Quantum field theory on curved low-dimensional space embedded in three-dimensional space

Recently, the quantum mechanics on a curved low-dimensional space was studied. There is an embedded effect when the space embedded in three-dimensional Cartesian space has some curvature. In this paper, we will consider second quantization of the spinless Schr\"odinger field there at finite temperature and show that there is also an embedded effect even though the low-dimensional space has no curvature as a manifold. This effect appears as an effective chemical potential.