Vacuum Energy?

Although the classical vacuum is a void, the quantum vacuum is a virtual “soup” of particle-antiparticle pairs that interact with real atoms to produce the Lamb shift (slight energy shift in atomic levels) and the Casimir effect (attraction of two plates in a vacuum). Does the quantum vacuum have energy content, or does the energy in the “soup” average out to zero?

Answer

There is always the zero-point energy in the vacuum. Whatever QM model for the vacuum is considered, all can be reduced in a first approximation to a large number of harmonic oscillators, which have a zero-point energy value that is non-zero. At present, QM calculations of the energy density of the vacuum seem to be too large by at least 30 orders of magnitude! The vacuum energy density should be about 10^–11 J m^–3 if this vacuum energy is the source of the accelerated expansion of the universe determined by the Type 1a Supernova measurements in 1998.

One can do an energy estimate using the Heisenberg uncertainty principle. Or, if the vacuum has an effective potential for a scalar field, the product of the visible matter density and the potential will give the energy density for an assumed radius of the universe. In either case, the assumptions necessary to estimate this energy density would take us too far astray.

However, we can determine whether an electrically neutral particle of mass Δm popping into existence for a time interval Δt can be detected by its gravitational field. We use the uncertainty relation ΔEΔt ≥ h/4π in the form c² ΔmΔt > h/4π. Suppose we have the most sensitive detector, a free particle of mass M initially at a distance R away from Δm; then in the Newtonian approximation the detector will receive a pulse P = FΔt. Substituting F = GMΔm/R² into the uncertainty relation produces GMΔmΔt/R² ≥ GMh/(4πR²c²).The initial state of the detector also obeys the uncertainty relation ΔPΔX ≥ h/4π, so that Δm to be noticeable requires the impulse P to be greater than about 2ΔP, or ΔX ≥ 4R (R/r_g), where the Schwarzschild radius of the detector r_g = GM/c². For objects ranging in size from protons to planets, r_g lies within the object itself. So the momentum transferred by the impulse will not be detected!