Abstract
In this chapter we would like to describe, briefly, the current topics of interest in the topos quantum theory field of research. Some of these topics have been partly investigated and are slowly developing, while others are still open issues which need to be addressed.
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Notes
- 1.
When we consider \(C^{\infty}(S, \mathbb {R})\) to be a Lie algebra we will use the notation \(C^{\infty}_{\mathit{Lie}}(S, \mathbb {R})\).
- 2.
There is always an infinite-dimensional transitive group, namely, the group of symplectic transformations of S.
- 3.
Note that the use of PG raises the interesting question as to the extent to which the non-commutative structure of the Lie algebra L(G) can be recovered from knowing the poset structure of its abelian Lie sub-algebras. Work in this direction has been done in [34], where it was shown that the Jordan structure of certain von Neumann algebras (without type I 2 summand) is determined by the poset of its abelian sub-algebras.
References
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K. Nakayama, Sheaves in quantum topos induced by quantization. arXiv:1109.1192 [math-ph]
C. Flori, Concept of quantization in a topos. In preparation
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C. Heunen, N.P. Landsman, B. Spitters, A topos for algebraic quantum theory. Commun. Math. Phys. 291, 63 (2009). arXiv:0709.4364 [quant-ph]
A. Doering, Generalised gelfand spectra of Nonabelian unital C ∗-algebras II: flows and time evolution of quantum systems (2012). arXiv:1212.4882 [math.OA]
A. Doering, Generalised gelfand spectra of Nonabelian unital C ∗-algebras I: categorical aspects, automorphisms and Jordan structure (2012). arXiv:1212.2613 [math.OA]
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Flori, C. (2013). Future Research. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_21
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