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I am reading a PhD thesis that considers the Lagrangian $$\mathcal{L}=\partial_\mu\phi\partial^\mu\phi^\star-U(|\phi^2|)$$ where $\phi$ is a complex scalar field and $U(|\phi|^2)$ is an arbitrary potential.

The thesis states that there is a conserved energy $E$ associated with this Lagrangian given by $$E=\int d^2x\;\left(\frac{1}{2}|\dot{\phi}|^2 +\frac{1}{2}|\nabla\phi|^2 +U(|\phi|^2)\right).$$

This is stated without proof/derivation (probably because it is trivial) but I can't determine the justification for this form. How would I go about deriving this myself?

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  • $\begingroup$ You may check it is time independent by use of the E-L equations, no? Have you learned about the energy function in transitioning to the Hamiltonian? $\endgroup$ – Cosmas Zachos May 31 at 23:18
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Check out David Tong's notes on Quantum Field Theory, page 15.

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    $\begingroup$ It would be better if you summarised the points made there than force readers to follow a link (which may rot in future). $\endgroup$ – jacob1729 May 31 at 18:01
  • $\begingroup$ @jacob1729 I should have, but my understanding is too shallow to confidently summarize it just yet, as I have just begun studying QFT. I just remembered coming across it last night so thought I could point Superbee in the right direction. $\endgroup$ – aRockStr May 31 at 18:04
  • $\begingroup$ @aRockStr Thanks for linking these notes; they are quite helpful. If this answer is eventually expanded (either by you, myself, or someone else) to summarize the main points then I will accept as best answer. $\endgroup$ – Superbee Jun 8 at 18:31

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