Lorentz Symmetry of Supermembrane in Light Cone Gauge Formulation

We prove the Lorentz symmetry of supermembrane theory in the light cone gauge to complete the program initiated by de Wit, Marquard and Nicolai. We give some comments on extending the formulation to the M(atrix) theory. After the discovery of string duality, our perception of string theory was dras­ tically changed. What used to be the obscure inhabitants of the string theory, the p-branes, turned out to be the key ingredients of the non-perturbative physics. M­ theory is believed to be one of the most symmetric forms of the "string" theory. However, because of our ignorance of the quantization of the p-branes, the very definition of the theory has been unknown. By critical use of the simplification due to the infinite momentum frame, BFSS 1) proposed a constructive definition of the M-theory. The momentum along the eleventh dimension is identified with the zero-brane charge. The infinite boost kills the degree of freedom which has zero (fundamental string) and negative (anti-zero brane) charges. The resulting Lagrangian is made up of only the zero-branes de­ scribed by the large N limit of the SU(N) Yang-Mills theory. BFSS have indicated two major pieces of evidence which support their idea. 1. The matrix theory Lagrangian coincides with that of supermembrane proposed by de Wit, Hoppe and Nicolai (dWHN) 2) if one replaces the gauge group from SU(N) to the area preserving diffeomorphism (APD) in two dimensions. 2. The scattering of the zero-branes coincides with the prediction of eleven­ dimensional supergravity. As usual, the subtlety in the infinite momentum frame is the Lorentz symmetry. This problem is very difficult to analyze in the matrix theory since the momentum exchange in the eleventh dimension implies the exchange of zero-brane charge. We need to treat the quantum process which changes the size of matrices.**) On the other hand, the analysis of a similar problem in the dWHN model is