Calculation Rule for Aoyama–Tamra’s Prescription for Path Integral with Quantum Tunnelling

We derive a simple calculation rule for Aoyama–Tamra’s prescription for path integral with degenerated potential minima. Non-perturbative corrections due to the restricted functional space (fundamental region) can systematically be computed with this rule. It becomes manifest that the prescription might give Borel summable series for finite temperature (or volume) system with quantum tunneling, while the advantage is lost at zero temperature (or infinite volume) limit. ? e-mail: [email protected] In quantum mechanics a perturbative expansion of the partition function (or of Green functions, of energy levels) around a degenerated potential minimum is known to be non-Borel summable, due to quantum tunneling [1]. This is also true for scalar field theories in finite volume and non-Abelian gauge field theories. For such a system, therefore it is a fundamental problem how to relate the true value and the information of perturbation series. Concerning this problem, Aoyama, and Aoyama and Tamra [2] proposed a prescription for path integral with a degenerated potential. Their observation is following: Take an integral