P-adic quantum mechanics explained

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]

See also

• P-adic analysis § P-adic quantum mechanics

Notes and References

1. Volovich. I. V.. 1987-06-01. p-adic space-time and string theory. Theoretical and Mathematical Physics. en. 71. 3. 574–576. 10.1007/bf01017088. 0040-5779. 1987TMP....71..574V. 119936152.
2. Freund. Peter G.O.. Peter Freund. Witten. Edward. Edward Witten. Adelic string amplitudes. Physics Letters B. 199. 2. 191–194. 10.1016/0370-2693(87)91357-8. 1987. 1987PhLB..199..191F.
3. Marinari. Enzo. Parisi. Giorgio. Giorgio Parisi. 1988-03-24. On the p-adic five-point function. Physics Letters B. 203. 1–2. 52–54. 10.1016/0370-2693(88)91569-9. 1988PhLB..203...52M.
4. Freund. Peter G. O.. Peter Freund. 2006-03-29. p‐Adic Strings and Their Applications. AIP Conference Proceedings. 826. 1. 65–73. 10.1063/1.2193111. 0094-243X. 2006AIPC..826...65F. hep-th/0510192. 119086848.
5. Book: Khrennikov, Andrei. Interpretations of probability. Andrei Khrennikov. 2009. Walter de Gruyter. 9783110213195. second. Berlin. 370384640.
6. Dimitrijevic. D.d.. Djordjevic. G.s.. Nesic. Lj.. 2008-04-18. Quantum cosmology and tachyons. Fortschritte der Physik. en. 56. 4–5. 412–417. 0804.1328. 10.1002/prop.200710513. 1521-3978. 2008ForPh..56..412D. 118417071.
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. en. 2. 4. 322–340. 1011.6589. 10.1134/s2070046610040060. 119297562. 2070-0466.
8. Book: P-adic analysis and mathematical physics. Vladimirov. V. S.. Volovich. I. V.. Zelenov. E. I.. 1994. World Scientific. 9789814355933. Singapore. 841809611.
9. Djordjevic. Goran S.. Dragovich. Branko. 2000-05-26. On p-Adic Functional Integration. math-ph/0005025.
10. Dragovich. Branko. 1995-06-30. Adelic harmonic oscillator. International Journal of Modern Physics A. 10. 16. 2349–2365. hep-th/0404160. 10.1142/s0217751x95001145. 0217-751X. 1995IJMPA..10.2349D. 18786418.