# Can we derive most fundamental laws from the Action Principle? [duplicate]

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It is said in the book Fearful Symmetry - The Search for Beauty in Modern Physics that we can derive all basic laws in physics from a simple principle called Least Action Principle (although it may be better called Stationary Action Principle). I am very excited at the following statements that: (cite page 109):

The reader should understand that the entire physical world is described by one single action. As physicists master a new area of physics, such as electromagnetism, they add to the formula for the action of the world an extra piece describing that area of physics. Thus, at any stage in the development of physics, the action is a ragtag sum of disparate terms.

A picture taken from the book would best illustrate the point: • Question:

As the author (A.Zee, from UCSB) is a particle physicist, I have confidence that this is true. But no further detail is given in the book. With Google I could not find any book about Action Principle. Nor could I find any reference to the action shown in the book. Most physics book I read use Action Principle to describe the dynamics of a specific system in that field (e.g action of a free particle or action of the string etc) and then compute the equation of motion. But few of them use the Principle to derive the fundamental law in that area (Except the classic: Mechanics by Landau, in which The Newton's three law was shown to be implied by the Action Principle).

Can anyone explain that how each of the magic terms in the above Fig.7.4 B may turn into the fundamental laws in Gravity, Electromagnetism, Strong and Weak forces? Or are there books I haven't found yet to describe this exciting law?

Very sorry for my late response. After spending some days devouring in the books recommended by @hft , I realized that tje difficulty in understanding them. But still I am pleased to find that satisfying answers lie in those books.

Some more books could be added to the list by @hft For classical mechanics and classical field theory(treating relativistic motion), Landau's books are always my favourite: Mechanics, third edition, Landau (http://www.amazon.com/Mechanics-Third-Edition-Theoretical-Physics/dp/0750628960 The Classical Theory of Fields, fourth edition, Landau (http://www.amazon.com/Classical-Theory-Fields-Fourth-Theoretical/dp/0750627689/), where I found good derivation of the Maxwell's equation.

## marked as duplicate by ACuriousMind♦, Jim, JamalS, Qmechanic♦Feb 25 '15 at 23:28

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• The Standard Model of particle physics is the most general set of fundamental laws we know, which seems to describe all forces (except gravity). The representation of this theorem is in terms of a Lagrangian, or Lagrangian density. The equations of motion describing physical quantities arise from minimizing the action described by this Lagrangian. As for understanding how that works, well, you'd need to learn quantum field theory. But you can maybe get some idea by studying classical fields--Goldstein is a book I've used in the past, but it is graduate level. – zeldredge Feb 25 '15 at 17:52
• Actually, the principle of stationary action only holds exactly in classical mechanics; so you can use it to derive e.g. Einstein's field equations in general relativity, or Maxwell's equations in classical electromagnetism. Mathematically, the most convenient way to do this, is to take whatever is to the right of the integral sign in the action, and substituting it into the Euler-Lagrange field equations. – jabirali Feb 25 '15 at 17:52
• However, in quantum mechanics, a particle doesn't take the path of least action, but it takes all paths between points in spacetime, and each path is weighted by a phase factor $\exp(iS/\hbar)$, where $S$ is the action. This is also known as the path-integral formulation of quantum mechanics and quantum field theory, if you wish to read more about it. The path of stationary action in classical mechanics, is then simply the path with the maximal constructive interference in quantum mechanics. – jabirali Feb 25 '15 at 17:53
• This is too broad. Almost all of physics can be cast into a principle of stationary action. Classical mechanics has Lagrangian mechanics, and quantum mechanics has the path integral formalism which is essentially the "quantum extension" of the idea of least action. The entire field of quantum field theory may be (a bit flippantly) said to be just about extracting the fundamental laws and predictions from the action. – ACuriousMind Feb 25 '15 at 17:55
• Possible duplicates: physics.stackexchange.com/q/3500/2451 , physics.stackexchange.com/q/20298/2451 and links therein. – Qmechanic Feb 25 '15 at 18:32

## 1 Answer

Formulating dynamics in terms of least action is well known. See, e.g., wikipedia: http://en.wikipedia.org/wiki/Principle_of_least_action

For a nice quantum mechanical treatment see: Feynman and Hibbs (http://www.amazon.com/Quantum-Mechanics-Path-Integrals-Emended/dp/0486477223)

For a standard application in QFT see Peskin and Schroder (http://www.amazon.com/Introduction-Quantum-Theory-Frontiers-Physics/dp/0201503972), e.g., equation 2.1. This reference is probably what you really want. They give a few examples of actions, like scalar field theory, and QED, and QCD, etc, etc, etc...