The Theory of Positivist Mechanics

January 2005 update: Over the past year, I have refined the ideas contained within the TOPM much further into a much more generalized and complete scheme which I refer to as the "Relative Histories Formulation." A brief introduction to the RHF, as well as links to papers on the RHF (archived in my yahoo briefcase) in pdf format, can be found at Relative Histories Formulation. These papers will be updated periodically.

I welcome any comments, which can be sent to my yahoo! email address (straycat_md).

For an online discussion of the Relative Histories Formulation, check out my Yahoo! group: QM_from_GR

Two other excellent Yahoo! groups for discussing and learning quantum mechanics are qm2 and undernetphysics.

The ideas presented below are now outdated, and have been incorporated into the more generalized scheme of the RHF.

Yapquack, or yet another paradigm for quantum ducks, is an outgrowth of the ideas presented in TOPM and discussed in QM_from_GR. Currently (as of July 2002) a collaborative effort between Ed Huff and David Strayhorn.

Abstract and Section 1: Introduction
Section 2: The Measurement Problem of QM
Section 3: Non-Determinism in Classical Mechanics: the Classical Random-Effect Generator (C-REG)
Section 4: Conceptual Solution to the Measurement Problem
Section 5: Summary and Future Directions
References and footnotes

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I welcome any and all comments and suggestions on this theory. To contact me, email me at: David.Strayhorn@mcmail.Vanderbilt.edu. David Strayhorn

SECTION 1. Introduction

Quantum mechanics (QM) differs from classical mechanics in two major ways. First, QM is non-deterministic. That is to say, with the advent of QM, theoretical physicists have given up the attempt at predicting the outcomes of experiments with absolute certainty, and are content with dealing in probabilities. Second, the mathematics involved in the calculations of these probabilities - the "statistical nature" of QM - differs markedly from classical or "common sense" statistics. The statistical nature of QM differs from that of classical mechanics through the manifestation, in QM, of several phenomena which have no counterpart in classical mechanics: that is, the phenomena of coherence, decoherence, and the "which-way" measurement. The defining feature of a "which-way" measurement is that whether or not one makes the "which-way" measurement determines whether one observes a pattern of decoherence or coherence in the probability distribution of the associated "end-measurement." There is no classical analogue to the phenomena of coherence, decoherence, or the which-way measurement; classically, there is no basis for understanding how or why a "which-way" measurement could induce the change in probability distribution from one of coherence to one of decoherence. The inability to understand these phenomena, from more fundamental principles, is known as the measurement problem of QM, a problem which has prompted Roger Penrose to remark "that the theory [of quantum mechanics] is incomplete, wrong, or something else - it needs some further attention." (Penrose, 1997, p. 63). In this paper I make no claim that QM is "wrong" - rather, I put forth the conjecture that GR, in all its beauty, offers the "further attention" needed to solve the measurement problem.

