Casimir energy for the interaction of \(n\) \(\delta\)-plates with a massless real scalar field

Abstract

Within the Symanzik approach, we consider a massless real scalar field in Euclidean spacetime in the presence of \(n\) planar \(\delta\)-plates with an interaction potential \(V(z)=\sum_{i=1}^n \lambda_i\delta(z-z_i)\). Using functional integration and the Sylvester identity, the interaction energy density (per unit area) can be reduced to the logarithm of the determinant of an \(n\times n\) matrix. After subtracting the self-energy contribution, the geometry of the system allows the problem to be reduced to calculating the determinant of a tridiagonal matrix, which is evaluated using a recurrence relation. For a system of identical plates with equal separations, we obtain a closed expression in terms of Chebyshev polynomials. As an illustration of the efficiency of the method, an explicit form for \(n=20\) is written out, and the limiting case of infinitely strong interaction (\(\lambda\to\infty\)), which is associated with Dirichlet boundary conditions, is considered. In addition, we show that in the case where the number of plates \(n\to\infty\) and the separation \(L\to0\), at fixed thickness \(T=nL\) and with weak coupling constant \(\lambda\), the system of \(\delta\)-films approaches a finite-thickness slab. The results are verified for the particular cases \(n=2\) and \(n=3\).