In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. And for that he uses the condition ∂tΨ≪mΨ, which in fact I suppose he means |∂t˜Ψ|≪|m˜Ψ|, otherwise I don't get it. In any case, starting with the Lagrangian:
L=∂μ˜ψ∂μ˜ψ∗−m2˜ψ˜ψ∗
Using the inequation I think it's correct, I can only get to:
L=−∇˜ψ∇˜ψ∗−m2˜ψ˜ψ∗
And from that I've tried relating ˜ψ or ψ (as we can write the above Lagrangian with both, as it's invariant under multiplying by a pure phase), to ˙ψ
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