  The notion that this transformation must be a gauge transformation fails, however. That the two structures are perfect mathematical mirror images of each other is not sufficient to ensure that they must represent the same physical structures. They certainly may represent the same physical structures, but they also may not. To see this, consider a mathematical, three dimensional Euclidean space used to represent a three dimensional physical space with Euclidean properties. The mathematical space hosts many flat, two dimensional surfaces, each of which can be transformed perfectly into any other. But to say that these transformations are merely gauge transformations is to collapse the three dimensions of the physical space into two dimensions. Each two dimensional surface in the physical space is a perfect copy of every other one; they are not all the same surface. The transformations between them cannot be gauge transformations.  