# Why is the current in the nonrelativistic Dirac sea infinite?

Consider the model of a nonrelativistic noninteracting Dirac sea. I define the model as one with two infinite bands of single particle states:

$$E_\pm (k) = \pm \biggr(\Delta + \frac{k^2}{2m}\biggr)$$

The single particle wave functions corresponding to this dispersion are the free-particle plane wave eigenfunctions $$e^{i \mathbf{k} \cdot \mathbf{x}}$$. We define our vacuum as the state where the lower "valence" band is completely filled and the upper "conduction" band is completely empty. Now consider the current operator at long wavelength:

$$j=\sum_\mathbf{k} \mathbf{k}c_\mathbf{k}^\dagger d^\dagger_\mathbf{k}$$ where $$c_\mathbf{k}^\dagger$$ creates an electron in the conduction band and $$d_\mathbf{k}^\dagger$$ creates a hole in the valence band. If we operate this current operator on the vacuum, the result is infinite due to the possibility of the creation of infinite momentum electron-hole pairs.

What have I done wrong in defining my model? One problem with this model is the infinite mean-squared current in the ground state. Any suggestions on how I can make this model more physical without truncating the two parabolic dispersions? Thanks!

• Where have you got this current operator from? – By Symmetry Oct 25 '18 at 22:45
• You can derive this from the current operator definition seem in Mahan's Many Particle Physics or starting from the real space definition in Table 1 of this PDF and taking the Fourier transform. google.com/url?sa=t&source=web&rct=j&url=https://… – Ian Oct 26 '18 at 1:05