The Ithaca interpretation of quantum mechanics

I list several strong requirements for what I would consider a sensible interpretation of quantum mechanics and discuss two simple recent theorems which have important implications for such an interpretation. My talk will not clear everything up; indeed, you may conclude that it has not cleared anything up. But I hope it will provide a different perspective from which to view some old and vexing puzzles (or, if you believe nothing needs to be cleared up, some ancient verities.)     

Decoherence in quantum mechanics

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 system’s total entropy increase. We also discuss the relevance of the results in our quantum mechanical model for interacting field theories.     

Langevin formulation of quantum mechanics

Summary We present in a rather pedagogical way a new formulation of quantum mechanics. Our starting point is the path integral representation of the quantum-mechanical propagator analytically continued to imaginary timeW(X″, s″|X′, s′). We view the set of random paths contributing toW(X″, s″|X′, s′) as the manifold of solutions of a Langevin equation with a Gaussian white noise. We thus obtainW(X″, s″|X′, s′) as the noise-average of a suitable functional of the solution of the Langevin equation. The standard quantum-mechanical propagator is finally recovered by analytically continuingW(X″, s″|X′, s′) back to real time. The present approach allows for a straightforward application of standard methods of classical stochastic processes to quantum-mechanical problems and offers a new promising way to perform computer simulations of quantum-dynamical systems. To speed up publication, the author has agreed not to receive proofs which have been supervised by the Scientific Committee.     

The Wigner function in the relativistic quantum mechanics

A detailed study is presented of the relativistic Wigner function for a quantum spinless particle evolving in time according to the Salpeter equation.     

Graph reconstruction and quantum statistical mechanics

We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to quantum statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.     

About the notion of truth in quantum mechanics

The meaning of truth in quantum mechanics is considered in order to respond to some objections raised by B. d'Espagnat against a logical interpretation of quantum mechanics recently proposed by the author. A complete answer is given. It is shown that not only can factual data be said to be true, but also some of their logical consequences, so that the definition of truth given by Heisenberg is both extended and refined. Some nontrue but reliable propositions may also be used, but they are somewhat arbitrary because of the complementarity principle. For instance, the propositions expressing wave packet reduction can be either true or reliable, according to the case under study. Separability is also discussed: as far as the true properties of an individual system are concerned, quantum mechanics is separable.     

Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.     

The thermal Green functions in nonextensive quantum statistical mechanics

The thermal Green functions of the quantum-mechanical harmonic oscillator are constructed within the framework of nonextensive statistical mechanics with normalized q -expectation values. For the Tsallis index q greater than unity, these functions are found to be expressed analytically in terms of the Hurwitz zeta function. It is found that influence of the nonextensivity on the time-ordered thermal propagator is relevant only at the “on-shell” states. In particular, the finite-temperature contribution to the thermal propagator becomes enhanced for the strong nonextensivity. Received 30 September 1998     

On the consistent histories approach to quantum mechanics

We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omnès, and Gell-Mann and Hartle, and describe the classification of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omnès' characterization of true statements—statements which can be deduced unconditionally in his interpretation—is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction, and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as-yetunknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the need to supplement quantum mechanics by some selection principle in order to produce a fundamental theory capable of unconditional predictions.     

Logical reformulation of quantum mechanics. II. Interferences and the Einstein-Podolsky-Rosen experiment

The consistent quantum interpretations of logic that were introduced in a previous paper are applied to four experiments: (1) ordinary interferences, (2) the Badurek-Rauch-Tuppinger neutron interferometry experiment, (3) the Einstein-Podolsky-Rosen experiment, and (4) the detection far away of the origin of a nonrelativistic particle initially near the origin. In the first two cases, the proposition calculus excludes the possibility of observing interferences and of asserting together through which path the particles went. It is used to provide a somewhat complete discussion of the Badurek-Rauch-Tuppinger experiment. The possibility of using logical implication allows a rather complete discussion of the EPR experiment, including the question of causality, although the lack of a relativistic version of the theory does not allow a complete discussion of causality. The last experiment leads to the following result: Detecting the position of a particle at timet sometimes allows one to determine with a finite uncertainty what its momentum was just before the position measurement, even when it is infinitely precise.     

