Abstract
Quantum systems with positions in \(\mathbb {\hat{Z}}\) and momenta in \(\mathbb {Q}/\mathbb {Z}\), are discussed. The Schwartz-Bruhat space of wavefunctions in these systems, is presented. The Heisenberg- Weyl group as a locally compact and totally disconnected topological group, is discussed. Wigner and Weyl functions in this context, are also discussed.
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References
Vourdas, A. (2011). Journal of Mathematical Physics, 52, 062103.
Vourdas, A. (2013). Journal of Physics A, 46, 043001.
Gel’fand, I. M., & Graev, M. I. (1963). Russian Mathematical Surveys, 18, 29–109.
Taibleson, M. H. (1975). Fourier analysis on local fields. Princeton: Princeton University Press.
Vladimirov, V. S. (1988). Russian Mathematical Surveys, 43, 19.
Gel’fand, I. M., Graev, M. I., & Piatetskii-Shapiro, I. I. (1990). Representation theory and automorphic functions. London: Academic.
Dragovoch, B. (1994). Theoretical and Mathematical Physics, 101, 1404.
Dragovich, B. (1995). International Journal of Modern Physics A, 10, 2349.
Bump, D. (1998). Automorphic forms and representations. Cambridge: Cambridge University Press.
Ramakrishnan, D., & Valenza, R. J. (1999). Fourier analysis on number fields. Berlin: Springer.
Vladimirov, V. S., Volovich, I. V., & Zelonov, E. I. (1994). p-adic analysis and mathematical physics. Singapore: World Scientific.
Dragovich, B., Khrenikov, A., Kozyrev, S. V., & Volovich, I. V. (2009). P-adic numbers. Ultrametic Analysis and Applications, 1, 1.
Albeverio, S., Khrennikov, A., & Shelkovich, V. M. (2010). Theory of p-adic distributions. Cambridge: Cambridge U. P.
Weil, A. (1973). Basic number theory. Berlin: Springer.
Vourdas, A. (2012). Journal of Mathematical Analysis and Applications, 394, 48.
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Vourdas, A. (2017). A Quantum System with Positions in the Profinite Group \({\widehat{\mathbb Z}}\) . In: Finite and Profinite Quantum Systems. Quantum Science and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-59495-8_12
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DOI: https://doi.org/10.1007/978-3-319-59495-8_12
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