Relative Histories Formulation of Quantum Mechanics
David Strayhorn ( http://webspace.webring.com/people/xy/yapquack/RelativeHistories.html )
Saint Louis, MO
A novel formulation of quantum mechanics, the “relative histories” formulation (RHF), is proposed based upon the assumption that the physical state of a closed system – e.g., the universe – can be represented mathematically by an ensemble E of fourdimensional manifolds W. In this scheme, any real, physical object – be it macroscopic or microscopic – can be assigned to the role of the quantum mechanical “observer.” The state of the observer is represented as a 3dimensional manifold, O. By using O as a boundary condition, we obtain E as the unique set of all W that satisfy the boundary condition defined by O. The evolution of the “wavefunction of the universe” E is therefore determined by the movement of the observer O through state space.
In a sense, the RHF is a specific instance of the consistent histories formulation, in which a single “history” is equated with a single W. However, the RHF can also be interpreted as an implementation of Einstein’s ensemble (or statistical) interpretation, which is based upon the notion that the wavefunction is to be understood as the description not of a single system, but of an ensemble of systems: in our case, an ensemble of W’s. We could, in addition, think of the RHF in terms of Everett’s relative state formulation (i.e., the multiple worlds interpretation (MWI)), in which each “world” is equated with one of the fourmanifolds W.
The mathamatical underpinnings of the RHF have been developed to an extent sufficient to demonstrate that the RHF gives rise to quantum statistics. However, the mathematical structure of the RHF is far from complete. At this stage, the primary motivation for the development of the RHF centers on the fact that it offers a constellation of interpretational advantages that are not found in any other single formulation of QM. One of these advantages is its versatility; as discussed above, the RHF is a sort of unification of many other standard formulations of QM, such as the consistent histories formulation, the MWI, Einstein’s statistical interpretation, and the FPI. In addition, the RHF is possessed of the following interpretational features: (1) a clear definition of the observer and of its space of states, and a lack of the fundamental split between observer and system that is characteristic of the Copenhagen Interpretation; (2) movement of the observer through state space that obeys classical notions of locality and probability; (3) a derivation of quantum statistics and quantum nonlocality from classical notions of probability and locality; (4) a demonstration of compatibility with general relativity; (5) an adherence to the notion that “all is geometry;” (6) the absence of a requirement for any extra “unphysical” dimensions of spacetime beyond the four of our everyday experience; and (7) a spacetime structure that is causal on the large scale. In addition, the RHF provides a simple conceptualization of the EPR argument that QM cannot be considered both local and complete. In exchange for these interpretational features, we are forced merely to accept that the structure of spacetime is acausal on the small scale.
As of January 2005, the RHF is summarized in a series of three papers that are available for download from the following site:
RHF papers.
These papers have been formatted using LaTeX to make them as readerfriendly as possible. There are three parts. Part I – Basics.pdf presents an introduction to the basic mathematical elements of the RHF, including the 4geon model of fundamental particles, and demonstrates in explicit terms that the RHF makes the same predictions as the Feynman path integral (FPI) approach. Since the FPI is an independent formulation of quantum mechanics — for example, the FPI is wellknown to provide a basis for the derivation of the Schrodinger equation — the equivalence between the RHF and the FPI implies equivalence between the RHF and quantum mechanics in general. Part II – Interpretation.pdf provides an overview of the interpretational issues surrounding the RHF, as summarized above. Part III – Further Mathematical Development.pdf is not (as of January 2005) yet ready for download and review, as I anticipate that it will undergo significant revision as I learn more about Morse theory and its application to the RHF.
I am currently (as of January 2005) in the process of soliciting an informal “peerreview” of the RHF prior to any attempt at submission for publication, even to arXiv. I anticipate this to be a slow process: the complexity of the RHF technique is about on a par as, say, the Feynman path integral (FPI) technique itself. Furthermore, an understanding of the RHF, especially its interpretational implications, requires a broad knowledge of the foundations of QM — especially regarding the inner workings of the FPI — that even many practicing physicists lack. I welcome any comments, be they from an expert or a layman, which should 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 (http://groups.yahoo.com/group/QM_from_GR)

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