On the geometrization of quantum mechanics

Nonrelativistic quantum mechanics is commonly formulated in terms of wavefunctions (probability amplitudes) obeying the static and the time-dependent Schrödinger equations (SE). Despite the success of this representation of the quantum world a wave–particle duality concept is required to reconcile the theory with observations (experimental measurements). A first solution to this dichotomy was introduced in the de Broglie–Bohm theory according to which a pilot-wave (solution of the SE) is guiding the evolution of particle trajectories. Here, I propose a geometrization of quantum mechanics that describes the time evolution of particles as geodesic lines in a curved space, whose curvature is induced by the quantum potential. This formulation allows therefore the incorporation of all quantum effects into the geometry of space–time, as it is the case for gravitation in the general relativity.     

Upper bounds for the energy expectation in time-dependent quantum mechanics

We consider quantum systems driven by Hamiltonians of the formH+W(t), where the spectrum ofH consists of an infinite set of bands andW(t) depends arbitrarily on time. Let H (t) denote the expectation value ofH with respect to the evolution at timet of an initial state . We prove upper bounds of the type H (t)=O(t ), >0, under conditions on the strength ofW(t) with respect toH. Neither growth of the gaps between the bands nor smoothness ofW(t) is required. Similar estimates are shown for the expectation value of functions ofH. Sufficient conditions to have uniformly bounded expectation values are made explicit and the consequences on other approaches to quantum stability are discussed.     

Measuring the energy level repulsion in quantum dots

Energy level repulsion is one of the remnants of classical chaos in quantum mechanics. Measurements of the distribution of nearest neighbor spacings in quantum dots reveal, in contrast to other classically chaotic systems, deviations from the predictions made by random matrix theory. Here, we survey possible contributions to these deviations from experimental peculiarities present in measurements on quantum dots, and discuss the methods to eliminate or reduce such distortions.     

Quantum Hamilton mechanics: Hamilton equations of quantum motion, origin of quantum operators, and proof of quantization axiom

This paper gives a thorough investigation on formulating and solving quantum problems by extended analytical mechanics that extends canonical variables to complex domain. With this complex extension, we show that quantum mechanics becomes a part of analytical mechanics and hence can be treated integrally with classical mechanics. Complex canonical variables are governed by Hamilton equations of motion, which can be derived naturally from Schrödinger equation. Using complex canonical variables, a formal proof of the quantization axiom p → pˆ = −i, which is the kernel in constructing quantum-mechanical systems, becomes a one-line corollary of Hamilton mechanics. The derivation of quantum operators from Hamilton mechanics is coordinate independent and thus allows us to derive quantum operators directly under any coordinate system without transforming back to Cartesian coordinates. Besides deriving quantum operators, we also show that the various prominent quantum effects, such as quantization, tunneling, atomic shell structure, Aharonov–Bohm effect, and spin, all have the root in Hamilton mechanics and can be described entirely by Hamilton equations of motion.     

Correlated electron states at level crossings of bilayer two-dimensional electron systems in tilted magnetic fields

We have measured the energy-level structure of high mobility, strongly coupled bilayer two-dimensional electron systems in tilted magnetic fields by means of magnetotransport experiments. At tilt angles where single-particle levels with opposite spin and symmetry cross, we observe a surprising sudden broadening of the quantum Hall plateaus and a deepening of the Shubnikov–de Haas minima. This observation is explained by an interaction-induced rearrangement of the energy level structure which strongly increases the energetic splitting of two (anti-)crossing levels.     

Maximally causal quantum mechanics

We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics     

The infinite well and Dirac delta function potentials as pedagogical,mathematical and physical models in quantum mechanics

The infinite square well and the attractive Dirac delta function potentials are arguably two of the most widely used models of one-dimensional bound-state systems in quantum mechanics. These models frequently appear in the research literature and are staples in the teaching of quantum theory on all levels. We review the history, mathematical properties, and visualization of these models, their many variations, and their applications to physical systems.