Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function ψ(x1,x2) with x1,x2 ∈ ℝ4 is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half-space, meaning that the Minkowski spacetime is cut off before t = 0. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at t = 0 can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in 1 + 3 dimensions.

中文翻译：

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两个直接相互作用的 1 + 3 维狄拉克粒子的相对论动力学的存在

在这里，我们证明了一类积分方程解的存在性和唯一性，该积分方程描述了具有直接相互作用的 1+3 维中的两个狄拉克粒子。这类积分方程自然地作为两粒子薛定谔方程积分版本的相对论推广而出现。多时间波函数的关键使用ψ(X1,X2)和X1,X2 ∈ ℝ4制作。一个核心特征是交互的时间延迟。我们的主要结果是 Minkowski 半空间的存在唯一性定理，这意味着 Minkowski 时空在吨 = 0. 我们进一步表明，解决方案是由柯西数据在初始时间确定的；然而，在其他时候没有柯西问题是可以接受的。第二个结果是将第一个结果扩展到具有大爆炸奇点的特定 FLRW 时空，在无质量情况下使用狄拉克方程的共形不变性。这表明截止在吨 = 0可以自然产生并与相对论完全兼容。因此，我们在1 + 3方面。