Ordinary Quantum Theory Research Articles

N-Dimensional non-commutative GUP quantization and application to the Bianchi I model

We analyse a n-dimensional Generalized Uncertainty Principle (GUP) quantization framework, characterized by a non-commutative nature of the configurational variables. First, we identify a set of states which are maximally localized only along a single direction, at the expense of being less localized in all the other ones. Subsequently, in order to recover information about localization on the whole configuration space, we use the only state of the theory which exhibits maximal localization simultaneously in every direction to construct a satisfactory quasi-position representation, by virtue of a suitable translational operator. The resultant quantum framework is then applied to model the dynamics of the Bianchi I cosmology. The corresponding Wheeler–DeWitt equation is reduced to Schrödinger dynamics for the two anisotropy degrees of freedom, using a WKB representation for the volume-like variable of the Universe, in accordance with the Vilenkin scenario. The main result of our cosmological implementation of the constructed quantum theory demonstrates how the dynamics of a wave packet peaked at some point in the configuration space represented in the quasi-position variables favours as the most probable configuration exactly the initial one for a relatively long time, if compared with the ordinary quantum theory. This preference arises from the different dynamical behavior exhibited by wave packets in the two quantum theories.

Negations and Meets in Topos Quantum Theory

The daseinisation is a mapping from an orthomodular lattice in ordinary quantum theory into a Heyting algebra in topos quantum theory. While distributivity does not always hold in orthomodular lattices, it does in Heyting algebras. We investigate the conditions under which negations and meets are preserved by daseinisation, and the condition that any element in the Heyting algebra transformed through daseinisation corresponds to an element in the original orthomodular lattice. We show that these conditions are equivalent, and that, not only in the case of non-distributive orthomodular lattices but also in the case of Boolean algebras containing more than four elements, the Heyting algebra transformed from the orthomodular lattice through daseinisation will contain an element that does not correspond to any element of the original orthomodular lattice.

Stochastic emergent quantum gravity

We present a stochastic framework for emergent quantum gravity coupled to matter. The Hamiltonian constraint in diffeomorphism-invariant theories demands the identification of a clock relative to which dynamics may be defined, and other degrees of freedom can play the role of rulers. However, a global system of clock and rulers is generally not available. We provide evidence that stochasticity associated with critical points of clock and ruler fields can be related to the emergence of both a probabilistic description consistent with ordinary quantum theory, and gravitation described by general relativity at long distances. We propose a procedure for embedding any Lorentz-invariant field theory, including the Standard Model and its Lorentz-invariant extensions, in this framework.

Can the wave function in configuration space be replaced by single-particle wave functions in physical space?

The ontology of Bohmian mechanics includes both the universal wave function (living in 3N-dimensional configuration space) and particles (living in ordinary 3-dimensional physical space). Proposals for understanding the physical significance of the wave function in this theory have included the idea of regarding it as a physically-real field in its 3N-dimensional space, as well as the idea of regarding it as a law of nature. Here we introduce and explore a third possibility in which the configuration space wave function is simply eliminated—replaced by a set of single-particle pilot-wave fields living in ordinary physical space. Such a re-formulation of the Bohmian pilot-wave theory can exactly reproduce the statistical predictions of ordinary quantum theory. But this comes at the rather high ontological price of introducing an infinite network of interacting potential fields (living in 3-dimensional space) which influence the particles’ motion through the pilot-wave fields. We thus introduce an alternative approach which aims at achieving empirical adequacy (like that enjoyed by GRW type theories) with a more modest ontological complexity, and provide some preliminary evidence for optimism regarding the (once popular but prematurely-abandoned) program of trying to replace the (philosophically puzzling) configuration space wave function with a (totally unproblematic) set of fields in ordinary physical space.

