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Albeverio, S.; Cianci, R.; Khrennikov, A. Yu.

Representation of a quantum field Hamiltonian in \(p\)-adic Hilbert space. (English. Russian original) Zbl 0968.46519

Theor. Math. Phys. 112, No. 3, 1081-1096 (1997); translation from Teor. Mat. Fiz. 112, No. 3, 355-374 (1997).
Summary: Gaussian measures on infinite-dimensional \(p\)-adic spaces are defined 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 such spaces and the formal analogy with the usual Segal representation is discussed. It is found that the parameters of the \(p\)-adic infinite-dimensional Weyl group are defined only on some balls. In \(p\)-adic Hilbert space, representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. The Hamiltonians with singular potentials are realized as bounded symmetric operators in \(L_2\)-space with respect to a \(p\)-adic Gaussian measure.

Cited in 11 Documents

MSC:

46N50 Applications of functional analysis in quantum physics
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
81T05 Axiomatic quantum field theory; operator algebras
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\textit{S. Albeverio} et al., Theor. Math. Phys. 112, No. 3, 1081--1096 (1997; Zbl 0968.46519); translation from Teor. Mat. Fiz. 112, No. 3, 355--374 (1997)
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References:

[1] H. S. Snyder,Phys. Rev.,7, 38 (1947). · Zbl 0035.13101
[2] A. Schild,Phys. Rev.,73, 414 (1948). · Zbl 0034.27602
[3] E. J. Hellund and K. Tanaka,Phys. Rev.,94, 192 (1954). · Zbl 0056.22605
[4] D. I. Blokhintsev,Space and Time in the Microcosm [in Russian], Nauka, Moscow (1982), English translation of previous edition: Dordrecht, Kluwer (1973). · Zbl 0254.53009
[5] Yu. Manin, in:Lect. Notes in Math., Vol. 1111,Quantum Field Theory (1985), p. 59.
[6] I. V. Volovich, ”Number theory as the ultimate physical theory,” Preprint TH.4781/87 (1987). · Zbl 1258.81074
[7] D. Amati, ”On space-time at small distances,” inSakharov Memorial Lectures (L. Keldysh and V. Fainberg, eds.), Nova Science, New York (1992).
[8] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, ”A Liouville string approach to microscopic time in cosmology,” Preprint CERN-TH 7000/93. · Zbl 1001.83503
[9] A. Ashtekar, ”Quantum gravity: a mathematical physics perspective,” Preprint CGPG-93/12-2. · Zbl 0972.83541
[10] S. Doplicher, K. Fredenhagen, and J. E. Roberts,Commun. Math. Phys.,172, 187 (1995). · Zbl 0847.53051
[11] E. Beltrametti and G. Cassinelli,Found. Phys.,2, 1 (1972).
[12] V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov,p-Adic Numbers in Mathematical Physics, World Scientific, Singapore (1993). · Zbl 0812.46076
[13] Yu. Manin, ”Reflections on arithmetical physics,” in:Conformal Invariance and String Theory (P. Dita and V. Georgescu, eds.), Acad. Press, Boston (1989), p. 293.
[14] S. Albeverio, J. E. Fenstad, R. Höegh-Krohn, and T. Lindström,Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Acad. Press, New York (1986). · Zbl 0605.60005
[15] A. Yu. Khrennikov,p-Adic Valued Distributions in Mathematical Physics, Kluwer, Dordrecht-Boston-London (1994).
[16] A. Robinson,Nonstandard Analysis, North-Holland, Amsterdam (1966).
[17] S. Albeverio, ”Nonstandard analysis in mathematical physics,” inNonstandard Analysis and its Applications. LMS Student Text 10 (N. J. Cutland, ed.), Cambridge Univ., Cambridge (1988), p. 182. · Zbl 0658.03047
[18] K. Mahler,Introduction to p-Adic Numbers and their Functions, Cambridge Univ., Cambridge (1973). · Zbl 0249.12015
[19] W. Schikhof,Ultrametric Calculus, Cambridge Univ., Cambridge (1984). · Zbl 0553.26006
