Wednesday, August 30, 2017

The annotated math of (almost) everything

Have you heard of the principle of least action? It’s the most important idea in physics, and it underlies everything. According to this principle, our reality is optimal in a mathematically exact way: it minimizes a function called the “action.” The universe that we find ourselves in is the one for which the action takes on the smallest value.

In quantum mechanics, reality isn’t quite that optimal. Quantum fields don’t have to decide on one specific configuration; they can do everything they want, and the action then quantifies the weight of each contribution. The sum of all these contributions – known as the path-integral – describes again what we observe.

This omniscient action has very little to do with “action” as in “action hero”. It’s simply an integral, usually denoted S, over another function, called the Lagrangian, usually denoted L. There’s a Lagrangian for the Standard Model and one for General Relativity. Taken together they encode the behavior of everything that we know of, except dark matter and quantum gravity.

With a little practice, there’s a lot you can read off directly from the Lagrangian, about the behavior of the theory at low or high energies, about the type of fields and mediator fields, and about the type of interaction.

The below figure gives you a rough idea how that works.

I originally made this figure for the appendix of my book, but later removed it. Yes, my editor is still optimistic the book will be published Spring 2018. The decision about this will fall in the next month or so, so stay tuned.


  1. This comment has been removed by the author.

  2. Do you have a clearer scan of it? Hard to read.

  3. If you click on the image you should get a high-res version.

  4. That is terrific! But how on Earth could the entire Universe be so utterly SIMPLE?

  5. Let me guess that anybody who can make the slightest sense out of your infographics is well acquainted with the principle of least action.

  6. I hope there will be a kindle version of your book!

  7. My QFT courses were long time ago, but it seems to me strong interaction is missing from this Lagrangian?

  8. Thomas,

    Well, that, in a nutshell was the reason I took it out. Though I had several pages of explanation accompanying this image.

  9. Without the labels I wouldn’t have known the meaning of the terms for Lagrangian of General Relativity, seeing what they were and how they are related was fascinating and a little frustrating; the relationship appeared to be directly relatable to the question/picture I couldn’t successfully communicate in the August 15th blog. Nevertheless you to continue to expand my knowledge and understanding, thank you.

  10. Milan,

    I haven't broken down the different interactions. The terms always have the same structure, which is the one shown above. I've avoided the indices because then it's easier to see which terms do what. So, no, this wouldn't pass any exam, if that's the question. I made it merely to demonstrate that understanding what's in a Lagrangian isn't black magic.

  11. The action of the Lagrangian doesn't have to be seen in terms of path integrals (though of course it now often is). The interaction component of the Lagrangian can also be seen as a way to generate a (time-ordered) unitary deformation of the dynamics of free quantum fields (or it would if it was well-defined). Classically, one can see the Lagrangian as not much more than a way to generate a system of differential equations, and as much can be said of the Lagrangian in QFT (again, up to worries about renormalization). In particular, symmetries can as much be found in a system of differential equations that is generated by a Lagrangian as they can be in the Lagrangian.
    That's too terse to be comprehensible, so I've posted a derivation that I find suggestive (and that I'm not aware of appearing elsewhere) on my very occasional blog, at (but feel free to moderate this into the vacuum).

  12. And the other way round, path integrals aren't necessarily for quantum systems, and the thing in the exponent isn't a priori an integral over the Lagrangian. Link is fine, very on topic. Thanks!

  13. bee:

    "Though I had several pages of explanation accompanying this image." can you make those pages available to us?


  14. I can't make heads or tails of this; I only have studied engineering and business.

    It's gonna make a killer t-shirt for a small subset of people, however. Trademark it.

  15. Hi Bee,
    A Little bit of nit-picking. Shouldn't this be called pr. of stationary action? As you know very often it is maximum and not least.

