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Local tomography and the Jordan structure of quantum theory
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural
constraints on what constitutes a system, and on what constitutes a composite
system), orthodox finite-dimensional complex quantum mechanics with
superselection rules is the only non-signaling probabilistic theory in which
(i) individual systems are Jordan algebras (equivalently, their cones of
unnormalized states are homogeneous and self-dual), (ii) composites are locally
tomographic (meaning that states are determined by the joint probabilities they
assign to measurement outcomes on the component systems) and (iii) at least one
system has the structure of a qubit. Using this result, we also characterize
finite dimensional quantum theory among probabilistic theories having the
structure of a dagger-monoidal category
Minisuperspaces: Observables and Quantization
A canonical transformation is performed on the phase space of a number of
homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian)
constraint. Using the new canonical coordinates, it is then easy to obtain
explicit expressions of Dirac observables, i.e.\ phase space functions which
commute weakly with the constraint. This, in turn, enables us to carry out a
general quantization program to completion. We are also able to address the
issue of time through ``deparametrization'' and discuss physical questions such
as the fate of initial singularities in the quantum theory. We find that they
persist in the quantum theory {\it inspite of the fact that the evolution is
implemented by a 1-parameter family of unitary transformations}. Finally,
certain of these models admit conditional symmetries which are explicit already
prior to the canonical transformation. These can be used to pass to quantum
theory following an independent avenue. The two quantum theories --based,
respectively, on Dirac observables in the new canonical variables and
conditional symmetries in the original ADM variables-- are compared and shown
to be equivalent.Comment: 34 page
Symmetry and Self-Duality in Categories of Probabilistic Models
This note adds to the recent spate of derivations of the probabilistic
apparatus of finite-dimensional quantum theory from various axiomatic packages.
We offer two different axiomatic packages that lead easily to the Jordan
algebraic structure of finite-dimensional quantum theory. The derivation relies
on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit
spaces having homogeneous, self-dual cones, and formally real Jordan algebras.Comment: In Proceedings QPL 2011, arXiv:1210.029
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