Casimir Energy in Contact-Interaction Models
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EMENTARY PARTICLES AND FIELDS Theory
Casimir Energy in Contact-Interaction Models Yu. V. Grats* Faculty of Physics, Moscow State University, Moscow, 119991 Russia Received November 7, 2017
Abstract—The problem of Casimir interaction between two δ d -like (d = 1, 2, and 3) sources in Minkowski space is examined on the basis of the ln det formalism. The result obtained for the case of two semitransparent plates (d = 1) coincides with the earlier result based on an alternative approach. The earlier assertion that there is no vacuum interaction between linear (d = 2) sources is disproved. An expression for the Casimir energy for two pointlike (d = 3) sources is obtained. DOI: 10.1134/S1063778818020096
1. INTRODUCTION The dynamics of fields in spaces featuring boundaries depends on the material and shape of the boundaries, and this has a substantial influence on the properties of the quantum vacuum and on their observational manifestations. A Casimir effect is the most widely known result of this influence. In the simplest form, the Casimir effect consists in the emergence of an experimentally observed force of attraction between two parallel conducting planes in a vacuum that are not charged. This is due to a change in the spectrum of electromagnetic-field zero-point oscillations because of the presence of boundaries. Further investigations revealed that a similar effect may arise not only owing to a change in the geometry and physical properties of boundaries but also owing to a nontrivial global structure of spaces being considered. As a result, the Casimir effect has now become an interdisciplinary object of investigations, playing an important role in phenomena differing strongly in its characteristic spacetime scales—from atomic, molecular, and nanophysics to gravitation and cosmology. In tackling relevant problems, the spacetime geometry of the boundaries, their physical properties, and so on are encoded by imposing respective additional conditions on field-equation solutions. Below, we examine the Casimir effect for the case where the encoding in question reduces to introducing δ-like potentials in the respective field equation. Such models have been the subject of vigorous investigations in theoretical and mathematical physics for a rather long time. Various names, including the models of pointlike interaction, contact interaction, zero-range interaction, and Fermi pseudopotentials, were used for them in the literature. They found *
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applications in solid-state physics (Kronig–Penney model [1]), atomic physics, and nuclear physics (description of short-range nuclear forces [2]). The introduction of a δ-function potential in the field equation made it possible to describe the vacuum interaction of two parallel perfectly conducting plates [3–5]. A similar situation arises in considering field-theory processes in the vicinity of topological defects. In the last case, it was found that two parallel infinitely thin cosmic strings are attracted to each other with a strength inversely proportional to the
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