SUPERSPACE DESCRIPTION OF QUASI-PERIODIC STRUCTURES AND THE NONUNIQUENESS OF SUPERSPACE EMBEDDING

Some of the features of the superspace description of quasiperiodic st ructures, of the so-called superspace embedding, can be arbitrarily ch osen. The superspace approach is reviewed in such a way that these par ticular features, which are not intrinsic to the theory, are explicitl y indicated and separated from the fundamental formalism. Although the superspace density as a scalar function of n variables is uniquely de fined, the embedding is only fully determined once the internal subspa ce is chosen. As the internal subspace is usually represented perpendi cular to the ''parallel'' subspace, the different possible embeddings can be considered as different choices for the metric associated with the superspace and, therefore, for the n-dimensional representation of the superspace density. This freedom on the superspace description of a quasiperiodic system is discussed and interpreted through the follo wing examples: a modulated incommensurate structure, a composite incom mensurate structure, the Fibonacci chain, and an icosahedral quasicrys tal. For each case, the standard embedding usually considered in the l iterature is compared with other possible alternative choices. In gene ral, the standard embedding is clearly distinguished by the greater si mplicity it conveys, However, there are cases where a unique ''better' ' embedding choice does not exist. The composite incommensurate struct ures and the Fibonacci chain are two clear examples of this fact. A pa rticular superspace embedding implies a particular election for the so -called phason degrees of freedom in the system. The existence of diff erent equivalent superspace embedding evidences the impossibility in t hese cases of totally determining the phason modes from purely static considerations.