论文标题
波算子的复杂力量和洛伦兹散射空间的光谱作用
Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces
论文作者
论文摘要
我们考虑了Minkowski空间的扰动以及更一般的空间，在该空间上，Wave Operator $ \ square_g $本质上是自我偶数。我们通过功能计算来定义复杂的powers $（\ square_g-i \ varepsilon）^{ - α} $，并表明痕量密度作为$α$的meromormorthic函数存在。我们将其极点与几何量相关，特别是标量曲率。结果使我们能够制定一个光谱作用原理，该原理是Chamseddine-Connes动作的玻色子部分的简单洛伦兹模型。我们的证明结合了微局部分解估计值，包括径向传播估计值，以及对Hadamard参数的均匀估计。该论点直接在洛伦兹签名中运行，不依赖于从欧几里得环境的过渡。在超级空间的情况下，结果也是如此。
We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator $\square_g$ is known to be essentially self-adjoint. We define complex powers $(\square_g-i\varepsilon)^{-α}$ by functional calculus, and show that the trace density exists as a meromorphic function of $α$. We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine-Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments operate in Lorentzian signature directly and do not rely on a transition from the Euclidean setting. The results hold also true in the case of ultrastatic spacetimes.
