The Renormalization Of Scalar Fields In Curved Spacetime And The A-theorem In Even Dimensions

Renormalization is an approach to eliminate the divergence in quantum field theory,it is an important part of quantum field theory.In the renormalization program,the infinite part is decided by the counter terms,but the finite part can be changed,depending on the choice of the renormalization scale.The coupling constants' evolution along the renormalization scale ? follow the renormalization group equation.Renormalization group's evolution is a basic concept of modern physics,which has important applications in high energy physics,condensed matter physics and other disciplines.For two and four dimensional spacetime of unitary,Poincare invariant quantum field theory system,renormalization group equations follow a-theorem: there exists a non-negative function a(g)of the coupling constant g,which is equal to the Euler density's coefficient of Weyl anomaly at fixed point in curved space-time,a(g)function monotonically decreases when the system according to the renormalization group equations flows from high energy to low energy scale.The two and four dimensional spacetime a-theorem has been nonperturbative proof strictly.As a fundamental theorem of quantum field theory,a-theorem has important physical meaning.Its establishment shows that the system along a field theory renormalization group evolution from high energy to low energy is irreversible,in addition to a(g)function itself can also serve as an effective measurement to the system's degrees of freedom,which can be understand by Wilson's renormalization group theory.But the six dimensional a-theorem syet is not nonperturbative proved.In this paper,we calculate and give the two loop results of the renormalization for ?4field in four-dimensional curved spacetime and ?3field in six-dimensional curved space-time by using heat kernel method,background field method and the dimensional regularization method.Then the proof to four-dimensional a-theorem have been described,the six dimensional a-theorem were also discussed,the calculation results show a result of an anti-a-theorem.And we try study the tability of the model and a-theorem by the effective potential in six dimensional spacetime.