    1. Not really kashap .... it is the least or lowest energy. What you are probably doing is envisioning the principle globally. If you do that the constraint on energy to lowest 'lowest' does vanish, but that's because all concept of change abstracts out and along with it obviously any spectrally arranged final status attributes like least/most. This is the context in which 'optimal' is a good way to go.
      But that is not the same as saying the optimal may not be the least. It doesn't translate to sometimes a maximum save trivially where the minimum is also the maximum because a single value resolves numerically.
      Everything comes clear when the non global nature of least energy at the applied level is recovnized.
      Given a specific problem space that least action princuple may be brought to bear culminating in a constrained solution, such as 'what path does the comet take?' Then it is legitimate to ay the principle directly applies to resolution. If you can't say that you can't talk about about outcomes like 'max' or anything else. If you can say it, and a path resolves that path is always least energy

  16. richard,

    no, because I have distributed them throughout the book and I'm currently not in the mood to try and reformulate them in order to not plagiarize myself.

  17. kiemjp,

    Too much too small font I suspect. There a lot of terminology in this description which I haven't added here: Fermions and gauge fields, coupling constants and kinetic terms and all such. Don't worry if you didn't understand it. I had a reason to remove it from the book.

  18. kashyap,

    Yes. It's why I usually write 'optimal' instead of 'smallest' but the problem is that the technical definition of 'optimal' isn't as widely known. It doesn't really matter to get an idea of how it works though.

  19. "It's gonna make a killer t-shirt for a small subset of people, however. Trademark it."

    Too involved. Better use the formulation where you combine the first two formulas (delta S = 0 where you write out S, and maybe the definition of the Lagrangian L(q(t), q\dot(t), t)), and add to it
    "The Theory of Everything"

    You could write it down in LaTeX and order such T-shirts now.

  20. Already exists. I've seen it many times.

  21. "Already exists. I've seen it many times."

    Quite likely I have seen it too and simply forgot where I saw the idea before. I had no intention to be original, btw.

  22. I spent 2 mins looking but I can't find your book. Which one is it?

  23. Maro,

    The book isn't yet published. It's supposed to be published Spring 2018.

  24. Practically every pop-sci book I have ever read treats the "Principle of Least Action" with a solemnity that borders on religious reverence. Yet the concept seems so ludicrously obvious, that to elevate it to the status of a canon in the body of science seems like overkill, as in the following example: You are at a corner of a square-shaped field. You wish to get to the opposite corner by the shortest route. Simple, just take the diagonal.

    Surely, the deep significance of this concept must lie in the mathematical formulation of the Standard Model, as per Bee's diagram. But my eyes have not gazed upon the pages of a calculus book in what seems like eternity. Well, it's high time to rouse those sleepy neurons, which like domestic cats prefer the principle of least effort.

    I look forward to Bee's upcoming book, which I hope will bridge the gap between a layperson's perception of fundamental science principles, versus that of a professional physicist. But that will only come about if we the readers devote the necessary effort to come to an appreciation of the subtleties of the mathematical foundations of modern science.

  25. @David Schroeder "" You are a lifeguard. Somebody is foundering in the ocean 50 yards offshore and 50 yards down the beach, At what point do you stop running (fast) and start swimming (slow) to get there as soon as possible? Snell's law across the phase discontinuity is principle of least action. A naked geometric construction is non-optimal.

  26. Thanks for this! I'm very sad that you had to take it out of your book. Sure, most people won't understand it all, even though it seems to me like you've abstracted it to just about the simplest form possible. This is the language where these ideas are best expressed, and it's important for everyone to see examples like this of math-as-a-language. It's not necessary for readers to understand this. Even without understanding it illustrates the essential challenge of physicists to both communicate ideas in English to others, and to even understand for themselves the ideas suggested by the formulae.

    Mathematics is an important tool for dealing with real, concrete problems. But you don't learn to use it by becoming better at multiplying large numbers on paper or even by learning to evaluate difficult integrals. At certain times those skills are useful but never without the much more important skill of crossing the gap between concrete and abstract, and from ideas to the math representing them.

