素数、量子场论和哥德巴赫猜想
Prime numbers, quantum field theory and the Goldbach conjecture
摘要：
基于数论中的哥德巴赫猜想和二维圆柱时空中的阿贝尔玻色化机制，我们研究了将实标量场重构为两个实费米子场（即所谓的\textit{prime}）的乘积，其中两个场的傅里叶展开只包含素模.通过引入作用在真空上的创造算子$b_{p}^†$ --用素数$p$标记--我们对这样的素场进行正则量子化并构造相应的Fock空间。基于量子场论的标准规则和黎曼假设的假设，我们对模型的分析使我们能够证明该理论是不可重整化的。我们还评论了这一结果对大整数哥德巴赫猜想的有效性或崩溃的潜在后果。
英文摘要：
Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^†$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.
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