# AG Mathematische Physik, Leonardo Sangaletti (Leipzig): An L4 quantum energy inequality in the thermal sector

**Leonardo Sangaletti (Leipzig)**

**An L4 quantum energy inequality in the thermal sector**

Abstract:

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 of 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 obtain a general

result concerning the expectation value of an operator affiliated to a

von Neumann algebra. Finally, the proven results are used to derive an

L4 state dependent non-trivial QEI.