Consider L0=iψ∗∂0ψ+b∇ψ∗⋅∇ψ as defining a classical field theory. Find the Euler-Lagrange equations. Find the plane-wave solutions, those for which ψ=ei(k⋅x−ωt), and find ω as a function of k. Although this theory is not Lorentz-invariant, it is invariant under space-time translations and an internal U(1) symmetry transformation. Thus it possesses a conserved energy, a conserved linear momentum and a conserved charge associated with the internal symmetry. Find these quantities as integrals of the fields and their derivatives. Fix the sign of b by demanding the energy be bounded below. (As explained in class, in dealing with complex fields, you just turn the crank, ignoring the fact that ψ and ψ∗ are complex conjugates. Everything should turn out all right in the end: the equation of motion for ψ will be the complex conjugate of that for ψ∗, and the conserved quantities will all be real.) (WARNING: Even though this is a non-relativistic problem, our formalism is set up with relativistic conventions; don't miss minus signs associated with raising and lowering spatial indices.)