The study of the axiomatic quantum field theory using ultrahyperfunctions

E. Bruening和S. Nagamachi在2004年《Journal of Mathematical Physics》上发表的论文《具有基本长度的相对论量子场论》表明，如果我们使用超函数理论，我们可以表述具有基本长度的量子场论。在这里，基本长度1具有这样的性质，即在长度1内发生的两个事件是无法区分的。这样的理论不能用分布或超函数来表述。基本长度理论从20世纪30年代就有了历史。光速c是相对论的基本常数，普朗克常数h是量子力学的基本常数。由于量子场论是相对论与量子力学的结合，因此狄拉克场方程既有常数c，也有常数h，其中c的维数为[L/T]， h的维数为[MLL/T]。W. Heisenberg认为物理学的基本方程还必须包含一个维度为…的常数[L]。如果引入这样一个常数1，那么任何其他量的维数都可以用基本常数c、h和1的组合来表示。1958年，海森堡和泡利引入了一个后来被称为宇宙方程的方程。这个方程有一个常数1，维度为[L]但不幸的是，没人能解出这个方程。我们考虑这个方程的线性化版本，它继承了具有维数[L]的重要常数1。E. Bruening教授和S. Nagamachi教授撰写的论文《HEISENBERG’s FUNDAMENTAL EQUATION AND QUANTUM FIELD THEORY WITH A FUNDAMENTAL LENGTH》表明，我们不能在标准框架内使用分布或超函数来求解这个模型，而可以在我们之前的论文框架内求解：具有基本长度的相对论量子场论。并且，模型的维数为[L]的常数1即为本文引入的基本长度。少

It is shown in the paper of E. Bruening and S. Nagamachi "Relativistic quantum field theory with a fundamental length" published in Journal of Mathematical Physics in 2004 that if we use the theory of ultrahyperfunctions we can formulate a quantum field theory with a fundamental length. Here the fundamental length 1 has the property that the two events occurring within the length 1 cannot be distinguished. Such a theory cannot be formulated by using distributions or hyperfunctions.The theory with a fundamental length has a history since 1930's. The speed c of light is a fundamental constant in relativity theory and Planck's constant h is the fundamental constant of quantum mechanics. Since the quantum field theory is the combination of relativity and quantum mechanics, the Dirac field equation has both constants c and h. The dimension of c is [L/T] and that of h is [MLL/T]. W. Heisenberg thought that a fundamental equation of Physics must also contain a constant 1 with the dimension of … More length [L]. If such a constant 1 is introduced, then the dimensions of any other quantity can be expressed in terms of combinations of the basic constants c, h, and 1. In 1958, Heisenberg and Pauli introduced an equation which was later called the equation of the universe. This equation has a constant 1 with the dimension [L] but unfortunately, nobody has been able to solve this equation.We consider the linearized version of this equation which inherits the important constant 1 with the dimension [L]. The paper "HEISENBERG'S FUNDAMENTAL EQUATION AND QUANTUM FIELD THEORY WITH A FUNDAMENTAL LENGTH" written by Prof. E. Bruening and S. Nagamachi shows that one cannot solve this model within the standard framework using distributions or hyperfunctions but can solve within the framework of our previous paper: Relativistic quantum field theory with a fundamental length. Moreover, the constant 1 with the dimension [L] of the model is shown to be the fundamental length introduced in this paper. Less