The problem of infinities in quantum field theory (QFT) is a longstanding problem in particle physics. To solve this problem, different renormalization techniques have been suggested but the problem persists. Here we suggest another approach to the elimination of infinities in QFT, which is based on non-Diophantine arithmetics – a novel mathematical area that already found useful applications in physics, psychology, and other areas. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. In addition, non-Diophantine integration is developed in these arithmetics. These constructions allow using constructed non-Diophantine arithmetics for computing integrals associated with Feynman diagrams. Although in the conventional QFT such integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined. As the result, QFT becomes consistent with quantum experiments.

量子场论 (QFT) 中的无穷大问题是粒子物理学中长期存在的问题。为了解决这个问题，已经提出了不同的重整化技术，但问题仍然存在。在这里，我们提出了另一种消除 QFT 中无穷大的方法，它基于非丢番图算术——一个已经在物理学、心理学和其他领域找到有用应用的新数学领域。为了实现这一目标，构建了新的非丢番图算法并研究了它们的性质。此外，在这些算法中还发展了非丢番图积分。这些构造允许使用构造的非丢番图算术来计算与费曼图相关的积分。尽管在传统的 QFT 中，这样的积分发散，它们的非丢番图对应物是收敛的且定义严格。结果，QFT 变得与量子实验一致。