Chapter 3 E2
the two particle operator can be defined at zero momentum
\[ O_2=\phi(0)\phi(0) \] We can also construct an operator that interpolates the same state
3.0.0.1 FT
we define the discrete Fourier Tansform in space
\[ \phi(p,t)=\frac{1}{V}\sum_x e^{i2\pi \vec{p}\cdot \vec{x}/L}\phi(x,t) \] with \(\vec{p}=(n_x,n_y,n_z)\) and \(n_i=0,1,...,L-1\) since the field \(\phi(x,t)\) is real we have that \[ \phi(-p,t)=\phi^*(p,t) \]
with back to back momentum. In the latter case to have an operator which projected on the \(A1\) irrep of \(O(24)\) we need to construct the combination \[ O_2^{A1}=\frac{1}{\sqrt{3}}\left[\phi(p_x)\phi^*(p_x)+\phi(p_y)\phi^*(p_y)+\phi(p_z)\phi^*(p_z)\right] \] where \(p_x=(2\pi /L,0,0)\) and we use the fact that \(\phi(-p)=\phi(p)^*\).
with these Operator we can construct the correlator \[ \frac{1}{T}\sum_{t_1}\langle O_2^{A1}(t_1) O_2^{A1*}(t_1+t)\rangle = |A_{0\to2}| \left( e^{-E_2 t}+e^{-E_2 (T-t)}\right)+ |A_{1\to1}| e^{-E_1 t} \]
\(E_2\) is the energy of the two particle state at rest;
\(E_1\) is the energy of the single particle with momentum \(|p|=2\pi/L\);
\(O_2^{A1*}=O_2^{A1}\) since it is real.
3.0.0.2 E irrep
The other irrep possible is \(E\) which is two dimensional \[ O_2^{E1}=\frac{1}{\sqrt{2}}\left[\phi(p_x)\phi(p_x)^*-\phi(p_y)\phi(p_y)^*\right] \] which should interpolate an exited state, the other state of this irrep is \[ O_2^{E2}=\frac{1}{\sqrt{2}}\left[\phi(p_x)\phi(p_x)^*+\phi(p_y)\phi(p_y)^*-2\phi(p_z)\phi(p_z)^*\right] \]
3.0.0.3 two particle with momentum operator
3.0.0.4 p=(1,0,0)
we can construct the two particle operator with momentum as \[ O_{2p_x}=\phi(p_x)\phi(0) \,,\quad p_x=(1,0,0) \] this operator is already in the irrep \(A1\). The two particle correlator can be define as \[ \langle O_{2p_x} O_{2p_x}^\dagger(t)\rangle = |A_{0\to2}|^2 \left( e^{-E_{2p_x} t}+e^{-E_{2p_x} (T-t)}\right)+ |A_{\phi(0)\to\phi(p_x)}|^2 \left(e^{-\omega_{p_x} T-m t+\omega_{p_x}t}+e^{-m T-\omega_{p_x} t+mt} \right)\,. \] We can average over all the direction to have a more precise correlator \[ \frac{1}{3}\sum_{i=x,y,z} \langle O_{2p_i} O_{2p_i}^\dagger(t)\rangle \]
3.0.0.5 p=(1,1,0)
\[ O_{2p_{xy}}=\phi(p_{xy})\phi(0) \,,\quad p_{xy}=(1,1,0) \] To construct the correlator we can average over all the direction to have a more precise correlator \[ \frac{1}{3}\sum_{i=xy,yz,zx} \langle O_{2p_i} O_{2p_i}^\dagger(t)\rangle \]
3.0.0.6 p=(1,1,1)
The operator \[ O_{2p_{xyx}}=\phi(p_{xyz})\phi(0) \,,\quad p_{xyz}=(1,1,1)\,, \] and the correlator \[ \langle O_{2p_{x,y,z}} O_{2p_{x,y,z}}^\dagger(t)\rangle \]