The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced. The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.

"relative State" Formulation of Quantum Mechanics

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

Effect algebras and unsharp quantum logics

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

http://www.math.pitt.edu/~troy/vlad/cooper.pdf

Dynamical reduction models

The interpretational problems of quantum mechanics are considered. The way in which the standard Copenhagen interpretation of quantum mechanics deals with these problems is reviewed. A new interpretation of the formalism of quantum mechanics, the transactional interpretation, is presented. The basic element of this interpretation is the transaction describing a quantum event as an exchange of advanced and retarded waves, as implied by the work of Wheeler and Feynman, Dirac, and others. The transactional interpretation is explicitly nonlocal and thereby consistent with recent tests of the Bell inequality, yet is relativistically invariant and fully causal. A detailed comparison of the transactional and Copenhagen interpretations is made in the context of well-known quantum-mechanical Gedankenexperimente and "paradoxes." The transactional interpretation permits quantum-mechanical wave functions to be interpreted as real waves physically present in space rather than as "mathematical representations of knowledge" as in the Copenhagen interpretation. The transactional interpretation is shown to provide insight into the complex character of the quantum-mechanical state vector and the mechanism associated with its "collapse." It also leads in a natural way to justification of the Heisenberg uncertainty principle and the Born probability law ($P=\ensuremath{\psi}{\ensuremath{\psi}}^{*}$), basic elements of the Copenhagen interpretation.

The transactional interpretation of quantum mechanics

The amazing accuracy in verifying quantum effects experimentally has recently renewed interest in quantum mechanical measurement theory. In this book the authors give within the Hilbert space formulation of quantum mechanics a systematic exposition of the quantum theory of measurement. Their approach includes the concepts of unsharp objectification and of nonunitary transformations needed for a unifying description of various detailed investigations. The book addresses advanced students and researchers in physics and philosophy of science. In this second edition Chaps. II-IV have been substantially rewritten. In particular, an insolubility theorem for the objectification problem has been formulated in full generality, which includes unsharp object observables as well as unsharp pointers.

The quantum theory of measurement

1. Introduction 2. The quantum theory of measurement 3. The probability interpretation 4. The problem of objectification 5. Universality and self-referentiality in quantum mechanics Appendices Bibliography Index.

The interpretation of quantum mechanics and the measurement process

In a quantum mechanical two-slit experiment one can observe a single photon simultaneously as particle (measuring the path) and as wave (measuring the interference pattern) if the path and the interference pattern are measured in the sense of unsharp observables. These theoretical predictions are confirmed experimentally by a photon split-beam experiment using a modified Mach—Zehnder interferometer.

Unsharp particle-wave duality in a photon split-beam experiment

Philosophische Probleme der modernen Physik

I/Space and Time- 1 Formulation of the Problem- 2 The Special Theory of Relativity- 3 The Structure of Physics in the Theory of Relativity- 4 Space and Time in the Philosophy of Kant- 5 Critique of the Concept of Time in the Theory of Relativity- II/Euclidean And Riemannian Geometry- 1 Formulation of the Problem- 2 The Foundation of Euclidean Geometry- 3 The Theory of the Metric Field- 4 The Foundation of a Physical Geometry- 5 Summary- III/The Quantum-Mechanical Measurement Process- 1 The Uncertainty Relation- 2 Quantum Theory- 3 The Measuring Process- 4 The Cut- 5 The Function of the Observer in Quantum Theory- IV/The Concept of Substance- 1 The Concept of Substance in Classical Physics- 2 The Concept of Substance in Quantum Theory- 3 Objectifiability- 4 Critique of the Concept of Substance in Quantum Theory- V/The Causal Law- 1 The Concept of Causality in Philosophy- 2 The Concept of Causality in Physics- 3 The Invalidity of the Causal Law in Quantum Theory- 4 The Problem of Hidden Parameters- VI/Logic and Quantum Logic- 1 Formulation of the Problem- 2 Classical Logic- 3 The Logic of Commensurable Properties- 4 The Logic of Incommensurable Properties- 5 Probability and Quantum Logic- 6 Summary

Peter Mittelstaedt

Papers

The quantum theory of measurement

The interpretation of quantum mechanics and the measurement process

Unsharp particle-wave duality in a photon split-beam experiment

Philosophische Probleme der modernen Physik

Philosophical Problems of Modern Physics