Note on the BlochNordsieck'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 BlochNordsieck'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 BN's method will be unsuccessful if the current perturbational treatment is uncorrect for mesonic systems. Since the BlochNordsieck'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 BN'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!ochNordsieck~s Method
we can make eight equations
(HE)IJf=O,
l±T. l±S (HE)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· rP",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!ochNordsieck'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 BN's method, where
and, in the perturbation methods,
Initial state
Energy difference
Aspect of bound field
Final state
Note on the EtochNordsieck'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.