The Orbit Space Approach to the Theory of Phase Transitions: The Non-Coregular Case
نویسنده
چکیده
We consider the problem of the determination of the isotropy classes of the orbit spaces of all the real linear groups, with three independent basic invariants satisfying only one independent relation. The results are obtained in the P̂ -matrix approach solving a universal differential equation (master equation) which involves as free parameters only the degrees da of the invariants. We begin with some remarks which show how the P̂ -matrix approach may be relevant in physical contexts where the study of invariant functions is important, like in the analysis of phase spaces and structural phase transitions (Landau’s theory). 1 The Orbit Space (OS) Approach Invariant functions under the transformations of a Compact Linear Group (CLG) acting in 1R can be expressed in terms of functions defined in the OS of the group, i.e. as functions of a finite set of basic invariant polynomials p(x) ≡ (p1(x), . . . , pq(x)), x ∈ 1R , which may be chosen to form a Minimal Integrity Basis (MIB) for the group G. Such an observation, originally due to Gufan (1971), simplifies the determination of the patterns of spontaneous symmetry breaking (SSB) in theories in which the ground state is determined by the minimum of an invariant potential V (x). When p ranges in the domain spanned by p(x), x ∈ 1R, the function V̂ (p) has the same range as V (x), but is not plagued by the same degeneracies. A correct exploitation of this idea required, however, the determination of the ranges of the functions pi(x), a problem which was completely solved only using
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