A Theorem on the Power of Supersymmetry in Matrix Theory

For the so-called source-probe configuration in Matrix theory, we prove the following theorem concerning the power of supersymmetry (SUSY): Let $\delta$ be a quantum-corrected effective SUSY transformation operator expandable in powers of the coupling constant $g$ as $\delta = \sum_{n\ge 0} g^{2n} \delta^{(n)}$, where $\delta^{(0)}$ is of the tree-level form. Then, apart from an overall constant, the SUSY Ward identity $\delta \Gamma=0$ determines the off-shell effective action $\Gamma$ uniquely to arbitrary order of perturbation theory, provided that the $ SO(9)$ symmetry is preserved. Our proof depends only on the properties of the tree-level SUSY transformation laws and does not require the detailed knowledge of quantum corrections.