The Symmetry of Antiparticle Isospin Doublets in Particle Physics

[orochi]

TL;DR

Isospin duplets for proton-neutron system

In Quarks & Leptons: An Introductory Course in Modern Particle Physics by Halzen and Martin page 42 reads:

The construction of antiparticle isospin multiplets requires care. It is well illustrated by a simple example. Consider a particular isospin transformation of the nucleon doublet, a rotation through π about the 2-axis. We obtain:

We define antinuclear states using the particle-antiparticle conjugation operator C,

Applying C therefore gives:

However, we want the antiparticle doublet to transform in exactly the same way as the particle doublet, so that we can combine particle and antiparticle states using the same Clebsch-Gordan coefficients, and so on. We must therefore make two changes...

I do not understand what the issue is, however. What do they mean by "we want... to transform in exactly the same way"?
Didn't they just show that they do transform in exactly the same way?

 

[vanhees71]

The point is that the charge-conjugation transformation involves complex conjugation, i.e., if particles are defined to transform under the fundamental representation of the isospin SU(2), like $(p,u)^{\text{T}}$ (with p having $t_3=+1/2$ and n having $t_3=-1/2$) the charge conjugate state transforms like the conjugate complex spinor $(p^*,u^*)^{\text{T}}$.

Now there's only one 2D representation of SU(2), the fundamental representation, up to isomorphism, i.e., the conjugate-complex representation must be isomorphic to the fundamental representation.

Now the fundamental representation is given with help of the Pauli matrices by
$$\hat{U}(\vec{n},\phi)=\exp \left (\frac{\mathrm{i}}{2} \hat{\vec{\sigma}} \cdot \vec{n} \phi \right).$$
The conjugate complex spinor transforms with the conjugate complex matrix,
$$\hat{U}^*(\vec{n},\phi) = \exp \left (-\frac{\mathrm{i}}{2} \hat{\vec{\sigma}}^* \cdot \vec{n} \phi \right).$$
To see that this is indeed a representation isomorphic to the fundamental representation, we need a unitary matrix $\hat{T}$ such that
$$\hat{T} \hat{U}(\vec{n},\phi) \hat{T}^{\dagger} = \hat{U}^*(\vec{n},\phi).$$
For that, obviously we must have
$$\hat{T} \hat{\vec{\sigma}} \hat{T}^{\dagger} = -\vec{\sigma}^*.$$
Now we note that
$$\hat{\sigma}_1^*=\hat{\sigma}_1, \quad \sigma_2^*=-\hat{\sigma}_2, \quad \sigma_3^*=\hat{\sigma}_3.$$
It's easy to see that we can set $\hat{T}=-\mathrm{i} \sigma_2$. Indeed this is a unitary matrix,
$$\hat{T} \hat{T}^{\dagger}=\hat{\sigma}_2 \hat{\sigma}_2^{\dagger}=\hat{\sigma_2}^2=\hat{1},$$
and
$$\hat{T} \hat{\sigma}_1 \hat{T}^{\dagger}=\hat{\sigma}_2 \hat{\sigma}_1 \hat{\sigma}_2 = -\hat{\sigma}_2^2 \sigma_1=-\hat{\sigma}_1=-\hat{\sigma}_1^*,$$
$$\hat{T} \hat{\sigma}_2 \hat{T}^{\dagger}=\hat{\sigma}_2^3 =\hat{\sigma}_2=-\hat{\sigma}_2^*,$$
$$\hat{T} \hat{\sigma}_3 \hat{T}^{\dagger}=\hat{\sigma}_2 \hat{\sigma}_3 \hat{\sigma}_2 =-\hat{\sigma}_2^2 \hat{\sigma}_3 = -\hat{\sigma}_3=-\hat{\sigma}_3^*.$$
So to get a spinor for antiprotons and antineutrons we have to use
$$\hat{T} \begin{pmatrix} \bar{p} \\ \bar{n} \end{pmatrix} = \begin{pmatrix}-\bar{n} \\ \bar{p} \end{pmatrix}.$$
It's important to note that the antiparticle isospin representation is isomorphic to the particle isospin representation holds true only for SU(2). For SU(3) flavor transformations (when taking into account also strange hadrons/quarks as in Gell-Mann's "eightfold-way model" for hadrons) the quark states transform under the fundamental representation but the antiquark states under the conjugate complex of the fundamental representation, which is not isomorphic to the fundamentale representation.

 

[ohwilleke]

I do not understand what the issue is, however. What do they mean by "we want... to transform in exactly the same way"?

Didn't they just show that they do transform in exactly the same way?

The author is writing in what is sometimes colloquially called "nerd view", i.e. from the perspective of some "nerd" that is different than the true perspective of the reader.

In this case, the author is writing from the perspective of someone trying to come up with a law of physics or process that has the desired properties mathematically, before the proof has been completed. The educational purpose for this approach is the hope that if you view the process of proving something in the first person ("we") that it will help you to identify better as a scientist and make you more likely to understand what is going on properly.

If you are following alone with the presentation, you don't experience the proof as something that has actually been proven to you until you get to the QED at the end.

 

[vanhees71]

The point is that you want to build invariant Lagrangians, equations of motion, etc. to fulfill some symmetry, in this case isospin symmetry (valid approximately as long as only strong interactions between light-quark (u,d) hadronx are concerned). It's easier to build such invariant Lagrangians when you group the antiparticles into isospin-spinor doublets such that these transform under the isospin transformations as the particles. Since the proton and neutron make up a isospin doublet transforming under the fundamental representation, it's convenient to define $(-\bar{n},\bar{p})$ as the corresponding isospin doublet for the anti-neutron and the anti-proton that transforms also under the fundamental representation.