Problem of time: facets and Machian strategy

The problem of time is that "time" in each of ordinary quantum theory and general relativity are mutually incompatible notions. This causes difficulties in trying to put these two theories together to form a theory of quantum gravity. The problem of time has eight facets in canonical approaches. I clarify that all but one of these facets already occur at the classical level, and reconceptualize and re-name some of these facets as follows. The frozen formalism problem becomes temporal relationalism, the thin sandwich problem becomes configurational relationalism, via the notion of best matching. The problem of observables becomes the problem of beables, and the functional evolution problem becomes the constraint closure problem. I also outline how each of the global and multiple-choice problems of time have their own plurality of facets. This article additionally contains a local resolution to the problem of time at the conceptual level and which is actually realizable for the relational triangle and minisuperspace models. This resolution is, moreover, Machian, and has three levels: classical, semiclassical, and a combined semiclassical-histories-timeless records scheme. I end by delineating the current frontiers of this program toward resolution of the problem of time in the cases of full general relativity and of slightly inhomogeneous cosmology.

Negative Entropy and Black Hole Information

Based on negative entropy in entanglement, it is shown that a single-system Copenhagen measurement protocol is equivalent to the two-system von Neumann scheme with the memory filling up the system with negative information similar to the Dirac sea of negative energy. After equating the two quantum measurement protocols, we then apply this equivalence to the black hole radiation. That is, the black hole evaporation corresponds to the quantum measurement process and the two evaporation approaches, the observable-based single-system and the two-system entanglement-based protocols, can be made equivalent using quantum memory. In particular, the measurement choice, \theta, with the memory state inside the horizon in the entanglement-based scheme is shown to correspond to the observable of the measurement choice, \theta, outside the horizon in the single-system protocol, that is, O_{\theta}^{out} = Q_{\theta}^{in}. This indicates that the black hole as quantum memory is filling up with negative information outside the horizon, and its entropy corresponds to the logarithm of a number of equally probable measurement choices. This shows that the black hole radiation is no different than ordinary quantum theory.

The pilot-wave perspective on quantum scattering and tunneling

The de Broglie-Bohm “pilot-wave” theory replaces the paradoxical wave-particle duality of ordinary quantum theory with a more mundane and literal kind of duality: each individual photon or electron comprises a quantum wave (evolving in accordance with the usual quantum mechanical wave equation) and a particle that, under the influence of the wave, traces out a definite trajectory. The definite particle trajectory allows the theory to account for the results of experiments without the usual recourse to additional dynamical axioms about measurements. Instead, one need simply assume that particle detectors click when particles arrive at them. This alternative understanding of quantum phenomena is illustrated here for two elementary textbook examples of one-dimensional scattering and tunneling. We introduce a novel approach to reconcile standard textbook calculations (made using unphysical plane-wave states) with the need to treat such phenomena in terms of normalizable wave packets. This approach allows for a simple but illuminating analysis of the pilot-wave theory's particle trajectories and an explicit demonstration of the equivalence of the pilot-wave theory predictions with those of ordinary quantum theory.

Problem of time in quantum gravity

AbstractThe Problem of Time occurs because the ‘time’ of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the conditions of both apply, e.g. in black holes or in the very early universe. Emphasis in this Review is on the Problem of Time being multi‐faceted and on the nature of each of the eight principal facets. Namely, the Frozen Formalism Problem, Configurational Relationalism Problem (formerly Sandwich Problem), Foliation Dependence Problem, Constraint Closure Problem (formerly Functional Evolution Problem), Multiple Choice Problem, Global Problem of Time, Problem of Beables (alias Problem of Observables) and Spacetime Reconstruction/Replacement Problem. Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets. Particular emphasis is placed upon A) relationalism as an underpinning of the facets and as a selector of particular strategies (especially a modification of Barbour relationalism, though also with some consideration of Rovelli relationalism). B) Classifying approaches by the full ordering in which they embrace constrain, quantize, find time/history and find observables, rather than only by partial orderings such as “Dirac‐quantize”. C) Foliation (in)dependence and Spacetime Reconstruction for a wide range of physical theories, strategizing centred about the Problem of Beables, the Patching Approach to the Global Problem of Time, and the role of the question‐types considered in physics. D) The Halliwell‐ and Gambini–Porto–Pullin‐type combined Strategies in the context of semiclassical quantum cosmology.

