For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

对于自然时间以量子状态向量的相对论粒子位置表示形式自然产生的多次波函数，薛定er方程的类似物由多个方程组成，每个方程对应一个时间变量。这就引出了一个问题，即如何证明这种PDE系统的一致性。对于具有粒子创建的理论来说，这个问题变得更加困难，因为波动函数的不同部分具有不同数量的时间变量。Petrat和Tumulka（2014）给出了这样一个模型的示例，并对其一致性进行了非严格的论证。在将紫外线截止值引入多次演化方程的创建和an灭项之后，我们在这里给出了一个严格的论点版本。这些方程式形成了耦合的PDE的无限系统。它们基于狄拉克方程，但不是完全相对论的（部分是由于截断）。我们证明了该系统的光滑解的存在性和唯一性，该光滑解针对某个类的每个初始波动函数，该类对应于适当的希尔伯特空间中的一个密集子空间。