可觀測量的非交換值在理論和實驗中的應用(2/2)

Project Details

Description

Our long term grand program on Quantum Relativity and Quantum Spacetimehas reached a comprehensive success of a first stage, giving an intuitivenoncommutative geometric picture of quantum mechanics. We illustrate that thenoncommutative geometry of the quantum observable algebra is exactly thesymplectic geometry of the usual quantum phase space. The latter as the infinite(real) dimensional projective Hilbert space can be described by the six positionand momentum operators as noncommutative coordinates, with the coordinatetransformation structure given explicitly for the full differential structure. Themathematical consistency is grounded on a key new conceptual notion of thenoncommutative value of a quantum observable, which can be described by a setof infinite number of real numbers, giving full information about the physicalquantity on a fixed state beyond that of the full statistical distribution on repeatedvon Neumann measurements. The product of the noncommutative values for twoobservables is exactly the noncommutative value for the product observable. It isthe first time in human science we go beyond the modeling notion of a physicalquantity having a value as a real number, and illustrate a noncommutativegeometric picture of the physical space, for a well-established theory instead ofjust some speculative models. It also gives an intuitive picture for quantummechanics. The grant project here applying focuses on using the revolutionary notion to understand better the theoretical and experimental aspects of quantummechanics, including our newly formulated Lorentz covariant version The latter isa new theory of quantum mechanics, for a spin zero particle, of pseudo-Hermitiannature with a Minkowski metric operator reflecting a consistent notion of theMinkowski nature of spacetime seen from the noncommutative coordinateobservables. As a representation of the symmetry, it is pseudo-unitary exactly inthe same way the classical Minkowski spacetime is. New physics predictionsdistinguishing the theory from others in the literature may be obtained. However,we would have to obtain the corresponding theories for the spin 1/2 and 1 casesfirst, before going on to study meaningful applications. The quantum observable algebra has been taken as essentially a deformationof the classical one, hence with a one-to-one correspondence between theelements. Each distinct quantum observable has a unique classical counterpart.One can however argue that this should not be the case. As the classical theoryis only an approximation to the quantum one, it is logically possible that twodistinguishable quantum ‘observables’ may have the same classical limit. Thedifference would be a kind of ultra-quantum 'observables'. The picture of theobservables as functions of real number coordinates of the projective Hilbertspace with noncommutative values gives a handle for us to study the problem. In relation to experiments, composite systems have to be studied. Studying thetwo particle system with our approach, for example, may lead to an answer to theEPR-paradox and a plausible notion of noncommutative locality to be clarified.Possible collaborative efforts with experimental groups will also be looked into.
StatusFinished
Effective start/end date1/08/2231/07/23

Keywords

  • Quantum Relativity
  • Quantum Spacetime
  • Fundamental (Particle) Dynamics
  • Symmetry Contraction
  • Noncommutative Geometry
  • Quantum Geometry
  • Observable Algebra
  • New Conceptual Picture of Quantum Mechanics
  • Lorentz Covariant Quantum Mechanics
  • Noncommutative Values of Physical Quantities

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