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Since our fundamental laws are invariant under rotations. Hence, we say that spacetime isotropic.

Now Quantum Mechanics is invariant under (global) complex rotations ($U(1)$ transformations. Hence, we say that $X$ is isotropic.

In other words, what is the correct analogous space $X$ here and is "isotropic" the correct term used?

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First of all, the space $X$ here is the Hilbert space $\mathcal H$ of the system, since the $U(1)$ transformations apply to the state vector $|\psi\rangle \in \mathcal H$ of the system. "Isotropic" is not the correct term to describe the symmetry. The term "isotropic" could maybe be used if $\mathcal H \cong \mathbb C^N$ were symmetric under $SU(N)$ transformations, but I don't think this is a common terminology (also, such a system would be very boring). Instead, this is called a (global) $U(1)$ gauge symmetry.

Let me make two more comments. First, the two symmetries here have an important fundamental difference. The $SO(3)$ symmetry of mechanics compares actual, physically different system states (different rotations of the physical system). However, the $U(1)$ symmetry of quantum mechanics is a gauge symmetry, meaning: The two states $|\psi\rangle$ and $e^{i\varphi} |\psi\rangle$ are the exact same physical state, but our description of the system allows that same state to be described with many different Hilbert space vectors. (This is for convenience only, the actual state space is $\mathcal H / \mathbb C^\ast$, but it is easier to work with $\mathcal H$.)

Second, the Noether theorem states that every symmetry corresponds to a conserved quantity. The conserved quantity corresponding to the $SO(3)$ symmetry of mechanics is angular momentum, but what is that of the $U(1)$ gauge symmetry? Once you learn about QED, you will see that it is the total electrical charge.

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  • $\begingroup$ The global phase freedom of QM is not a gauge symmetry and is distinct from the electromagnetic $\mathrm{U}(1)$ gauge symmetry, see physics.stackexchange.com/a/433501/50583 and physics.stackexchange.com/questions/433457/… $\endgroup$ – ACuriousMind Oct 9 '18 at 16:47
  • $\begingroup$ @ACuriousMind Why would you say it is not a gauge symmetry? I would have defined a gauge symmetry exactly like this, that one physical state has several representations. I don't understand your comment about the distinction from the EM $U(1)$ either: I would have thought that the extra factor of $e$ in the exponent is just a redefinition / change of units of $\varphi$. $\endgroup$ – Noiralef Oct 9 '18 at 18:50
  • $\begingroup$ 1. A gauge symmetry has a technical definition in terms of the Hamiltonian being constrained or the relation between canonical momenta and generalized velocities being non-invertible. You may use it differently, but this strikes me as confusing. 2. The point is that the space of states can conceivably contain states of different charge, e.g. in a many-body space of states, it can have states $|n\rangle$ of n electrons. Then the gauge symmetry acts as $|0\rangle +|1\rangle +|2\rangle\mapsto |0\rangle +e^{i\phi}|1\rangle +e^{2i\phi}|2\rangle$, which is obviously different from a global phase. $\endgroup$ – ACuriousMind Oct 9 '18 at 19:01
  • $\begingroup$ Note also that the electromagnetic gauge transformation does not act on uncharged states at all, while the global phase choice of course does. $\endgroup$ – ACuriousMind Oct 9 '18 at 19:03
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In analogy with the more common notion "isospin space", I would say the corresponding space for $U(1)$ gauge symmetry is "charge space".

For example, one book which uses this notion is Particle Astrophysics by Donald H. Perkins.

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