Chapter 5
Stable Structures, Hilbert Sectors, Phases
The mathematical investigation of the existence of solutions for the non-linear equation (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.
For quantum field theory the realization of the basic mathematical structure which
renders the theory physically sound is due to Wightman
16 and it is nowadays standard
to accept as “solutions” of the quantum field equations those which satisfy Wightman’s axioms. A similar problem arises in Statistical Mechanics and the basic structure has been clarified,
17 with the realization that the same dynamics may describe
different physical realizations or phases of a given system.
Since not all solutions of (4.6) describe physically acceptable configurations, one
has to look for those subsets S which satisfy a few (additional) basic requirements.
In particular, we are interested in characterizing possible disjoint subsets S (of solutions), which may be interpreted as describing disjoint realizations or “phases” of the
system, in strict analogy with the disjoint inequivalent representations of the algebra
of local observables in Statistical Mechanics and in the Quantum Theory of infinitely
extended systems.
With these motivations, general considerations, to be further discussed below,
suggest to look for structures S, in the set of solutions of (4.6), characterized by the
following properties
16 R.F. Streater and A.S. Wightman, PC T , Spin and Statistics and All That, Benjamin-Cumming
Publ. C. 1980.
17 See, e.g. D. Ruelle, Statistical Mechanics, Benjamin 1969; R. Haag, Local Quantum Theory,
Springer-Verlag 1992.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_5
21
Stable Structures, Hilbert Sectors, Phases
The mathematical investigation of the existence of solutions for the non-linear equation (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.
For quantum field theory the realization of the basic mathematical structure which
renders the theory physically sound is due to Wightman
16 and it is nowadays standard
to accept as “solutions” of the quantum field equations those which satisfy Wightman’s axioms. A similar problem arises in Statistical Mechanics and the basic structure has been clarified,
17 with the realization that the same dynamics may describe
different physical realizations or phases of a given system.
Since not all solutions of (4.6) describe physically acceptable configurations, one
has to look for those subsets S which satisfy a few (additional) basic requirements.
In particular, we are interested in characterizing possible disjoint subsets S (of solutions), which may be interpreted as describing disjoint realizations or “phases” of the
system, in strict analogy with the disjoint inequivalent representations of the algebra
of local observables in Statistical Mechanics and in the Quantum Theory of infinitely
extended systems.
With these motivations, general considerations, to be further discussed below,
suggest to look for structures S, in the set of solutions of (4.6), characterized by the
following properties
16 R.F. Streater and A.S. Wightman, PC T , Spin and Statistics and All That, Benjamin-Cumming
Publ. C. 1980.
17 See, e.g. D. Ruelle, Statistical Mechanics, Benjamin 1969; R. Haag, Local Quantum Theory,
Springer-Verlag 1992.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_5
21