Rigorous proof of the attractive nature for the Casimir force of a \(p\)-odd hypercube (Q2773141)

A Hermitian massless scalar field confined to the interior of a \((D-1)\)-dimensional rectangular cavity \(\Omega\) with \(p\) edges of finite lengths, \(L_1,L_2,\dots,L_p\), and \(D-p-1\) edges with lengths of orders \(\lambda_i \gg L_1\), \(L_2,\dots,L_p\), \(i=p+1\), \(p+2,\dots,D-1\), respectively, are considered, with Dirichlet boundary conditions for the field on the faces of the hypercube. The authors make use of powerful zeta function regularization techniques for the relevant Epstein zeta function, with several formulas taken after some papers by the reviewer and his former student \textit{A. Romeo} [Phys. Rev. D 40, 436 (1989); Rev. Math. Phys. 1, 113-128 (1989; Zbl 0711.11021)] (further self-contained and much more detailed references are: \textit{E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko} and \textit{S. Zerbini}, Zeta regularization techniques with applications. World Scientific, Singapore (1994); \textit{E. Elizalde}, Ten physical applications of spectral zeta functions, Lecture Notes in Physics (Springer-Verlag, Berlin) (1995; Zbl 0855.00002)).NEWLINENEWLINENEWLINEThe authors manage to demonstrate that the Casimir effect gives rise to an attractive force within the configuration that confines the massless scalar field inside the Dirichlet boundaries, provided an odd dimensional hypercube is chosen, for an otherwise arbitrary number of spacetime dimensions \(D\) which is less than a critical value, \(D_c\), which on its turn the authors obtain in terms of \(p\). For small values of \(p\), \(D_c\) is equal to \(p+4\), and \(D_c\) tends to be just \(p\), when \(p\) grows sufficiently.