Relativistic Wave Equations

Seiler, R.

Les rencontres physiciens-mathématiciens de Strasbourg -RCP25, Conférences de : J. Leray, J.P. Ramis, R. Seiler, J.M. Souriau et A. Voros, Tome 18 (1973), Exposé no. 4, 24 p.

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3) A.S. Wightman Relativistic Wave Equations as Singular Hyperbolic Systems, Preprint. 1972. | MR | Zbl

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A. Einstein Ann. d. Phys. 17, 132 (1905) | JFM

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E. Schrödinger Dokumente der Naturwissenschaft Bd. 3, Leipzig 1927.

E. Schrödinger Dokumente der Naturwissenschaft Bd. 3, Stuttgart 1963.

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for a review, W. Pauli, Handbuch der Physik 2, Band V, Teil 1, Springer 1958

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P.A.M. Dirac For a detailed account of the history of the Dirac equation we refer to A. S. Wightman in Aspects of Quantum Theory, Edited by A. Salam and E.P. Wigner (Cambridge University Press, 1972).

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C. S. Morawetz and W. A. Strauss, Comm. Pure Appl. Math. 25, 1 (1972) | MR | Zbl

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John M. Chadam Indiana University Preprint, Bloomington, Indiana (1972)

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H. Joos Fortschr. Physik 10, 65 (1962) | Zbl

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20) A. S. Glass Princeton Thesis 1971, unpublished

21) A. Capri Jour. Math. Phys. 10, 575 (1969)

22) For a most complet review on relativistic wave equations without interaction see ref. 20)

23) see ref. 2) and ref. 30)

P. Minkowski and R. Seiler, Phys. Rev. D 4, 359 (1971) | MR

25) J. Bellissard and R. Seiler, to be published in Lettere al Nouovo Cimento

26) A. S. Wightman ref. 9) p. 107

27) J. Bellissard Université de Provence-Centres Saint-Charles, Thèse; unpublished

28) J. Leray and Y. Ohya Systèmes lineaires hyperboliques non stricts Colloques CBM Louvain (1964) ; | Zbl

J. Leray and Y. Ohya Reprint in Battelle Rencontres on Hyperbolic Equations and Waves, Seattle 1968

29) R. Seiler, in preparation

30) B. Schroer, A. Swieca and R. Seiler, Phys. Rev. D 2, 2927 (1970) | Zbl

31) Spin zero particle in external field, ref. 30).

32) Spin 1/2 particle in external field R. Seiler, Comm. Math. Phys. 25, 127 (1972)

33) Existence already follows from classical theorems by Scale resp. Scale and Stinespring

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 $S{=}_{\text{in}}{\langle 0\left|S\right|0\rangle }_{\text{in}}:exp\int dxdy{\varphi }_{\text{in}}^{+}\left(x\right)I(x,y){\varphi }_{\text{in}}\left(y\right):$ 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) $S_{\genfrac{}{}{0pt}{}{R}{A}}(x,y;A)=S_{\genfrac{}{}{0pt}{}{R}{A}}^0(x-y)-i\int dz{S}_{\genfrac{}{}{0pt}{}{R}{A}}^0(x-z)\neg A\left(z\right)S_{\genfrac{}{}{0pt}{}{R}{A}}(z,y;A)$.

35) A. S. Wightman, private communication