Von Neumann algebras, subfactors, topology and quantum physics

给定因子 M 的有限索引的子因子 N，标准不变量是 n 从 0 到无穷大的 n 的第零上同调的直和。 M 对 N 的张量幂，被视为 N-N 双模。这个分级向量空间有很多结构，特别是来自简单张量积的环的结构。这与代数几何的“规范环”有着惊人的相似之处。事实上，已经证明，这个规范环的完成给出了一个规范子因子，其标准不变量与原来的不变量相同！我们将详细探讨该结构，特别是用于描述和分析规范环的标准不变量上的平面代数结构。各种组合结构（例如哈达玛矩阵和拉丁方）产生了这些规范环，但存在严重的计算复杂性问题。 随机 nxn 矩阵的大 n 限制也给出了这样的结构。从技术上讲，上面提到的 N 上的张量积是 Connes 张量积，我们正在研究这个张量积如何与量子物理直接相关。 福克空间是第二量子化的数学引擎，这一过程对于使量子力学与狭义相对论兼容是必要的。有费米子和玻色子版本。在我们的研究中，我们正在追求一个基于子代数和代数的更通用的福克空间。小代数对应于第二量子化中的标量，大代数对应于第一量子化希尔伯特空间。人们不能再像费米子和玻色子情况下那样自由地交换粒子。但湮灭算子和创造算子仍然存在，并形成一个类似于射影簇的规范环的环。在我们的情况下，规范环/福克空间具有更多的结构，即平面代数的结构。 粗略地说，这给出了每个地图的操作。想象一下像美国这样的国家的地图。将福克空间的一个元素放入地图上的每个州后，我们将得到整个国家的输出，这是福克空间的另一个元素。说我们的结构是平面代数意味着福克空间上的这种运算对于每一个可以在纸上绘制的可想象的地图都存在。

Given a subfactor N of finite index of a factor M, the standard invariant is the direct sum for n from 0 to infinity of the zeroth cohomology of the nth. tensor power of M over N, viewed as an N-N-bimodule. This graded vector space has a lot of structure, in particular that of a ring coming from the simple tensor product. This bears a striking resemblance to the "canonical ring" of algebraic geometry.Indeed it has been shown that the completion of this canonical ring gives a canonical subfactor whose standard invariant is the same as the original one! This structure will be explored in all its detail, especially the planar algbra structure on the standard invariant which is used to describe and analyse the canonical ring. Various combinatorial structures such as Hadamard matrices and Latin squares give rise to these canonical rings but there are serious problems of computational complexity. The large n limit of random nxn matrices also gives such structures. Technically the tensor product over N referred to above is the Connes tensor product and we are looking at how this tensor product may be directly relevant to quantum physics. Fock space is the mathematical engine of second quantization, a process that, among other things, is necessary to make quantum mechanics compatible with special relativity. There are fermionic and bosonic versions. In our research we are pursuing a more general Fock space based on a subalgebra and an algebra.The small algebra corresponds to the scalars in second quantization and the large one corresponds to the first quantized Hilbert space.One can no longer freely exchange particles as in the fermionic and bosonic situations. But annihilation and creation operators still exist and form a ring analogous to the canonical ring of a projective variety. In our situation the canonical ring/Fock space has much more structure, namely that of a planar algebra. Roughly speaking this gives operations for every map. Imagine the map of a country like the United States. Upon putting an element of the Fock space into each State on the map one would get an output for the whole country, which is another element of the Fock space. To say that our structure is a planar algebra means that such an operation on Fock space exists for every conceivable map that can be drawn on a piece of paper.