A Statistical Model of Mesons
Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 33 (1983), Exposé no. 1, 48 p.
@article{RCP25_1983__33__1_0,
author = {Nagasawa, M. and Yasue, K.},
title = {A {Statistical} {Model} of {Mesons}},
journal = {Les rencontres physiciens-math\'ematiciens de Strasbourg -RCP25},
note = {talk:1},
publisher = {Institut de Recherche Math\'ematique Avanc\'ee - Universit\'e Louis Pasteur},
volume = {33},
year = {1983},
language = {en},
url = {http://www.numdam.org/item/RCP25_1983__33__1_0/}
}
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PB  - Institut de Recherche Mathématique Avancée - Université Louis Pasteur
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Nagasawa, M.; Yasue, K. A Statistical Model of Mesons. Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Tome 33 (1983), Exposé no. 1, 48 p. http://www.numdam.org/item/RCP25_1983__33__1_0/

1. J. C. Taylor, Gauge Theories of Weak Interactions, (Cambridge University Press, 1976). | MR 468756

2. L. D. Faddeev and A. A. Slavnov, Gauge Fields Introduction to Quantum Theory, (Benjamin, Reading 1980). | MR 618649 | Zbl 0486.53052

3. W. Marciano and H. Pagels, Phys. Rep. 36C, 137 (1978).

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5. H. Yukawa (Editor), Theory of Elementary Particles Extended in Space-Time, Prog. Theor. Phys. Suppl. No 67 (1979). | MR 575671

6. J. L. Gervais and A. Neveu (Editors), Phys. Rep. 23C, 237 (1976).

7. One of $\sigma$ and k (respectively a' and k') is a redundant parameter.

8. E. C. Tichmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations, I (Oxford, 1962). | Zbl 0099.05201

9. We do not adopt the notion of "mixture of quark states" in this paper. It seems, however, plausible that the decay mode of $S*\left(975\right)\to \pi \pi \left(78±3%\right)$, $KK\left(22±3%\right)$ indicates the necessity of introducing this notion. If we do so, $\left(\left(s+d\right)/2,{\phi }_{8,0},\left(\overline{s}+\overline{d}\right)/2\right)\left(980\right)$ is a candidate for $S*\left(975\right)$.

10. Compare two functions: $J=1.43{x}^{3/4}$ and $J=0.5+0.86x$, where $x={M}^{2}$, $0. It is difficult to judge which function approximates the given experimental data better.

11. We are assuming that $\left(b,{\phi }_{74},\overline{b}\right)$ is the smallest $\left(b,\overline{b}\right)$-meson, since $\gamma \left(1\right)\left(9458\right)$ is the smallest one ever observed by now, although a smaller $\left(b,\overline{b}\right)$-meson with spin zero is expected by our composite model.

12. For details and other applications of the model, see Nagasawa, M., Segregation of a population in an environment. J. Math. Biology (1980), 9, 213-235 | Zbl 0447.92017

Nagasawa, M. An application of the segregation model for septation of Escherichia coli. J. Theor. Biol. (1981), 90, 445-455

Nagasawa, M. A statistical model of systems of interacting diffusion-particles (in preparation).

Albeverio, S., Blanchard, Ph., & Høegh-Krohn, R., A stochastic model for the orbits of planets and satellites : An interpretation of Titius-Bode law (preprint). | Zbl 0533.70006

13. In higher dimensions we need duality arguments, which will not come across in one-dimension. See Nagasawa (1980).

14. For stochastic differential equations see, e.g. K. Itô and S. Watanabe, Introduction to stochastic differential equations, Proc. of Intern. Symp. SDE Kyoto, 1976 (Ed. by K. Itô) i-xxx, Kinokuniya Book-Store, Co. LTD, Tokyo. | Zbl 0405.60058

15. Mckean, H. P. (1966) A class of Markow processes associated with non-linear parabolic equations, Proc. Nat. Acad. Sci. 56, 1907-1911. | MR 221595 | Zbl 0149.13501

