Thermal Time as a POVM
R.J. Pieters (TU Delft - Electrical Engineering, Mathematics and Computer Science)
J.M.A.M. Van Neerven – Mentor (TU Delft - Analysis)
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Abstract
This thesis explores the mathematical foundations of quantum measurements for relativistic particles, focusing on the construction of positive operator-valued measures (POVMs) within the framework of functional analysis. Motivated by recent developments in mathematical physics, particularly the construction of a POVM for a onedimensional relativistic massless particle, we develop an analogous POVM for a relativistic particle with mass. The mathematical framework is firmly grounded in the Tomita-Takesaki modular theory, and its connection to Connes-Rovelli thermal time is established.
A central component of the thesis is a detailed, self-contained proof of Tomita’s theorem, a cornerstone result in the modular theory of von Neumann algebras. This theorem plays a critical role in understanding the structure of operator algebras associated with quantum systems and underpins the rigorous formulation of covariant observables. Furthermore, it is of crucial importance to formulating Rovelli’s thermal time hypothesis.
Building upon this theoretical groundwork, we construct a POVM for a relativistic massive particle in one spatial dimension. The resulting POVM satisfies temporal covariance relations and provides a mathematically consistent description of time measurements for massive particles, addressing key challenges in quantum field theory. The thesis concludes by relating the newly found time observable to Connes-Rovelli thermal time.