Theoretical Physics(DTP)
Physique théorique (DPT)
Ariel EDERY
Bishop's University
Compact formulas for Casimir energies in D-dimensions via operator technique
An operator technique is derived for the multi-dimensional application of the Euler-Maclaurin formula to the Casimir energy problem. We obtain compact formulas for the Casimir energy of a scalar field confined to a D-dimensional hypercube with von Neumann or Dirichlet boundary conditions. The formulas are conveniently expressed as a finite sum of the well-known gamma and Riemann zeta functions and allow for quick numerical calculations at higher values of D. The case of the Dirichlet energy reveals a critical dimension at D = 36. We briefly discuss the connection between the Casimir energy in D-dimensions and the rD(n) arithmetic function (where rD(n) represents the number of ways a positive integer n can be expressed as a sum of D integer squares).