Table of contents

This special issue of Journal of Physics A: Mathematical and Theoretical entitled 'The Quantum Universe' is dedicated to Professor Giancarlo Ghirardi on the occasion of his 70th birthday. Giancarlo Ghirardi has made many important contributions to the foundations of quantum mechanics including the celebrated Ghirardi–Rimini–Weber (GRW) model of spontaneous wavefunction collapse.

However, although Professor Ghirardi's birthday is the inspiration for this issue, it has a much broader scope than the area traditionally known as Foundations of Quantum Mechanics. All invited authors are experts in areas of physics in which quantum theory is fundamental: non relativistic quantum mechanics, quantum computation and information, quantum field theory, quantum gravity, quantum cosmology and philosophy of science. The issue was conceived as an opportunity for workers in these diverse areas to share with the widest possible readership their views on quantum theory. Authors were encouraged to give their personal assessment of the role of quantum theory in their work particularly as it pertains to a vision of the global aims of their research. The articles are accessible to any physicist with a solid knowledge of quantum mechanics, and many contain an emphasis on conceptual developments, both those achieved and those hoped for.

One theme that runs throughout Giancarlo Ghirardi's contributions to science is the unity of physics: the development of the GRW model itself was motivated by the conviction that the same physics should govern microscopic and macroscopic systems. However, readers of this special issue will clearly see that there is no unity as yet in the views of workers on fundamental quantum theories. Indeed the diversity of the articles, ranging from technical developments in well defined approaches, to new proposals for interpretations of quantum mechanics, indicates the state of fundamental physics: healthily active and yet lacking the consensus we seek in science.

Much work clearly remains and this volume has the potential to be a lasting reference recording the achievements of, and crucial further challenges for, our quantum community.

By taking advantage of my long lasting research activity on the conceptual foundations of quantum mechanics I reconsider some of the basic problems which have been at the centre of the recent debate on this theory. Specific attention is given to topics like quantum nonlocality, the impossibility of faster than light communication and the so-called measurement or macro-objectification problem. A large part of the paper deals with the dynamical reduction program and discusses its merits and achievements, as well as its limitations. The above considerations lead me in a natural way to express my personal views on the present status of the foundational studies.

Delivered at Trieste on the occasion of the 25th Anniversary of the International Centre for Theoretical Physics, 2 November 1989

The video of this lecture is available here. Please see the PDF for the transcript of the lecture.

General remarks by Angelo Bassi and GianCarlo Ghirardi

During the autumn of 1989 the International Centre for Theoretical Physics, Trieste, celebrated the 25th anniversary of its creation. Among the many prestigious speakers, who delivered extremely interesting lectures on that occasion, was the late John Stewart Bell. All lectures have been recorded on tape. We succeeded in getting a copy of John's lecture.

In the lecture, many of the arguments that John had lucidly stressed in his writings appear once more, but there are also extremely interesting new remarks which, to our knowledge, have not been presented elsewhere. In particular he decided, as pointed out by the very choice of the title of his lecture, to call attention to the fact that the theory presents two types of difficulties, which Dirac classified as first and second class. The former are those connected with the so-called macro-objectification problem, the latter with the divergences characterizing relativistic quantum field theories. Bell describes the precise position of Dirac on these problems and he stresses appropriately how, contrary to Dirac's hopes, the steps which have led to a partial overcoming of the second class difficulties have not helped in any way whatsoever to overcome those of the first class. He then proceeds to analyse the origin and development of the Dynamical Reduction Program and draws attention to the problems that still affect it, in particular that of a consistent relativistic generalization.

When the two meetings Are there quantum jumps? and On the present status of Quantum Mechanics were organized in Trieste and Losinj (Croatia), on 5–10 September 2005, it occurred to us that this lecture, which has never been published, might represent an extremely interesting historical record for all the participants who certainly shared with us a great admiration for this outstanding scientist and deep thinker. Accordingly, with the permission of the Abdus Salam International Centre for Theoretical Physics, and with thanks to the financial support of the Consorzio per la Fisica of the Trieste University, we have produced from the original record a DVD which has been given to all participants although, unfortunately, the video tape of the event was not particularly good.

Taking into account that the participants to the meetings represented only a very small subset of those scientists who might be interested in hearing what John Bell said in probably his last lecture, we considered that it would be useful for the scientific community interested in foundational problems to publish the text of this lecture in order to make it accessible to everybody. The lecture was preceded by a presentation by the Chairman, Alain Aspect, which we have also included.

