What is the meaning of local field theory in classical field theory?

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Local field theory refers to a framework in which physical fields are defined at every point in space and time, allowing for interactions to occur locally. In this context, the Lagrangian density is a function of fields and their derivatives, enabling the formulation of dynamics through local interactions. This approach emphasizes that the behavior of fields can be described by equations that depend only on values at nearby points, adhering to the principle of locality. The discussion clarifies that while the text is from a quantum field theory perspective, the principles of local field theory are applicable to classical field theories as well. Understanding local field theory is essential for grasping the foundational concepts in both classical and quantum physics.
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In page of 15 of 'An Introduction to Quantum Field Theory,' Peskin and Schroeder writes

In a local field theory the Lagrangian can be written as the spatial integral of a Lagrangian density, ... , which is a function of one of more fields and their derivatives.

Can you explain what the term local field theory means in this context?

P.S. This question is in the 'Classical Physics' site of the forum because it is a question about classical (and not quantum) field theory.
 
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