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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
    Commented Jul 27, 2017 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. $\endgroup$ Commented Dec 16, 2019 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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For a compact Lie group, every representation can be made unitary.

Say you have a vector space $V$ and a group representation $\rho(g)$ which acts on $v \in V$.

Now say you have some hermitian inner product $\langle u, v \rangle$ for which $\rho$ is not a unitary representation, i.e.

$$ \langle \rho(g) u, \rho (g) v \rangle \neq \langle u, v \rangle. $$

We can use this inner product to construct one for which $\rho$ is actually unitary. We just average over the inner product with Haar measure:

$$ \langle u ,v \rangle_{\rm new} = \int_G d g \langle\rho(g)u, \rho(g)v \rangle. $$

Therefore, at least for compact Lie groups, any representation on a complex vector space can be considered to be unitary. The word "unitary" has no effect on the representation theory itsef.

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Motivation: the symmetries of a system as unitary operators and representations

In quantum mechanics, we usually consider two different kind of symmetries, as explained in Valter Moretti's answer there: (Coming from Wigner's Theorem): What is a Symmetry in QFT?, namely Wigner symmetries and Dynamical symmetries. Wigner symmetries are transformations acting on states, which are rays in the projective space of the Hilbert space of your system.

  • A unitary operator automatically defines a Wigner symmetry transformation (on your projective space). This is explained in @ZeroTheHero reply.
  • A converse of this fact is given by Wigner's theorem: we can associate to every symmetry transformation (in ray space) a unitary transformation on your Hilbert space (which is defined up to a complex phase factor).

This is the physical motivation, and is explained in What are unitary representations used for in physics? and Needs of unitary representation for QFT.

Decomposition of these representations

Now, the set of these symmetry operators forms a group, and one is often interested in the representation of this symmetry group, and more specifically in the simple representations, or irreducible representations of these representations. And in a way we are lucky, because unitary representations are easy to decompose, both for finite groups and compact groups. This is shown using the following theorems.

For finite groups, Maschke theorem is usually stated in two different ways:

  1. Every representation is equivalent to a unitary representation, i.e is related to a unitary representation by a change of basis. This is what @user1379857 says in his answer. This is actually a lemma combined with the fact that:
  2. Every unitary representation of a finite group is semisimple/decomposable/completely reducible, i.e is a direct sum of irreducible/simple representations (the orthogonal complement of every closed invariant subspace under the representation is again a closed invariant subspace). Giving Maschke theorem, telling us that every representation of a finite group is semisimple, or completely reducible. This is what @Janusz Przewocki explains in his reply.

An equivalent but more general statement applies for compact groups (not necessarily finite), Peter-Weyl theorem: A unitary representation of a compact group on a complex Hilbert space splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations.

I am quite unsure about how I formulated these ideas, and I invite people to correct me wherever I might be wrong, and to add any extra useful information.

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