论文标题
Feynman-kac公式用于订单$ \ leq 1 $和非交换几何形状的扰动公式
Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry
论文作者
论文摘要
让$ q $是复杂的公制矢量捆绑包$ \ mathscr {e} \ to \ mathscr {m} $的订单$ \ leq 1 $的差异操作员。在非常轻微的规律性假设上，$ q $保证$ \ nabla^{\ dagger} \ nabla/2+q $生成一个holomorphic semigroup $ \ mathrm {e}^{ - zh^{ - zh^{\ nabla} $γ_{l^2}（\ Mathscr {m}，\ Mathscr {e}）$（其中$ z $通过一个包含$ [0，\ infty）$的复杂扇区运行，我们证明了明显的Feynman-kac类型式的$ \ \ \ \ m i}} $ t> 0 $，概括标准的自我伴侣理论，其中$ q $是自动追捕零订单运算符。对于紧凑型$ \ Mathscr {m} $，我们将此公式与berezin集成相结合，以推导Feynman-kac类型公式用于运算符$$的操作员痕迹\ Mathrm {tr} \ left（\ widetilde {v} \ int^t_0 \ mathrm {e}^{ - sh^{ - sh^{\ nabla} _ {v}} p \ mathrm {e} s \ right），$ $，其中$ v，\ widetilde {v} $是零订单，$ p $是订单$ \ leq 1 $。然后，这些公式用于获得近端chern特征（JLO-Cocycle的差异分级扩展）的概率表示，即紧凑型甚至二维的Riemannian Spin歧管，与duclistermaat-Heckmann neriald and loop and loop and On an loop and On no loop and On no loop and On no sone confucial come cy the Cyclity of Cyclit sinology结合起来起着至关重要的作用。
Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$ generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in $Γ_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard self-adjoint theory where $Q$ is a self-adjoint zeroth order operator. For compact $\mathscr{M}$'s we combine this formula with Berezin integration to derive a Feynman-Kac type formula for an operator trace of the form $$ \mathrm{Tr}\left(\widetilde{V}\int^t_0\mathrm{e}^{-sH^{\nabla}_{V}}P\mathrm{e}^{-(t-s)H^{\nabla}_{V}}\mathrm{d} s\right), $$ where $V,\widetilde{V}$ are of zeroth order and $P$ is of order $\leq 1$. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat-Heckmann localization formula on the loop space of such a manifold.
