非交換（代數）幾何、基礎物理和量子資訊

Our long term grand research project, with its innovative perspectives and approach, has attained a first-stage comprehensive success in the past years. The grand project is based on a serious symmetry theoretical formulation much beyond the level of what has been done before, retrieving from the representation theories of the fundamental/relativity symmetries at the different levels essentially every aspects of the dynamical theories, with a full theory at a lower level derivable from a symmetry contraction to one above. The Lie type symmetry naturally gives a representation space with a symplectic structure and the noncommutativity of basic observables. That is complemented by our improved perspective of algebraic geometry with the revolutionary idea of the noncommutative/q-number values of physical quantities replacing the traditional real/c-number values, and identification of a piece of quantum information as such a q-number. We also have a new theory of Lorentz covariant quantum mechanics within the grand scheme, featuring a Minkowski metric operator defining a nondefinite inner product among state vectors and giving a noncommutative (1+3)-(Gel’fand-Kirillov)dimensional Minkowski spacetime. As the starting point to explore beyond the theoretical success of the first stage, the plan for the current grant proposal has a more diverse scope pushing our studies in a few different but complementary directions. One main direction is to further explore the applications of the notion of the qnumber (value) both in theoretical and practical physics. On the theoretical side, we have already picked up on the recently popular topic of quantum reference frame transformations, presented a systematic formulations of them in quite general setting and used the q-numbers to uniquely successfully described the notion of the amount transformed that encodes the reference frame dependence of ‘uncertainties’ and entanglement. We will push on with the studies to the extent of getting an optimal full formulation of the whole symmetry system of quantum reference frame transformations. This is a challenge that has not been attempted by others. Another key issue is the locality of quantum information in the Heisenberg picture we have clarified in terms of the q-numbers theoretically. Further development we are particularly interested in include the analysis of (noncommutative) spatial locality in projective measurements, like the EPR problem, in relation to a ‘relativistic’ decoherence theory. We also plan on exploring plausible picture of quantum gravity based on the notion of a nontrivial metric operator. If that works out, even just for a relatively simple case, we have a revolutionary approach to the subject matter. Other topics include exploring the physical meaning of the class of 'c-number 'observables'/functions of the quantum phase space that cannot be matched to an operator (results may give new insight about the quantum theory), extending noncommutative algebraic geometry with the q-numbers as scalars as a handle to the tangent space of a noncommutative geometry useful for pursuing quantum gravity, experimental applications of the q-number values, and the formulations of theories at the deep microscopic level (with X-X and P-P noncommutativity) symmetry contractions of which give our Lorentz covariant quantum theory.