Relativistic Wave Equations

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1. 1) B. KlaiberLectures in Theoretical Physics, Boulder1967 
2. 2) A.S. WightmanProceedings of Fifth Coral Gables Conference 1968 p. 291 
3. 3) A.S. WightmanRelativistic Wave Equations as Singular Hyperbolic Systems, Preprint. 1972. Zbl0261.35054MR342075
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34. 22) For a most complet review on relativistic wave equations without interaction see ref. 20) 
35. 23) see ref. 2) and ref. 30) 
36. P. Minkowski and R. Seiler, Phys. Rev. D 4, 359 (1971) MR342072
37. 25) J. Bellissard and R. Seiler, to be published in Lettere al Nouovo Cimento 
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39. 27) J. BellissardUniversité de Provence-Centres Saint-Charles, Thèse; unpublished 
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42. 29) R. Seiler, in preparation 
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44. 31) Spin zero particle in external field, ref. 30). 
45. 32) Spin 1/2 particle in external field R. Seiler, Comm. Math. Phys.25, 127 (1972) 
46. 33) Existence already follows from classical theorems by Scale resp. Scale and Stinespring 
47. 34) There is a close connection between the kernel $I(x,y)$ comming up in the formal expression for the $S$-matrix in perturbation theory $S{=}_{\text{in}}{\langle 0\left|S\right|0\rangle }_{\text{in}}:exp\int dxdy{\phi }_{\text{in}}^{+}\left(x\right)I(x,y){\phi }_{\text{in}}\left(y\right):$ and the fundamental solution $S_R(x,y;A)$ resp. $S_A(x,y;A)$ of the wave equation (11). $G=G^0+G^{0}I{G}^0$ where $G^0$ denotes the fundamental solution of the free equation with Feynman boundary conditions. Then $G$ is a solution of the equation $G=G^0-i{G}^0\neg AG$ On the other hand $S_R$ resp. $S_A$ are solutions of the Yang-Feldman equation (12) $S_{\genfrac{}{}{0pt}{}{R}{A}}(x,y;A)=S_{\genfrac{}{}{0pt}{}{R}{A}}^0(x-y)-i\int dz{S}_{\genfrac{}{}{0pt}{}{R}{A}}^0(x-z)\neg A\left(z\right)S_{\genfrac{}{}{0pt}{}{R}{A}}(z,y;A)$. 
48. 35) A. S. Wightman, private communication