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I understand that unitary representations arise naturally in quantum mechanics when groups act on the Hilbert space in a way that preserves probability.

I don't understand what details make unitary representations different from other representations. It seems as though physicists talk explicitly about "unitary representations" all the time.

Are there some theorems or examples that show why working with a "unitary" representation ought to be notable?

I've heard that compact Lie groups have only finite dimensional unitary representations, but seeing as so many important Lie groups in physics are non-compact this doesn't seem like the most vital reason.

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  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Jul 27 '17 at 18:11
  • $\begingroup$ Every (finite-dimensional) unitary representation of a group on the Hilbert space of a quantum system is fully reducible, i.e. is a direct sum of irreducible representations. This oftentimes allows one in practice to focus on one irreducible multiplet of states at a time. For the same reason, the classification of (finite-dimensional) representations of compact Lie groups is much easier than for non-compact ones. $\endgroup$ – Tomáš Brauner Dec 16 '19 at 11:18
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Since the Hamiltonian is hermitian, and the time evolution of a system is $U(t)=e^{-itH/\hbar}$, $U(t)$ is automatically unitary. Moreover, unitary transformations play the role of rotations in 3d space, in the sense that they preserve the inner product: $$ \langle \phi\vert \psi\rangle = \langle \phi'\vert\psi'\rangle\, ,\qquad \vert\psi'\rangle=U\vert\psi\rangle\, ,\quad \vert\phi'\rangle=U\vert\phi\rangle $$ and thus they preserve the physical predictions of quantum theory, which depends on $\vert \langle \phi\vert \psi\rangle\vert^2 = \vert \langle \phi'\vert \psi'\rangle\vert^2$. This makes the predictions independent of the choice of initial basis vectors, much like the predictions of classical physics are independent of the initial choice of directions of the basis vectors.

This conclusion is applicable to compact or non-compact groups.

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One property I like about unitary representations is that every invariant subspace has a complementary invariant subspace (namely the orthogonal complement).

From that it follows, that when we find an invariant subspace $V$ of a unitary representation, the representation can be decomposed into a direct sum of $V$ and its orthogonal complement.

Moreover, from that we see that every finite-dimensional unitary representation is a direct sum of irreducible representations.

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