Due to the aforementioned low quality of the recording it has not been easy to pass from the tape to the text we are presenting below, and we have to thank, for her precious collaboration, Dr Julia Filingeri who did most of the work, as well as Mrs Anne Gatti from ICTP, Professors Detlef Düurr and Sheldon Goldstein, and the staff of IOP Publishing who contributed in an essential way in deciphering some particularly difficult passages. Obviously, we take full responsibility for any possible inappropriate rendering of the original talk. We thank the Abdus Salam International Centre for Theoretical Physics for authorizing IOP Publishing to publish this important document.

Some final remarks are in order. Firstly, we have put in square brackets parenthetical remarks that John made while reading sentences from his transparencies. We have also indicated by parenthetical ellipsis (...) very short parts of the speech (usually one word) which we have not been able to decipher. We have included a picture (figure 1) shown by him, taking it from the tape image which is of rather poor quality (we apologize for this) and three figures taken from his transparencies. Moreover, to help the reader in grasping the various points John Stewart Bell brilliantly raised in his talk we have divided the paper into six sections whose titles have been chosen by us to summarize the most crucial points of his argument.

Presentation by the Chairman, Alain Aspect

It is a great pleasure and an honour to introduce Professor Bell. When looking to my old papers I discovered that this 25th anniversary of the ICTP also coincides with the famous paper in which appeared, for the first time, inequalities that are now known as Bell's inequalities so it's a very good opportunity to have a talk by John Bell here. Many of us have been strongly influenced by this work of John Bell because he has shown us that quantum mechanics is much more difficult to understand that we thought it was. I am sure that today he will again raise some questions which are very embarrassing but that we have definitely to face.

We study lower and upper bounds on the parameters for stochastic state vector reduction, focusing on the mass-proportional continuous spontaneous localization (CSL) model. We show that the assumption that the state vector is reduced when a latent image is formed, in photography or etched track detection, requires a CSL reduction rate parameter λ that is larger than conventionally assumed by a factor of roughly 2 × 10^9±2, for a correlation length r_C of 10⁻⁵cm. We reanalyse existing upper bounds on the reduction rate and conclude that all are compatible with such an increase in λ. The best bounds that we have obtained come from a consideration of heating of the intergalactic medium (IGM), which shows that λ can be at most ∼10^8±1 times as large as the standard CSL value, again for r_C = 10⁻⁵cm. (For both the lower and upper bounds, quoted errors are not purely statistical errors, but rather are estimates reflecting modelling uncertainties.) We discuss modifications in our analysis corresponding to a larger value of r_C. With a substantially enlarged rate parameter, CSL effects may be within range of experimental detection (or refutation) with current technologies.

The basics of quantum mechanics with spontaneous localization (GRW model) are rediscussed in the framework of a quantum stochastic process introduced by Ford and Lewis and originated by instantaneous fuzzy space-localization processes superimposed upon an otherwise reversible Schrödinger time evolution.

This paper comprises a few notes illustrating the impact of GianCarlo Ghirardi's achievements, even on the thinking of a 'non-realist'.

We investigate the multi-photon quantum superposition state generated by the quantum-injected high-gain optical parametric amplification of a single photon. The physical configurations based on the optimal universal and on the phase-covariant quantum cloning have been adopted. The theoretical results are supported by a set of experiments leading to the generation of an average number of clones in excess of 10³.

Both an additional nonlinear term in the Schrödinger equation and an additional non-Hamiltonian term in the von Neumann equation, proposed to ensure localization and decoherence of macro-objects, resp., contain the same Newtonian interaction potential formally. We discuss certain aspects that are common for both equations. In particular, we calculate the enhancement of the proposed localization and/or decoherence effects, which would take place if one could lower the conventional length-cutoff and resolve the mass density on the interatomic scale.

We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wavefunctions on the universal covering space of the configuration space.

Within the framework of the theory of interacting classical and quantum gases, it is shown that the atomistic constitution of gases can be understood as a consequence of (second) quantization of a continuum theory of gases. In this paper, this is explained in some detail for the theory of non-relativistic interacting Bose gases, which can be viewed as the second quantization of a continuum theory whose dynamics is given by the Hartree equation. Conversely, the Hartree equation emerges from the theory of Bose gases in the mean-field limit. It is shown that, for such systems, the time evolution of 'observables' commutes with their Wick quantization, up to quantum corrections that tend to zero in the mean-field limit. This is an Egorov-type theorem.

Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the coloured Jones polynomial. The construction is based on the complete solution of the Chern–Simons topological quantum field theory and its connection to Wess–Zumino–Witten conformal field theory. The coloured Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network quantum automaton. A quantum circuit is constructed capable of simulating the automaton and hence of computing such an expectation value. The latter is efficiently approximated using a standard sampling procedure in quantum computation.

Central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment, such as that of Joos and Zeh. Such master equations enjoy a close relationship with spontaneous localization models, like the GRW model. The aim of this paper is to present two derivations of the master equation. The first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible, focusing on physical principles and without the complications of S-matrix theory. This derivation may serve as a useful tutorial example for students attempting to learn this subject area. The second is the opposite: a very general derivation using non-relativistic many-body field theory. It reduces to the equation of the type given by Joos and Zeh in the one-particle sector, but correcting certain numerical factors which have recently become significant in connection with experimental tests of decoherence. This master equation also emphasizes the role of local number density as the 'preferred basis' for decoherence in this model.

General relativity is a deterministic theory with non-fixed causal structure. Quantum theory is a probabilistic theory with fixed causal structure. In this paper, we build a framework for probabilistic theories with non-fixed causal structure. This combines the radical elements of general relativity and quantum theory. We adopt an operational methodology for the purposes of theory construction (though without committing to operationalism as a fundamental philosophy). The key idea in the construction is physical compression. A physical theory relates quantities. Thus, if we specify a sufficiently large set of quantities (this is the compressed set), we can calculate all the others. We apply three levels of physical compression. First, we apply it locally to quantities (actually probabilities) that might be measured in a particular region of spacetime. Then we consider composite regions. We find that there is a second level of physical compression for a composite region over and above the first level physical compression for the component regions. Each application of first and second level physical compression is quantified by a matrix. We find that these matrices themselves are related by the physical theory and can therefore be subject to compression. This is the third level of physical compression. The third level of physical compression gives rise to a new mathematical object which we call the causaloid. From the causaloid for a particular physical theory we can calculate everything the physical theory can calculate. This approach allows us to set up a framework for calculating probabilistic correlations in data without imposing a fixed causal structure (such as a background time). We show how to put quantum theory in this framework (thus providing a new formulation of this theory). We indicate how general relativity might be put into this framework and how the framework might be used to construct a theory of quantum gravity.

Human languages employ constructions that tacitly assume specific properties of the limited range of phenomena they evolved to describe. These assumed properties are true features of that limited context, but may not be general or precise properties of all the physical situations allowed by fundamental physics. In brief, human languages contain 'excess baggage' that must be qualified, discarded, or otherwise reformed to give a clear account in the context of fundamental physics of even the everyday phenomena that the languages evolved to describe. The surest route to clarity is to express the constructions of human languages in the language of fundamental physical theory, not the other way around. These ideas are illustrated by an analysis of the verb 'to happen' and the word 'reality' in special relativity and the modern quantum mechanics of closed systems.

We stress the notion of statistical experiment, which is mandatory for quantum mechanics, and recall Ludwig's foundation of quantum mechanics, which provides the most general framework to deal with statistical experiments giving evidence for particles. In this approach particles appear as interaction carriers between preparation and registration apparatuses. We further briefly point out the more modern and versatile formalism of quantum theory, stressing the relevance of probabilistic concepts in its formulation. At last we discuss the role of macrosystems, focusing on quantum field theory for their description and introducing objective state parameters for them.

A class of theories alternative to standard quantum mechanics, including that of Ghirardi et al ('GRWP'), postulates that when a quantum superposition becomes amplified to the point that the superposed states reach some level of 'macroscopic distinctness', then some non-quantum-mechanical principle comes into play and realizes one or other of the two macroscopic outcomes. Without specializing to any particular theory of this class, I ask how far such 'macrorealistic' theories are generically constrained, if one insists that the physical reduction process should respect Einstein locality, by the results of existing EPR-Bell experiments. I conclude that provided one does not demand that the prescription for reduction respects Lorentz invariance, at least some theories of this type, while in principle inevitably making some predictions that conflict with those of standard quantum mechanics, are not refuted by any existing experiment.

In 1935, Einstein, Podolsky and Rosen raised the issue of the completeness of the quantum description of a physical system. What they had in mind is whether or not the quantum description is informationally complete, in that all physical features of a system can be recovered from it. In a collapse theory such as the theory of Ghirardi, Rimini and Weber, the quantum wavefunction is informationally complete, and this has often been taken to suggest that according to that theory the wavefunction is all there is. If we distinguish the ontological completeness of a description from its informational completeness, we can see that the best interpretations of the GRW theory must postulate more physical ontology than just the wavefunction.

