1 INTRODUCTION: THE CURIE PRINCIPLE

The Curie symmetry principle (Curie, 1894) is the causality relation between the symmetry of the cause and that of the effect. The principle is composed of three parts:

• If certain causes yield the known effects, the symmetry elements of the causes should be contained in the generated effects.
• If the known effects manifest certain dissymmetry (absence of symmetry elements), this latter should be contained in the causes which have generated those effects.
• The converse to these two previous propositions is not true, at least in practical: i.e., the effects may have higher symmetry than the causes which generate these effects.

2 NONLINEAR FUNCTIONAL ANALYSIS

Sattinger (1979) described the method to analyze functional in association with group theory. Nonlinear functional equations are generally represented in the form (1) F(m , w) = 0 where F: R´B®B and w = w(m) (mÎR,wÎ B). It denotes that w is defined on a Banach space B and depends on a real parameter (bifurcation parameter) and that F maps w to new w with the change of m. Sattinger investigated the functional equations which mapping F is covariant under group G, i.e. (2) F(m , gw) = gF(m , w) where G and g are set and its element, respectively. In this paper, we consider for unitary group that preserves the inner product invariant. This assumption doesn’t loose generality in considering most of real phenomena. Since unitary group is isomorphic with general linear group that is a set of linear mapping, we can take advantage of knowledge and techniques in linear algebra. With application of group representation theory, we can define "standard decomposition" (Sattinger, 1979). We decompose Hilbert space H here since we have to analyze functional equations on H to deal with unitary group. Then H is decomposed into some H^a which corresponds to each irreducible element g_aÎG and H^a intersects orthogonally one another. Therefore, we can now treat variables by its symmetrical compositions, e.g. w = (w_{1f 1}, w_{2f 2}, ¼ , w_{qf q}) where (f₁, f_{2, ¼ ,} f_q) is a symmetrical basis. Each projection P^a onto H^a can also be defined (Sattinger, 1979; Fujii and Yamaguti, 1980). Especially, a= 1 is assigned to isotoropy group and T_gP¹ = P¹ for the transformations T_g (gaÎG). Furthermore, we define Fréchet derivative F´= ¶ F/¶ w to investigate local mapping of functional in the neighborhood of a point (m_{0, w0)} at which F´ is defined. If there exists reducible F´(m₀, w₀), implicit function theorem assures that eq.(1) has a unique solution in the neighborhood of (m_{0, w0)}. If F is irreducible at a certain point, this point is singular and several solutions exist in its neighborhood. For simplicity, we consider about only simple, i.e. not multiple singular points in this paper.
 
 

3 SYMMETRY IN FUNCTIONAL SYSTEMS

First we consider the case thatF is reducible. Since T_gP¹ = P¹, T_gw = w for w = w₁ÎH¹. With this G-invariant property, symmetry preserving of G-covariant mapping in the subspace is proved. This is to say that for all w₁ÎH¹,^F(m, w₁) = P^aF(m, w₁) is derived from eq.(2). Therefore, P^aF(m, w₁) = 0 (a = 2, 3, ¼ , q). Since P¹ reflects isotoropy group, both relations indicate that the solution path of eq.(1) is enclosed in the subspace H¹ unless F becomes irreducible. In other words, the symmetry of the system does not change on each solution path. Next we consider the case that F is irreducible. From eq.(2), Fréchet derivative F´ satisfies the following equation F´(m, T_gw)×T_g = T_gF´(m, w) ("g ÎG, "w ÎH). Multiplying by the group characters and summing for "g ÎG, we get F´(m, w₁)P^a = P^aF´(m, w₁) (a = 1, ¼ , q; "w_{1 Î}H¹). According to linear algebra, it denotes that F´ can be represented as a diagonal matrix form, i.e. F´(m ,w) = (F₁´, F₂´, ¼, F_q´). Since the basis represents the symmetry, each eigenvalue Fa´ corresponds to a certain symmetrical element of the system. According to Sattinger (1979), the symmetry corresponding Fa´ = 0 dominates the new solution path, e.g. the path bifurcating from the singular point. Because the new solution path is more stable (Fujii and Yamaguti, 1980), the system follows the new path. Therefore, if eigenfunction of Fa´ = 0 doesn’t correspond the symmetry dominated the system until reaching the singular point, the symmetry of the system has not been held yet. Since the old path belongs to the space corresponding the isotoropy group, the new path belongs in most case to the space corresponding the other subgroup with the lower symmetry than that of decomposed original space. The system is "dissymetrical" at the singular point in this sense.
 
 

4 RECONSIDERATION OF CURIE PRINCIPLE

Finally, we consider the Curie principle in functional analysis. We can derive another Fréchet derivative = ?F/?m in the space with symmetrical basis by the same procedures described above. Because eq.(1) denotes the relation w = w(m), we can refer m and w as the cause and the effect, respectively. Then, we can refer  and F´ as representing the symmetries of the cause and the effect, respectively. Once F_a´ not corresponding to the existing component of  becomes to 0, the symmetry of the cause is no longer preserved in the symmetry of the effect. This case is just the "breaking of the Curie symmetry principle". In contrast, even if F_a´ corresponding the existing component of becomes to 0, the symmetry of the cause is held in the effect. The Curie symmetry principle is preserved in this case. Of course, the principle is preserved if F´ is reducible. In this sense, both preserving and breaking of the Curie symmetry principle occur in a same nonlinear system.
 
 

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