具有无穷多导数的动力学：初值问题
Dynamics with Infinitely Many Derivatives: The Initial Value Problem
摘要：
无穷阶微分方程是理论物理中一类越来越重要的方程。这样的方程在弦场论中无处不在，最近也吸引了宇宙学家的极大兴趣。虽然这些方程已经在经典数学文献中进行了研究，但物理学界似乎在很大程度上没有意识到相关的形式主义。特别重要的是初始值问题的命运。在什么情况下无穷阶微分方程有一个明确的初值问题，需要多少初始数据？本文在形式算子微积分的数学框架下，研究了具有解析初值的无穷阶微分方程初值问题。这种形式主义使我们能够同时处理各种不同的非局部方程在一个单一的框架，也承认一个透明的物理解释。我们证明了无穷阶微分方程一般不容许无穷多个初始值。相反，传播算子的每个极点为最终解贡献两个初始数据。虽然可以找到无穷阶微分方程，它只允许两个初始值的定义良好的初值问题，但p-adic弦论和弦场论的动力学方程似乎都不属于这一类。然而，这两个理论都可以通过适当定义形式伪微分算子的作用而变得无重影。这个规定将理论限制在复平面中的某些轮廓内的频率，因此可以被认为是一种紫外截止。
英文摘要：
Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists. Though these equations have been studied in the classical mathematical literature, it appears that the physics community is largely unaware of the relevant formalism. Of particular importance is the fate of the initial value problem. Under what circumstances do infinite order differential equations possess a well-defined initial value problem and how many initial data are required? In this paper we study the initial value problem for infinite order differential equations in the mathematical framework of the formal operator calculus, with analytic initial data. This formalism allows us to handle simultaneously a wide array of different nonlocal equations within a single framework and also admits a transparent physical interpretation. We show that differential equations of infinite order do not generically admit infinitely many initial data. Rather, each pole of the propagator contributes two initial data to the final solution. Though it is possible to find differential equations of infinite order which admit well-defined initial value problem with only two initial data, neither the dynamical equations of p-adic string theory nor string field theory seem to belong to this class. However, both theories can be rendered ghost-free by suitable definition of the action of the formal pseudo-differential operator. This prescription restricts the theory to frequencies within some contour in the complex plane and hence may be thought of as a sort of ultra-violet cut-off.
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