Two-dimensional theory of chirality. II. Relative chirality and the chirality of complex fields

Journal of Mathematical Physics

Volumn 41, Issue 9, 2000, Pages 5986-6006

  Le Guennec, Patrick ^a,b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034345840     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1285981     Document Type: Article

References (14)

1. P. Le Guennec, "Two dimensional theory of chirality. I. Absolute chirality," J. Math. Phys. 41, 5954 (2000). Section II A of I will here be referred to as I, Sec. II A, the equation (4.31) of I as I, (4.31).
2. We call analytic the fields representable by power series in the coordinates x, y understood as two independent real or complex variables. Fields are not assumed representable by power series in the single variable x + iy.
3. n is (everywhere) dense. H. Cartan, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes (Hermann, Paris, 1961) (in French); cf. also Ref. 9.
4. A Baire space is a topological space in which countable intersections of dense open sets are dense. The idea of using the Baire property in this question is due to M. Balabane (private communication). The properties of Baire spaces are discussed in: J. Dugundji, Topology (Allyn and Bacon, Boston, 1966); J. L. Kelley, General Topology (Van Nostrand, New York, 1955).
5. A Baire space is a topological space in which countable intersections of dense open sets are dense. The idea of using the Baire property in this question is due to M. Balabane (private communication). The properties of Baire spaces are discussed in: J. Dugundji, Topology (Allyn and Bacon, Boston, 1966); J. L. Kelley, General Topology (Van Nostrand, New York, 1955).
6. ∝ projections can have compact supports, so examples in which there is no circle on which all projections (and even all pairs of projections) are nonzero are trivially built.
7. The strategy followed in this article can clearly be transposed to similar discrete symmetry matters (similar to the presence or the absence of indirect elements in symmetry groups) in the context of other Lie groups, which may not be compact, as the Lorentz or Poincaré groups of interest to the relativistic generalization of this theory. In such cases it is to be expected that neither indices need be discrete.
8. n are dμ-null. From this unified point of view chirality is a property distributed over Ω̄ and defined up to a dμ̄-null subset. This argument points again at the relevance of integration theory to the theory of chirality.
9. Only uniform bases, i.e., bases that do not vary with r, will be considered in this article. "Local" bases depending on r can be introduced. These bases and the associated coordinates are related to each other by local phase changes analogous to second order gauge transformations. The first order gauge transformations are the rotations.
10. W. Rudin, Real and Complex Analysis (Mc-Graw Hill, New York, 1970).
11. (a) J.-P. Serre, Linear Representations of Finite Groups (Springer, New York, 1977);
12. (b) A. Robert, Introduction to the Representation Theory of Compact and Locally Compact Groups (Cambridge University Press, Cambridge, 1983).
13. R. Bracewell, The Fourier Transform and its Applications (McGraw Hill, New York, 1965).
14. Statements of the theorems of the Appendix were guessed by us, but proofs were established by H. Randriambololona (private communication).