p-adic quantum mechanics

p-adic quantum mechanics is a collection of related research efforts in quantum physics that replace real numbers with p-adic numbers. Historically, this research was inspired by the discovery that the Veneziano amplitude of the open bosonic string, which is calculated using an integral over the real numbers, can be generalized to the p-adic numbers.[1] This observation initiated the study of p-adic string theory.[2][3][4] Another approach considers particles in a p-adic potential well, with the goal of finding solutions with smoothly varying complex-valued wave functions. Alternatively, one can consider particles in p-adic potential wells and seek p-adic valued wave functions, in which case the problem of the probabilistic interpretation of the p-adic valued wave function arises.[5] As there does not exist a suitable p-adic Schrödinger equation,[6][7] path integrals are employed instead. Some one-dimensional systems have been studied by means of the path integral formulation, including the free particle,[8] the particle in a constant field,[9] and the harmonic oscillator.[10]

References
  1. Volovich, I. V. (1987-06-01). "p-adic space-time and string theory". Theoretical and Mathematical Physics. 71 (3): 574–576. Bibcode:1987TMP....71..574V. doi:10.1007/bf01017088. ISSN 0040-5779.
  2. Freund, Peter G.O.; Witten, Edward (1987). "Adelic string amplitudes". Physics Letters B. 199 (2): 191–194. Bibcode:1987PhLB..199..191F. doi:10.1016/0370-2693(87)91357-8.
  3. Marinari, Enzo; Parisi, Giorgio (1988-03-24). "On the p-adic five-point function". Physics Letters B. 203 (1–2): 52–54. Bibcode:1988PhLB..203...52M. doi:10.1016/0370-2693(88)91569-9.
  4. Freund, Peter G. O. (2006-03-29). "p‐Adic Strings and Their Applications". AIP Conference Proceedings. 826 (1): 65–73. arXiv:hep-th/0510192. Bibcode:2006AIPC..826...65F. doi:10.1063/1.2193111. ISSN 0094-243X.
  5. Khrennikov, Andrei (2009). Interpretations of probability (second ed.). Berlin: Walter de Gruyter. ISBN 9783110213195. OCLC 370384640.
  6. Dimitrijevic, D.d.; Djordjevic, G.s.; Nesic, Lj. (2008-04-18). "Quantum cosmology and tachyons". Fortschritte der Physik. 56 (4–5): 412–417. arXiv:0804.1328. Bibcode:2008ForPh..56..412D. doi:10.1002/prop.200710513. ISSN 1521-3978.
  7. Dragovich, Branko; Rakić, Zoran (2010-12-01). "Path integrals for quadratic lagrangians on p-adic and adelic spaces". P-Adic Numbers, Ultrametric Analysis, and Applications. 2 (4): 322–340. arXiv:1011.6589. doi:10.1134/s2070046610040060. ISSN 2070-0466.
  8. Vladimirov, V. S.; Volovich, I. V.; Zelenov, E. I. (1994). P-adic analysis and mathematical physics. Singapore: World Scientific. ISBN 9789814355933. OCLC 841809611.
  9. Djordjevic, Goran S.; Dragovich, Branko (2000-05-26). "On p-Adic Functional Integration". arXiv:math-ph/0005025.
  10. Dragovich, Branko (1995-06-30). "Adelic harmonic oscillator". International Journal of Modern Physics A. 10 (16): 2349–2365. arXiv:hep-th/0404160. Bibcode:1995IJMPA..10.2349D. doi:10.1142/s0217751x95001145. ISSN 0217-751X.
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