Emergence of a classical Universe from quantum gravity and cosmology

I describe how we can understand the classical appearance of our world from a universal quantum theory. The essential ingredient is the process of decoherence. I start with a general discussion in ordinary quantum theory and then turn to quantum gravity and quantum cosmology. There is a whole hierarchy of classicality from the global gravitational field to the fluctuations in the cosmic microwave background, which serve as the seeds for the structure in the Universe.

The Quantum as an Emergent System

Double slit interference is explained with the aid of what we call "21st century classical physics". We model a particle as an oscillator ("bouncer") in a thermal context, which is given by some assumed "zero-point" field of the vacuum. In this way, the quantum is understood as an emergent system, i.e., a steady-state system maintained by a constant throughput of (vacuum) energy. To account for the particle's thermal environment, we introduce a "path excitation field", which derives from the thermodynamics of the zero-point vacuum and which represents all possible paths a particle can take via thermal path fluctuations. The intensity distribution on a screen behind a double slit is calculated, as well as the corresponding trajectories and the probability density current. Further, particular features of the relative phase are shown to be responsible for nonlocal effects not only in ordinary quantum theory, but also in our classical approach.

Towards Einstein's dream of a unified field theory: Reports from a journey on a long and winding road

With general arguments it is motivated that we are still in search of a hidden-variables theory or sub-quantum mechanics. Aspects of a solvable model for quantum measurements are discussed, and a solution of the quantum measurement problem within ordinary quantum theory. The related statistical interpretation of quantum mechanics is advocated. Soliton approaches for elementary particles are proposed and some classical looking aspects of the relativistic hydrogen atom are mentioned. The integration of these solitons with stochastic electrodynamics is sketched.

Purification of mixed states with closed timelike curve is not possible

In ordinary quantum theory, any mixed state can be purified in an enlarged Hilbert space by adding an ancillary system. The purified state does not depend on the state of any extraneous system with which the mixed state is going to interact or on the physical interaction. Here, we prove that it is not possible to purify a mixed state that traverses a closed timelike curve (CTC) and is allowed to interact in a consistent way with a causality-respecting (CR) quantum system in the same manner. In other words, if a CTC quantum system with a mixed state has to undergo an arbitrary interaction with an arbitrary CR system then it has to be in a ``proper'' mixture, as it cannot be regarded as a subsystem of a pure entangled state. Thus, in general for arbitrary interactions between CR and CTC systems, there is no universal purification by embedding in a larger Hilbert space for mixed states with CTCs. This shows that in quantum theory with CTCs there can exist proper and improper mixtures.

John S. Bell’s concept of local causality

John Stewart Bell’s famous theorem is widely regarded as one of the most important developments in the foundations of physics. Yet even as we approach the 50th anniversary of Bell’s discovery, its meaning and implications remain controversial. Many workers assert that Bell’s theorem refutes the possibility suggested by Einstein, Podolsky, and Rosen (EPR) of supplementing ordinary quantum theory with “hidden” variables that might restore determinism and/or some notion of an observer-independent reality. But Bell himself interpreted the theorem very differently—as establishing an “essential conflict” between the well-tested empirical predictions of quantum theory and relativistic local causality. Our goal is to make Bell’s own views more widely known and to explain Bell’s little-known formulation of the concept of relativistic local causality on which his theorem rests. We also show precisely how Bell’s formulation of local causality can be used to derive an empirically testable Bell-type inequality and to recapitulate the EPR argument.

The Theory of (Exclusively) Local Beables

It is shown how, starting with the de Broglie–Bohm pilot-wave theory, one can construct a new theory of the sort envisioned by several of QM’s founders: a Theory of Exclusively Local Beables (TELB). In particular, the usual quantum mechanical wave function (a function on a high-dimensional configuration space) is not among the beables posited by the new theory. Instead, each particle has an associated “pilot-wave” field (living in physical space). A number of additional fields (also fields on physical space) maintain what is described, in ordinary quantum theory, as “entanglement.” The theory allows some interesting new perspective on the kind of causation involved in pilot-wave theories in general. And it provides also a concrete example of an empirically viable quantum theory in whose formulation the wave function (on configuration space) does not appear—i.e., it is a theory according to which nothing corresponding to the configuration space wave function need actually exist. That is the theory’s raison d’etre and perhaps its only virtue. Its vices include the fact that it only reproduces the empirical predictions of the ordinary pilot-wave theory (equivalent, of course, to the predictions of ordinary quantum theory) for spinless non-relativistic particles, and only then for wave functions that are everywhere analytic. The goal is thus not to recommend the TELB proposed here as a replacement for ordinary pilot-wave theory (or ordinary quantum theory), but is rather to illustrate (with a crude first stab) that it might be possible to construct a plausible, empirically viable TELB, and to recommend this as an interesting and perhaps-fruitful program for future research.

Causal quantum theory and the collapse locality loophole

Causal quantum theory is an umbrella term for ordinary quantum theory modified by two hypotheses: state vector reduction is a well-defined process, and strict local causality applies. The first of these holds in some versions of Copenhagen quantum theory and need not necessarily imply practically testable deviations from ordinary quantum theory. The second implies that measurement events which are spacelike separated have no non-local correlations. To test this prediction, which sharply differs from standard quantum theory, requires a precise theory of state vector reduction. Formally speaking, any precise version of causal quantum theory defines a local hidden variable theory. However, causal quantum theory is most naturally seen as a variant of standard quantum theory. For that reason it seems a more serious rival to standard quantum theory than local hidden variable models relying on the locality or detector efficiency loopholes. Some plausible versions of causal quantum theory are not refuted by any Bell experiments to date, nor is it obvious that they are inconsistent with other experiments. They evade refutation via a neglected loophole in Bell experiments -- the {\it collapse locality loophole} -- which exists because of the possible time lag between a particle entering a measuring device and a collapse taking place. Fairly definitive tests of causal versus standard quantum theory could be made by observing entangled particles separated by $\approx 0.1$ light seconds.

Wave Equations with Energy-Dependent Potentials

We study wave equations with energy dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which conditions such equations can be handled as evolution equation of quantum theory with an energy dependent potential. Once these conditions are met, such theory can be transformed into ordinary quantum theory.

What is the therapeutically active ingredient of homeopathic potencies ?

The nature of the ‘active ingredient’, in homeopathic high dilutions is investigated. A model for every degree of dilution is introduced; within this the active ingredient can be dealt with in physical terms. In mathematical terms this model has features which correspond to the axioms of weak quantum theory. Features which are similar to entanglement in ordinary quantum theory are discussed in particular.

Weak quantum theory: Complementarity and entanglement in physics and beyond

The concepts of complementarity and entanglement are considered with respect to their significance in and beyond physics. A formally generalized, weak version of quantum theory, more general than ordinary quantum theory of physical systems, is outlined and tentatively applied to two examples.

Cosmic quantum measurement

Hardy's theorem states that the hidden variables of any realistic theory of quantum measurement, whose predictions agree with ordinary quantum theory, must have a preferred Lorentz frame. This presents the conflict between special relativity and any realistic dynamics of quantum measurement in a severe form. The conflict is resolved using a `measurement field', which provides a timelike function of spacetime points and a definition of simultaneity in the context of a curved spacetime. Locally this theory is consistent with special relativity, but globally, special relativity is not enough; the time dilation of general relativity and the standard cosmic time of the Robertson-Walker cosmologies are both essential. A simple but crude example is a relativistic quantum measurement dynamics based on the nonrelativistic measurement dynamics of L\"uders.

DOUBLE WAVE FUNCTION APPROACHES TO SYSTEMS OF PLANAR AND SPHERICAL ROTATOR

In this paper, we give the double wave function theory for systems of planar and spherical rotator, obtain rigorously the classical mechanical equations of motion for these two systems study the related problems in the ordinary quantum theory for these sfstems.