Energy density and its positivity properties represent a fundamental subject in classical and quantum physics. In this talk, we will investigate this topic in the thermal representation of a free massive quantum scalar field. After a brief review of the fundamental mathematical tools at the base of this work, we will construct the GNS representation of our QFT induced by a state at thermal equilibrium (KMS). Therein, we will identify the generator of the time evolution and its spatial density. The symmetry between the "particles" and "holes" makes evident the impossibility for a lower bound for the expectation value of the energy density in this representation. In order to tackle this problem, we will investigate and extend some results of modular theory and non-commutative Lp spaces. In this way, we will obtain a general result concerning the expectation value of operators affiliated to a von Neumann algebra. Finally, the proven results will be applied to our setting to derive an L4 state dependent non-trivial QEI. The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0