Estimates on Functional Integrals of Quantum Mechanics and Non-relativistic Quantum Field Theory

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• 作者：Gonzalo A. Bley ; Lawrence E. Thomas
• 刊名：Communications in Mathematical Physics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：350
• 期：1
• 页码：79-103
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0916
• 卷排序：350

文摘

We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form \({E[\rm {exp}(\mathcal{A}_T)]}\), the (effective) action \({\mathcal{A}_T}\) being a function of particle trajectories up to time T. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman–Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schrödinger operator with \({1/|x|^2}\) potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models.