Nonrelativistic particle field ($\mathcal{L}=i{\Psi }^{†}{\partial }_t\Psi -|\nabla \Psi {|}^2/(2m)$)

This first-order-in-time field theory has an annihilation-field expansion $\Psi =\int a_{\mathbf{p}}{e}^{-i{\mathbf{p}}^{2}t/(2m)+i\mathbf{p}\cdot \mathbf{x}}{d}^{3}p/(2\pi {)}^3$. Its adjoint creates particles; an antiparticle creation term is not required by a relativistic mass shell or Lorentz-invariant microcausality. It is often an effective low-energy sector of a relativistic charged theory after the antiparticle sector is omitted. This does not mean that antimatter cannot be treated nonrelativistically: one may introduce a separate nonrelativistic field for that species.