Abstract
The mathematical investigation of the existence of solutions for the nonlinear (4.6) does not exhaust the problem of the physical interpretation of the corresponding classical field theory. For infinitely extended systems, in general not every solution is physically acceptable; one has to supplement the analysis of the possible solutions by a list of mathematical properties which the solutions must share in order to allow a physical interpretation.
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References
R.F. Streater and A.S. Wightman, PCT, Spin and Statistics and All That, Benjamin-Cumming Publ. C. 1980.
See e.g. D. Ruelle, Statistical Mechanics, Benjamin 1969; R. Haag, Local Quantum Theory, Springer-Verlag 1992.
It is not difficult to recognize the analogies with the stability theory, which plays a crucial role in the theory of non-linear phenomena, in the finite dimensional case; see e.g. G. Sansone and R. Conti, Non-Linear Differential Equations, Pergamon Press 1964, Chap. IX.
C. Parenti, F. Strocchi and G. Velo, Phys. Lett. 62B, 83 (1976); Comm. Math. Phys. 53, 65 (1977); Lectures at the Int. School of Math. Phys. Erice 1977, in Invariant Wave Equations, G. Velo and A.S. Wightman eds., Springer-Verlag 1978.
See footnote 12.
S. Coleman, Phys. Rev. D 15, 2929 (1977).
See footnotes 11, 12.
F. Strocchi, Lectures at the Workshop on Recent Advances in the Theory of Evolution Equations, ICTP Trieste 1979, published in Topics in Functional Analysis 1980–81, Scuola Normale Superiore, Pisa 1982; contribution to the Workshop on Hyperbolic Equations (1987), published in Rend. Sem. Mat. Univ. Pol. Torino, Fascicolo speciale 1988, pp. 231–250.
The physical relevance of locality has been emphasized by R. Haag and D. Kastler, J. Math. Phys. 5, 848 (1964), see also R. Haag, loc. cit. (see footnote 17).
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Strocchi, F. (2008). Stable Structures, Hilbert Sectors, Phases. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_5
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