This paper is another step in our pursuit of nonperturbative Minkowskian quantum field theory. Instead of studying the Schwinger functions arising from the Euclidean approach, we study the Wightman distributions directly. We use the non-rigorous Feynman integral for a given quantum field theory as a guide for the rigorous definition of the characteristic functional corresponding to the Lagrangian, but this approach limits us to time-ordered Wightman distributions. We use the idea that if a classical action is perturbed by an interaction involving only N excitations, then one should be able to express the resulting characteristic functional rigorously in terms of the previous characteristic functional. Implementation requires the modification of the classical action with imaginary terms involving the excitation amplitudes. If the previous characteristic functional has a measure-theoretic representation of some kind, then one can subsequently remove these auxiliary terms with a limiting argument. We continue to consider the scalar Minkowskian quantum field theory with the free, massive case as our starting point. The characteristic functional can be defined in terms of the Minkowskian Feynman propagator, and previously, we constructed a probabilistic representation of the functional. In the appendix of that previous paper, we introduced the scalar quartic field interaction with only N space–time excitations. Under the condition that their Fourier transforms be all supported on one side of the mass hyperboloid, we constructed the resulting characteristic functional. In this paper, we construct the characteristic functional in the case where both regions of energy–momentum space contribute to the finite collection of excitations.

中文翻译：

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跨越质量双曲面

这篇论文是我们追求非微扰闵可夫斯基量子场论的又一步。我们没有研究由欧几里得方法产生的 Schwinger 函数，而是直接研究 Wightman 分布。我们使用给定量子场论的非严格费曼积分作为严格定义对应于拉格朗日量的特征泛函的指南，但这种方法将我们限制在时间顺序的怀特曼分布上。我们使用的想法是，如果一个经典动作受到仅涉及 N 个激发的相互作用的干扰，那么人们应该能够根据先前的特征泛函严格地表达所产生的特征泛函。实现需要用涉及激励幅度的虚项修改经典动作。如果先前的特征泛函具有某种测度理论表示，则可以随后使用限制参数删除这些辅助项。我们继续考虑标量闵可夫斯基量子场论，以自由的、有质量的情况为起点。特征泛函可以根据闵可夫斯基费曼传播子来定义，之前，我们构建了泛函的概率表示。在前一篇论文的附录中，我们介绍了只有 N 个时空激发的标量四次场相互作用。在质量双曲面的一侧都支持它们的傅立叶变换的条件下，我们构建了由此产生的特征泛函。在本文中，