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In a , the Theory Was Developed for the Case When the Electromagnetic Interaction Alone Is Involved. in the First Part of the Pr 1941] _On Yukawa's Theory of the Beta-Disintegration. 291 (the first term in (28) being ignored), which settles an upper limit for the proper lifetime of the meson. This in good agreement with the value 2•~10-6sec. obtained from the cosmic ray observations, if we choose correspoding to a cutting off Wave number of the order of 3ƒÈ I think, this result must be considered as very satisfaetory, since through outthe quantum theory reasonable results can only be obtained by the perturbation method if we cut off at wave length of this order of In conelusion, I wish to express my cordinal thanks to Professor H Yukawa for his kind interest in this work. Institute of Theoretical Physics, Kyoto Imperial University. (Reecived Feb. 12, 1941.) On Yukaua's Theory of the Beta-Disintegration and the Lifetime of the Meson. By Shoichi SAKATA. (Read Jan 18, 1941.) Introduction. Ina recent paper by M. Taketani and the present author(1) a matrix formulation of the vector meson theory was described. Though this formalism is inadequate for the diseussion of the general character (e.g., Lorentz invarianey, ete.) of the theory, it proves to be very useful for the practical calenlation. Especially, the spin sunmmation of the meson can then be carried out automatically by using the orthogonality , the theory was developed for the case when the electromagnetic interaction alone is involved. In the first part of the present paper. an extension to the generalnc case when the mesons interact. with the. (1) S. Sakata and M. Taketani, Scient.Pap. I.P.C.R.38. No. 984 (1940),1, referred as A. See also M. Taketani and S. Sakata, Proc. Phys.-Math. Soc.Japan. 22 (1940), 757 Inmagnitude. A 292 Shoichi SAKATA. [Vol. 23 heavy and light particles is given(2). It is further shown that the present formalism can be easily extended so as to include all the cases of the mesons with spin 0 or 1 considered by Kemmer(3). In the subsequem parts of this paper, a new derivation of thebeta -disintegration formula and the meson lifetime, resulting from each of the four types of the meson theories, is given by using the above formalism. The well-known discrepancy(4) between the calculated and experimental values of the meson lifetime is also discussed. I. General Theory. 1. Vector Theory. In the vector theory of the meson developed by Yukawa and others(5) the total Hamiltonian for the system consist ingof nusons, heavy particles and light particles was given byH= Hu+Hh+Hl+H'q+H'q+H'q2+H'qq'+H'q'2, (1) with Hu=•ç1/4ƒÎ(FF)+1/ƒÈ2divFdivF+(UU)+1/ƒÈ2(curlU)(curlU)/dv . ( 2)Hh=•çƒµ{-ihcƒÏ(h)1ƒÐ<•¨>hgrad+grad+ƒÏ(h)3Mre2}ƒµdv+•çƒµ{-ihcƒÏ1hƒÐ<•¨>(h)grad+ƒÏ(h)3MNc2}ƒµdr, (3)+•çƒÓ{-ihcƒÏ(h)1ƒÐ<•¨>(l)grad+ƒÏl3ƒÊ_??_2}ƒÓdv, (4)H'_??_=•ç{-g1/ƒÈ1M_??