![]() ![]() This kind of propagator leads to a wide range of physical phenomena and is richer than the corresponding one found for the Lee–Wick gauge field, where we have just one massive pole and one massless pole. The propagator ( 7) exhibits a singular configuration, in which we have two terms with opposite signs, each one with a different massive pole. It is used Minkowski coordinates with the diagonal metric with signature ( , −, −, −). Throughout the paper we shall deal with a scalar model with higher order derivatives, in a 3 1 dimensional space-time. 5 is devoted to our final remarks and conclusions. 4 we consider the case where the charge is placed on the Dirichlet mirror. We show that the interaction energy between the Dirichlet plane and a point-like charge is finite, even in the limit when the charge is placed near on the plane. We also compare the results obtained along the paper with the corresponding ones computed for the abelian Lee–Wick model (for the gauge field) as well as with the ones obtained for the standard Klein–Gordon field theory. ![]() We compare the interaction forces with the ones obtained in the free theory (theory without the Dirichlet plane) and verify that the image method is not valid in our model for the Dirichlet boundary condition. We obtain exact results only for some special situations and we perform numerical analysis for more general cases. 3 from the propagator founded previously, we calculate the interaction energy as well as the interaction force between the point-like scalar charge and the Dirichlet plane. 2 we compute the propagator for the Lee–Wick scalar field in the presence of a Dirichlet plane. This paper is devoted to this subject, where we investigate some aspects of the Lee–Wick-like scalar model in the vicinity of a single Dirichlet plane. Are these divergences still controlled with the presence of boundary conditions? Investigations of this type have not been explored in the literature as it should. A related question to this subject regards on the fact that some divergences in theories with higher order derivatives, but without boundary conditions, are naturally controlled. A natural question which arises in this scenario concerns on the modifications in which the Lee–Wick scalar propagator undergoes due to the presence of a single Dirichlet plane, and the influence of this kind of surface on the dynamics of point-like field sources. However, the Lee–Wick-like scalar model have not been considered in the presence of boundary conditions up to now, as far as the authors know. Regarding the presence of boundary conditions in the Lee–Wick Electrodynamics, we can mention the study of the Casimir effect and the Lee–Wick Electrodynamics in the vicinity of a perfectly conducting plate, which is an example of a linear field theory where the image method is not valid. An alternative approach to this kind of problem is the study of models with fields coupled to external potencials defined along surfaces. We highlight, the phenomena related to the presence of material boundaries in field theories, just to mention a few examples. It is well known that field theories with the presence of boundary conditions is a subject with a wide range of application in several branches of physics. In fact, there is a vast literature concerning Lee–Wick models. Some aspects of Lee–Wick models have been investigated, for example, in interactions between external sources, issues related with the ghosts problems, radiative corrections, propagation of waves, dual symmetry, phenomenological and cosmological implications, the connection with Pauli–Villars regularization, and so on. Nowdays, models similar to the one proposed Podolsky, Lee and Wick are called Lee–Wick models. The simplest model of this type was proposed by Podolsky, Lee and Wick, where the self energy for a point-like charge in \(3 1\) dimensions is finite. This is the main reason why these models are intensively studied in the literature. One of the most remarkable features of models with higher order derivatives concerns on the fact that they tame some ultraviolet divergences in field theories, both at classical as well as at quantum levels. ![]()
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