ABSTRACT
We examine under this heading the transition amplitude in the Feynman path integral, which we will discuss shortly and will be reduced to an integral over the continuum degrees of freedom, such as a set of p i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/inline4_1.jpg"/> and q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/inline4_2.jpg"/> . This will involve harmonic (non-interacting) systems that are reduced to the problem of multiple Gaussian integrals over potentially coupled degrees of freedom, q i i = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/inline4_3.jpg"/> that we develop appropriate Feynman path integral tools for their solution. Considering that the harmonic oscillator is one of the central objects of theoretical physics then Gaussian integrals are likely to be the key tools of theoretical physics. This will be demonstrated utilizing Feynman path-integral formulation of quantum mechanics where Gaussian integrals will be pivotal for computation in quantum mechanics, quantum statistical mechanics and generally in quantum field theory. It will be interesting to introduce the Maslov correction [11,12] to the wave function which is a jump of π 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/inline4_4.jpg"/> in the phase when a system passes through a caustic point, a phenomenon related to the second variation as well as to the geometry of paths. This phenomenon is applicable to any system using the quasi-classical approximation. The harmonic oscillator will be one of the objects amongst others to illustrate this.
