Math Chalkboard

Do Quantum Systems Exist?

A Summary of Dr. Gaebler’s Dissertation

Written by Kokko Tso


Dr. David Gaebler is an assistant professor of mathematics at Hillsdale College who specializes in functional analysis. Below follows a general overview of his dissertation, “Unital Dilations of Completely Positive Semigroups.”


In both classical and quantum thermodynamics, all open systems must be contained within a closed system.

Open systems are those that exchange mass and energy with their environment (e.g. planet Earth), while closed systems do not (e.g. the universe). Quantum thermodynamics differs from the classical version in that open systems can also “entangle” with their environment. The key to validating the concept of quantum open systems is proving the existence of their enclosing closed systems. To do this, Dr. Gaebler examines two properties of dilation: continuity and unitality.

Dilation is a term used to describe the relationship between the time evolution of a given open system and its enclosing closed system.

Continuity is a concept with roots in physics and refers to the idea that a physical system will experience a small change within a small period of time, resulting in a gradual, smooth transition from one state to another.

Unitality rests heavily in the study of probability and refers to systems which contain more than one particle in motion, as opposed to systems with the same particles but more available states for them to occupy.

In order to describe an open quantum system, it is desirable for the mathematical model to posses both unital and continuous dilation. Previous research by American mathematics professor William Arveson proved the existence of continuous dilations, but only non-unital ones. A different approach by French mathematician Jean-Luc Sauvageot produced unital, but discontinuous, dilations.

Dr. Gaebler’s achievement comes from the successful unification of these two theories to produce a mathematical model of a system that possesses a dilation which is both unital and continuous, thus confirming the existence of quantum systems.


Kokko Tso graduated from Hillsdale College in 2013 with majors in music, Latin, and history. He currently works for his alma mater as the Digital Content Manager.