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Compactified Yang-Mills theories with one universal extra dimension werefound [arXiv:1008.4638] to exhibit two types of gauge invariances: the standardgauge transformations (SGTs) and the nonstandard gauge transformations (NSGTs).In the present work we show that these transformations are not exclusive tocompactified scenarios. Introducing a notion of hidden symmetry, based on thefundamental concept of canonical transformation, we analyse three differentgauge systems, each of which is mapped to a certain effective theory that isinvariant under the so-called SGTs and NSGTs. The systems under discussion are:(i) four dimensional pure $SU(3)$ Yang-Mills theory, (ii) four dimensional$SU(3)$ Yang-Mills with spontaneous symmetry breaking, and (iii) pureYang-Mills theory with one universal compact extra dimension. The canonicaltransformation, that induces the notion of hidden symmetry, maps objects withwell defined transformation laws under a gauge group $G$ to well definedobjects under a non-trivial subgroup $H \subset G$. In the case wherespontaneous symmetry breaking is present, the set of SGTs corresponds to thegroup into which the original gauge group is broken into, whereas the NSGTs areassociated to the broken generators and can be used to define the unitarygauge. For the system (iii), the SGTs coincide with the gauge group $SU(N,\cal{M}^{4}) $, whereas the NSGTs do not form a group; in this system the'fundamental' theory and the effective one are shown to be classicallyequivalent.

Hidden symmetries induced by a canonical transformation and gauge structure of compactified Yang-Mills theories