Comment on “Quantum Statistical Mechanics of Nonrelativistic Membranes: Crumpling Transition at Finite Temperature” by M.E.S. Borelli, H. Kleinert, and A.M.J. Schakel

Borelli et.al. [Phys. Lett. A 267 (2000) 201] study a 2-dimensional fluid membrane in a 2+1-dimensional space, subject to quantum fluctuations. Unfortunately, they use an unphysical kinetic-energy term which is not invariant under coordinate transformations. Moreover, they expand the resulting field theory around D = 2, which is not its lower critical dimension. We study the corrected field theory to one loop order in a D = ǫ expansion and find a quantum transition even at T = 0. Borelli et.al. [1] investigate the quantum fluctuations of a membrane with curvature stiffness and surface tension. They start from the action S0 = ∫ dtdσ √ g ( 1 2ν (∂tX) 2 + r + 2 α H )