_divF-g1/ƒÈ(MU)+g2/ƒÈ2(SeurlU)-g2/ƒÈ(TF)}dv +comp. conj., (5) H'0'=•ç{-g'1/ƒÈ2M'0divF-g'1/ƒÈ(M'U)+g'2/ƒÈ2(S'eurlU)-g'2/ƒÈ(T'F)}dv,+comp. conj., (6) H'_??_2=4ƒÎ/ƒÈ2•ç{g2_??_M_??_M_??_+g2/2(SS)}dv.(7) 2) An equivalent formulation based on Kemmer's theory (Proc. Roy Soc. A 173(1939 ), 91)has been given by A.H. Wilson (Proc. Camb, Phil. Soc. XXXVI (1940), 363),b ut this is not so convenient as ours owiny to the singular character of the ƒÀ-matrices. (3) N. Kemmer. Proc Roy. Soc. A 166 (1938), 127. (4) L. Nordheim, Phys. Rev. 55 (1939), 506.(5) H . Yokawa, S. Sakata, M. Kobayasi and M. Taketani, Proc. Phys.-Math. Soc. Japan, 20 (1938), 720 which will be referred as IV . In the following, we shall use thesame notations as in IV. 1941] On Yukawa's Theory of the Beta-Disintegration293H'g_??_=4ƒÎ/ƒÈ2•ç{g1g'1(M0M'0+M'0M0)+g2g'2((SS')+(S'S))}dv, (8)H'g'2=4ƒÎ/ƒÈ2•ç{g'12M'0M'0+g'2S'S')}dv, (9) Amongthese expressions. (2), (3) and (4) are the energies of the mes ons, heavy particles and light particles in the absemce of external fields respectively. while the rest represents the interaetion energies betweenthese partieles. U<•¨>, U<•¨>. F<•¨> and F<•¨> are the field variables of the meson field satisfying commutation relations[Fi(r<•¨>,t).Uk(r'<•¨>,t)]=i4ƒÎhckƒÂ(r<•¨>-r'<•¨>)ƒÂik,[Fi(r<•¨>,t).Uk(r'<•¨>,t)]=i4ƒÎhckƒÂ(r<•¨>-r'<•¨>)ƒÂik,} (10)* a nd ƒµ. ƒ³, ƒÕ and ƒÓ denote Dirae's functions for the proton, (M<•¨>.M0) and (M'<•¨>,M'0), and the six vectors, (S,<•¨>T<•¨>) and (S'<•¨>,T'<•¨>), are defined by the following equations:M<•¨>=ƒ³ƒÏ(h)1ƒÐ<•¨>(h)ƒµ, M0-ƒ³ƒµ,S<•¨>=ƒ³ƒÏ(h)_??_ƒÐ<•¨>(h)ƒµ, T<•¨>=-ƒ³ƒÏ(h)2ƒÐ<•¨>(h)ƒµ,} (11)M'<•¨>=ƒµƒÏ(l)1ƒÐ<•¨>(l)ƒÓ. M'0=ƒµƒÓ.S'<•¨>=ƒµƒÏ(l)3ƒÐ<•¨>(l)ƒÓ. T<•¨>=-ƒµƒÏ(l)2ƒÐ<•¨>(l)ƒÓ.} (12)** where1, ƒÏ2,ƒÏ ƒÏ3. and ƒÐ<•¨> are the usual Dirae operators and the superscript (h) or (l) refers to the heavy particle or the light particle. The constants (g1. g2) and (g'1. g'2), which have each the dimension of an electric charge, determine the strength of interactions of the mesons with the heavy and light particles, respectively.F inally, m_??_=hƒÈ_??_e, M_??_, M_??_, m and ƒÊ denote the masses of the , proton, neutron, electron and neutrino, respectivelu.Recently Taketani •and the anthor(1) developed a matrix formulation of the vector meson theory for the case when the mesons interact only with an electromagnetic field Introducing a six-componentwave functionƒÔ-1/•ã<AƒÎhck>-F<•¨>U<•¨>) (13)*** A more general assmuption involving the derivatives of the wave function for the light particle, which was made in IV order to obtain energy distribution of the beta-ray of Konopinski-Uhlenbeck type, leads to a very much too small value for the meson lifetime and is therefore, not considered here. (***) In order to avoid confusion, we write ƒÔ instead of ƒµ in A, all other notations being the same as in A meson(*)neutron. [A, elcetron B]=AB-B].(**) and neutrino, respectively. Further, the four vectors. 