Geometric invariant theory: a model-independent approach to spontaneous symmetry and/or supersymmetry breaking

Sartori, G.

doi:10.1007/bf02810048

Cited by 27 publications

(42 citation statements)

References 129 publications

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“…It has been shown in [14,15] that, besides the constraints listed in Subsection 2.1 under P1, every P -matrix has to satisfy some additional conditions, that can be put in the form of a set of differential relations, so that one can try to determine the P -matrices generated by CCLG's as particular solutions of a system of differential equations. Let us briefly recall the derivation of these results.…”

Section: Boundary Conditionsmentioning

confidence: 99%

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Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

Sartori

Valente

2005

Acta Appl Math

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We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix P (p), which is defined only in terms of the scalar products between the gradients ∂p 1 (x), . . . , ∂p q (x) of the elements of a minimal integrity basis (MIB) for the ring R[R n ] G of G-invariant polynomials. The domain of semi-positivity of P (p) is known to realize the orbit space R n /G of G as a semi-algebraic variety in the space R q spanned by the variables p 1 , . . . , p q .The matrices P (p) can be obtained from the solutions of a universal differential equation (master equation), which satisfy convenient initial conditions. The master equation and the initial conditions involve as free parameters only the degrees d a of the p a (x)'s. This approach tries to bypass the actual impossibility of explicitly determining a set of basic polynomial invariants for each group.Our results may be relevant in physical contexts where the study of covariant or invariant functions is important, like in the determination of patterns of spontaneous symmetry breaking in quantum field theory, in the analysis of phase spaces and structural phase transitions (Landau's theory), in covariant bifurcation theory, in crystal field theory and so on. (2000): 14L24, 13A50, 14L30. Mathematics Subject Classifications

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Section: Boundary Conditionsmentioning

confidence: 99%

“…The matter will be presented in the following order. In Section 2 we shall recall some known results concerning the characterization of orbit spaces (see, for instance, [26,1,2,12,15,4]) in the P -matrix approach. In Section 3 we recall the derivation of the boundary conditions and of the master conditions in the case of CCLG's.…”

Section: Introductionmentioning

confidence: 99%

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

Sartori

Valente

2005

Acta Appl Math

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“…We will try to keep to as simple an exposition as possible; there are expositions designed for physicists rather than for mathematicians [17,18,19,21,22], and the reader desiring more details is referred to these, see in particular the first part of [17] and the first chapter of [22]. A readable introduction to the mathematical point of view is provided by [31].…”

Section: Basic Notations and Landau Theorymentioning

confidence: 99%

“…In the present note we want to clarify the meaning of this criterion (see also [9,10]), which should be seen in the context of the orbit space approach to variational problems [17,18,19,21,22], and discuss it in the light of the theory of (symmetric) Poincaré-Birkhoff normal forms [11,12,13,14,15]. This will enable us to understand in simple terms the origin of the criterion, and how it should be modified when considering a range of values for the control parameter(s)…”

Section: Introductionmentioning

confidence: 99%

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Lie-Poincaré transformations and a reduction criterion in Landau theory

Gaeta

2004

Annals of Physics

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Abstract. In the Landau theory of phase transitions one considers an effective potential Φ whose symmetry group G and degree d depend on the system under consideration; generally speaking, Φ is the most general G-invariant polynomial of degree d. When such a Φ turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e. to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincaré theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group G as examples, and study in detail the application to the Sergienko-Gufan-Urazhdin model for highly piezoelectric perovskites.

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Practical normal form computations for vector fields

2004

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We present a method to compute a Poincaré-Dulac normal form of a vector field, as well as a corresponding reduced vector field, near a stationary point with arbitrary linearization. Explicit knowledge of the eigenvalues of the linearization is not necessary, and only rational operations are required. Some examples are presented, including a normal form computation relevant for Hopf bifurcations, and coupled nonlinear oscillators.

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Geometric invariant theory: a model-independent approach to spontaneous symmetry and/or supersymmetry breaking

Cited by 27 publications

References 129 publications

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

Lie-Poincaré transformations and a reduction criterion in Landau theory

Practical normal form computations for vector fields

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