A Statistical Model of Mesons

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7. 7. One of σ and k (respectively a' and k') is a redundant parameter. 
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9. 9. We do not adopt the notion of “mixture of quark states” in this paper. It seems, however, plausible that the decay mode of $S*\left(975\right)\to \pi \pi (78\pm 3\%)$, $KK(22\pm 3\%)$ indicates the necessity of introducing this notion. If we do so, $((s+d)/2,{\varphi }_{8,0},(\overline{s}+\overline{d})/2)\left(980\right)$ is a candidate for $S*\left(975\right)$. 
10. 10. Compare two functions: $J=1.43x^{3/4}$ and $J=0.5+0.86x$, where $x=M^2$, $0lt;x\leqq 5$. It is difficult to judge which function approximates the given experimental data better. 
11. 11. We are assuming that $(b,{\varphi }_{74},\overline{b})$ is the smallest $(b,\overline{b})$-meson, since $\gamma \left(1\right)\left(9458\right)$ is the smallest one ever observed by now, although a smaller $(b,\overline{b})$-meson with spin zero is expected by our composite model. 
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16. 13. In higher dimensions we need duality arguments, which will not come across in one-dimension. See Nagasawa (1980). 
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23. 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. 
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30. 17. Nagasawa, M., Interacting diffusions and Schrödinger equation. Journées de Probabilités, 1983, Bern. 
31. 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 
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34. 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). 
35. 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. 
36. 19. The following arguments are based on discussions with H. Föllmer. 
37. 20. For example take $\varphi =c{x}^2{e}^{-x^2}$, then $\frac{1}{2}\frac{1}{\varphi }{\varphi }^{\text{'}}=-x+\frac{1}{x}$. 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 $x^{-4}$. 
38. 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.