Mechanics and the Bohmian Interpretation

No Heading Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions.     

Embedding Quantum Mechanics into an Objective Framework

An elementary model is given which shows how an objective (hence local and noncontextual) picture of the microworld can be constructed without conflicting with quantum mechanics (QM). This contradicts known no-go theorems, which however do not hold in the model, and supplies some suggestions for a broader theory in which QM can be embedded.     

Classical Mechanics and Quantum Mechanics

The Newton equation of motion is derived from quantum mechanics.     

Spatial Degrees of Freedom in Everett Quantum Mechanics

Stapp claims that, when spatial degrees of freedom are taken into account, Everett quantum mechanics is ambiguous due to a “core basis problem.” To examine an aspect of this claim I generalize the ideal measurement model to include translational degrees of freedom for both the measured system and the measuring apparatus. Analysis of this generalized model using the Everett interpretation in the Heisenberg picture shows that it makes unambiguous predictions for the possible results of measurements and their respective probabilities. The presence of translational degrees of freedom for the measuring apparatus affects the probabilities of measurement outcomes in the same way that a mixed state for the measured system would. Examination of a measurement scenario involving several observers illustrates the consistency of the model with perceived spatial localization of the measuring apparatus.This work was sponsored by the Air Force under Air Force Contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the U.S. Government.     

Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics

Bell's theorem depends crucially on counterfactual reasoning, and is mistakenly interpreted as ruling out a local explanation for the correlations which can be observed between the results of measurements performed on spatially-separated quantum systems. But in fact the Everett interpretation of quantum mechanics, in the Heisenberg picture, provides an alternative local explanation for such correlations. Measurement-type interactions lead, not to many worlds but, rather, to many local copies of experimental systems and the observers who measure their properties. Transformations of the Heisenberg-picture operators corresponding to the properties of these systems and observers, induced by measurement interactions, label each copy and provide the mechanism which, e. g., ensures that each copy of one of the observers in an EPRB or GHZM experiment will only interact with the correct copy of the other observer(s). The conceptual problem of nonlocality is thus replaced with a conceptual problem of proliferating labels, as correlated systems and observers undergo measurement-type interactions with newly-encountered objects and instruments; it is suggested that this problem may be resolved by considering quantum field theory rather than the quantum mechanics of particles.     

An Algebra of Effects in the Formalism of Quantum Mechanics on Phase Space

Defining an addition of the effects in the formalism of quantum mechanics on phase space, we obtain a new effect algebra that is strictly contained in the effect algebra of all effects. A new property of the phase space formalism comes to light, namely that the new effect algebra does not contain any pair of noncommuting projections. In fact, in this formalism, there are no nontrivial projections at all. We illustrate this with the spin-1/2 algebra and the momentum/position algebra. Next, we equip this algebra of effects with the sequential product and get an interpretation of why certain properties fail to hold. PACS: 02.10.Gd, 03.65.Bz. This paper was a submission to the Fifth International Quantum Structure Association Conference (QS5), which took place in Cesena, Italy, March 31–April 5, 2001.     

Foundations of Quantum Mechanics: The Connection Between QM and the Central Limit Theorem

In this paper we unravel the connection between the quantum mechanical formalism and the Central limit theorem (CLT). We proceed to connect the results coming from this theorem with the derivations of the Schrödinger equation from the Liouville equation, presented by ourselves in other papers. In those papers we had used the concept of an infinitesimal parameter x that raised some controversy. The status of this infinitesimal parameter is then elucidated in the framework of the CLT. Finally, we use the formal apparatus developed in our previous papers and the results of the present one to advance an alternative objective interpretation of quantum mechanics in which its relations with the classical framework are made explicit. The relations between our approach and those using the Wigner–Moyal transformation are also addressed.     

The Ithaca Interpretation of Quantum Mechanics, and Objective Probabilities

The Ithaca interpretation of quantum mechanics, proposed in 1996 by David Mermin, seeks to reduce the interpretive puzzles of quantum mechanics to the single puzzle of interpreting objective quantum probabilities. Some suggestions are made as to how the numerical values of quantum probabilities could be ontologically based in a world containing all the possible outcomes of all probabilistic processes. It is then shown that Hardy's paradox, discussed by Mermin, can be resolved when probabilities are interpreted in this way.     

Generalization of Drinfeld Quantum Affine Algebras

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.     

A Perspectival Version of the Modal Interpretation of Quantum Mechanics and the Origin of Macroscopic Behavior

We study the process of observation (measurement), within the framework of a perspectival (relational, relative state) version of the modal interpretation of quantum mechanics. We show that if we assume certain features of discreteness and determinism in the operation of the measuring device (which could be a part of the observer's nerve system), this gives rise to classical characteristics of the observed properties, in the first place to spatial localization. We investigate to what extent semi-classical behavior of the object system itself (as opposed to the observational system) is needed for the emergence of classicality. Decoherence is an essential element in the mechanism of observation that we assume, but it turns out that in our approach no environment-induced decoherence on the level of the object system is required for the emergence of classical properties.     

Quantum Mechanics is About Quantum Information

I argue that quantum mechanics is fundamentally a theory about the representation and manipulation of information, not a theory about the mechanics of nonclassical waves or particles. The notion of quantum information is to be understood as a new physical primitive---just as, following Einsteins special theory of relativity, a field is no longer regarded as the physical manifestation of vibrations in a mechanical medium, but recognized as a new physical entity in its own right.     

Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics

We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the von Neumann uncertainty error-disturbance principle. Moreover, as it was shown in the 1980s, this Markov model induces all stochastic linear and nonlinear equations of the phenomenological informational dynamics such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac-type boundary-value problem for the second quantized input offer waves from future in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the trajectories of the output particles. This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.     

The World According to Quantum Mechanics (Or the 18 Errors of Henry P. Stapp)

Several errors in Stapp's interpretation of quantum mechanics and its application to mental causation (Henry P. Stapp, Quantum theory and the role of mind in nature, Foundations of Physics 31, 1465–1499 (2001)) are pointed out. An interpretation of (standard) quantum mechanics that avoids these errors is presented.