The Principle of Least Action as a Theory of Everything?

Question

Theoretical physicist Sabine Hossenfelder views the Principle of Least Action as " the closest thing we have to a theory of everything." It works in classical physics as well as in quantum mechanics.

If I am not mistaken, Feynman derived Schrödinger's formalism from it, in his Ph.D. thesis. How to interpret the principle? That is the question.

Why should the time integral of T-V be stationary, from a physical stand-point? What does it tell us about nature, except that the math just seems to work, in physics? If one regards the principle as a kind of theory of everything, then it may be regarded as a lot more than a mere computational device.

Answer 1

It's wrong to call PLA a theory by itself. It becomes a theory only when you write what the action is and what the observables are. So until you specify what the action of ToE is, you have no theory.

Besides, quantum theories are not formulated in terms of PLA. They do have an action, but the universe does not minimize that action. Instead, the purpose of that action is to serve as a function to be substituted in the path integral (this is what Feynman did). We still call it the "action" because it's the same function we use as the action in the PLA of the corresponding classical theory. In quantum theory, the function remains the same, but its purpose isn't to be minimised anymore, but to be path integrated.

So you could say that the closest we have got to a theory of everything is the path integral formalism. But that's basically the same as saying that the universe is quantum.

That said, it is indeed intriguing that all the classical theories, which are only an approximation of the universe, have some PLA with some specific action.

Answer 2

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In this answer I first give a link to a post by me on physics.stackexchange in which the question 'why should the time integral of T-V be stationary?' is addressed. The rest of the answer is focused on philosophy of physics.

Hamilton's stationary action demonstrated with visualizations. The diagrams visualize a trajectory, variation sweep of that trajectory, and how kinetic energy and potential energy respond to the variation sweep.

Does the PLA tell us something about the Universe?

Let me make a comparison:

As far as we can tell the entire domain of physics has the following in common: the physics taking place can be described with differential equations. There is something universal about differential equations.

It seems to me that differential equations are ubiquitous in physics because a recurring theme is that some physical quantity A relates to the rate of change of some quantity B. In theory of motion there is the two-step cascade of position, velocity, acceleration; a cascade of first derivative with respect to time and second derivative with respect to time.

In and of itself that universal aspect is not enough to constitute a Principle of Physics. There is no such thing as offering 'the Principle of Differential Equations' as 'the closest thing we have to a theory of everything.'

We have that every phenomenon that is modeled with a differential equation is modeled with a bespoke differential equation.

In parallel with that: whenever a stationary action approach is applied the quantities that go into the equations and the relations between those quantities are specific to the phenomenon that the approach seeks to solve for.

Differential equations are a mathematical tool, and likewise a stationary action equation is a mathematical tool.

At this point I want to emphasize the following: the name 'least action' is awkward because it represents an over-interpretation. The purpose of the stationary action equation is to identify at what point in variation space the derivative of the action is zero. Whether that point-where-the-derivative-is-zero is a minimum or a maximum is irrelevant. (There may be cases where it can only be a minimum, or can only be a maximum; that is on a case-by-case basis. It's just never relevant whether it is min./max.)

The name 'stationary action' expresses the idea that when framing a theory it is better to not let the assumptions go beyond necessity. As a matter of principle the name 'stationary action' says only: look for the point in variation space where the derivative is zero.

In classical mechanics there is the concept of Energy. A force is categorized as a 'conservative force' when it has a well defined corresponding potential energy. (The potential is well defined if the difference in potential between points A and point B is independent of how an object moves from A to B)

With a force that is conservative: acceleration due to such a force has the property that the sum of potential energy and kinetic energy is conserved.

That is where the stationary of stationary action comes in: during the entire time the rate of change of kinetic energy must match the rate of change of potential energy.

When those two rates of change match during the entire trajectory then the derivative of the corresponding action is zero.

A recurring theme in physics is that of conserved quantities.

In classical mechanics: the point in variation space where the derivative of the action is zero identifies the point in variation space (the true trajectory) which has the property that everywhere along the trajectory the rate of change of kinetic energy matches the rate of change of potential energy.

Answer 3

The Principle of Least Action (PLA) is only an approach for the mathematical formulation of physical theories, where some quantities evolve that can be related to physical entities. But each particular physical theory requires making many choices compatible with the PLA, so the PLA itself does not determine the evolution of any physical system, because additional axioms/hypotheses are always required.

On the other hand, technically a Theory of Everything (ToE) is something more concrete, than PLA. A ToE can be formulated in the form of a PLA, probably, but in itself the PLA tells us nothing about a possible ToE suitable to describe our universe. For that reason, PLA is not strictly like a ToE but something of a different nature.

Not that Sabine Hossenfelder is wrong, she almost always makes pertinent and accurate observations, but at the same time she likes to use a certain irony and sense of humor. I remember having seen precisely the YouTube video where she makes assertion, and it seems perfectly clear to me that she is speaking metaphorically, to point out the omnipresence of the PLA in many physical theories, but it is clear to me that she never meant to assert something like PLA = ToE.