Time — Empirical Mathematics — Quantum Theory

Michael E. Miller

Quantum field theory accurately describes the world on the finest scales to which we have empirical access. There has been significant disagreement, however,about which mathematical structures ought to be taken as constitutive of the theory, and thus over which structures should serve as the basis for its interpretation. Perturbative methods allow for successful empirical prediction but require mathematical manipulations that are at odds with the canonical approach to interpreting physical theories that has been passed down from the logical positivists. Axiomatic characterizations of the theory, on the other hand, have not been shown to admit empirically interesting models. This dissertation shows how to understand the empirical success of quantum field theory by reconsidering widely held commitments about how physical meaning accrues to mathematical structure.

H. D. Zeh

C. K. Raju

George Jaroszkiewicz

Introduction In this chapter we apply the concepts presented in Chapter 22 to a topic that has been of much theoretical interest in recent decades: causal set theory (CST). It is interesting and useful to review the status of CST relative to other foundational topics in mathematical physics that have been and remain of prime significance to empirical physics, namely, special relativity (SR), general relativity (GR), and quantum mechanics (QM). Those great frameworks are built on foundations that presuppose the continuity of space and time. Despite this, it is an empirical fact that we are surrounded by discreteness in the form of atoms, so it is natural to see how that discreteness fits in with the continuity of space and time. We have attempted in this book to show that the application of discrete principles to QM can go far beyond the basic Planck quantum of action. We have gone as far as discretizing empirical time, in the form of the stage concept. However, as far as relativity in either of its manifestations is concerned, the case for any form of discreteness has not yet been established empirically, although great efforts are being made in that direction in the program known as quantum gravity . We have particularly in mind the discretization approaches to GR of Regge (Regge, 1961) and Penrose (Penrose, 1971). Regge's approach is a classically based discretization of the Riemannian manifold concept, so has at best a generalized propositional classification (GPC) of two. Penrose's spin network paradigm started as a quantum description of spacetime and has led to much work in the area of spin networks in the hope of unifying GR and QM. However, workers in spin networks and related branches of quantum gravity have focued their attention on the mathematical complexities and hardly any on the role of observers and their apparatus, leading to a GPC classification of quantum gravity as one, that is, as mathematical physics and not physics. Claims made that such programs represent empirical physics have not been empirically substantiated to date; quantum gravity at this time remains empirically vacuous, despite the best efforts of many theorists over many decades.

Łászló A. Székely

Through the analysis of the interpretations of quantum mechanics and the theory of relativity, the paper aims to demonstrate the following thesis: the Duhem-Quine thesis \non empirical underdetermination can be extended to the claim that not only the empirical data, but also the empirical data together with the mathematical formalism of a physical \ntheory do not suffice to determine completely our theories of physical reality. As a consequence, beyond observational data and mathematical physics, cultural and social factors \nas well as the value-preferences of physicists also play role as constitutive elements of physical theories. For example, the indeterminist interpretations of quantum mechanics are not neutral but value-laden, and deterministic interpretations, which insist on classical determinism as a value, are also possible. On the other hand, the thesis of the underdetermination of physical reality by mathematical physics confirms neither radical relativism, nor extreme versions of sociology of science. On the contrary, the mathematical formalism to be interpreted is autonomous – and in this sense “objective” – to a great extent with respect to cultural, social and other non-scientific factors and, therefore, there is a firm consensus regarding it in science. \n