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AIP conference proceedings

The general boundary formulation of quantum theory and its relevance for the problem of quantum gravity

Daniele Colosi, Luis Arturo Ureña-López, Hugo Aurelio Morales-Técotl, Román Linares, Elí Santos-Rodríguez, Sendic Estrada-Jiménez

Published Jan 1, 2011• 3 citations
Publisher

Abstract

We present a non‐technical introduction to the general boundary formulation (GBF) of quantum theory. We emphasize the main motivation for the GBF: Formulating the quantum theory in way compatible with the principles of general relativity. We give an overview of the basic structures of the GBF, defining the notions of state space, amplitude, probability interpretation and expectation value of observables in the GBF. Finally, we briefly mention some of the results obtained so far and indicate future lines of research.

DOI
10.1063/1.3647532

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We present a non‐technical introduction to the general boundary formulation (GBF) of quantum theory. We emphasize the main motivation for the GBF: Formulating the quantum theory in way compatible with the principles of general relativity. We give an overview of the basic structures of the GBF, defining the notions of state space, amplitude, probability interpretation and expectation value of observables in the GBF. Finally, we briefly mention some of the results obtained so far and indicate future lines of research.
We give a brief introduction to the so-called general boundary formulation (GBF) of quantum theory. This new axiomatic formulation provides a description of the quantum dynamics which is manifestly local and does not rely on a metric background structure for its definition. We present the basic ingredients of the GBF, in particular we review the core axioms that assign algebraic structures to geometric ones, the two quantisation schemes so far developed for the GBF and the probability interpretation which generalizes the standard Born rule. Finally we briefly discuss some of the results obtained studying specific quantum field theories within the GBF.
We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds.Temporal transition amplitudes are generalized to amplitudes for arbitrary space-time regions.State spaces are associated to general (not necessarily spacelike) hypersurfaces.We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms.We explain how standard quantum mechanics arises as a special case.We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework.We formulate vacuum axioms and incorporate space-time symmetries into the framework.We show how the Schrödinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form.We discuss the role of operators.
The geometric form of standard quantum mechanics is compatible with the two postulates: (1) the laws of physics are invariant under the choice of experimental setup and (2) every quantum observation or event is intrinsically statistical. These postulates remain compatible within a background independent extension of quantum theory with a local intrinsic time implying the relativity of the concept of a quantum event. In this extension the space of quantum events becomes dynamical and only individual quantum events make sense observationally. At the core of such a general theory of quantum relativity is the three-way interplay between the symplectic form, the dynamical metric and non-integrable almost complex structure of the space of quantum events. Such a formulation provides a missing conceptual ingredient in the search for a background independent quantum theory of gravity and matter. The crucial new technical element in our scheme derives from a set of recent mathematical results on certain infinite-dimensional almost Kahler manifolds which replace the complex projective spaces of standard quantum mechanics.
Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics sought to give a unified picture of physical systems based on its underling geometrical structures, e.g., now, the states are represented by points of a symplectic manifold with a compatible Riemannian metric, the observables are real-valued functions on the manifold, and the quantum evolution is governed by a symplectic flow that is generated by a Hamiltonian function. In this work we will give a compact introduction to main ideas of geometric formulation of quantum mechanics. We will provide the reader with the details of geometrical structures of both pure and mixed quantum states. We will also discuss and review some important applications of geometric quantum mechanics.
Introduction Indefinite causal structure poses particular problems for theory formulation since many of the core ideas used in the usual approaches to theory construction depend on having definite causal structure. For example, the notion of a state across space evolving in time requires that we have some definite causal structure so we can define a state on a space-like hypersurface. We will see that many of these problems are mitigated if we are able to formulate the theory in a formalism-local (or F-local) fashion. A formulation of a physical theory is said to be F-local if, in making predictions for any given arbitrary space-time region, we need only refer to mathematical objects pertaining to that region. This is a desirable property both on the grounds of efficiency and since, if we have indefinite causal structure, it is not clear how to select some other space-time region on which our calculations may depend. The usual ways of formulating physical theories (the time-evolving state picture, the histories approach, and the local-equations approach) are not F-local. We set up a framework for probabilistic theories with indefinite causal structure. This, the causaloid framework, is F-local. We describe how quantum theory can be formulated in the causaloid framework (in an F-local fashion). This provides yet another formulation of quantum theory. This formulation, however, may be particularly relevant to the problem of finding a theory of quantum gravity. The problem of quantum gravity is to find a theory that reduces in appropriate limits to general relativity and quantum theory (including, at least, those situations in which those two theories have been experimentally confirmed).