The overarching concern of my research program is the form and character of formulations of quantum theory general enough both to pose quantum questions in space and time (as opposed to merely at a moment of time), and to encompass a quantum theory of spacetime — quantum gravity.
A natural framework for such investigations is provided by the generalized decoherent-histories quantum mechanics first introduced by J.B. Hartle and further developed by C.J. Isham and others (including myself.) A distillation of the predictive structure of quantum theory to only its most essential features — superposition of states, and consistent assignment of probabilities to possible physical histories — the mathematical structure of generalized quantum theory is in essence a generalization of the algebraic formulation of quantum mechanics and quantum logic to encompass an explicit measure of the quantum interference between possible histories: the decoherence functional. The formulation of any specific quantum theory then amounts to a specification of the possible histories of a system, and of the system's decoherence functional. It is in these choices that all the important physical questions lie.
My work in this area has several distinct threads. The first is an interest in the formal mathematical structure of generalized quantum mechanics in a C*-algebra setting. Initiated by Chris Isham and collaborators, this research has placed generalized quantum theory on a firm mathematical footing. Work I have already completed in this field includes an extensive study of "the geometry of decoherence," a mathematical analysis of the geometric structure associated with the decoherence functional, the generalization of the algebraic notion of quantum state that is the central structural element of generalized quantum theory. (An earlier version of this research has been posted to arXiv.org, and is currently being readied for publication [II].) The tools developed in the course of that study provide the foundation necessary for planned further investigations into the character of physically reasonable choices of decoherence functional. Preliminary work on some of these questions has already begun. For example, my collaborators and I have demonstrated within the framework of generalized quantum theory the existence of an inequality constraining the existence of "quantal hidden variables" that parallels and generalizes the Bell inequalities, which constrain the existence of classical hidden variables, and indeed are violated experimentally. Experimental violation of this generalized inequality would falsify quantum mechanics itself, as well as a wide class of generalizations of it — specifically, any theory described by a strongly positive decoherence functional. We have also explored the relations between the existence of such an inequality with the causality and locality properties of quantum theory [IV].
More recently, I have been exploring the connection between the consistency of histories and Werner Heisenberg's famous uncertainty principle, possibly the central meme of quantum theory. The essence of the uncertainty principle is that it is not possible to know everything at once about a physical system that our experience with classical, macroscopic physics suggest we should be able to know. There is a deep connection between this general principle and the "decoherence" or "consistency" of the corresponding histories — the condition that determines (via the system's decoherence functional) whether or not physically meaningful probabilities can be assigned to those histories. Much of this work has been in collaboration with Le Moyne College undergraduate students Elliot Connors and Adam Lemke.
The second major thread of my research activity is the application of these ideas to quantum cosmology. In a collaboration with the University of California's J.B. Hartle, we have developed the complete predictive framework for a fully four-dimensional quantum theory of homogeneous cosmologies, establishing rigorous versions of traditional heuristics for assessing the physical content of recollapsing quantum cosmological models [III]. Building on this foundation, my collaboration with Parampreet Singh — formerly of the Perimeter Institute for Theoretical Physics, now in the Physics Department at Louisiana State University — has concerned itself with framing model theories of quantum gravity in this way. So far we have succeeded in constructing the decoherence functional for a homogeneous, isotropic, Friedmann-Robertson-Walker cosmological model in a Wheeler-DeWitt quantization [V]. We have used the new decoherence functional to demonstrate conclusively that these models are inevitably singular — that is, they will invariably begin or end in what is colloquially termed "the big bang".
We have similarly constructed the decoherence functional for the same physical model, but in a so-called "loop quantization" rooted in the approach to the quantization of general relativity known as "loop quantum gravity". In this quantization we rigorously confirm previous strong evidence from other workers that the singularity is actually removed — in this quantization, there is no "big bang". We plan to apply the same techniques are now being to other models of physical interest, in particular to study the robustness of the prediction of removal of the cosmological singularity. Work is also underway to construct the decoherence functional for a spin foam quantization of symmetric cosmological models along similar lines. An additional investigation will be a much deeper pursuit of the relationship between histories-type frameworks and so-called "relational observables" first revealed by work on the Wheeler-DeWitt model.
This program is of importance because it is actively developing the foundation for an explicit, rigorous, internally consistent framework for extracting predictions from quantum gravitational theories — theories for which an appeal to external classical measurements is simply not available since such theories are putatively capable of modeling the universe as a whole. Robust strategies will therefore be essential for making internally consistent sense of emerging candidates for a quantum theory of gravity, be that theory loop quantum gravity, string theory, causal dynamical triangulations, causal sets, or something else. Indeed, as we have shown, in the absence of a coherent predictive framework it is easy to be misled by a superficial reading of a theory's amplitudes. Something like a decoherent histories framework will ultimately prove to be an essential element of any fundamental quantum theory of gravity. The models worked out so far provide the template for future development of the theory and applications to more complex models.
An additional thread of my research program is an investigation into the nature of the partitions of all possible spacetime paths which are suitable to characterize approximately classical behaviour in a functional integral formulation of non-relativistic quantum mechanics, and a concomitant theoretical and numerical exploration of the properties of how various classes of paths contribute to the value of path integral representations of particle propagators. (This includes in part exploration of the relationship between decoherence and the uncertainty principle noted above.) This is a problem inspired by the corresponding problem in the more complicated quantum-cosmological case. Work on aspects of this problem has included projects with former Hamilton College thesis student Andrew Yue, and Le Moyne College students Adam Lemke and Christopher Carson, all undergraduates.
You can find me on the arXiv and (somewhat) more completely at INSPIRE or Google Scholar — though the citation counts are a bit wonky. Further scientific entertainment and enlightenment may be found at xxx. Watch your step.
Last modified Monday, March 13, 2017 11:23 AM
Page URL: http://web.lemoyne.edu/~craigda/research.html
The views and opinions expressed on this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Le Moyne College. © David A. Craig.