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Robert Oeckl proposed this new formulation of QT in December of last year. http://arxiv.org/abs/1212.5571 I think it's important and worth learning about. It could replace Dirac form of QT for some (especially general covariant) quantizations.
Historically it derives from 1980s work by Witten Atiyah Segal on TQFT (topological qft)
The basic objects don't even need a smooth structure. What we are dealing with are not differential manifolds (or Riemannian, or pseudo Riemannian), that would be needlessly elaborate. It's more elegant and bare than that. Unless otherwise specified, the mappings only need to be continuous! The basic objects could for instance simply be a collection of 4d and 3d topological manifolds. Think of a 4d manifold as a spacetime region where a PROCESS takes place, and a 3d manifold as the hypersurface bounding the region.
Given a set of topological manifolds, for starters, he proceeds to define stuff. In particular he assigns quantum mechanical state spaces to these objects. Each manifold gets assigned a Hilbertspace (or a slight generalization thereof called a Krein space.) So far this is the same setup as Oeckl's previous General Boundary Formulation of QT. The earlier GBF can be distinguished by calling it amplitude GBF because adapted to give quantum amplitudes.
At this point Oeckl takes a new tack and focuses on the lattice of projection operators rather than the Hilbertspace itself. Positive unit-trace operators on said Hilbertspace. The resulting new GBF is distinguished by calling it positive GBF.
From what i can tell the result of this adjustment of mathematical focus is fortunate. I'll quote his introduction which explains his motives.
==quote Oeckl http://arxiv.org/abs/1212.5571 ==
The standard formulation of quantum theory (as laid out for example in von Neumann’s book [1]) relies on a fixed a priori notion of time. While this is unnatural from a special relativistic perspective, it is not irreconcilable, as shown by the success of quantum field theory. It flatly contradicts general relativistic principles, however. This has been a key difficulty in bringing together quantum theory and general relativity.
In contrast, the general boundary formulation (GBF) of quantum theory [2], relies merely on a weak (topological) notion of spacetime and is thus compatible from the outset with general relativistic principles. The GBF is indeed motivated by the problem of providing a suitable foundation for a quantum theory of gravity [3]. However, it is also motivated by the stunning empirical success of quantum field theory. In particular, the GBF is an attempt to learn about the foundations of quantum theory from quantum field theory [4].
So far, the GBF has been axiomatized in analogy to the pure state formalism of the standard formulation. In particular, Hilbert spaces are basic ingredients of the formalism in both cases.* The elements of these Hilbert spaces have been termed “states” in the GBF as in the standard formulation, although they do not necessarily have the same interpretation. In fact, projection operators play a more fundamental role in the probability interpretation of the GBF than “states”. This suggests to construct a formalism for the GBF where this fundamental role is reflected mathematically. This is somewhat analogous to the transition from a pure state to a mixed state formalism in the standard formulation, where in the latter also (positive normalized trace-class) operators play a more fundamental role.
There are various motivations for introducing such a formalism. One is operationalism. As already alluded to, the new formalism is intended to bring to the forefront the objects of more direct physical relevance, eliminating operationally irrelevant structure and information. It turns out that this leads to a more simple and elegant form of the probability interpretation. At the same time, the positivity of probabilities becomes imprinted on the formalism in a rather direct way, via order structures on vector spaces. Therefore we term the new formalism the positive formalism. In contrast we shall refer to the usual formalism as the amplitude formalism. A consequence of the elimination of superfluous structure is a corresponding widening of the concept of a quantum theory. We expect this to be beneficial both in understanding quantum field theories from a GBF perspective as well as in the construction of completely new theories, including approaches to quantum gravity. Another motivation is the opening of the GBF to quantum information theory. In particular, the positive formalism should facilitate the implementation of general quantum operations in the GBF as well as the introduction of information theoretic concepts such as entropy.
In Section 2 we expand on the motivation for the positive formalism by drawing on the analogy to the mixed state formalism in the standard formulation...
