我們一直致力於一個宏偉計劃，其最終目標在於尋找微觀量子時空的模型，並建立其動力學理論，並已取得成功的第一套突破性成果，將量子力學描述為在量子物理空間上的粒子動力學，且其各個方面都正確給出牛頓極限。我們獨特的方法基於量子相對論的觀點。時空模型是相對論群的表示，C*-群代數的相應表示給出了可觀測量的代數。時間演化自然地作為可觀察代數的自同構流獲得，它是希爾伯特空間上的酉流匹配的海森堡圖像。相對論的對稱收縮在所有方面都準確地給出牛頓結果近似。這基於C*-群代數的方法自然適用於我們的量子相對論圖像的任何更深或更高層次的設置。我們的結果建立了量子相空間的位形部分和動量部分，就像閔可夫斯基時空中的空間和時間一樣，只能有在牛頓近似才可單獨處理。而且即使不考慮重力，物理時空也存在非交換性。我們還基本上完成了相應的洛倫茲協變量子力學的理論表述及其量子時空模型，它的對稱收縮近似給出我們對量子力學和量子物理空間以及洛倫茲協變的和牛頓的古典理論。另一方面，我們成功地給出了量子可觀量代數的非交換幾何圖像，證明它正是量子相空間的辛幾何，六個位置和動量算符是作為無限（實數）維投影希爾伯特空間的一套非交換坐標，並明確給出了坐標變換描述。通過成功引入可觀測量的非交換值及其可以看作一組無限數量的實數的描述，建立的其概念一致性。非交換值包含關於固定態下可觀測量的超過重複馮·諾依曼測量的整個統計分佈之完整信息，並且可以通過實驗判定。當前申請計劃重點放在兩個方面。一方面是將我們的理論新觀點和建構應用於某些實驗環境，包括研究自旋的複合系統或粒子，嘗試提取有用的新信息甚至新的物理預測。後者對於我們的洛倫茲協變理論將特別有趣，該理論在理論結構上和應用上都與通常的“相對論”量子力學有所不同。第二個方面是開始探索具有更多更全面非交換性的所謂普朗克尺度物理的相應表述。我們將使用我們的對稱群表示框架進行向上的工作，以找到正確的表示形式，該表示形式將通過嚴格的收縮近似得出當前的結果。 ;We have been working on a very aggressive grand program with the big final goal of finding the right model for the deep microscopic quantum spacetime and build the theory of its dynamics, and achieved the first set of breakthrough results successfully formulating quantum mechanics as a theory of particle dynamics on a quantum model of the physical space all aspects of which go to the Newtonian limit. Our unique approach is based on a quantum relativity perspective. The spacetime model is a representation of the relativity group. The corresponding representation of the group C*-algebra gives the algebra of observables. Time evolution is obtained naturally as an automorphism flow of the observable algebra as the Heisenberg picture matched to the unitary flow on the Hilbert space generated by the energy observable. The contraction limit exactly retrieves the Newtonian results in all aspects. More importantly, the approach as based on the group C*-algebra is naturally applicable to settings at any deeper or higher up levels of our quantum relativity picture. Our results establish the configuration part and the momentum part of the quantum phase space as like space and time in Minkowski spacetime, parts which can be handled separately only as the Newtonian approximation. And the noncommutativity nature of the physical space manifests itself even without considering gravity. We have also essentially finished the formulation of a Lorentz covariant version of quantum mechanics and its quantum spacetime model along the line, giving as its contraction limits our formulation of the quantum mechanics and quantum physical space as well as the Lorentz covariant and Newtonian classical theories. On the complementary side, we have succeeded in giving a picture of the noncommutative geometry of the quantum observable algebra as exactly the symplectic geometry of the usual quantum phase space. The position and momentum operators serve as a set of six noncommutative coordinates of the otherwise infinite (real) dimensional projective Hilbert space with the coordinate transformation description explicitly presented. Conceptual consistency is established by the successful introduction of the notation of a noncommutative value of an observable with an explicit description of it as a set of infinite number of real numbers. The noncommutative value contains the full information about an observable on a fixed state beyond the full statistical distribution of repeated von Neumann measurements, and is experimentally accessible.The current grant project applied plans on focusing on two aspects. One is to apply our formulations and new perspectives to some experimental setting, including looking into composite system or particle with spin, trying to extract useful new information or even new physics predictions. The latter will be particularly interesting for our Lorentz covariant theory which is somewhat different from the usual ‘relativistic’ quantum mechanics both theoretically and practically.The second aspect is to start pursuing the corresponding formulation towards so-called Planckian physics with more noncommutativity. We will use our group representation framework to work backwards or upwards finding the right representation that has our current results as approximations through the rigorous contraction limits.
