Thermodynamic formulation of vacuum energy density in flat spacetime and potential implications for the cosmological constant

Abstract

We propose a thermodynamical definition of the vacuum energy density $ρ_{\rm vac}$, defined as $\langle 0| T_{μν} |0\rangle = - ρ_{\rm vac} \, g_{μν}$, in quantum field theory in flat Minkowski space in $D$ spacetime dimensions, which can be computed in the limit of high temperature, namely in the limit $β= 1/T \to 0$. It takes the form $ρ_{\rm vac} = {\rm const} \cdot m^D$ where $m$ is a fundamental mass scale and ${\rm "const"}$ is a computable constant which can be positive or negative. Due to modular invariance $ρ_{\rm vac}$ can also be computed in a different non-thermodynamic channel where one spatial dimension is compactifed on a circle of circumference $β$ and we confirm this modularity for free massive theories for both bosons and fermions for $D=2,3,4$. We list various properties of $ρ_{\rm vac}$ that are generally required, for instance $ρ_{\rm vac}=0$ for conformal field theories, and others, such as the constraint that $ρ_{\rm vac}$ has opposite signs for free bosons verses fermions of the same mass, which is related to constraints from supersymmetry. Using the Thermodynamic Bethe Ansatz we compute $ρ_{\rm vac}$ exactly for 2 classes of integrable QFT's in $2D$ and interpreting some previously known results. We apply our definition of $ρ_{\rm vac}$ to Lattice QCD data with two light quarks (up and down) and one additional massive flavor (the strange quark), and find it is negative, $ρ_{\rm vac} \approx - ( 200 \, {\rm MeV} )^4$. Finally we make some remarks on the Cosmological Constant Problem since $ρ_{\rm vac}$ is central to any discussion of it.