The scale of the action integral?

[pellman]

Since the Euler-Lagrange equation is homogeneous, the equation of motion of a system of particles or a field is independent of the scale of the Lagrangian. That is, we can multiply the Lagrangian by any constant and arrive at the same equation of motion.

But does the scale of Lagrangian or the action integral ever have any absolute meaning?

Suppose we have two fields A and B with respective free field action integrals S_A and S_B. The action for the full system would be of the form S_A + S_B + S_interaction. But how do I know this is correct? Since kS_B, where k is a constant, is an equivalent free-field action for field B, the correct full action might be S_A + kS_B + S_interaction.

Is the determination of k a purely empirical question? Or are there theoretical considerations which could narrow the possible values of k?

 

[Echows]

I think that you already answered to your first question (i.e. does the scale of the Lagrangian have any absolute meaning) in you next paragraph. That is, the scale indeed plays a role if we have a situation where the total action is sum of two or more actions. But even in that case, what matters is the relative scale of the actions since we can always scale the total action with an arbitrary constant.

To your second question, I think it depends on the system you are considering. For example in QED the scales are set by fundamental constants such as the unit charge, which cannot (at least to current knowledge) be derived from elsewhere. On the other hand, there are effective theories (especially in condensed matter physics) where the constants in the action can sometimes be derived using an underlying microscopic theory.

 

[pellman]

Thanks, Echows.