If the system is invariant in time (time translation symmetry) and invariant in space (space translation symmetry), then we can simplify because σ σ ( x , x ′ , t , t ′ ) = σ σ s y m ( x − − x ′ , t − − t ′ ) for some convolution kernel σ σ s y m . We can also consider plane wave solutions for E and J like so: