Ciro Santilli
🔗
🔗
When we particles particles, the action is obtained by integrating the Lagrangian over time:
🔗
In the case of fields however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.
🔗
Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.
🔗
E.g. for a 2-dimensional field :
🔗
Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with :
🔗
And in the context of special relativity, people condense that even further by adding to the spacetime Four-vector as well, so you don't even need to write that separate pesky .
🔗
The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.
🔗
🔗

Ancestors

🔗