非对易几何中的Monge-Kantorovich度量
Towards a Monge-Kantorovich metric in noncommutative geometry
摘要：
我们调查是否之间的识别Connes的谱距离在非交换几何和Monge-Kantorovich距离的理论中的1阶最优运输-已指出的Rieffel在交换的情况下-仍然是有意义的非交换框架。为此，给定一个谱三元组（A，H，D），其中A是非对易的，我们在A的态空间上引入一个“Monge-Kantorovich”类距离W_D，取纯态间的谱距离d_D作为代价函数.我们在充分的一般性，d_D是从来没有大于W_D，并展示了几个例子中的平等实际上成立，特别是在单位两球视为状态空间的代数复杂的2 × 2矩阵。我们还讨论了双片模型（流形与C^2的乘积）中的W_D，指出了希格斯场作为一个在对角线上不为零的代价函数的可能解释。
英文摘要：
We investigate whether the identification between Connes' spectral distance in noncommutative geometry and the Monge-Kantorovich distance of order 1 in the theory of optimal transport - that has been pointed out by Rieffel in the commutative case - still makes sense in a noncommutative framework. To this aim, given a spectral triple (A, H, D) with noncommutative A, we introduce a "Monge-Kantorovich"-like distance W_D on the space of states of A, taking as a cost function the spectral distance d_D between pure states. We show in full generality that d_D is never greater than W_D, and exhibit several examples where the equality actually holds true, in particular on the unit two-ball viewed as the state space of the algebra of complex 2-by-2 matrices. We also discuss W_D in a two-sheet model (product of a manifold by C^2), pointing towards a possible interpretation of the Higgs field as a cost function that does not vanish on the diagonal.
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