[20] A. van Rooij,Non-Archimedean Functional Analysis, Marcel Dekker, New York (1974).
[21] S. Albeverio and J.-L. Wu, ”Nonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interaction,” Preprint SFB-237, No. 246, Bochum (1995). · Zbl 0864.46046
[22] M. Capinski and N. J. Cutland,Nonstandard Methods for Stochastic Fluid Dynamics, World Scientific, Singapore (1995).
[23] I. V. Volovich,Class. Quantum Gravit.,4, L83 (1987).
[24] P. G. O. Freund and M. Olson,Phys. Lett. B,199, 186 (1987).
[25] P. G. O. Freund, M. Olson, and E. Witten,Phys. Lett. B,199, 191 (1987).
[26] I. Ya. Aref’eva, B. Dragovich, and I. V. Volovich,Phys. Lett. B,200, 512 (1988).
[27] I. Ya. Aref’eva, B. Dragovich, P. H. Frampton, and I. V. Volovich,Int. J. Mod. Phys. A,6, 4341 (1991). · Zbl 0733.53039
[28] S. Albeverio and W. Karwowski,Stoch. Proc. Appl.,53, 1 (1994). · Zbl 0810.60065
[29] A. Yu. Khrennikov,Phys. Lett. A,200, 119 (1995). · Zbl 1020.81534
[30] R. Cianci and A. Yu. Khrennikov,Int. J. Theor. Phys.,18, 1217 (1994). · Zbl 0842.60096
[31] A. Yu. Khrennikov,Russ. Math. Surv.,45, No. 4, 87 (1990). · Zbl 0722.46040
[32] V. S. Varadarajan, ”Quantization of semisimple groups and some applications,” Preprint No. 127, Department of Mathematics, University of California at Los Angeles (1995).
[33] V. S. Varadarajan,Lett. Math. Phys.,34, 319 (1995). · Zbl 0830.22006
[34] S. Albeverio and A. Yu. Khrennikov, ”Representation of the Weyl group in spaces of square integrable functions with respect top-adic-valued Gaussian distributions”, Preprint SFB-237, No. 263, Inst. Math., Bochum (1995).
[35] H. Weyl,Theory of Groups and Quantum Mechanics, Dover, New York (1931). · JFM 58.1374.01
[36] J. Glimm and A. Jaffe,Quantum Physics: A Functional Integral Point of View, Springer, Berlin-Heidelberg New York (1987). · Zbl 0461.46051
[37] S. Albeverio and R. Höegh-Krohn, ”Quasi-invariant measures, symmetric diffusion processes, and quantum fields,” in:Proc. of Int. Colloquium on Mathematical Methods of Quantum Field Theory. Colloques Internationaux du Centre Nationale de la Recherche Scientifique, No. 248. Editions CNRS (1976), p. 11.
[38] J. C. Baez, I. E. Segal, and Z. F. Zhou,Introduction to Algebraic and Constructive Quantum Field Theory, Princeton Univ., Princeton (1992). · Zbl 0760.46061
[39] B. Simon, TheP({\(\phi\)})2Euclidean (Quantum) Field Theory, Princeton Univ., Princeton (1974).
[40] J. Glimm and A. Jaffe, ”Boson quantum field models,” in:Mathematics in Contemporary Physics. Proc. Instructional Conf., London, 1972 (R. F. Streater, ed.), Acad. Press, New York (1974), p. 77. · Zbl 0191.27101
[41] S. Albeverio and R. Höegh-Krohn, ”Quantum fields and fields with values in groups,” in:Stochastic Analysis and Applications (M. Pinsky and M. Dekker, eds.), Acad. Press, New York (1984), p. 1. · Zbl 0575.60078
[42] S. Albeverio, Ph. Blanchard, Ph. Combe, R. Höegh-Krohn, and M. Sirugue,Commun. Math. Phys.,90, 329 (1983). · Zbl 0538.46053
[43] S. Albeverio and M. Röckner,Prob. Theory Relat. Fields,89, 347 (1991). · Zbl 0725.60055
[44] A. Yu. Khrennikov,Theor. Math. Phys.,66, 223 (1986). · Zbl 0622.35082
[45] T. Hida, H. H. Kuo, J. Potthoff, and L. Streit,White Noise, Kluwer, Dordrecht (1993).
[46] K. R. Parthasarathy,An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel (1992). · Zbl 0751.60046
[47] A. Yu. Khrennikov and Zhiyuan Huang,Quantum Probability and Related Topics,9, 273 (1994).
[48] J. von Neumann,Mathematical Foundations of Quantum Mechanics, Princeton (1955). · Zbl 0064.21503
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