Mckean, H. P. (1967) Propagation of chaos for a class of non-linear parabolic equations. Lecture series in differential equations 7, Catholoc Univ. 41-57. | MR 233437 | Zbl 0181.44401

Brown, W. and Hepp, K. (1977), The Vlasov dynamics and its fluctuations in 1/N limit of interacting classical particles, Comm. Math. Phys. 56, 101-113. | MR 475547 | Zbl 1155.81383

Tanaka, H. (to appear), Limit theorems for certain diffusion processes with interaction, Taniguchi International Symposium, July 1982. | MR 780770 | Zbl 0552.60051

Dawson, D. A. Critical dynamics and fluctuations for a mean field model of cooperative behavior. J. of Statistical Physics (1983), 31, 29-85. | MR 711469

16. Time reversal plays an important role in this model, although it is hidden in one dimension. For time reversal of diffusion processes see : Schrödinger, E., Ueber die Umkehrung der Naturgesetze. Berliner Berichte (1931), Sitzung der physikalisch-mathematischen Klasse, 144-153.

Kolmogoroff, A., Zur Umkehrbarkeit der Statistischen Naturgesetze, Math. Ann. 113 (1937), 766-772. | MR 1513121 | Zbl 0015.26004

Nagasawa, M., Time reversions of Markov processes, Nagoya Math. Journal 24 (1964), 177-204. | MR 169290 | Zbl 0133.10702

Chung, K. L. and Walsh, J. B., To reverse a Markov process, Acta Math. 123, (1969), 225-251. | MR 258114 | Zbl 0187.41302

Mayer, P. A., Le returnement du temps, d'aprés Chung et Walsh, Lecture Notes in Math. 191, (1971), 213-245 (Springer). | Numdam | MR 383549

Nagasawa, M. and Maruyama, T., An application of time reversal of Markov processes to a problem of population genetics, Advances in Appl. Probability 11, (1979), 457-478. | MR 533054 | Zbl 0406.60070

Föllmer, H., On local time and time reversal, Journées de probabilités, 1983, Bern.

17. Nagasawa, M., Interacting diffusions and Schrödinger equation. Journées de Probabilités, 1983, Bern.

Interrelation between Schrödinger équation and diffusion processes has been discussed by Fényes and Nelson, see: Fenyes, I., Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantemechanik, Z. für Physik, 132 (1952), 81-106

Nelson, E., Derivation of Schrödinger equation from Newtonian Mechanics, Phys. Rev. 150 (1966), 1076-1085

Yasue, K. Stochastic Quantization : A Review, International Journal of Theor. Phys. 18 (1979), 861-913. | MR 574682 | Zbl 0435.60003

The interpretation of a diffusion process as a typical partiale of a system of interacting particles is different from theirs and was given in Nagasawa (1980).

18. See Theorem 6.1 of Nagasawa (1980) (in the proof, (6.11) should be read as $\left|\psi \right|{∥\nabla \beta ∥}^{2}=0\left(1\right)$, and also Nelson, E., Critical diffusions. Journées de Probabilités, 1983, Bern.

19. The following arguments are based on discussions with H. Föllmer.

20. For example take $\phi =c{x}^{2}{e}^{-{x}^{2}}$, then $\frac{1}{2}\frac{1}{\phi }{\phi }^{\text{'}}=-x+\frac{1}{x}$. Hence, ${b}_{1}\left(x\right)=-x$ and ${b}_{2}\left(x\right)=\frac{1}{x}$. The solution of (46) for this ${b}_{1}\left(x\right)$ is $h\left(x\right)=-x$. The solution ${h}_{2}\left(x\right)$ of (47) for the ${b}_{2}\left(x\right)$ has a singularity of ${x}^{-4}$.

21. This is the so called "piecing together (or revival) technique" of the theory of Markov processes. Cf. Theorem 1 and 2 of Nagasawa, M., Basic models of Branching Processes, Proc. of 41st Session of ISI, New Delhi, 1977, XLVII (2), 423-445.