Let me point out some differences between the relativistic and non-relativistic case. The most striking is that the relativistic theory is quadratic in time derivative, while the non-relativistic theory is linear in time derivative. Thus, in the non-relativistic theory the momentum density conjugate to the field φ. In condensed matter physics it is often illuminating to write φ =√ρ exp(iθ), so that in Lagrangian, the first term is a total divergence. The second term tells us something of great importance [See P. Anderson, Basic Notions of Condensed Matter Phycics, p. 235] in condensed matter physics: in the canonical formalism, the momentum density conjugate to the phase field θ(x). Integrating ρ(x, t) and defining N = the total number of bosons, we find one of the most important relations in condensed matter physics [N, θ] = i. Number is conjugate to phase angle, just as momentum is conjugate to position. Marvel at the elegance of this! You would learn in a condensed matter course that this fundamental relation underlies the physics of Josephson junction. You may know that a system of bosons with a “hard core” repulsion between them is a superfluid at zero temperature. In particular, Bogoliubov showed that the system contains an elementary excitation obeying a linear dispersion relation.