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

伪微分算子理论在费曼路径积分中的应用

• 批准号: 13640161
• 负责人: ICHINOSE Wataru
• 金额: $ 2.18万
• 依托单位: Shinshu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640161/
• 关键词: path integral; Schroedinger equation; generating function; functional derivative; pseudo differential operators; 量子観測理論

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

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

项目成果

Y.Morimoto, C.J.Xu: "Logarithmic Sobolev inequality and semi-linear Dirichlet problem for infinitely degenerate elliptic operators"Asterisque, Soc.Math.France Inst.Henri Poincare. 284. 245-264 (2003)

Y.Morimoto、C.J.Xu：“无限简并椭圆算子的对数 Sobolev 不等式和半线性 Dirichlet 问题”Asterisque，Soc.Math.France Inst.Henri Poincare。

H.Kozono, T.ogawa, Y.Taniuchi: "Navier Stokes equations in the Besov space L^∞-and BHO"Kyuohu J.Math.. 57・2. 303-324 (2003)

H.Kozono、T.okawa、Y.Taniuchi：“贝索夫空间 L^∞-和 BHO 中的纳维斯托克斯方程”Kyuohu J.Math.. 57・2 (2003)。

M.Hirokawa: "Ground state transition for two level system coupled with pose field"Physics Letters A. 294. 13-18 (2002)

M.Hirokawa：“与位姿场耦合的两能级系统的基态转换”Physics Letters A. 294. 13-18 (2002)

M.Hirokawa: "Ground state transition for two level system coupled with Bose field"Physics letter A. 294. 13-18 (2002)

M.Hirokawa：“与玻色场耦合的两能级系统的基态转变”物理学信件 A. 294. 13-18 (2002)

W.Ichinose: "On convergence of the Feynman path integral in phase space"Osaka J.Math.. 39. 181-208 (2002)

W.Ichinose：“论相空间中费曼路径积分的收敛性”Osaka J.Math.. 39. 181-208 (2002)