×

zbMATH — the first resource for mathematics

\(p\)-adic valued quantization. (English) Zbl 1187.81137
Summary: This review covers an important domain of \(p\)-adic mathematical physics – quantum mechanics with \(p\)-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of \(p\)-adic numbers \(\mathbb Q_p\) , operators – symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the \(p\)-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg and Weyl forms, spectral properties of the operator of \(p\)-adic coordinate.We also present postulates of \(p\)-adic valued quantization. Here observables as well as probabilities take values in \(\mathbb Q_p\). A physical interpretation of \(p\)-adic quantities is provided through approximation by rational numbers.

MSC:
81Q65 Alternative quantum mechanics (including hidden variables, etc.)
81T10 Model quantum field theories
81S10 Geometry and quantization, symplectic methods
81S05 Commutation relations and statistics as related to quantum mechanics (general)
PDF BibTeX Cite
Full Text: DOI
References:
[1] V. S. Vladimirov and I.V. Volovich, ”Superanalysis. I. Differential calculus,” Theor. Math. Phys. 59, 317–335 (1984). · Zbl 0552.46023
[2] V. S. Vladimirov and I. V. Volovich, ”Superanalysis. II. Integral calculus,” Theor. Math. Phys. 60, 743–765 (1985). · Zbl 0599.46068
[3] I. V. Volovich. ”p-Adic string,” Class. Quant. Grav. 4, 83–87 (1987). · Zbl 0636.12015
[4] I. Ya. Aref’eva, B. Dragovich and I. V. Volovich, ”On the p-adic summability of the anharmonic ocillator,” Phys. Lett. B 200, 512–514 (1988).
[5] B. Dragovich, ”On signature change in p-adic space-time,” Mod. Phys. Lett. 6, 2301–2307 (1991). · Zbl 1020.81506
[6] A. Yu. Khrennikov, ”Mathematical methods of the non-archimedean physics,” Uspekhi Mat. Nauk 45(4), 79–110 (1990). · Zbl 0722.46040
[7] A. Yu. Khrennikov, ”Quantum mechanics over Galois extensions of number fields,” Dokl. Akad. Nauk USSR 315, 860–864 (1990).
[8] A. Yu. Khrennikov, ”p-Adic quantum mechanics with p-adic valued functions,” J.Math. Phys. 32, 932–937 (1991). · Zbl 0746.46067
[9] A. Yu. Khrennikov, ”Real-non-Archimedean structure of space-time,” Theor. Math. Phys. 86(2), 177–190 (1991). · Zbl 0753.46042
[10] A. Yu. Khrennikov, ”p-Adic probability and statistics,” Dokl. Akad. Nauk 322, 1075–1079 (1992). · Zbl 0768.60002
[11] A. Yu. Khrennikov, ”p-Adic probability theory and its applications. The principle of statistical stabization of frequencies,” Theor. Math. Phys. 97(3), 348–363 (1993).
[12] A. Yu. Khrennikov, p-Adic Valued Distributions in Mathematical Physics (Kluwer, Dordrecht, 1994).
[13] R. Cianci and A. Yu. Khrennikov, ”p-Adic numbers and the renormalization of eigenfunctions in quantum mechanics,” Phys. Lett. B 328, 109–112 (1994). · Zbl 0910.46060
[14] R. Cianci and A. Yu. Khrennikov, ”Energy levels corresponding to p-adic quantum states,” Dokl. Akad. Nauk 342(5), 603–606 (1995). · Zbl 0879.46038
[15] A. Yu. Khrennikov, ”On probablity distributions on the field of p-adic numbers,” Theory of Probab. and Appl. 40(1), 189–192 (1995). · Zbl 0848.60011
[16] A. Yu. Khrennikov and H. Zhiyan, ”Generalized functionals of p-adic white noise,” Dokl. Akad. Nauk 344(1), 23–26 (1995).
