The Azimuth Project
Quantum techniques for stochastic mechanics (course)

Quantum Techniques for Stochastic Mechanics

Welcome to the homepage for the course

The course is based on the Azimuth Network Theory Project so it is fitting that the homepage be kept on this site. It has some advantages: the lecture content will be kept here on this wiki.



Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of chemical reaction networks, which describes the interactions of molecules in a stochastic rather than quantum way. Computer science and population biology use the same ideas under a different name: stochastic Petri nets. But if we look at these theories from the perspective of quantum theory, they turn out to involve creation and annihilation operators, coherent states and other well-known ideas—but in a context where probabilities replace amplitudes. In this course we will explain this connection as part of a detailed analogy between quantum mechanics and stochastic mechanics. We will study the overlap of quantum mechanics and stochastic mechanics, which involves Hamiltonians that can generate either unitary or stochastic time evolution. These Hamiltonians are called Dirichlet forms, and they arise naturally from electrical circuits made only of resistors. The area is ripe to be further connected with modern topics in quantum computation and quantum information theory. This lecture series is part of QIC 890/891 held at the University of Waterloo, Spring 2011 and organized by Michele Mosca. The content is primarily based on joint work with John Baez.

Location and Times

Dates: July 7, 12 and 14; Time: Tuesdays and Thursday, 1:00pm - 2:20pm; Location: RAC1 2009 (IQC main building) Recitations: Friday July 8th and 15th, 1:00 pm in RAC1 2009.

Lecture content

Course Book


It is my pleasure to acknowledge great conversations and meaningful collaborations with several researchers which have influenced the course material. I work with some of them at the Centre for Quantum Technologies. In alphabetical order, a partial list includes

Suggested Additional Reading and Key References

The main course content is from the blog series Network Theory which appeared on Azimuth in 2011 and 2012. The full series is kept on John Baez’s homepage: Network Theory.

The series is being turned into a book, which can be downloaded free online:

Brendan Fong contributed to the topic, particularly in the following post on Noether’s theorem:

The book contains many references as do the blog articles. Among these, there is the notable lecture series by Feinberg which would be of interest for those who want to dig deeper into chemical reaction networks (we only covered the basics in the course).

The bigger picture of what’s going on here is explained well in the first entry of the network theory series.

We are also compiling an annotated list of references on diagrammatic notations for networks various types.

Not directly related to the lecture content of this course, but related to the larger goal of creating a unified theory of network science would be the work on categorical physics.