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 TMF, 1997, Volume 112, Number 3, Pages 355–374 (Mi tmf1048)

A representation of quantum field Hamiltonian in a $p$-adic Hilbert space

S. A. Albeverioa, A. Yu. Khrennikovb, R. Ciancic

a Ruhr-Universität Bochum, Mathematischer Institut
b Växjö University
c University of Genova, Department of Mathematics

Abstract: Gaussian measures on infinite-dimensional $p$-adic spaces are introduced and the corresponding $L_2$-spaces of $p$-adic valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in $p$-adic $L_2$-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the $p$-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). $p$-Adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in $L_2$-spaces with respect to a $p$-adic Gaussian measure.

DOI: https://doi.org/10.4213/tmf1048

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English version:
Theoretical and Mathematical Physics, 1997, 112:3, 1081–1096

Bibliographic databases:

Citation: S. A. Albeverio, A. Yu. Khrennikov, R. Cianci, “A representation of quantum field Hamiltonian in a $p$-adic Hilbert space”, TMF, 112:3 (1997), 355–374; Theoret. and Math. Phys., 112:3 (1997), 1081–1096

Citation in format AMSBIB
\Bibitem{AlbKhrCia97} \by S.~A.~Albeverio, A.~Yu.~Khrennikov, R.~Cianci \paper A representation of quantum field Hamiltonian in a $p$-adic Hilbert space \jour TMF \yr 1997 \vol 112 \issue 3 \pages 355--374 \mathnet{http://mi.mathnet.ru/tmf1048} \crossref{https://doi.org/10.4213/tmf1048} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1486794} \zmath{https://zbmath.org/?q=an:0968.46519} \transl \jour Theoret. and Math. Phys. \yr 1997 \vol 112 \issue 3 \pages 1081--1096 \crossref{https://doi.org/10.1007/BF02583040} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000071403900001} 

• http://mi.mathnet.ru/eng/tmf1048
• https://doi.org/10.4213/tmf1048
• http://mi.mathnet.ru/eng/tmf/v112/i3/p355

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. A. Albeverio, P. E. Kloeden, A. Yu. Khrennikov, “Human memory as a $p$-adic dynamic system”, Theoret. and Math. Phys., 117:3 (1998), 1414–1422
2. Kochubei A.N., “Non-Archimedean normal operators”, Journal of Mathematical Physics, 51:2 (2010), 023526
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8. Saburov M., bin Ismail M.J., “On Square Root Function Over Q(P) and Its Application”, 37Th International Conference on Quantum Probability and Related Topics (Qp37), Journal of Physics Conference Series, 819, eds. Accardi L., Mukhamedov F., Hee P., IOP Publishing Ltd, 2017, UNSP 012028
9. Ahmad Mohd Ali Khameini, Liao L., Saburov M., “Periodic P-Adic Gibbs Measures of Q-State Potts Model on Cayley Trees i: the Chaos Implies the Vastness of the Set of P-Adic Gibbs Measures”, J. Stat. Phys., 171:6 (2018), 1000–1034
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