What Are the Key References and Concepts in Algebraic Quantum Theory?

[jarek]

Hi all,

what would be your standard Ref. for algebraic approach to quantum theory (algebras, states, folia, etc.)? Haag or there is something better?

Beyond the standard things, I'm interested in two aspects of algebraic theory when applied to non-relativistic MECHANICS:

i) Classical limit. Is there anything like that? Commutative algebras => Gelfand-Naimark theorem and measures?

ii) Relation to noncommutative topology (states and noncommutative measures, etc.)

Thanks for any input,
jarek

 

[Haelfix]

Yea the defacto reference is Spin, Statistics, PCT and all that by Wightman et al.

After that there are a few more modern books on constructive approaches, but alas I don't have my references with me atm. I personally have lecture notes from a conference in 1999, that more or less do the job. I know there are arxiv reference papers as well.

 

[denian]

Hello Jarek,

Thank you for your question about algebraic quantum theory. The standard reference for this approach is indeed the book "Local Quantum Physics" by Rudolf Haag. However, there are also other excellent resources such as the book "Algebraic Quantum Mechanics" by Joachim Kupsch and the article "Quantum Theory without Observers" by Carlo Rovelli.

To address your two specific interests, I would first like to mention that the algebraic approach to quantum theory is typically applied to relativistic quantum field theory rather than non-relativistic mechanics. However, there are some attempts to extend it to non-relativistic systems as well.

Regarding the classical limit, the algebraic approach does not necessarily provide a direct route to classical mechanics, as it is primarily concerned with quantum systems. However, there have been some attempts to derive classical mechanics from quantum mechanics using the algebraic approach, such as the work of Ola Bratteli and Derek Robinson. Additionally, the Gelfand-Naimark theorem can indeed be used to construct classical states from commutative algebras in the context of quantum mechanics.

As for the relation to noncommutative topology, the algebraic approach does not explicitly deal with topology. However, there have been connections made between noncommutative geometry and algebraic quantum theory, particularly in the work of Alain Connes.

I hope this provides some insight into the algebraic approach to quantum theory and its applications to classical limit and noncommutative topology. Let me know if you have any further questions.

Best regards,