I begin with the assumption that classical mechanics (footnote 1), like QM, is non-deterministic; that is, contrary to the currently accepted view, classical mechanics injects a certain degree of "randomness" into nature, so that the outcomes of future measurements of a system cannot always be predicted with absolute certainty, even when the initial conditions of the system are specified in full. As described in more detail in Section 3 below, this assumption is supported by recent theoretical work in GR by Kip Thorne and colleagues analyzing the trajectories of billiard balls through wormholes. This work makes the surprising discovery that in certain situations, there is more than one solution to the billiard ball's trajectory which is fully consistent with the laws of GR. This is true even when the system's initial conditions are fully specified. The number of trajectories is called the "multiplicity" of the problem and may be finite or infinite, depending on the system under consideration. For a problem with multiplicity > 1, the best that classical mechanics can do is to predict that one of the solutions will be observed, without being able to predict which solution will be observed. It thus becomes necessary to retreat into a statistical interpretation of classical mechanics regarding problems with multiplicity > 1, where the best we can do is to predict the probabilities of observing certain outcomes. In order to calculate these probabilities - that is, in order to generate an understanding of the "statistical nature" of classical non-determinism - novel principles must be devised for the intrepretation of problems with multiplicity > 1. The goal of this paper is to provide a conceptual framework for the interpretation of the statistical nature of such classical non-deterministic problems. This is accomplished through careful application of several "common-sense" principles, which I have named 1) the principle of equal probability and 2) the principle of logical positivism. Briefly stated, the principle of equal probability argues that, by symmetry, each solution to a problem with multiplicity = N > 1 is as likely to be observed as any other, i.e., each solution has odds of being observed = 1/N. Thus, when determining the odds of observing an outcome or set of outcomes to a problem with multiplicity > 1, it is necessary to calculate the multiplicity accurately. To determine the multiplicity of a problem, therefore, I propose the principle of logical positivism, which is founded upon the concept that what cannot be measured or observed should be excluded from the laws of physics. This concept has clearly served as a guide in the formulation of QM and GR. However, it has not to my knowledge been previously formulated as an explicit postulate or principle, and it is the further development of this concept upon which the Theory of Positivist Mechanics rests. The principle of logical positivism states that in order for a possible solution to a problem to count toward the multiplicity of that problem, it must be observationally distinct from the other outcomes. That is to say, the principle of logical positivism draws a distinction between the number of observationally distinct solutions O to a problem, and the number of mathematically distinct solutions M to a problem, since it is possible to have O < M, by stating that the multiplicity N = O rather than M. However, when one introduces a measurement capable of distinguishing between various mathematical solutions to a problem - mathematical solutions which cannot be distinguished, by observation, in the absence of that measurement - then one increases the number O of the problem, resulting in an increase in the problem's multiplicity and a change in the resulting probability distribution. Thus, a measurement of this type demonstrates the characteristics of a "which-way" measurement, with the change in probability distribution serving the function of the destruction of the interference pattern. In broad outline, this is the essence of the argument which I propose as a solution to the measurement problem.

In Section 2 I offer a brief restatement/synopsis of the measurement problem of QM. In Section 3 I discuss the assumption that GR is a non-deterministic theory, and reformulate it by positing the existence of a "classical random-effect generator" (C-REG). In Section 4, in order to analyze the statistical behavior of non-deterministic problems arising from classical mechanics, I perform thought experiments involving one or more C-REGs. Two principles, the principle of equal probability and the principle of logical positivism, are introduced for this analysis, the results of which predict phenomena that may be interpreted as the measurement-induced shift from coherence to decoherence; in this way the measurement problem of QM is solved. In Section 5 I summarize these findings and propose future avenues for the further development of this theory. One such avenue which is discussed in Section 5 is the possibility that the mathematical formalism of stochastic quantum mechanics and/or scale relativity may be adopted and integrated into the Theory of Positivist Mechanics in order to derive the mathematical formalism of QM, including the Feynman path integral, the Schrödinger equation, the uncertainty principle, and bose and fermi statistics.

Abstract and Section 1: Introduction
Section 2: The Measurement Problem of QM
Section 3: Non-Determinism in Classical Mechanics: the Classical Random-Effect Generator (C-REG)
Section 4: Conceptual Solution to the Measurement Problem
Section 5: Summary and Future Directions
References and footnotes

Note on recent updates: I have recently come across the published papers of Dr. Mark Hadley, who proposes a theory that is VERY similar to mine. I have included references to his work in my references - the main one is (Hadley, 1997). Dr. Hadley's work is similar to mine in that it is inspired by Kip Thorne's work involving closed timelike curves (CTCs), in which Dr. Thorne discovered that spacetimes with CTCs may evolve in a non-unique fashion. I find Dr. Hadley's proposition of modelling particles as 4-geons to be particularly appealing. However, the principle of logical positivism, which is the centerpiece of my theory, sets the TOPM apart from Dr. Hadley's work. See section 5.5 for some extensive recent additional work (still in progress).

I am in the process of archiving the current work on my theory while I attempt to formulate some of the mathematics necessary for its further development. Let me present here some of what I am attempting to do.