FOOTNOTE *In the presence of fermionic degrees of freedom the GBF requires the slight generalization from Hilbert spaces to Krein spaces [5].
Historically it derives from 1980s work by Witten Atiyah Segal on TQFT (topological qft)
The basic objects don't even need a smooth structure. What we are dealing with are not differential manifolds (or Riemannian, or pseudo Riemannian), that would be needlessly elaborate. It's more elegant and bare than that. Unless otherwise specified, the mappings only need to be continuous! The basic objects could for instance simply be a collection of 4d and 3d topological manifolds. Think of a 4d manifold as a spacetime region where a PROCESS takes place, and a 3d manifold as the hypersurface bounding the region.
Given a set of topological manifolds, for starters, he proceeds to define stuff. In particular he assigns quantum mechanical state spaces to these objects. Each manifold gets assigned a Hilbertspace (or a slight generalization thereof called a Krein space.) So far this is the same setup as Oeckl's previous General Boundary Formulation of QT. The earlier GBF can be distinguished by calling it amplitude GBF because adapted to give quantum amplitudes.
At this point Oeckl takes a new tack and focuses on the lattice of projection operators rather than the Hilbertspace itself. Positive unit-trace operators on said Hilbertspace. The resulting new GBF is distinguished by calling it positive GBF.
From what i can tell the result of this adjustment of mathematical focus is fortunate. I'll quote his introduction which explains his motives.
==quote Oeckl http://arxiv.org/abs/1212.5571 ==
The standard formulation of quantum theory (as laid out for example in von Neumann’s book [1]) relies on a fixed a priori notion of time. While this is unnatural from a special relativistic perspective, it is not irreconcilable, as shown by the success of quantum field theory. It flatly contradicts general relativistic principles, however. This has been a key difficulty in bringing together quantum theory and general relativity.
In contrast, the general boundary formulation (GBF) of quantum theory [2], relies merely on a weak (topological) notion of spacetime and is thus compatible from the outset with general relativistic principles. The GBF is indeed motivated by the problem of providing a suitable foundation for a quantum theory of gravity [3]. However, it is also motivated by the stunning empirical success of quantum field theory. In particular, the GBF is an attempt to learn about the foundations of quantum theory from quantum field theory [4].
So far, the GBF has been axiomatized in analogy to the pure state formalism of the standard formulation. In particular, Hilbert spaces are basic ingredients of the formalism in both cases.* The elements of these Hilbert spaces have been termed “states” in the GBF as in the standard formulation, although they do not necessarily have the same interpretation. In fact, projection operators play a more fundamental role in the probability interpretation of the GBF than “states”. This suggests to construct a formalism for the GBF where this fundamental role is reflected mathematically. This is somewhat analogous to the transition from a pure state to a mixed state formalism in the standard formulation, where in the latter also (positive normalized trace-class) operators play a more fundamental role.
There are various motivations for introducing such a formalism. One is operationalism. As already alluded to, the new formalism is intended to bring to the forefront the objects of more direct physical relevance, eliminating operationally irrelevant structure and information. It turns out that this leads to a more simple and elegant form of the probability interpretation. At the same time, the positivity of probabilities becomes imprinted on the formalism in a rather direct way, via order structures on vector spaces. Therefore we term the new formalism the positive formalism. In contrast we shall refer to the usual formalism as the amplitude formalism. A consequence of the elimination of superfluous structure is a corresponding widening of the concept of a quantum theory. We expect this to be beneficial both in understanding quantum field theories from a GBF perspective as well as in the construction of completely new theories, including approaches to quantum gravity. Another motivation is the opening of the GBF to quantum information theory. In particular, the positive formalism should facilitate the implementation of general quantum operations in the GBF as well as the introduction of information theoretic concepts such as entropy.
In Section 2 we expand on the motivation for the positive formalism by drawing on the analogy to the mixed state formalism in the standard formulation...
FOOTNOTE *In the presence of fermionic degrees of freedom the GBF requires the slight generalization from Hilbert spaces to Krein spaces [5].
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