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Typically, when a course examines symmetries of space-time and their consequences (e.g. symmetries of Lagrangians and conserved quantities), either the Lorentz group or the Poincaré group are considered.

But what about scaling? Shouldn't physics be invariant under a scaling of my coordinate system? Or is scaling similar to coordinate transforms like cartesian to cylindrical, where the physics stays the same but the form of the equations changes?

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    $\begingroup$ Physics is not necessarily invariant under scaling. Basically if there is any length scale in your system, it is not scale-invariant. You can certainly consider scale transformation together with Poincare group, which usually become the conformal group. Field theories invariant under conformal transformations are called conformal field theories. $\endgroup$
    – Meng Cheng
    Commented Mar 22, 2022 at 11:51
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    $\begingroup$ I see. And because of that, a constant (eg some coupling constant) is not constant under scaling when you ignore the units. And when you keep the units (eg metres for the constant and kilometres for the transformed coordinates), you don't really transform the problem but just rewrite the same one. Thanks. $\endgroup$
    – Uroc327
    Commented Mar 22, 2022 at 12:07

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