Fully off-shell effective action and its supersymmetry in matrix theory

As a step toward clarification of the power of supersymmetry (SUSY) in matr ix theory, a complete calculation, including all the spin effects, is perfo rmed of the effective action of a probe D-particle, moving along an arbitra ry trajectory in interaction with a large number of coincident source D-par ticles, at one-loop at order four in the derivative expansion. Furthermore, exploiting the SUSY Ward identity developed previously, the quantum-correc ted effective supersymmetry transformation laws are obtained explicitly to the relevant order and are used to verify the SUSY-invariance of the effect ive action. Assuming that the agreement with Ii-dimensional supergravity pe rsists, our result can be regarded as a prediction for the supergravity cal culation, which, as yet unavailable, is known to be highly non-trivial.