# References for examples of $\ast$-algebra approach to QM and QFT

In studying QFT on curved spacetime I've found the $\ast$-algebra approach as one viable approach to the subject on the paper Quantum Fields in Curved Spacetime by Wald.

The $\ast$-algebra approach seems like one quite nice and general approach to both QM and QFT, but it is quite abstract, so that it seems hard on the beginning of the study of this approach to see how it is used in practice and how, in the end, it is just a generalization of usual QM.

The point is that in QM books, like Cohen's, after presenting the postulates, examples are given to emphasize what is the physical meaning of everything and how one works with it, like spin 1/2 systems, the harmonic oscillator and so forth.

Although Wald shows some examples, I believe more examples and more details would be nice to get started.

What I'm looking for here are references showing examples of the $\ast$-algebra approach in practice for both QM and QFT. In other words: some simple examples showing how to connect the abstractness of the approach with the underlying physics and the usual approaches.

Any kind of reference is good: books, papers, lecture notes, video lectures, etc.

## 3 Answers

Lecture notes exposing standard perturbative quantum field theory this way are on PF-Insights A first Idea of Quantum Field Theory. The star algebra perspective ("quantum probability theory") comes alive with the introduction of the free field vacuum state in section 4 of chapter 14. Free quantum fields.

The books by Bratteli and Robinson (the link is for volume 1 that is more on the abstract theory of C* and W* algebras, there is also volume 2 that has physical applications) are a standard reference on C*-algebraic methods in physics, even if the applications they develop are mostly in statistical mechanics. Nonetheless, the concepts introduced have important applications also in relativstic quantum field theories.

The book is not introductory, and a bit heavy to read, but contains many very interesting results.

Also local quantum physics by Rudolf Haag is a standard reference, written by the pioneer of algebraic QFT (that introduces, e.g., to the concept of nets of algebras of local observables).

Let me add a pair of references.

QFT: I. Khavkine and V. Moretti Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction

published as the chapter 5 of Advances in Algebraic Quantum Field Theory, R. Brunetti et al. (eds.), Springer, 2015