[613style]

Definitions are created to be useful or interesting or both, and sometimes those specifiers overlap with physical reality. But that doesn't mean they're created to match reality or to tell you anything about it.

I remember struggling as an undergrad to come up with a metaphor for visualizing what a group homomorphism is, aiming to develop better intuition when working with them. It's hard because the real world doesn't contain any, and all the "it's kind of like X" types of examples you can think of aren't really useful when mapped back to the mathematical domain. If math described the real world, I'd expect this type of thing to be much easier.

[tutfbhuf]

An example of a group homomorphism in physics is the relationship between the rotational symmetry group of a physical system (often the group SO(3) for three-dimensional rotations or SU(2) for spin systems) and the representation of that group by the angular momentum operators in quantum mechanics. The angular momentum operators in quantum mechanics form a representation of the symmetry group because they satisfy the same commutation relations as the infinitesimal generators of the group. This means that the algebraic structure of the rotational symmetry group is mirrored by the algebraic structure of the angular momentum operators, and the way these operators transform states in the Hilbert space of the quantum system reflects the symmetry properties of the system.

[613style]

I'm not sure if you intend this as a helpful analogy for an undergrad or as an argument in favor of groups/homomorphisms being created to match reality, but I don't think it applies very well to either.

(Though it is very interesting!)