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There are three kinds of representations: real, complex, and pseudoreal. A complex representation is not equivalent to its conjugate, and a real one is, which is pretty straightforward. A pseudoreal representation is also equivalent to its conjugate, but the change of basis matrix that relates them has some funny properties. (Note that these definitions are independent of the terms 'real' and 'complex' in math; all representations in quantum mechanics are 'complex' in the math sense.)

There's a clear physical meaning of a complex representation, i.e. that particles that transform in these representations are not the same as their antiparticles. But I can't find any simple physical meaning for pseudoreality that distinguishes it from reality; it looks to me to be a fairly arbitrary distinction and I don't even know why we would want to make that distinction on mathematical grounds. How should I think about reality and pseudoreality physically?

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    $\begingroup$ Mathematically speaking, all representations of a group $G$ are defined the same way, $\pi: G \rightarrow Aut V$. So the reps you're referring to must be to additional conditions over and above the general definition of a rep. $\endgroup$ – Mozibur Ullah Jan 18 '18 at 21:10
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    $\begingroup$ The standard prototype for applications is all reps of $SU(2)$ being pseudoreal, which dictate the vanishing of anomalies for it--the symmetric $d$-coeffs in the Lie algebra vanish. Perhaps you'd find a satisfactory meaning in this. $\endgroup$ – Cosmas Zachos Jan 19 '18 at 1:18
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The definition of a real representation is not merely that it is isomorphic to its conjugate.

Both real and pseudoreal representations are isomorphic to their conjugates. This isomorphy forces the existence of an equivariant anti-linear map $$ J : V \to V, $$ where $V$ is the representation space, which simply is the isomorphism $\phi : V\to V^\ast$ concatenated with complex conjugation.

Such an equivariant map necessarily squares to a multiple of the identity by Schur's lemma, i.e. $J^2 = c\cdot \mathrm{id}_V$ for some $c\in \mathbb{R}$. If $c>0$, then $J$ is a real form and the representation is real, if $c<0$, then $J$ is a quaternionic form and the representation is pseudoreal = quaternionic.

You can reduce a real representation to a literal representation on a real vector space, i.e. restrict the representation to the subspace with $J(v) = v$, i.e. the complex representation $V$ splits naturally into the direct sum of two real representations $V_\mathbb{R}\oplus \mathrm{i}\mathbb{R}$. This is e.g. what Majorana spinors are.

You cannot reduce a quaternionic representation in this way (although there is some funny business with pseudo-Majoranas which is not entirely clear to me).

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    $\begingroup$ Okay, I see. So if I’m getting this straight, this distinction is never important in quantum mechanics (because the Hilbert space is always complex) but can be in quantum field theory (because fields can be real-valued), right? $\endgroup$ – knzhou Jan 18 '18 at 19:41
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    $\begingroup$ @knzhou Indeed, this matters only for representations on complex vector spaces which are not quantum-mechanical spaces of states. $\endgroup$ – ACuriousMind Jan 18 '18 at 19:52

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