Von Neumann algebras, subfactors, topology and quantum physics

给定一个因子M的有限指数的子因子N，标准不变量是n从0到无穷大的第n个的零上同调的直和。M在N上的张量幂，被视为N-N-双模。这个分次向量空间有很多结构，特别是来自简单张量积的环。这与代数几何中的“正则环”有着惊人的相似之处。事实上，它已经表明，这个正则环的完备化给出了一个正则子因子，其标准不变量与原来的标准不变量相同！这个结构将探讨其所有的细节，特别是平面代数结构的标准不变，这是用来描述和分析的标准环。各种组合结构，如阿达玛矩阵和拉丁广场产生这些典型的环，但有严重的问题，计算复杂性。 随机nxn矩阵的大n极限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。 福克空间是二次量子化的数学引擎，这个过程是使量子力学与狭义相对论相容的必要条件。有费米子和玻色子的版本。在我们的研究中，我们在子代数和代数的基础上寻求一个更一般的Fock空间。小代数对应于第二量子化的标量，大代数对应于第一量子化的Hilbert空间。人们不再能像在费米子和玻色子情况下那样自由交换粒子。但湮灭和创造算子仍然存在，并形成一个类似于投射簇的典范环的环。在我们的情况下，标准环/Fock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Fock空间的一个元素放入地图上的每个州后，将得到整个国家的输出，这是Fock空间的另一个元素。说我们的结构是一个平面代数，意味着对于可以画在纸上的每一个可能的映射，都存在这样一个在Fock空间上的运算。

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.