Causality in Schwinger’s Picture of Quantum Mechanics

This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

Position in Minimal Length Quantum Mechanics

Several approaches to quantum gravity imply the presence of a minimal measurable length at high energies. This is in tension with the Heisenberg Uncertainty Principle. Such a contrast is then considered in phenomenological approaches to quantum gravity by introducing a minimal length in quantum mechanics via the Generalized Uncertainty Principle. Several features of the standard theory are affected by such a modification. For example, position eigenstates are no longer included in models of quantum mechanics with a minimal length. Furthermore, while the momentum-space description can still be realized in a relatively straightforward way, the (quasi-)position representation acquires numerous issues. Here, we will review such issues, clarifying aspects regarding models with a minimal length. Finally, we will consider the effects of such models on simple quantum mechanical systems.

On the Quantum Mechanical Description of the Interaction between Particle and Detector

Quantum measurements of physical quantities are often described as ideal measurements. However, only a few measurements fulfil the conditions of ideal measurements. The aim of the present work is to describe real position measurements with detectors that are able to detect single particles. For this purpose, a detector model is developed that can describe the time dependence of the interaction between a non-relativistic particle and a detector. The example of a position measurement shows that this interaction can be described with the methods of quantum mechanics. At the beginning of a position measurement, the detector behaves as a target consisting of a large number of quantum mechanical systems. In the first reaction, the incident particle interacts with a single atom, electron or nucleus, but not with the whole detector. This reaction and all following reactions are quantum mechanical processes. At the end of the measurement, the detector can be considered as a classical apparatus. A detector is neither a quantum mechanical system nor a classical apparatus. The detector model explains why one obtains a well-defined result for each individual position measurement. It further explains that, in general, it is impossible to predict the outcome of an individual measurement.

Quantum gravity corrections to the mean field theory of nucleons

In this paper, we analyze the correction to the mean field theory potential for a system of nucleons. It will be argued that these corrections can be obtained by deforming the Schrödinger’s equation describing a system of nucleons by a minimal length in the background geometry of space-time. This is because such a minimal length occurs due to quantum gravitational effects, and modifies the low energy quantum mechanical systems. In fact, as the mean field potential for the nucleons is represented by the Woods–Saxon potential, we will explicitly analyze such corrections to this potential. We will obtain the corrections to the energy eigenvalues of the deformed Schrödinger’s equation for the Woods–Saxon potential. We will also construct the wave function for the deformed Schrödinger’s equation.

Color confinement and Bose-Einstein condensation

We propose a unified description of two important phenomena: color confinement in large-N gauge theory, and Bose-Einstein condensation (BEC). We focus on the confinement/deconfinement transition characterized by the increase of the entropy from N0 to N2, which persists in the weak coupling region. Indistinguishability associated with the symmetry group — SU(N) or O(N) in gauge theory, and SN permutations in the system of identical bosons — is crucial for the formation of the condensed (confined) phase. We relate standard criteria, based on off-diagonal long range order (ODLRO) for BEC and the Polyakov loop for gauge theory. The constant offset of the distribution of the phases of the Polyakov loop corresponds to ODLRO, and gives the order parameter for the partially-(de)confined phase at finite coupling. We demonstrate this explicitly for several quantum mechanical systems (i.e., theories at small or zero spatial volume) at weak coupling, and argue that this mechanism extends to large volume and/or strong coupling. This viewpoint may have implications for confinement at finite N, and for quantum gravity via gauge/gravity duality.

Compactification, T-duality and quantum erasers

Using T-duality, we will argue that a zero point length exists in the low-energy effective field theory of string theory on compactified extra dimensions. Furthermore, if we neglect all the oscillator modes, this zero point length would modify low quantum mechanical systems. As this zero length is fixed geometrically, it is important to analyze how it modifies purely quantum mechanical effects. Thus, we will analyze its effects on quantum erasers, because they are based on quantum effects like entanglement. It will be observed that the behavior of these quantum erasers gets modified by this zero point length. As the zero point length is fixed by the radius of compactification, we argue that these results demonstrate a deeper connection between geometry and quantum effects.

On the Stochastic Mechanics Foundation of Quantum Mechanics

Among the famous formulations of quantum mechanics, the stochastic picture developed since the middle of the last century remains one of the less known ones. It is possible to describe quantum mechanical systems with kinetic equations of motion in configuration space based on conservative diffusion processes. This leads to the representation of physical observables through stochastic processes instead of self-adjoint operators. The mathematical foundations of this approach were laid by Edward Nelson in 1966. It allows a different perspective on quantum phenomena without necessarily using the wave-function. This article recaps the development of stochastic mechanics with a focus on variational and extremal principles. Furthermore, based on recent developments of optimal control theory, the derivation of generalized canonical equations of motion for quantum systems within the stochastic picture are discussed. These so-called quantum Hamilton equations add another layer to the different formalisms from classical mechanics that find their counterpart in quantum mechanics.