Is There a Quantum Mechanics Framework Without Hilbert Space?

[CarlB]

Is anyone out there working on a theory of elementary particles that is basic quantum mechanics without the Hilbert space? The reason I'm asking is because I found this article by B. J. Hiley:

Algebraic Quantum Mechanics, Algebraic Spinors and Hilbert Space.
The orthogonal Clifford algebra and the generalised Clifford algebra, $$C^n$$, (discrete Weyl algebra) is re-examined and it is shown that the quantum mechanical wave function (element of left ideal), density operator (element of a two sided ideal) and mean values (algebraic trace) can be constructed from entirely within the algebra. No appeal to Hilbert space is necessary. We show how the GNS construction can be obtained from within both algebras. The limit of $$C^n$$ as $$n \to \infty$$ is shown to be the extended Heisenberg algebra. Finally the relationship to the usual Hilbert space approach is discussed.
http://www.bbk.ac.uk/tpru/BasilHiley/Algebraic Quantum Mechanic 5.pdf

Carl

 

[AndyW]

, thank you for bringing up this interesting topic. As a fellow scientist, I am also intrigued by the possibility of developing a theory of elementary particles without the use of Hilbert space. The article by B. J. Hiley that you mentioned seems to present a promising approach using the orthogonal Clifford algebra and the generalised Clifford algebra. It is interesting to see how the quantum mechanical wave function, density operator, and mean values can be constructed solely within the algebra without the need for Hilbert space.

I am not personally working on this specific theory, but I am familiar with the concept of algebraic quantum mechanics. This approach has been gaining attention in recent years as an alternative to traditional Hilbert space formulations of quantum mechanics. It offers a more abstract and algebraic framework for understanding and describing quantum systems.

One advantage of this approach is that it allows for a more general description of quantum systems, including those with infinite dimensions. This is particularly useful in the study of quantum field theory, where traditional Hilbert space methods encounter difficulties.

However, it is important to note that the use of Hilbert space in quantum mechanics is deeply rooted in the theory and has been successful in making accurate predictions. Therefore, any new approach must be able to explain and account for the success of Hilbert space methods.

I am curious to see how this theory develops and how it compares to other approaches in the field. Thank you for bringing it to our attention.

 

[Entropia]

, thank you for bringing this article to my attention. It is intriguing to see that there are researchers exploring the possibility of constructing quantum mechanics without the use of Hilbert space. While the use of Hilbert space has been a fundamental aspect of quantum mechanics, it is always valuable to explore alternative approaches and see if they can provide new insights or improvements to the theory.

The work presented in this article by B.J. Hiley offers an interesting perspective on algebraic quantum mechanics and its potential to provide a framework for understanding elementary particles. By utilizing the orthogonal Clifford algebra and the generalised Clifford algebra, C^n, the author shows that the key elements of quantum mechanics such as the wave function, density operator, and mean values can all be constructed within the algebra itself, without the need for Hilbert space.

This approach may have significant implications for our understanding of quantum mechanics and its applications. It offers a different perspective on the mathematical foundations of the theory and may lead to new insights and developments in the field. Furthermore, the relationship between this approach and the traditional Hilbert space approach is also worth exploring, as it may reveal new connections and connections between the two.

Overall, I believe that the work presented in this article is a valuable contribution to the field of quantum mechanics and I look forward to seeing further developments and research in this area. It is exciting to see scientists pushing the boundaries and exploring alternative approaches to established theories, and I believe that this work has the potential to lead to new discoveries and advancements in our understanding of the fundamental nature of the universe.