[17] S. Albeverio and A. Yu. Khrennikov, ”Representation of the Weyl group in spaces of square integrable functions with respect to p-adic valued Gaussian distributions,” J. Phys. A: Math. Gen. 29, 5515–5527 (1996). · Zbl 0903.46073
[18] S. Albeverio and A. Yu. Khrennikov, ”p-Adic Hilbert space representation of quantum systems with an infinite number of degrees of freedom,” Int. J. Mod. Phys. B 10, 1665–1673 (1996). · Zbl 1229.81283
[19] A. Yu. Khrennikov, ”Ultrametric Hilbert space representation of quantum mechanics with a finite exactness,” Found. Physics 26(8), 1033–1054 (1996).
[20] A. Yu. Khrennikov and Z. Huang, ”A model for white noise analysis in p-adic number fields,” Acta Math. Scientia (China) 16(1), 1–14 (1996). · Zbl 0867.60021
[21] S. Albeverio, R. Cianci and A. Yu. Khrennikov, ”On the spectrum of the p-adic position operator,” J. Phys. A: Math. Gen. 30, 881–889 (1997). · Zbl 0992.81022
[22] S. Albeverio, R. Cianci and A. Yu. Khrennikov, ”A representation of quantum field Hamiltonian in a p-adic Hilbert space,” Theor. Math. Phys. 112(3), 355–374 (1997). · Zbl 0968.46519
[23] S. Albeverio, R. Cianci and A. Yu. Khrennikov, ”On the Fourier transform and the spectral properties of the p-adic momentum and Schrodinger operators,” J. Phys. A:Math. Gen. 30, 5767–5784 (1997). · Zbl 0927.46060
[24] S. Albeverio and A. Yu. Khrennikov, ”A regularization of quantum field Hamiltonians with the aid of p-adic numbers,” Acta Appl. Math. 50, 225–251 (1998). · Zbl 0896.46047
[25] A. Yu. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models (Kluwer, Dordrecht, 1997).
[26] A. Yu. Khrennikov, ”Description of experiments detecting p-adic statistics in quantum diffraction experiments,” Doklady Mathematics 58(3), 478–480 (1998). · Zbl 1065.81504
[27] A. Yu. Khrennikov, Interpretations of Probability (VSP, Utrecht, 1999).
[28] A. Monna, Analyse non-Archimédienne (Springer-Verlag, Berlin-Heidelberg-New York, 1970).
[29] A. van Rooij Non-Archimedean Functional Analysis (Marcel Dekker, New York, 1978).
[30] W. H. Schikhof, Ultrametric Calculus. An Introduction to p-Adic Analysis (Cambridge University Press, Cambridge, 1984). · Zbl 0553.26006
[31] G. K. Kalisch, ”On p-adic Hilbert spaces,” Ann.Math. 48, 180–192 (1947). · Zbl 0029.14102
[32] J. M. Bayod, Productos Internos en Espacios Normados non-Arquimedianos (Univ. de Bilbao Press, Bilbao, 1976).
[33] S. Albeverio, J. M. Bayod, C. Perez-Garcia, A. Yu. Khrennikov and R. Cianci, ”Non-Archimedean analogues of orthogonal and symmetric operators,” Izv. Akademii Nauk 63(6), 3–28 (1999). · Zbl 0943.46044
[34] V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific Publ., Singapore, 1995). · Zbl 0812.46076
[35] M. Endo and A. Yu. Khrennikov, ”The unboundedness of the p-adic Gaussian distribution,” Izvestia Akad. Nauk SSSR: Ser. Mat. 56(4), 456–476 (1992). · Zbl 0819.46063
[36] A. N. Kochubei, ”p-Adic commutation relations,” J. Phys. A: Math. Gen. 29, 6375–6378 (1996). · Zbl 0905.46051
[37] H. Keller, H. Ochsenius and W. H. Schikhof, ”On the commutation relation AB - BA = I for operators on nonclassical Hilbert spaces,” Lecture Notes in Pure and Appl. Math. 222, 177–190 (2003). · Zbl 0993.47045
[38] B. Dragovich, ”Adelic harmonic oscillator,” Int. J. Mod. Phys. A 10, 2349–2365 (1995). · Zbl 1044.81585
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.