Quantum interest for scalar fields in Minkowski spacetime

The quantum interest conjecture of Ford and Roman states that any negative energy flux in a free quantum field must be preceded or followed by a positive flux of greater magnitude, and the more the surplus of positive energy grows the further the positive and negative fluxes are apart. In addition, the maximum possible separation between the positive and negative energy decreases the larger the amount of negative energy. We prove that the quantum interest conjecture holds for arbitrary fluxes of noninteracting scalar field energy in 4D Minkowski spacetime, and discuss the consequences in attempting to violate the second law of thermodynamics using negative energy. We speculate that quantum interest may also hold for the electromagnetic and Dirac fields, and might be applied to certain curved spacetimes.