In most multiple-input multiple-output (MIMO) communication systems, the amount of information that can be transmitted reliably depends on the effective degrees of freedom (EDoF) of the wireless channel. Conventionally, one can model the channel matrix and study the EDoF, based on an electromagnetic (EM) channel model that is built with the free space Green's function. However, the EDoF of free-space channel model may not fit the practical scenario when EM waves only transmit above the ground. In this paper, we analyze the EDoF for both discrete and continuous aperture MIMO systems in the half space. We also propose an approach to quickly calculate the Green function in the half space from the Sommerfeld identity. Simulation results show that the difference between the EDoF in the half space and that in the free space is non-negligible for the near field communications, which indicates that the ground exerts noticeable influence on EDoF. The proposed study establishes a fundamental electromagnetic framework for MIMO wireless communication in the half space.