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Can the Lagrangian density of a field be derived from the stress-energy tensor?

Physics Asked by Smitology on January 22, 2021

I have been learning some field theory and learning about Lagrangian and Hamiltonian density. In classical mechanics, the Hamiltonian is the energy of the system in terms of position and momentum. I understand that similarly, the Hamiltonian density is the energy density in terms of the field and the conjugate momentum field. In that way, the Hamiltonian density is $T_{00}$, where $T$ is the stress-energy tensor.

Since the Lagrangian density is relativistically invariant, is there an invariant relation between Lagrangian density and the stress-energy tensor?

One Answer

Indeed, see Classical field theory Landau Lifschitz pgs 110-11; there it is carried out for conservative fields (that is, its quadrivergence is null); The definition of the momentum energy tensor is not unique because it can be added the quadrivergence in relation to the subscript i, of a tensor of rank 3 (Tikl) antisymmetric in relation to the indices k and l. Also you can consult QuantumField Theory Ryder pg 85 ec 3-20 I hope I have been able to collaborate

Answered by oscar cepeda giraldo on January 22, 2021

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