Casimir Energy of a Relativistic Perfect Fluid Confined to a D-dimensional Hypercube

TL;DR

This paper derives compact formulas for the Casimir energy of a relativistic perfect fluid in a D-dimensional hypercube, revealing oscillatory behavior and a critical dimension at D=36, with implications for high-dimensional quantum field theories.

Contribution

It provides new explicit formulas for Casimir energy in D-dimensional hypercubes using gamma and zeta functions, and analyzes their behavior across dimensions.

Findings

Casimir energy formulas expressed as finite sums of gamma and zeta functions.

Oscillatory pattern of energy with respect to dimension D.

Critical dimension at D=36 where behavior changes.

Abstract

Compact formulas are obtained for the Casimir energy of a relativistic perfect fluid 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. Emphasis is placed on the mathematical technique used to extract the Casimir energy from a $D$-dimensional infinite sum regularized with an exponential cut-off. Numerical calculations show that initially the Dirichlet energy decreases rapidly in magnitude and oscillates in sign, being positive for even $D$ and negative for odd $D$. This oscillating pattern stops abruptly at the critical dimension of D=36 after which the energy remains negative and the magnitude increases. We show that numerical calculations performed with 16-digit precision are inaccurate at higher values of $D$.