Some Theorems upon Negative Energy Density of a Quantum Free Scalar Field in an Inertial World Line of the Minkowski Space-Time

Abstract

Abstract. After presenting a lemma, two theorems on negative energy density associated with a quantum free scalar field are established. The first theorem provides a lower bound for a non-negative weight function whose existence is guaranteed by the lemma. The above energy density is evaluated over an inertial world line of the Minkowski space-time. The second theorem provides an upper bound for the averaged (with respect to the sampling function) absolute expectation value of the negative energy-density function. In particular, a complex-valued sampling function is introduced by the first time so a more generalized formulation is proposed. IntroductionIn quantum field theory, one can find very interesting examples on negative energy density (see, for instance, ref.[1]). Referring to the general context of quantum physics, there are cases in which energy density can be negative. In this respect, the Casimir effect (see, for example, ref.[1]) is certainly relevant: the quantum fluctuations of the electromagnetic field in the vacuum give rise to a macroscopic effect (Casimir effect) by which the electromagnetic-energy density is negative in the cavity formed by two conducting plates separated by a certain distance. One can think that, in principle, the energy density associated with a quantum field can be made arbitrarily negative at any point of the space-time by choosing a suitable state. Despite considering a classical field in which energy densities are positive in every point of the space-time, the renormalized energy density of the field quantized from the above classical field can be zero and can take either strictly positive or strictly negative values (note that the above density must be continuous). In principle, the renormalized energy density in question may even be made arbitrarily negative at a given point of the space-time. As a matter of fact, arbitrarily negative expectation values of the energy density can be measured after quantization even though the spatially integrated density holds non-negative. The aim of the present paper lies on proving two useful theorems about negative energy density of a free scalar field on an inertial world line of the Minkowski space-time. In this context, we may consider, for example, Dirac or Klein-Gordon fields. On the other hand, a key ingredient in the hypotheses of the above theorems is assuming realistically that the energy density of the aforementioned free scalar field cannot take on negative values arbitrarily. Really, negative energy density is constrained to certain conditions so that the negativeness of the density is not arbitrary; those conditions have been formulated currently by means of quantum inequalities. Before establishing the theorems in question, we will enunciate a lemma which certainly contributes enough to the clarification of the state of the art. In addition, a complex-valued sampling function will be introduced.