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Abstract We propose a thermodynamical definition of the vacuum energy density ρ vac, defined as 〈vac|T μν |vac〉 = − ρ 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 → 0. It takes the form ρ vac = const ∙ m D 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 ρ 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 vac that are generally required, for instance ρ vac = 0 for conformal field theories, and others, such as the constraint that ρ 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 vac exactly for 2 classes of integrable QFT’s in 2D and interpreting some previously known results. We apply our definition of ρ 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 vac ≈ − (200 MeV)4. Finally we make some remarks on the Cosmological Constant Problem since ρ vac is central to any discussion of it. |
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