The Dirichlet Casimir effect for ϕ⁴theory in (3 + 1) dimensions: a new renormalization approach

Abstract

We calculate the next to the leading order Casimir effect for a real scalar field, within φ 4 theory, confined between two parallel plates in three spatial dimensions with the Dirichlet boundary condition. In this paper we introduce a systematic perturbation expansion in which the counterterms automatically turn out to be consistent with the boundary conditions. This will inevitably lead to nontrivial position dependence for physical quantities, as a manifestation of the breaking of the translational invariance. This is in contrast to the usual usage of the counterterms in problems with nontrivial boundary conditions, which are either completely derived from the free cases or at most supplemented with the addition of counterterms only at the boundaries. Our results for the massive and massless cases are different from those reported elsewhere. Secondly, and probably less importantly, we use a supplementary renormalization procedure, which makes the usage of any analytic continuation techniques unnecessary.

A new renormalization approach to the Dirichlet Casimir effect for ϕ4 theory in 1+1 dimensions

“…We share the point of view with some authors such as the ones in [14,18] that the analytic continuation techniques are not always completely justified physically. Moreover, like the first of the aforementioned authors, we have found counterexamples, which we point out in this paper and elsewhere [29]. The counterexamples show that it alone might not yield correct physical results, and sometimes even gives infinite results [30].…”

“…See for example Eq. (29). What is almost invariably done is to adjust the regulators so that the singular terms exactly cancel each other, i.e.…”

The radiative corrections to the mass of the kink using an alternative renormalization program

“…We finally obtain the following general result, which is also obtained in [19,20] using analogous general arguments but a slightly different method, δ m (x, t) = −3λG(x, t; x, t).…”

“…solitons, have important manifestations in the physical properties of the systems. In particular, an alternative renormalization program has been proposed [18] which is fully consistent with the boundary conditions and its use has led to new results for the Next to Leading Order (NLO) Casimir effect within φ 4 theory [19,20]. The main purpose of this paper is to explore another manifestation of this newly proposed renormalization program by presenting an analogous study for systems with non-trivial backgrounds.…”

Radiative correction to the Casimir energy for massive scalar field on a spherical surface

“…This statement necessities using the position-dependent counter-terms instead of free counter-terms in the renormalization program. This viewpoint on renormalization program has been explained in detail in previous studies [20][21][22][23][24][25][26][27] and all their physical aspects and advantages have been discussed. The same idea in the renormalization of interacting quantum field theory in the curved space time has been extensively investigated [28][29][30][31][32][33][34][35].…”