What is the meaning of manifest locality in Lagrangian formalism?

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Manifest locality in the Lagrangian formalism refers to the ability to express the Lagrangian as an integral of a Lagrangian density, which consists of sums and products of fields evaluated at the same spacetime point. This definition emphasizes the requirement of a finite number of derivatives in the formulation. While locality in Hamiltonian quantum field theory (QFT) is tied to Lorentz invariance through the commutation of spacelike field operators, manifest locality does not necessarily imply Lorentz covariance. This distinction is crucial for understanding the implications of locality in different theoretical frameworks.

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atyy
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The term "manifest locality" keeps being used, eg. in talks by Cachazo http://pirsa.org/10080013/ and Arkani-Hamed http://online.kitp.ucsb.edu/online/bblunch/arkanihamed/ .

In Hamiltonian QFT, "locality" means Lorentz invariance, and is implemented by requiring that spacelike field operators commute.

Is "manifest locality" in the Lagrangian formalism the same thing? If so, I guess it means that (i) the Lagrangian consists of sums and products of fields at the same place and time, and (ii) the action is Lorentz covariant? (Seems maybe not, since they always talk about "manifest locality" of the Feynman diagrams - what does that mean?)
 
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I'm pretty sure that it just means your point (i).
Manifest locality means that your lagrangian can be written as the integral of a lagrangian density, which is a sum of products of fields at the same space-time point. You also need to have only a finite number of derivatives.

You can have locality without Lorentz covariance, so I'm not sure about your Hamiltonian QFT statement.
 

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