Fluctuations of the vacuum energy density of quantum fields in curved spacetime via generalized zeta functions

For quantum fields on a curved spacetime with an Euclidean section, we derive a general expression for the stress energy tensor two-point function in terms of the effective action. The renormalized two-point function is given in terms of the second variation of the Mellin transform of the trace of the heat kernel for the quantum fields. For systems for which a spectral decomposition of the wave opearator is possible, we give an exact expression for this two-point function. Explicit examples of the variance to the mean ratio $\Delta' = (<\rho^2>-<\rho>^2)/(<\rho>^2)$ of the vacuum energy density $\rho$ of a massless scalar field are computed for the spatial topologies of $R^d\times S^1$ and $S^3$, with results of $\Delta'(R^d\times S^1) =(d+1)(d+2)/2$, and $\Delta'(S^3) = 111$ respectively. The large variance signifies the importance of quantum fluctuations and has important implications for the validity of semiclassical gravity theories at sub-Planckian scales. The method presented here can facilitate the calculation of stress-energy fluctuations for quantum fields useful for the analysis of fluctuation effects and critical phenomena in problems ranging from atom optics and mesoscopic physics to early universe and black hole physics.