Abstract
In the framework of Micro–Macro duality with quadrality scheme, the mutual relations between Micro and Macro are considered in a systematic way. On this basis, many interesting aspects of symmetry breaking are discussed, according to which the classifying space of sectors is shown to have the structure of symmetric spaces characterized by the condition similar to Maxwell and Einstein equations. Applying these results to the attempts for constructing a theory of dressed photons, we are naturally led to the idea of the relevance of automorphic forms which have close connections with number theory.
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Notes
- 1.
While the contents of [9] is the crossed products of von Neumann algebras, this book is very instructive for understanding the essence of the concept, either in C\(^{*}\)- or W\(^{*}\)-versions.
- 2.
Equality \(\mathrm{Im}(q)=\ker (p)\) means that the vanishing energy balance (\(\ker (p)\)) taken as the visible work is equivalent to the input-output of the heat (\(\mathrm{Im}(q)\)).
- 3.
This has led to the mathematical definition of “sectors” of \(\mathcal {A}\) by \(End(\mathcal {A})/Inn(\mathcal {A})\).
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Acknowledgements
I would like to express my sincere thanks to Professor Ohtsu for inviting me into the exciting and challenging research field of dressed photons. Without this extremely nice opportunity, I could not find the interesting directions to which the framework of Micro–Macro duality and quadrality scheme should be developed further in such a fascinating forms of connections with experimental situations. On this opportunity, let me express my thanks to Dr. Sakuma for inspiring discussions on Clebsch duality for many years. On many subjects, I have benefited very much from the collaborations and instructive discussions with all the members at RODreP. Last but not least, I would like to thank Professor Yatsui for all his patient helps in facilitating my slow writing process.
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Ojima, I. (2018). Control over Off-Shell QFT via Induction and Imprimitivity. In: Yatsui, T. (eds) Progress in Nanophotonics 5. Nano-Optics and Nanophotonics. Springer, Cham. https://doi.org/10.1007/978-3-319-98267-0_4
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