294 shoichi SAKATA. [Vol. 23 they rewrote (2) and (10) in the following, forms: Hu=•çƒÔƒÏ2HƒÔdv,(15) [{ƒÔ(r<•¨>,t)ƒÏ2{ƒÊ, ƒÔƒË(r'<•¨>,t)]=-ƒÂƒÊƒËƒÐ(r<•¨>r'<•¨>), (16) where the operator H is defined byƒÏ2H=hcƒÈ[1-graddiv/2ƒÈ+ƒÏ3{(ƒÏ<•¨>grad)2/ƒÈ2-graddiv/2ƒÈ2}], (17) ƒÐ<•¨> =(ƒÐ1, ƒÐ2, ƒÐ3). Further, they also showed that the wave function ƒÔ satisfies in the absence of the external fields the equation ih•ÝƒÔ/•Ýt-HƒÔ=0. (18) (*) No confusion between these matrices and the Dirac operators will arise if it is kept in mind that the latter have superscripts 1941] On Yukawa's Theory of the Beta-Disintegration. 295 This formulation can be easily extended to the general case when the interactions with the heavy and light particles are involved . To write the interaction energies (5) and (6) in terms of ƒÔ , we introduce rowmatrices ƒÌ and ƒÌ', which are defined byƒÌ=(-g1/ƒÈM0grad-g2T, g1M-g2/ƒÈS_??_grad), (19)ƒÌ=(-g'1/ƒÈM'0grad-g'2T', g'1M'-g'2/ƒÈS'_??_grad). (20) these notations, we obtain from (5) and (6) the expressionH'g=-•ã<4ƒÎhc/ƒÈ>•çƒÌƒÔdv+comp. conj., (21)H'g'=-•ã<4ƒÎhc/ƒÈ>•çƒÌ'ƒÔdv+comp. conj., (22) Now, we go to the momentum space of the mesons. By using the plane wave solutions of (18), we expand ƒÔ and ƒÔ as follows ((22) in A) ƒÔ=‡”<p<•¨>,ƒÉ,ƒÃ>a(p<•¨>,ƒÉ,ƒÃ,)u(p<•¨>,ƒÉ,ƒÃ,)etp<•¨>r<•¨>/hc,ƒÔ=‡”<p<•¨>,ƒÉ,ƒÃ>a*(p<•¨>,ƒÉ,ƒÃ,)u(p<•¨>,ƒÉ,ƒÃ,)e-tp<•¨>r<•¨>/hc,} (23) where u (p<•¨>, ƒÉ, ƒÃ), representing a column matrix with c-number components, ponents, is the solution of the equation and where the constant, E and p<•¨> satisfy the relation For a given momentum p<•¨>/c, there are six independent solutions of (25), which we have distinguished by ƒÉ and ƒÃ ƒÉ=1, 0, -1 and ƒÃ=E/|E|-1, or -1 correspond to three polarization states, and two charge states of the meson, respectively. The u's have to be normalized so that ((18) in A) u<•¨> (p<•¨>,ƒÉ,ƒÃ)ƒÏ2u(p<•¨>,ƒÉ',ƒÃ')=ƒÃƒÂAƒÉ'ƒÂƒÃƒÃ', (26) from which we obtain the converse orthogonality relations ‡”<ƒÉ>‡”<ƒÃ>ƒÃ{u(p<•¨>,ƒÉ,ƒÃ)ƒÏ2}ƒÊuƒË(p<•¨>,ƒÉ,ƒÃ)=ƒÂƒÊƒË,‡”<ƒÉ>‡”<ƒÃ>ƒÃuƒÊ(p<•¨>,ƒÉ,ƒÃ){ƒÏ2ƒÊ(p<•¨>,ƒÉ,ƒÃ)}ƒË=ƒÂƒÊƒË.} (27) The explicit form of u may be written as Using Shoichi SAKATA. [Vol. 23 )-p<•¨>/p. e<•¨>(p<•¨>, 1) and e<•¨>(p<•¨>. -1) are unit vectors perpendicular to each other. The as and a*s are_q-numbers satisfying the commutation rela tions((23) in A) [ a(p<•¨>,ƒÉ,ƒÃ), a*(p<•¨>ƒÉ',ƒÃ')]=ƒÃƒÂAƒÉ'ƒÂƒÃƒÃ'. (29)Further, N+(p<•¨>,ƒÉ)=a*(p<•¨>ƒÉ,1) a(p<•¨>ƒÉ,1) or N-(p<•¨>,ƒÉ)=a(p<•¨>,ƒÉ,-1) a*(p<•¨>ƒÉ,-1) c an be•interpreted as the number of the mesons with the charge. c or -c in the state of energy |E|, momentum p<•¨>/c or -p<•¨>/c and polarization ƒÉ. Hence, a*(p<•¨>,ƒÉ,1) and a(p<•¨>,ƒÉ,1) denote the operators which increase the number N+(p<•¨>,ƒÉ) and N-(p<•¨>,ƒÉ) by one res pectively, whereas a(p<•¨>,ƒÉ,1) and a*(p<•¨>,ƒÉ,-1) denote those whichdecrease them by one respectively. Inserting(23) in (15).
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