IDENTITY AND THE FOUNDATIONS OF QUANTUM THEORY

As shown in a previous paper 2, the combination of complementarity logic and probability calculus is sufficient to establish the essential features of the general formalism of quantum theory, known as statistical transformation theory. The advantages of this way of founding quantum theory as compared to other procedures lie in the following: (1) The complex-valuedness of the state vectors (unitary metric) is seen to be a direct consequence of complementarity logic.3 (2) The concept of physical quantity is reduced to the more general concept of physical property. Hence it can be proved that the operators representing physical quantities must be hypermaximal. (3) The existence of a Hamiltonian is not required.

For the last decade I have been demonstrating that many of the so-called paradoxes generated by the Copenhagen interpretation of quantum mechanics have less puzzling analogs in nonlinear dynamics and chaos theory. This raises questions about the possibility of nonlinearities inherent in the foundations of quantum theory. Since most of us do not think intuitively with nonlinear logic, I take this opportunity to dwell on several peculiarities of nonlinear dynamics and chaos: nonlinear logic itself and the possible connection of infinite nonlinear regressions with free will. Superficially, nonlinear dynamics can be every bit as counterintuitive as quantum theory; yet, its apparent paradoxes are more amenable to logical analysis. As a result, using nonlinear dynamics to resolve quantum paradoxes winds up being more straightforward than many of the alternative interpretations currently being formulated to replace the orthodox interpretation. Chaos theory could be a candidate for bridging the gap between the determinism so dear to Einstein and the statistical interpretation of the Copenhagen School: For deterministic chaos is indeed deterministic; however, intrinsic limitations on precision in measuring initial conditions necessitate analyzing it statistically. Einstein and Bohr both could have been correct in their debates.

The place of the concept of indistinguishability in the formation of quantum theory is examined, including particularly Planck’s earliest formulations of the quantum hypothesis, as well as the use of this concept in the developments surrounding the new quantum theory.

The universe is made of stories, not of atoms.1 —MURIEL RUKEYSER Graduate work in both physics and history taught me to use highly specialized research methods to rigorously search out truth and eradicate myths. In spring 2012, I brought this mindset with me as I sat down for lunch with physicist Pierre Hohenberg at the Apple Restaurant near Manhattan’s Washington Square Park. Pierre was a brilliant physicist and a family friend. Toward the end of his life, he was particularly invested in work on the foundations of quantum theory....

The multipartite correlations derived from local measurements on some composite quantum systems are inconsistent with those reproduced classically. This inconsistency is known as quantum nonlocality and shows a milestone in the foundations of quantum theory. Still, it is NP hard to decide a nonlocal quantum state. We investigate an extended question: how to characterize the nonlocal properties of quantum states that are distributed and measured in networks. We first prove the generic tripartite nonlocality of chain-shaped quantum networks using semiquantum nonlocal games. We then introduce a new approach to prove the generic activated nonlocality as a result of entanglement swapping for all bipartite entangled states. The result is further applied to show the multipartite nonlocality and activated nonlocality for all nontrivial quantum networks consisting of any entangled states. Our results provide the nonlocality witnesses and quantum superiorities of all connected quantum networks or nontrivial hybrid networks in contrast to classical networks.

Kirkwood-Dirac classical pure states

Bell’s inequality really beats all records on publications, citations, discussions and controversies. As an organizer of a few international conferences on foundations of quantum theory I was really disappointed by stormy debates on Bell’s inequality. I could not find any problem in quantum foundations which can be compared with Bell’s inequality by intensiveness of discussions and strongness of reactions to opponent’s views. I asked myself many times: “What is so special in Bell’s inequality?” It seems that the main source of these debates is mathematical imprecision of Bell’s considerations (and not at all physical importance of the problem).

We present a novel approach to represent ecological systems using reaction networks, and show how a particular framework called Chemical Organization Theory (COT) sheds new light on the longstanding complexity-stability debate. Namely, COT provides a novel conceptual landscape plenty of analytic tools to explore the interplay between structure and stability of ecological systems. Given a large set of species and their interactions, COT identifies, in a computationally feasible way, each and every sub-collection of species that is closed and self-maintaining. These sub-collections, called organizations, correspond to the groups of species that can survive together (co-exist) in the long-term. Thus, the set of organizations contains all the stable regimes that can possibly happen in the dynamics of the ecological system. From here, we propose to conceive the notion of stability from the properties of the organizations, and thus apply the vast knowledge on the stability of reaction networks to the Complexity-Stability debate. As an example of the potential of COT to introduce new mathematical tools, we show that the set of organizations can be equipped with suitable joint and meet operators, and that for certain ecological systems the organizational structure is a non-boolean lattice, providing in this way an unexpected connection between logico-algebraic structures, popular in the foundations of quantum theory, and ecology.

There exists a growing literature on the so-called physical Church-Turing thesis in a relativistic spacetime setting. The physical Church-Turing thesis is the conjecture that no computing device that is physically realizable (even in principle) can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, Istv\'an N\'emeti and Gyula D\'avid (2006) have shown how an appreciation of the physical Church-Turing thesis necessitates the confluence of mathematical, computational, physical, and indeed cosmological ideas. In this essay, I will honour Istv\'an's seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in N\'emeti and D\'avid (2006) might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo-Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church-Turing thesis.

Ergodic properties of one-parameter automorphism groups of operator algebras

SummaryWe explore the consequences of adjoining a symmetry group to a statistical model. Group actions are first induced on the sample space, and then on the parameter space. It is argued that the right invariant measure induced by the group on the parameter space is a natural non‐informative prior for the parameters of the model. The permissible sub‐parameters are introduced, i.e., the subparameters upon which group actions can be defined. Equivariant estimators are similarly defined. Orbits of the group are defined on the sample space and on the parameter space; in particular the group action is called transitive when there is only one orbit. Credibility sets and confidence sets are shown (under right invariant prior and assuming transitivity on the parameter space) to be equal when defined by permissible sub‐parameters and constructed from equivariant estimators. The effect of different choices of transformation group is illustrated by examples, and properties of the orbits on the sample space and on the parameter space are discussed. It is argued that model reduction should be constrained to one or several orbits of the group. Using this and other natural criteria and concepts, among them concepts related to design of experiments under symmetry, leads to links towards chemometrical prediction methods and towards the foundation of quantum theory.

Martin Jesse Klein

This final volume of papers by Lord Rayleigh covers the period from 1911 to his death in 1919. The first of the Solvay Conferences in 1911 played a key role in the foundation of quantum theory. Although invited, Rayleigh did not attend. His principal achievements lay in development and consolidation across classical physics, in which he continued to work. In a 1917 paper, he used electromagnetic theory to derive a formula for expressing the reflection properties from a regularly stratified medium. In 1919, he investigated the iridescent colours of birds and insects. Rayleigh continued his long-standing participation in the Society for Psychical Research, founded in 1882 for the study of 'debatable phenomena'. One of his last publications was his presidential address to that society, which considers several highly unorthodox views and practices. He concludes by asserting the importance to scientists of maintaining open minds in the pursuit of truth.

Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.

Quantum processes on phase space