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Given that Poincare transformation can mix different different directions of spacetime with each other, does this mean that in the case that some dimensions are compactified, large and compact dimensions mix?

Or compact dimensions and large ones mix separately among each other?

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It seems OP is asking for the isometry group $$G~=~\underbrace{[\mathbb{R}^{n,1}\times U(1)^m]}_{\text{remnant of transl. group}} \rtimes \underbrace{[O(n,1)\times SL(m,\mathbb{Z})]}_{\text{remnant of Lorentz group}}$$ of a partially compactified Minkowski spacetime $$M~=~\mathbb{R}^{n,1}\times (\mathbb{S}^1)^m.$$ In particular, the compactification of Minkowski spacetime $\mathbb{R}^{n+m,1}$ breaks mixed Poincare generators.

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