Two-dimensional theory of chirality. I. Absolute chirality

Chirality is a notion already familiar to undergraduates as the fact that an object is not superposable to its mirror image. Its dichotomous or "yes/no" character, implicit in this definition due to Lord Kelvin, has always been considered self-evident. We prove here and in the companion article that in two dimensions chirality-the very concept of chirality-is a continuous phenomenon in the special case of square-integrable wave functions. This conception of chirality is to Kelvin's definition what the continuous conception of door opening is to the closed/nonclosed dichotomy; hence it provides a continuous description of the discrete achiral symmetry breaking. This result is first extended to three and higher dimensions, then to the whole nonrelativistic quantum description of matter. Thus molecules are more or less chiral just as doors are more or less open, and molecular chirality changes continuously during chemical reactions. Chirality splits into two complementary forms-absolute and relative chirality. We present here the theory of absolute chirality. More generally, this unexpected and paradoxical breakthrough in symmetry theory is based on a geometrical description of wave functions that should find broad applications in molecular physics and in stereochemistry, where the notion of chirality has an overwhelming importance since long ago. (C) 2000 American Institute of Physics. [S0022-2488(00)02307-0].