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In Peskin & Schroder, Introduction to Quantum Field Theory equation (15.82) states that

$$ t^a_{\bar{r}} = -(t^a_{r})^* = (t^a_{r})^T $$

Why is the representation which satisfies

$$ t_{\overline{r}}^{a}=Ut_{r}^{a}U^{\dagger} $$

called real?

Does this real means something physically real?

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@user49115 : In particular, there is a wikipedia link in the question which Qmachanic mentions on Real Representations. You see the statement in the first sentence of the third paragraph. –  Flint72 Jun 8 at 12:19

1 Answer 1

This means mathematically real, that is, a real-valued representation.

Have you studied the Representation Theory of (Lie) Groups and of Lie Algebras?

A representation may be real, in which the matrices that represent the linear action of the group element are matrices with real-valued elements. We than say that the matrices are in some subgroup of $GL(n,\mathbb{R})$.

Converserly, a representation may be Complex-Valued, and have matrices which are in some subgroup of $GL(n,\mathbb{C})$.

In fact, if you had looked at equation (15.82) in Peskin & Schroeder, which is defining the Conjugate Representation $\bar{r}$ or some representation $r$

$$ t^a_{\bar{r}} = - (t^a_{r})^* = - (t^a_{r})T $$

you would see in the following paragraph he explains his definition of a real representation $r$ to be one for which

$$ r \equiv \bar{r} $$

That is, for $r$ to be real, there must be a unitary transformation

$$ t^a_{\bar{r}} = U t^a_r U^{\dagger} $$

as you say, and the complex-conjugate operation above is trivial. Equivalently, $r$ must be real-valued.

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Thank you for your reply. I would like to also ask the related question about sentence of the below (19.129) at p.678 (Peskin). He explains a gauge-invariant mass term by using the fact of real representation. Do you know how does this related to it? –  user49115 Jun 8 at 12:43
    
@user49115 : If it is a new question then please post it as such, and I'll take a look if I can. Also, if this answered your first question, don't be afraid to accept it! Although in general it is advised to wait a day or two in case anyone else comes along with something better! –  Flint72 Jun 8 at 12:46
    
Thank you for your advice. I use another post for the related question. –  user49115 Jun 8 at 12:58
    
@user49115 : Not at all, I'm happy to help! –  Flint72 Jun 8 at 13:01
    
If the representation r is real, how can we see the invariance of $G_{ab}\eta_{a}\xi_{b}$? –  user49115 Jun 8 at 13:30

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