Do Halzen and Martin use p and n to represent complex numbers?

Question

In Halzen and Martin's Quarks and Leptons, on page 42, the $SU(2)$ isospin transformation represented by $e^{i\boldsymbol{\theta}\cdot\boldsymbol{\tau}/2}$ is said to act on the column represented by $$|\psi\rangle=\begin{pmatrix}\mathrm{p} \\\mathrm{n}\end{pmatrix}\tag{1}$$ with $$\mathrm{p}=\begin{pmatrix}1\\0\end{pmatrix}\hspace{0.3cm} \text{and}\hspace{0.3cm}\mathrm{n}=\begin{pmatrix}0 \\1\end{pmatrix}\tag{2}$$

I think this notation i.e., Eq, (1) used in conjunction with Eq.(2), is confusing. In the basis $|\mathrm{p}\rangle=\begin{pmatrix}1\\0\end{pmatrix}$, $|\mathrm{n}\rangle=\begin{pmatrix}0 \\1\end{pmatrix}$, the state $|\psi\rangle$ should be represented by a 2-component column (1) with its entries $\mathrm{p}$ and $\mathrm{n}$ being complex numbers. So are they using the notation that $$|\psi\rangle=\mathrm{p}|\mathrm{p}\rangle+\mathrm{n}|\mathrm{n}\rangle?\tag{3}$$

In this regard, Griffith's Introduction to Elementary particles, I think, uses a clearer notation. They represent general nucleon state $|N\rangle$ as $$|N\rangle=\alpha|p\rangle+\beta|n\rangle\tag{4}$$ with $$|p\rangle=\begin{pmatrix}1\\0\end{pmatrix}\hspace{0.3cm} \text{and}\hspace{0.3cm}|n\rangle=\begin{pmatrix}0 \\1\end{pmatrix}\tag{5}$$ so that the state $|N\rangle$ is indeed represented by a 2-component column vector with two complex numbers $\alpha$ and $\beta$.

My question Do Halzen and Martin use the notation I wrote in Eq. (3) with $\mathrm{p}$, $\mathrm{n}$ being complex numbers? I ask this because it is not clear from their notation.

Answer 1

This is definitely bad (unclear) notation. There are a couple ways I can think of to interpret it: first, and most sensibly, they are doing exactly what you put forward in your question, namely reusing the same variables for both state labels and coefficients without distinguishing which are which.

The other possibility I can think of is that they are using the letters in $\begin{pmatrix}\mathrm{p}\\ \mathrm{n}\end{pmatrix}$ to represent particles. In this sense, you shouldn't think of $\begin{pmatrix}\mathrm{p}\\ \mathrm{n}\end{pmatrix}$ as a precisely defined mathematical expression; it's simply abusing the notation to convey how the isospin transformation mixes the particles (or really, their corresponding fields). Note that in this case, there are two almost entirely unrelated meanings for $\mathrm{p}$ and $\mathrm{n}$: $\mathrm{p}$ can mean either a proton, i.e. when it appears in the doublet, or the isospin state of a proton, i.e. when it appears in the definition $\mathrm{p} = \begin{pmatrix}1 \\ 0\end{pmatrix}$; and similarly for $\mathrm{n}$.