Hilbert Space Formulation Of Quantum Mechanics Research Articles

Ground state of nonassociative hydrogen and upper bounds on the magnetic charge of elementary particles

Formulations of magnetic monopoles in a Hilbert-space formulation of quantum mechanics require Dirac's quantization condition of magnetic charge, which implies a large value that can easily be ruled out for elementary particles by standard atomic spectroscopy. However, an algebraic formulation of non-associative quantum mechanics is mathematically consistent with fractional magnetic charges of small values. Here, spectral properties in non-associative quantum mechanics are derived, applied to the ground state of hydrogen with a magnetically charged nucleus. The resulting energy leads to new strong upper bounds for the magnetic charge of various elementary particles that can appear as the nucleus of hydrogen-like atoms, such as the muon or the antiproton.

Transitions without connections: quantum states, from Bohr and Heisenberg to quantum information theory

In his recent paper, L. Freidel noted that instead of representing the motion of electrons in terms of oscillators and predicting their future states on the basis on this representation, as in the previous, classical, electron theory of H. Lorentz, quantum theory was, beginning nearly with its inception, concerned with the probabilities of transitions between states of electrons, without necessarily representing how these transitions come about. Taking N. Bohr’s and then W. Heisenberg’s thinking along these lines in, respectively, Bohr’s 1913 atomic theory and Heisenberg’s quantum mechanics of 1925 as a point of departure, this article reconsiders, from a nonrealist perspective (which suspends or even precludes this representation of the mechanism behind these transitions), the concept of quantum state, as a physical concept, in contradistinction to the mathematical concept of quantum state, a vector in the Hilbert-space formalism of quantum mechanics. Transitions between quantum states appear, from this perspective, as “transitions without connections,” because, while one can register the change from one quantum phenomena to another, observed in measuring instruments, we have no means of representing or possibly even conceiving of how this change comes about. The article will also discuss quantum field theory and, in closing, briefly quantum information theory as confirming, and giving additional dimensions to, these concepts of quantum state and transitions between them.

A Hierarchy of Compatibility and Comeasurability Levels in Quantum Logics with Unique Conditional Probabilities

In the quantum mechanical Hilbert space formalism, the probabilistic interpretation is a later ad-hoc add-on, more or less enforced by the experimental evidence, but not motivated by the mathematical model itself. A model involving a clear probabilistic interpretation from the very beginning is provided by the quantum logics with unique conditional probabilities. It includes the projection lattices in von Neumann algebras and here probability conditionalization becomes identical with the state transition of the Lüders-von Neumann measurement process. This motivates the definition of a hierarchy of five compatibility and comeasurability levels in the abstract setting of the quantum logics with unique conditional probabilities. Their meanings are: the absence of quantum interference or influence, the existence of a joint distribution, simultaneous measurability, and the independence of the final state after two successive measurements from the sequential order of these two measurements. A further level means that two elements of the quantum logic (events) belong to the same Boolean subalgebra. In the general case, the five compatibility and comeasurability levels appear to differ, but they all coincide in the common Hilbert space formalism of quantum mechanics, in von Neumann algebras, and in some other cases.

Why Do the Quantum Observables Form a Jordan Operator Algebra?

The Jordan algebra structure of the bounded real quantum observables was recognized already in the early days of quantum mechanics. While there are plausible reasons for most parts of this structure, the existence of the distributive nonassociative multiplication operation is hard to justify from a physical or statistical point of view. Considering the non-Boolean extension of classical probabilities, presented in a recent paper, it is shown in this paper that such a multiplication operation can be derived from certain properties of the conditional probabilities and the observables, i.e., from postulates with a clear statistical interpretation. The well-known close relation between Jordan operator algebras and C*-algebras then provides the connection to the quantum-mechanical Hilbert space formalism, thus resulting in a novel axiomatic approach to general quantum mechanics that includes the types II and III von Neumann algebras.

Non-Boolean probabilities and quantum measurement

A non-Boolean extension of the classical probability model is proposed. The non-Boolean probabilities reproduce typical quantum phenomena. The proposed model is more general and more abstract, but easier to interpret, than the quantum mechanical Hilbert space formalism and exhibits a particular phenomenon (state-independent conditional probabilities) which may provide new opportunities for an understanding of the quantum measurement process. Examples of the proposed model are provided, using Jordan operator algebras.