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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.

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    If PLA were really a theory of everything, then it must be an answer to the seemingly self-reflecting question "why PLA is a theory of everything?" and all other well defined questions. I guess you mean a theory of everything in the field of physics for all well-defined physical phenomena. But even here such an abstract principle alone could hardly explain any physical theory such as QM since you inevitably need other related concrete terms and premises/principles... Aug 1 at 2:16
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    In this blog post, Hossenfelder says "In English they talk about a 'Theory of Everything'. In German we talk about the 'Weltformel', the world-equation. I’ve always disliked the German expression [...] because equations in and by themselves don’t tell you anything. [...] However, in physics we do have an equation that’s pretty damn close to a “world-equation” [...] and it’s called the principle of least action." She clearly states it isn't really a theory of everything.
    – benrg
    Aug 1 at 21:55
  • I can see that it was a mistake to refer to TOE introducing my question, " Why should the time integral of T-V be stationary, from a physical stand-point?" The TOE issue side tracked the real issue. I'm interested in making physical sense of the principle of least action.
    – Paul S.
    Aug 6 at 21:49

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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.

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  • It's also intriguing to ask whether this character of classical theories tells us something about the universe or whether it tells us something about our minds. Aug 1 at 8:42
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About my background:
I am mostly active on physics.stackexchange.
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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.

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  • Would it be appropriate to regard The Principle of Least Action universally as the law of conservation of action?
    – Paul S.
    Aug 6 at 22:04
  • @PaulS. In a comment (in response to another comment) you write: "I'm interested in making physical sense of the principle of least action". For that question specifically I referred to an answer by me on physics.stackexchange about Hamilton's stationary action That discussion is illustrated with visualizations. (Possibly you are not aware that 'Hamilton's stationary action' and 'principle of least action' are two names for the same thing.)
    – Cleonis
    Aug 7 at 5:39
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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.

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  • I just find it easier to understand the meaning of the minimization of T+V as a physical principle than the the Principle of Least Action. That the integral of T-V ought to be stationary, leaves me baffled. It leads to differential equations and path integral approaches that work mathematically, but this does not satisfy my intuition.
    – Paul S.
    Aug 6 at 21:57
  • Note that the extremes of T+V only correspond to solutions that are equilibrium points and, therefore, static solutions where nothing changes.
    – Davius
    Aug 7 at 7:12

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