Principle Of Least Action

Principle of Least Action is a very useful thing in physics, but with a little twist, that is usually overlooked.

In a second year or so of studying physics, in a classical mechanics (CM) class every student is told that it is a good idea to use this nice principle of least action to solve problems in classical mechanics that are almost impossible to solve if only Newtonian equations are used.
This principle was historically motivated by some talk about light paths and/or virtual work in 18-th century.

In the beginning of the 20-th century human kind discovered that no matter how good classical mechanics is for everyday objects, it does not work for electrons and other little things. As a result quantum mechanics (QM) was created. And since everything around consists of little quantum things, classical mechanics should arise as a limit, or as an effective picture on top of quantum mechanics. And so it does.

When students attend quantum mechanics class, they may do a little problem that involves path integration over possible particle positions. Result of this problem says that path with the highest probability will be the one that minimizes a certain integral quantity that has same shape and same physical units as a classical action.
This means that principle of least action follows from quantum mechanics. And so it should, cause CM is supposed to follow from QM. Very good.

In the same QM class students will be told that QM is not a final theory as it is not relativistic. And a relativistic theory is Quantum Field Theory (QFT). And our best understanding of matter, a Standard Model, is a QFT.

So, how does a usual QFT class starts? It starts with "let's take classical action, and let's do n'th quantization".

What? Are you kidding me? In QM class I am told that QM is not fundamental and it follows from QFT. In the same class I derive myself a classical action from QM. This way CM is deduced from QM, and QM is deduced from QFT. And now I have to use CM elements as a base for a more fundamental QFT. Doesn't it sound like cyclic reasoning? Frankly, it does.

But this approach to QFT did produce the best theory known to man. Yes, this is the little twist I am talking about.

By the way, if you happen to be into strings, you start with action as a basis of a theory of everything. And if you are into loops, you still use an action.

So, if QFT works, then the principle of least action must be legitimate. And the cyclical reasoning that one hears in school is a repetition of a historical development. Folks really tried to use action principle, and it happened to work. Which leads to exactly two honest options in laying down a fundamental theory.
First option is to explicitly require this huge complex of action, which will make it an incredibly non-trivial postulate, or an axiom. The second option is to provide a reason for using a principle of least action as a mechanism for extracting laws of motion.
I will try a second option, cause I happened to stumble upon a good reason while thinking about time and clocks.

So, here how the story goes.

There are two ways of thinking about spacetime, and time in particular. One is a "box-like" view, or a container view of spacetime, in which spacetime exists as a fundamental entity and stuff is present inside of it. Another view is a relational one. In this view spacetime is not a fundamental entity, but is an assembly of relations between matter, or matter states.
But here I want only to suppose that spacetime is nothing but relations between states of matter.

So, suppose that time and space are relations in matter. What sort of relations are these? How can they be quantified and expressed in some useful formal way? To answer these question we must look at experiment.

Well, we use clocks to measure time. We use GPS to find position in space. Both of these devices use events in matter for measurement and they both produce only relative results.
GPS system uses connection between space and time that is introduced by Special Relativity Theory (SR) to make measurements. So, trying to use GPS experience is complicated by the need to somehow introduce connection between space and time.
Clocks are much simpler devices. Clock is a local device, and it is only involved in measuring time. Ok, let us analyze clocks.

There are many types of clocks. But they are all similar in that they use some process(es) as “ticks” of time that are simply counted. And each subsequent "tick" represents some state of matter. This suggests to think about time as a sequence of matter states.

So, let's also suppose that time is a sequence of matter states.

How do we lay out a time measure on matter states?
We need a time measure to label all motions and all jiggling that matter can possibly do. And if some motions of matter happen between two elementary ticks of a clock, they all will be labeled as occurring at the same “tick”-count, or at the same time, which is not good, cause we loose time resolution.

Let’s set an operator L to represent all of matter jiggling within some time interval dt. For a more precise time measure we need matter states labeled with t so that there is as less matter jiggle as possible in any given infinitesimal period of time, over any extended period of time. So we look for matter states ordering or trajectories that minimize the following integral quantity:

S = \int L dt

From QM we know that an operator of infinitesimal change in time is an energy operator. Therefore, L shall have units of energy.
S is an integration of total matter jiggle, so, we’ll call it an action with physical units (energy × time).
In other words, sequences of matter states, or trajectories that define time t in the above equation, should minimize action S.

Let's recoup it:
  1. We assumed a relational view of space time.
  2. We assumed that time is a sequence of matter states.
  3. It forces conclusion that  best time-defining sequences of matter states should minimize action.
  4. At the same time, our best experimentally confirmed theory of matter, Standard Model, uses principle of least action to deduce equations of motion.
This suggests that our initial assumptions are correct, i.e. spacetime is relations between matter states, and in particular, time is a sequence of matter states.
And if these are used as a basis of a fundamental theory, then we may use a principle of least action as a method to find laws of motion.