相对论量子物理中一类新的Volterra型积分方程
A new class of Volterra-type integral equations from relativistic quantum physics
摘要：
本文研究相对论量子力学双粒子波函数$ψ(x_1,x_2)$的一类新的线性积分方程，其中$x_1,x_2$是时空点。在延迟相互作用的情况下，这些积分方程在时间变量上是Volterra型的，即，它们涉及从0到$t_i = x_i^0,~i=1,2$的时间积分。它们不仅在物理学中的应用方面很有趣，而且还因为以下数学特征：（a）时间和空间变量比正常的时间相关问题更加相互关联，（b）积分核是奇异的，并且这些奇异性的结构是非平凡的，（c）它们具有时间延迟。我们制定了一些这样的方程的标量波函数的例子，并证明他们的解决方案的存在性和唯一性。我们还指出开放的数学问题。
英文摘要：
Here we study a new kind of linear integral equations for a relativistic quantum-mechanical two-particle wave function $ψ(x_1,x_2)$, where $x_1,x_2$ are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to $t_i = x_i^0,~i=1,2$. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.
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