# What does 'emergent' mean in a physics context (split from Information Paradox)

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7 minutes ago, Markus Hanke said:

The principle of least action is a general principle of nature, which applies both in the classical and the quantum domain. It says that a given system will evolve such that the ‘action’ - a quantity which equals the time integral of the Lagrangian of the system (being the difference between kinetic and potential energy) - is extremal, usually taken as being at a minimum. This is equivalent to the Euler-Lagrange equation. Hence, to find the evolution equation of a system, you can first work out its Lagrangian, and then find the extremum of the corresponding action. For example, the Einstein equations emerge in this way from the Hilbert action.

This is amongst the most fundamental and most powerful known principles in physics.

Thank you very much for a very good explanation of what the principle of least action is. Unfortunately, it doesn't relate to my question. But it's OK. You don't necessarily know what my question relates to.

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What my question (see above) relates to, can be found in a variety of sources. A "simple", Feynman-style explanation on how the principle of least action emerges in the path integral picture, is in his book "QED: The Strange Theory of Light and Matter". On pp.42-45 he applies this picture in an example of reflection of light and "derives" the principle of least action, in this case. He concludes, "And that’s why, in approximation, we can get away with the crude picture of the world that says that light only goes where the time is least" (p.45).

More generally, on p.123, "This brings us all the way back to classical physics, which supposes that there are fields and that electrons move through them in such a way as to make a certain quantity least. (Physicists call this quantity “action” and formulate this rule as the “principle of least action.”) This is one example of how the rules of quantum electrodynamics produce phenomena on a large scale."

More formal derivation is in Zee, A.. Quantum Field Theory in a Nutshell. On p.12, "Applying the stationary phase or steepest descent method ... [to a path integral] we obtain ...  the “classical path” determined by solving the Euler-Lagrange equation ... with appropriate boundary conditions." (I've removed the math expressions.)

Edit: After re-reading these sections in the books I got the answer to my question. Thus I don't have any more open questions in this thread.

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Thank you very much for a very good explanation of what the principle of least action is. Unfortunately, it doesn't relate to my question. But it's OK. You don't necessarily know what my question relates to.

Indeed, I didn’t. Thanks for clarifying. And I had to make an edit to my post, as I was typing it in haste, and it got all muddled up and imprecise.

Yes, it is interesting that the path integral formalism in QFT gives the same results as the action principle; but I’m not sure if this can be considered a derivation. I rather think these are different formulations of the same principle (but I’m open to correction on this)?

Edited by Markus Hanke

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