The application of the theory of the pseudo-diffrential operators to the Feynman path integral

我们项目的目的是赋予物理中常用的Feynman路径积分严格的含义，该积分是通过折线路径确定的时间切片近似方法定义的。具体研究内容如下：(1)证明了当离散化参数趋于零时泛函的相空间Feynman路径积分的收敛性，并给出了它的算符表示式。(2)在量子力学和量子自由场理论中证明了源J的生成泛函Z(J)的存在性。其次，我们证明了Z(J)在J中的泛函可微性，并且它的导数给出了相关函数。(3)给出了路径积分微扰理论的数学证明，除了对量子自由场理论Z(J)的研究外，我们的目的已经完成。首先，我们研究了物理中常用的相空间Feynman路径积分的收敛问题。证明了它的路径积分收敛于…进一步得到了L^2空间中相应薛定谔方程的解。其次，我们证明了相空间路径积分不仅在L^2空间中收敛，而且在广义Sobolev空间B^a中收敛，并利用这一结果研究了泛函II^N_&lt；j=1&gt；q(T_J)p(T_J)的相空间路径积分的收敛问题。给出了它们的算符表示式，证明了这些路径积分给出了关联函数、正则换位子关系等，并将此结果推广到泛函II^N&lt；J=1&gt；z_j(q(T_J)，p(T_J))的相空间路径积分。给出了z_j(x，p)(j=1，2，[三重键]，N)上该相空间路径积分收敛的充要条件。给出了路径积分收敛时的算符表示式。根据这一结果，我们可以给出费曼1948年著名论文中启发式结果的严格证明。其次，证明了任意J∈X的生成泛函Z(J)f(f∈B^a)的存在性，其中X是[O，T]上所有R^n值连续函数的集合。我们还证明了泛函Z(J)f：X→B^a在J中是Frechet可微的，并且它的Frechet导数给出了相关函数。关于这一结果的论文正在准备中。在我们的项目中，每个研究者都研究了振荡型积分算符，玻色场的基态跃迁，BMO空间和Besov空间及其在微分方程组中的应用。

The aim of our project was to give the rigorous meaning to the Feynman path integral usually used in physics, which is defined by the method of the time-slicing approximation determined through broken line paths. In detail, we study : (1)We show the convergence of the phase space Feynman path integral of the functional as the discretization parameter tends to zero and give its expression in terms of the operator notation. (2)We show the existence of the generating functional Z(J) of the source J in quantum mechanics and also quantum free field theory. Next we show the functional differentiability of Z(J) in J and that its derivative gives the correlation function. (3)We give the mathematical proof to the perturbation theory of the path integral.Our aim has been completed except for the study of Z(J) of quantum free field theory and (3). First, we studied the convergence of the phase space Feynman path integral that is usually used in physics. We proved that its path integral converges … More to the solution of the corresponding Schrodinger equation in L^2 space. Next, we proved the convergence of the phase space path integral not only in L^2 space but also in the generalized Sobolev spaces B^a. By means of this result we studied the convergence of the phase space path integral of the functional II^N_<j=1> q(t_j)p(t_j). We gave their expressions in terms of the operator notation and showed that these path integrals give correlation functions, the canonical commutator relation and etc. In addition, we generalized this result to the phase space path integral of the functional II^N_<J=1> z_j(q(t_j),p(t_j)). We gave the necessary and sufficient condition on z_j(x, p) (j=1,2,【triple bond】,N) for this phase space path integral to be convergent. We also gave the expression in terms of the operator notation when the path integral was convergent. From this result we could give the rigorous proofs of the heuristic results in Feynman's celebrated paper 1948. Next, we showed the existence of the generating functional Z(J)f (f∈B^a) for any J∈X, where X is the set of all R^n-valued continuous functions on [O, T]. We also proved that the functional Z(J)f : X→B^a is Frechet differentiable in J and its Frechet derivatives give correlation functions. The paper on this result is in preparation. In relation to our project each investigator studied the integral operator of the oscillatory type, the ground state transition of Bose field, and BMO space and Besov space and its applications to the differential equations Less