The violation of the Noether relation between symmetries and charges is reduced to the time dependence of the charge associated with a conserved current. For the U(1) gauge symmetry a non-perturbative control of the charge commutators is obtained by an analysis of the Coulomb charged fields. From this, in the unbroken case we obtain a correct expression for the electric charge on the Coulomb states, its superselection and the presence of massless vector bosons; in the broken case, we obtain a general non-perturbative version of the Higgs phenomenon, i.e. the absence of massless Goldstone bosons and of massless vector bosons. The conservation of the (gauge-dependent) current associated with the U(1) axial symmetry in QCD is shown to be compatible with the time dependence of the corresponding charge commutators and a non-vanishing η' mass, as a consequence of the non-locality of the (conserved) current.

In this volume in honour of GianCarlo Ghirardi, I discuss my involvement with ideas of dynamical collapse of the state vector. Ten problems are introduced, nine of which were seen following my initial work. Four of these problems had a resolution in GianCarlo Ghirardi, Alberto Rimini and Tullio Weber's spontaneous localization (SL) model (which added one more problem). This stimulated a (somewhat different) resolution of these five problems in the continuous spontaneous localization (CSL) model, in which I combined my initial work with SL. In an upcoming volume in honour of Abner Shimony, I shall discuss the status of the remaining five post-CSL problems.

I share some thoughts on the future of unified models.

When suitably generalized and interpreted, the path integral offers an alternative to the more familiar quantal formalism based on state vectors, self-adjoint operators and external observers. Mathematically one generalizes the path-integral-as-propagator to a quantal measure μ on the space Ω of all 'conceivable worlds', and this generalized measure expresses the dynamics or law of motion of the theory, much as Wiener measure expresses the dynamics of Brownian motion. Within such 'histories-based' schemes new and more 'realistic' possibilities open up for resolving the philosophical problems of the state-vector formalism. In particular, one can dispense with the need for external agents by locating the predictive content of μ in its sets of measure zero: such sets are to be 'precluded'. But unrestricted application of this rule engenders contradictions. One possible response would remove the contradictions by circumscribing the application of the preclusion concept. Another response, more in the tradition of 'quantum logic', would accommodate the contradictions by dualizing Ω to a space of 'co-events' and effectively identifying reality with an element of this dual space.

Three postulates are discussed: first that well-defined properties cannot be assigned to an isolated system, second that quantum unitary evolution is atemporal, and third that some physical processes are never reversed. It is argued that these give useful insight into quantum behaviour. The first postulate emphasizes the fundamental role in physics of interactions and correlations, as opposed to internal properties of systems. Statements about physical interactions can only be framed in a context of further interactions. This undermines the possibility of objectivity in physics. However, quantum mechanics retains objectivity through the combination of the second and third postulates. A rule is given for determining the circumstances in which physical evolution is non-unitary. This rule appeals to the absence of spacetime loops in the future evolution of a set of interacting systems. A single universe undergoing non-unitary evolution is a viable interpretation.

I am concerned with two views of quantum mechanics that John S Bell called 'unromantic': spontaneous wavefunction collapse and Bohmian mechanics. I discuss some of their merits and report about recent progress concerning extensions to quantum field theory and relativity. In the last section, I speculate about an extension of Bohmian mechanics to quantum gravity.

The basic concept of the two-state vector formalism, which is the time symmetric approach to quantum mechanics, is the backward evolving quantum state. However, due to the time asymmetry of the memory's arrow of time, the possible ways to manipulate a backward evolving quantum state differ from those for a standard, forward evolving quantum state. The similarities and the differences between forward and backward evolving quantum states regarding the no-cloning theorem, nonlocal measurements and teleportation are discussed. The results are relevant not only in the framework of the two-state vector formalism, but also in the framework of retrodictive quantum theory.

We discuss several proposals for astrophysical and cosmological tests of quantum theory. The tests are motivated by deterministic hidden-variables theories, and in particular by the view that quantum physics is merely an effective theory of an equilibrium state. The proposed tests involve searching for nonequilibrium violations of quantum theory in: primordial inflaton fluctuations imprinted on the cosmic microwave background, relic cosmological particles, Hawking radiation, photons with entangled partners inside black holes, neutrino oscillations and particles from very distant sources.