Casimir theory of the relativistic composite string revisited, and a formally related problem in scalar QFT

Abstract

The main part of this paper is to present an updated review of the Casimir energy at zero and finite temperature for the transverse oscillations of a piecewise uniform closed string. We make use of three different regularizations: the cutoff method, the complex contour integration method, and the zeta-function method. The string model is relativistic, in the sense that the velocity of sound is for each string piece set equal to the velocity of light. In this sense the theory is analogous to the electromagnetic theory in a dielectric medium in which the product of permittivity and permeability is equal to unity (an isorefractive medium). We demonstrate how the formalism works for a two-piece string, and for a 2N-piece string, and show how in the latter case a compact recursion relation serves to facilitate the formalism considerably. The Casimir energy turns out to be negative, and the more so the larger the number of pieces in the string. The two-piece string is quantized in D-dimensional spacetime, in the limit when the ratio between the two tensions is very small. We calculate the free energy and other thermodynamic quantities, demonstrate scaling properties, and comment finally on the meaning of the Hagedorn critical temperature for the two-piece string.Thereafter, as a novel development we present a scalar field theory for a real field in three-dimensional space in a potential rising linearly with a longitudinal coordinate z in the interval 0 < z < 1, and which is thereafter held constant on a horizontal plateau. The potential is taken as a rough 1 iver.h.brevik@ntnu.no 1 model of the two-piece string potential under simplifying conditions, when the length ratio between the pieces is replaced formally with the mentioned length parameter z. 2

Finite temperature Casimir effect for charged massless scalars in a magnetic field

“…Since Casimir forces have many applications-from nanotubes and nanotechnology [6][7][8][9], to branes and compactified extra dimensions , to string theory [32][33][34][35]-a large effort has gone into studying the Casimir effect and its generalization to quantum fields other than the electromagnetic field: fermions were first considered by Johnson [36] in connection with the bag model [37], then investigated by many others; for example [38,39], bosons and other scalar fields have also been investigated extensively [4].…”

Critical Casimir Effect: Exact Results

“…29 Practically, in Monte Carlo studies one often makes a finite-size scaling Ansatz and fits the parameters there, considering even L as an effective parameter the value of which is to be determined from the best fit of the data. 30 As noted in Ref. [125] and commented on in Ref.…”

On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results

“…There is a vast amount of literature concerning research on the quantum Casimir effect. Here, we only mention the review articles in [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ], involving recent studies of the dynamical Casimir effect (in which actual photons can be created if a single mechanical mirror undergoes accelerated motion in vacuum) [ 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 ] and studies of the effects which emerge in systems out of thermodynamic equilibrium (in which the material bodies are characterized by different temperatures) [ 47 , 48 , 49 , 50 , 51 , 52 , 53 ]. Currently, the QED Casimir effect is a popular subjec...…”