In QM, the state of a system is specified completely by its wavefunction, which is an element of a Hilbert space. In the Theory of Positivist Mechanics, as well as in a similar theory published by Mark Hadley (see the references section for links to his work), the full general relativistic description of the state of a system requires, not just a single manifold (with a prescribed topology and metric), but rather an entire set of manifolds. Thus, I postulate that a set of manifolds should play the same role in GR as the wavefunction does in QM - that is, they are alternate (but complementary) ways of providing the full specification of the state of any given system. If this postulate is true, then it seems reasonable to suppose that the mathematical manipulations and constraints which can be imposed upon wavefunctions in QM would have their counterparts in GR. In particular, let us consider that in QM, the wavefunction is an element of a Hilbert space. This means, for instance, that wavefunctions can be subject to various operations (addition, scalar multiplication, etc), and that an inner product is defined which maps two wavefunctions onto the real numbers. Might it be possible to define analogous relationships (addition, scalar multiplication, inner product, etc.) which would apply to sets of manifolds? That is, would it be possible to construct a new type of space, similar in many respects to a Hilbert space, in which the individual element is, rather than a wavefunction, a set of manifolds (not an individual manifold, but a set)? And if so, would this new type of mathematics have application to physics? This is the question to which I am currently devoting my efforts. My plan is to learn more about how one goes about the construction of a Hilbert space, and then to do the same thing where the elements are sets of manifolds instead of square-integrable complex functions. Some questions immediately come to mind: How should I define "addition" of two sets of manifolds? How should I define "multiplication?" How should I define the "inner product" between two sets of manifolds? What is the analogue of a "hermitian operator" in my newly defined space? In what sense could two elements of this space (two sets of manifolds) be said to be "linearly independent?" More generally, what would be a basis of this space? (And don't forget the dual basis as well!)

I emailed Dr. Hadley and asked whether anyone has developed any mathematics along these lines, and if so, whether he thinks they would apply to his theory. His reply was that this idea "sounds fine in principle. I have not heard of anyone doing it." So, in accord with my own Quixotic nature, I will attempt the daunting task of developing a new branch of mathematics! Wish me well!

Preliminary thoughts on the inner product. Let us suppose that we have a system composed of a spin 1/2 particle, P, and a measurement apparatus (a Stern-Gerlach apparatus) set up to measure the x-component of the spin of particle P. Before the measurement, the state of the entire system (particle plus apparatus) is described by a set of manifolds M. This set M may be described as a union of subsets M+ and M-, such that M+ is the set of manifolds where the particle will be measured to be in the +1/2 spin state; likewise for the subset M-. What is the probability that the particle will be found to have spin +1/2? How do you calculate this? At first glance, one might suppose that you should count the number of manifolds in M+ and divide by the number of manifolds in the set M. (In this case, the answer should be 1/2, or 50% likelihood, but in the future we might pose more complex questions like, what is the likelihood that the particle will be found within a specified interval delta-x?) However, it may be readily imagined that the "number of manifolds" within a given set may be some high order of infinity, so that this process of counting manifolds and dividing them won't work. Let me make an analogy. Suppose we have a machine which produces numbers at random between 0 and 100, and we want to calculate the probability that any given number will be in the interval [0,10]. What we could do is count the number of numbers in the interval [0,10], and divide by the number of numbers in the interval [0,100], and say that that is our answer (10%, obviously). Well, this would work fine if the number generator produces only integers, or at least if it spits out numbers which can be mapped onto integers and are therefore countable. But what if it produces real numbers? In that case, the number of numbers between 0 and 10 is the same as the number of numbers between 0 and 100 (as demonstrated by the fact that the mapping y = 10x from [0,10] to [0, 100] is one-to-one onto). Thus, what we need to do is to define the length of an interval; this way, we can say that the probability of getting a result along the interval [0,10] is defined as the length of that interval divided by the length of the interval [0,100]. In an analogous fashion, we must define the length of a set of manifolds, so that - returning to my previous example - the odds of finding the particle in state P+ will be the length of the set M+ divided by the length of the set M. Or, more to the point, the probability will be the projection of M+ onto M. This will be the role of the inner product between two sets of manifolds. How can this be defined mathematically? I'm not sure! I need to get those creative juices flowing!

Last updated 05 July 2002.

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