Strict deformation quantization of a particle in external gravitational and Yang-Mills fields
Strict deformation quantization of a particle in external gravitational and Yang-Mills fields
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外部引力场和杨米尔斯场中粒子的严格形变量子化
DOI：
10.1016/0393-0440(93)90010-c
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发表时间：
1993
影响因子：
1.5
通讯作者：
N. P. Landsman
中科院分区：
文献类型：
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作者：
N. P. Landsman
Rieffel的“严格变形量子化”概念的一个修改被应用于在外部规范场中的任意黎曼流形Q上运动的粒子，即Q上的主H丛P上的联络。因此Poisson代数A0 = C 0（（T <$P）/H）被变形为L2（P）上H不变紧算子的C <$-代数A= K（L2（P））H，它同构于K（L2（Q））<$C <$（H），涉及H的群代数.普朗克常数h是一个真正的数，而不是一个形式的展开参数，并且在极限h → 0时，在一个定义明确的解析意义上，逆算子和逆算子分别收敛于泊松括号和逐点乘积。这种变形可以用李群胚和代数胚来解释，因为A 0是李代数胚（TP）/H的泊松代数，而A是丛（P，Q，H）的规范群胚的C-代数。我们从形式主义的角度讨论的其他主题是维格纳函数，量子化的哈密顿量以及位置和动量（包括它们的域）。
An adaptation of Rieffel's notion of “strict deformation quantization” is applied to a particle moving on an arbitrary Riemannian manifold Q in an external gauge field, that is, a connection on a principal H-bundle P over Q. Hence the Poisson algebra A 0= C 0 ((T∗ P)/H) is deformed into the C∗-algebra A= K (L 2 (P)) H of H-invariant compact operators on L 2 (P), which is isomorphic to K (L 2 (Q))⊗ C∗(H), involving the group algebra of H. Planck's constant h ̶ is a genuine number rather than a formal expansion parameter, and in the limit h ̶→ 0 commutators and anti-commutators converge to Poisson brackets and pointwise products, respectively, in a well-defined analytic sense. This deformation can be interpreted in terms of Lie groupoids and algebroids, as A 0 is the Poisson algebra of the Lie algebroid (TP)/H, whereas A is the C∗-algebra of the gauge groupoid of the bundle (P, Q, H. Other topics we discuss from the point of view of our formalism are Wigner functions, and the quantization of the Hamiltonian as well as position and momentum (including their domains).