This is related to Modern Supersymmetry: Dynamics and Duality by Terning.

Consider $N=1$ SUSY. $\{Q_a,\bar{Q}_{\dot{a}}\}=2\sigma_{a\dot{a}}^\mu P_\mu$. Sum over $a=\dot{a}=1,2$ and this yields $4P^0=\sum_{a=\dot{a}=1,2}\{Q_a,\bar{Q}_{\dot{a}}\}$.

Suppose vacuum $|\Omega>$ is supersymmetric.(According to the book, $Q_a|\Omega>=0$ pg 6 equation (1.24). The book did not say $\bar{Q}_{\dot{a}}|\Omega>=0$.) Here I treated $\bar{Q}_{\dot{a}}$ as creation operator by boosting to the rest frame and I am assuming massive states.

I compute $<\Omega|4P^0|\Omega>=\sum_{a=\dot{a}=1,2}<\Omega|Q_a\bar{Q}_{\dot{a}}|\Omega>$ where the other part drops out by $Q_a|\Omega>=0$. The computation can be further simplified. One can check $\{Q_2,\bar{Q}_2\}=0$ by rest frame and compute this anti-commutator. Hence $<\Omega|4P^0|\Omega>=<\Omega|Q_1\bar{Q}_{\dot{1}}|\Omega>$. And I still do not see this vanishes.

$\textbf{Q:}$ I do not have SUSY vacuum energy $0$ here. If I have $\bar{Q}_{\dot{a}}|\Omega>=0$, it goes through but $\bar{Q}_{\dot{a}}$ is creation operator for massive fermionic state. Where are mistakes with thought process?

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    $\begingroup$ The vacuum is invariant under all supersymmetries involved, and so it is, in fact, annihilated by all four Q s. $\endgroup$ Jan 2 '19 at 1:40

The definition of the vacuum in $N=2$ SUSY is from

$$| \Omega \rangle = Q_1 Q_2|m, j\rangle. $$

where $m$ is the mass eigenvalue of the casimir $P^2$ and the $j$s are the eigenvalues of the casimir $J^2$ where

$$J_i = S_i - \frac{1}{4m} \bar{Q}\bar{\sigma}_iQ. $$

From the SUSY algebra one can immediately see that

$$ Q_i \Omega \rangle = 0 \qquad i=1,2$$

and that if we act with the $\bar{Q}$s on the clifford vacuum that

\begin{align*} \{|\Omega \rangle, \bar{Q}_1 |\Omega\rangle, \bar{Q}_2 |\Omega\rangle, \bar{Q}_2\bar{Q}_1 |\Omega \rangle\} \end{align*}

are the only irreducible representations (for the massive case).


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