Friday, 18 October 2019

conservation laws - Noether Current when the Lagrangian depends on second derivative of the fields


Let a Lagrangian density for a field theory of N fields {ϕi}Ni=1 be given.


Assume that the Lagrangian density depends on the fields, their spacetime derivatives, and their second spacetime derivatives: L(ϕi,μϕi,νμϕi).



Then a short derivation shows that the Euler-Lagrange equations are given by:


δLδϕiμδLδμϕi+νμδLδμνϕi=0i{1,,N}


Using a similar derivation to the proof of the Noether theorem, I was able to show that the conserved Noether current is: jμ=i[δLδμϕiΔϕi+δLδμνϕiνΔϕi(νδLδμνϕi)Δϕi]


My question is: is this correct?


It feels fishy to me because there is a term νΔϕi and I would somehow expect all terms to be proportional to Δϕi alone.


I'm doing this to find the conserved Noether current (see this related question and this one which unfortunately had no answers yet) of the BRST transformation.



Answer



Yes it is correct. I derived and used the same expression in http://vixra.org/abs/1008.0051 page 5 (with one extra term to account for space-time transformations that is not needed for internal symmetries). The dependence on the derivatives νΔϕi is necessary and not a problem.


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