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

Abstract

Abstract We propose a thermodynamical definition of the vacuum energy density ρvac, defined as 〈vac|Tμν|vac〉 =  − ρvacgμν, 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 → 0. It takes the form ρvac = const ∙ mD where m is a fundamental mass scale and “const” is a computable constant which can be positive or negative depending on interaction couplings. Due to modular invariance ρvac 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 ρvac that are generally required, for instance ρvac = 0 for conformal field theories, and others, such as the constraint that ρvac 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 ρvac exactly for 2 classes of integrable QFT’s in 2D and interpreting some previously known results. We apply our definition of ρvac to Lattice QCD data with two light quarks (up and down) and one additional massive flavor (the strange quark), and find it is negative, ρvac ≈ − (200 MeV)4. Finally we make some remarks on the Cosmological Constant Problem since ρvac is central to any discussion of it.