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

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Teoreticheskaya i Matematicheskaya Fizika, 1997, Volume 112, Number 3, Pages 355–374
DOI: https://doi.org/10.4213/tmf1048 (Mi tmf1048)

This article is cited in 20 scientific papers (total in 20 papers)

A representation of quantum field Hamiltonian in a

$p$ -adic Hilbert space

S. A. Albeverio^a, A. Yu. Khrennikov^b, R. Cianci^c

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

Full-text PDF (342 kB) Citations (20)

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

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.

Received: 05.02.1997

English version:
Theoretical and Mathematical Physics, 1997, Volume 112, Issue 3, Pages 1081–1096
DOI: https://doi.org/10.1007/BF02583040

Bibliographic databases:

Language: Russian

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

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Linking options:

https://www.mathnet.ru/eng/tmf1048

https://doi.org/10.4213/tmf1048

https://www.mathnet.ru/eng/tmf/v112/i3/p355

This publication is cited in the following 20 articles:

Paolo Aniello, Sonia L'Innocente, Stefano Mancini, Vincenzo Parisi, Ilaria Svampa, Andreas Winter, “Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups”, Lett Math Phys, 114:3 (2024)
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “Quantum mechanics on a p-adic Hilbert space: Foundations and prospects”, Int. J. Geom. Methods Mod. Phys., 21:10 (2024)
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “Trace class operators and states in p-adic quantum mechanics”, Journal of Mathematical Physics, 64:5 (2023)
Paolo Aniello, Stefano Mancini, Vincenzo Parisi, “A p-Adic Model of Quantum States and the p-Adic Qubit”, Entropy, 25:1 (2022), 86
Mukhamedov F., Khakimov O., “Translation-Invariant Generalized P-Adic Gibbs Measures For the Ising Model on Cayley Trees”, Math. Meth. Appl. Sci., 44:16 (2021), 12302–12316
Farrukh Mukhamedov, Otabek Khakimov, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, Advances in Non-Archimedean Analysis and Applications, 2021, 115
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
Stepic A.I., Ognjanovic Z., “Logics to Formalise P-Adic Valued Probability and Their Applications”, Int. J. Parallel Emerg. Distrib. Syst., 33:3, SI (2018), 257–275
Dragovich B., Khrennikov A.Yu., Misic N.Z., “Summation of P-Adic Functional Series in Integer Points”, Filomat, 31:5 (2017), 1339–1347
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
A. Ilić Stepić, Z. Ognjanović, N. Ikodinović, A. Perović, “p-Adic probability logics”, P-Adic Num Ultrametr Anal Appl, 8:3 (2016), 177
Stepic A.I., Ognjanovic Z., “Logics For Reasoning About Processes of Thinking With Information Coded By P-Adic Numbers”, Stud. Log., 103:1 (2015), 145–174
Mukhamedov F., Dogan M., “On P-Adic Lambda-Model on the Cayley Tree II: Phase Transitions”, Rep. Math. Phys., 75:1 (2015), 25–46
Mansoor Saburov, Mohd Ali Khameini Ahmad, “On Descriptions of All Translation Invariant p-adic Gibbs Measures for the Potts Model on The Cayley Tree of Order Three”, Math Phys Anal Geom, 18:1 (2015)
Ilic-Stepic A., Ognjanovic Z., Ikodinovic N., “Conditional P-Adic Probability Logic”, Int. J. Approx. Reasoning, 55:9, SI (2014), 1843–1865
Ilic-Stepic A., Ognjanovic Z., Ikodinovic N., Perovic A., “A P-Adic Probability Logic”, Math. Log. Q., 58:4-5 (2012), 263–280
Andrei Yu. Khrennikov, Jan Harm Van der Walt, “On topological extensions of Archimedean and non-Archimedean rings”, P-Adic Num Ultrametr Anal Appl, 3:4 (2011), 326
Kochubei A.N., “Non-Archimedean normal operators”, Journal of Mathematical Physics, 51:2 (2010), 023526
Sergio Albeverio, Roberto Cianci, Andrei Yu. Khrennikov, “p-Adic valued quantization”, P-Adic Num Ultrametr Anal Appl, 1:2 (2009), 91
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

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

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