Perturbative self-interacting scalar field theory: a differential equation approach

摘要

We revisit the investigation about the partition function related to a\phi^4-scalar field theory on a n-dimensional Minkowski spacetime, which isshown to be a self-interacting scalar field theory at least in 4-dimensionalMinkowski spacetime. After rederiving the analytical calculation of theperturbative expansion coefficients and also the approximate values forsuitable limits using Stirling's formulae, which consists of Witten's proposedquestions, solved by P. Deligne, D. Freed, L. Jeffrey, and S. Wu, weinvestigate a spherically symmetric scalar field in a n-dimensional Minkowskispacetime. For the first perturbative expansion coefficient it is shown how itcan be derived a modified Bessel equation (MBE), which solutions areinvestigated in one, four, and eleven-dimensional Minkowski spacetime. Thesolutions of MBE are the first expansion coefficient of the series associatedwith the partition function of \phi^4-scalar field theory. All results are depicted.