Note on the Bloch-Nordsieck's Method

Progress of Theoretical Physics, Dec 1951

The Bloch-Nordsieck's method has been applied to mesonic systems by several authors. Here we examine the convergence character of this method and show that it is about the same with that of the current perturbation method. So the B-N's method will be unsuccessful if the current perturbational treatment is uncorrect for mesonic systems.

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Note on the Bloch-Nordsieck's Method

Gyo TAKEDA 0 1 Yasutaka TANlKAWA 0 1 Tosiya TANIUTI 0 1 0 Faculty of Education , Kobe Universit;y 1 Physical Department , Kobe Universit;y The Bloch-Nordsieck's method has been applied to mesonic systems by several authors. Here weelCamine the convergence character of this method and show that it is about the same with that of the current perturbation method. So the B-N's method will be unsuccessful if the current perturbational treatment is uncorrect for mesonic systems. Since the Bloch-Nordsieck's method1) was successfully applied to radiative processes in order to overcome the infrared catastrophe of electromagnetic systems, this same method has been applied also to mesonic systems.2) But, in latter cases, careful attention was never made about the applicability of it. So we want to inquire how is the case compared with the current perturbation method. Here, as an illustration, we take up the symmetrical pseudoscalar meson theory with pseudovector coupling. Employing the same notations with those of the previouspaper,3) the Hamiltonian becomes The B-N's method consists essentially in replacing uncommuting quantities a, {3" ~ T, 0" by their classical representatives. To do this, we introduce three operators A, T and S, given by where sand t are classical unit vectors. All of them have eigenvalues 1 and -1, so we can divide the wave function 1J! into eight parts according to signs of their eigenvalues and - H-- a· p + mid + 1/2 2]",k(tJk(P"~k + Q"~k) Note on the B!och-Nordsieck~s Method we can make eight equations (H-E)IJf=O, l±T. l±S (H-E)IJf=O. 2 2 {m+ (1/2) 2J",kWk(P"~k+ Q"~k)-E -r5(S' p)x++ - {r5(S'p) + (a.p)} X+- (g/p.) 2ja.k(lIJk/2Q) J/2(Q",k cosk· r-P",k sink·r) x {r5t"x+++r5(t,,+r,,)X-+} . x nct.,k ho (Qct.,k +Act.,k cosk·r) , (l/12n") (g2/471") (K./p)4 > 1~+-12, 1~-+12 1~--12 ~ (1/371") (g2/471") (K,,/pr, § 2. Comparison with pertnrbation method Note on the B!och-Nordsieck's Method and if it is admitted to developlh in g, g¢l,l + g2¢l,2+ ... , g2¢2,2+ .,. , § 3. Discussion of results in B-N's method, where and, in the perturbation methods, Initial state Energy difference Aspect of bound field Final state Note on the Etoch-Nordsieck's Method 1) F. Bloch and A. Nordsieck , Phys. Rev . 52 ( 1937 ), 54 . 2) Lewis, 9ppenheimer and Wouthuysen , Phys. Rev . 73 ( 1948 ), 127. S. D. Drell, Phys. Rev . 83 ( 1951 ), 555 . 3) ibid . S. D. Drell.


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Gyo Takeda, Yasutaka Tanikawa, Tosiya Taniuti, Keiiti Saeki. Note on the Bloch-Nordsieck's Method, Progress of Theoretical Physics, 1951, 994-999, DOI: 10.1143/ptp/6.6.994