Superspace description of quasiperiodic structures and the nonuniqueness of superspace embedding

Some of the features of the superspace description of quasiperiodic structures, of the so-called superspace embedding, can be arbitrarily chosen. The superspace approach is reviewed in such a way that these particular features, which are not intrinsic to the theory, are explicitly indicated and separated from the fundamental formalism. Although the superspace density as a scalar function of n variables is uniquely defined, the embedding is only fully determined once the internal subspace is chosen. As the internal subspace is usually represented perpendicular 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 following examples: a modulated incommensurate structure, a composite incommensurate structure, the Fibonacci chain, and an icosahedral quasicrystal. For each case, the standard embedding usually considered in the literature is compared with other possible alternative choices. In general, the standard embedding is clearly distinguished by the greater simplicity it conveys, However, there are cases where a unique ''better'' embedding choice does not exist. The composite incommensurate structures and the Fibonacci chain are two clear examples of this fact. A particular superspace embedding implies a particular election for the so-called phason degrees of freedom in the system. The existence of different equivalent superspace embedding evidences the impossibility in these cases of totally determining the phason modes from purely static considerations.