Exercise Sheet 13

(i) Show that $P^{2}=\Id$, and that $\Pi_\pm$ are projectors with $\Pi_+\Pi_-=0$ and $\Pi_++\Pi_-=\Id$. (ii) Show that $\tr P=d$, and deduce the dimensions of the symmetric and antisymmetric subspaces. (iii) For $d=2$, identify the two subspaces with the spin triplet and singlet.

Two particles with labels in $\C^{d}$ and a label-blind delta interaction, with $\psi(x_1,x_2)=\pm P\psi(x_2,x_1)$ and, on $x_1<x_2$, $\psi=Ae^{\ii(k_1x_1+k_2x_2)}+A'e^{\ii(k_2x_1+k_1x_2)}$. Lecture 13 shows that where $\psi$ is symmetric in the positions $A'=S(k)A$ with $k=k_1-k_2$, and where it is antisymmetric $A'=-A$. (i) For bosons the symmetric part is $\Pi_+$: show that $Y_{\mathrm B}=S\Pi_+-\Pi_-=\frac{kP-\ii c}{k+\ii c}$. (ii) For fermions show that $Y_{\mathrm F}=S\Pi_--\Pi_+=-\frac{kP+\ii c}{k+\ii c}$. (iii) For spin-$\tfrac12$ fermions, what does $Y_{\mathrm F}$ do in the triplet and in the singlet? Why do two fermions with parallel spins not feel the delta?

(i) With $k=cu$, show that $X_{\mathrm B}=PY_{\mathrm B}=\Rmat(u)/(u+\ii)$ and $X_{\mathrm F}=PY_{\mathrm F}=\Rmat(-u)/(u+\ii)=-(u+\ii P)/(u+\ii)$. (ii) Show that $\Rmat(u)\Rmat(-u)=-(u^{2}+1)\Id$, and deduce unitarity, $X(u)X(-u)=\Id$, for both.

On $(\C^{d})^{\otimes3}$: (i) Show that $P_{12}P_{13}=P_{13}P_{23}=P_{23}P_{12}=:C$, that $P_{13}P_{12}=P_{12}P_{23}=P_{23}P_{13}=C^{-1}$, and that $P_{12}P_{13}P_{23}=P_{13}=P_{23}P_{13}P_{12}$. (ii) Expand $\Rmat_{12}(u_{12})\Rmat_{13}(u_{13})\Rmat_{23}(u_{23})$ and $\Rmat_{23}(u_{23})\Rmat_{13}(u_{13})\Rmat_{12}(u_{12})$ by the number of $P$’s, and show that they are equal exactly because $u_{13}=u_{12}+u_{23}$ (for $d\ge2$, where $C\ne C^{-1}$). (iii) Deduce that $X_{\mathrm B}$ and $X_{\mathrm F}$ satisfy the Yang–Baxter equation.

Let $\Rmat^{\epsilon}(u)=u-\ii(1+\epsilon u)P$. (i) Repeat the expansion of Exercise 4, and show that left minus right is $-\Delta\,(C-C^{-1})$ with $\Delta=\epsilon\,u_{12}u_{23}(2+\epsilon u_{13})$. (ii) Deduce that $\Rmat^{\epsilon}$ fails the Yang–Baxter equation for every $\epsilon\ne0$ (and $d\ge2$); give $\Delta$ at $\epsilon=0.2$, $u_{12}=0.7$, $u_{23}=1.3$.

Lecture 13’s three-body theorem puts the six amplitudes of the factorised ansatz on the permutations of $S_3$ and the contact conditions on the edges $\sigma\to\sigma s_j$. (i) Draw the hexagon, and list the two routes from the incoming wave ($\mathrm{id}$) to the outgoing one ($w_0$), with the crossings each makes. (ii) (Computational.) For $d=2$, $u_{12}=0.7$ and $u_{23}=1.3$, build $X_{\mathrm B}$ on $(\C^{2})^{\otimes3}$ as $8\times8$ matrices and compare the two products. (iii) Do the same for $\Rmat^{\epsilon}(u)/(u+\ii)$ at $\epsilon=0.2$, and account for the size of the difference. (iv) Why is the equation automatic for particles without labels ($d=1$)?

The Yang–Baxter equation for $f\Id+gP$. Let $\Rmat(u)=f(u)\Id+g(u)P$ with $f,g$ continuous and $g$ never zero, and $d\ge2$. (i) Show that $\Rmat$ satisfies the Yang–Baxter equation, with $u_{13}=u_{12}+u_{23}$, if and only if $h(u_{12})+h(u_{23})=h(u_{12}+u_{23})$ for all $u_{12},u_{23}$, where $h=f/g$. (ii) Show that the continuous solutions of this equation are $h(u)=\kappa u$, and deduce that $\Rmat(u)=g(u)(\kappa u+P)$: for $\kappa\ne0$, up to a scalar and a rescaling of $u$, the rational R-matrix. (iii) Check $u-\ii P$ against (ii).