    I have two daughters in middle school and high school, and I can't tell you how frustrating it has been over the past several years to witness their awful math education. I haven't found a way to really help them through it, but what they have been learning is almost exclusively in order to score well on standardized tests. Augh! Tears. Let readers see math, if only to make this point.

  27. @Uncle Al, good point it didn't occur to me when I first read David's analogy that the shortest route in distance may not be the shortest route in time, that lead me to wonder what route would cause the lowest expenditure of energy.

  28. @Uncle Al, It would make sense to me to run till I'm perpendicular to the foundering swimmer, that is, as close as possible, then begin swimming. I've heard of Snell's Law, but will have to look it up.

  29. @Uncle Al, In my previous note I naively assumed getting as close to the swimmer from the shoreline, after a fast run, would minimize the time to reach the foundering swimmer (didn't have the thinking cap on). I neglected to consider the differential velocities attainable in the two mediums - water and land. Well the shortest path length would be the hypotenuse of the triangle formed by the initial point of the lifeguard, the point on shore, where the swimmer is at a right angle to the (assumed straight) shoreline, and the swimmers position.

    In the extreme case where I'm carrying jet powered flippers, enabling me to swim as fast as I can run, then a straight shot along the hypotenuse would take the least time. But with each incremental reduction in swimming speed possible, the optimal point to enter the water would be progressively further along the beach. If my swimming speed was, say, 1/10,000th my running speed, the optimal water entry point would be close to the closest point of the swimmer to the beach.

    I just glanced at the Wiki article on Snell's law and that jogged my memory, but will have to see how to formulate this problem with an equation, presumably incorporating Snell's law.

  30. @Uncle Al, I'm a little embarrassed not to have instantly realized that this problem setup is a straightforward mechanical analogue of a light ray cutting across the boundary between two mediums with different indexes of refraction, and thus Snell's law is applicable to it's solution. Now I've got to look at, or rearrange, the equation and put in the variables. Am taking a station break for a bit.

  31. The lowest energy pathway, or 'route of least resisistance' as it gets coined when the subject is human affairs is one of about half dozen alternate and svientifically equivalent conception of physical law. It's not the most important as there is no way to prefer that view over one of the others. Feynman thought a good physicist ought know them all since, he pointed out although they are exactly equivalent in the eyes of science, they were very different things psychologically which meant that given a goal a physicist may hit a brick wall using some preffered physics concept? But if he know others he can try on another which may prove much more intuitive when fovussef on that goal.
    The other thing is that it's not true that least energy translates to some mathematical perfection. Very large assumptioms are necessary to get to there. But as with the equivalence of that view and several others, there are several other lines that implications become available and these alternative 'lines' are all completely equivalent to one another. Which doesn't translate to they all say the same thinh in different words but that they say totally diffetent things and can't all be right but that there is no scientificic way to settle the matter. But the perfect maths one is the worst possoble because our brains respond to this perfection like its am opiate

  32. Optimal action Newton derives from infinite lightspeed; zero Planck's and Boltzmann's constants; Equivalence Principle. Defective postulates fail outside derivation.

    Massless boson photons observe achiral isotropic vacuum, Noether’s theorems’ demand exact angular momentum conservation. Baryogenesis; Tully-Fisher, non-classical gravitations, Chern-Simons repair of Einstein-Hilbert action; SUSY. Fermion hadron mirror images differentially embed within trace chiral anisotropic vacuum background. Noether’s theorems leak Milgrom acceleration.

    Compare vacuum free fall optimal action paths of self-similar atom-scale emergence enantiomorphs, opposite shoes embedded within a vacuum trace left foot, to repair empirical physics. Test spacetime geometry with geometry.

  33. Feynman on the principle of least action. Having many years ago done two years of calculus and two of physics before moving on to other studies, I felt a bit cheated to learn that F = dp/dt is not how "real" physics is done at all! There have apparently been some efforts of late to incorporate least action into the first-year physics curriculum, but I have no idea if they have been successful.

  34. For people with an undergraduate background who want a patient introduction to QFT and this notation please see…

    A Story of Light – A Short Introduction to Quantum Field Theory and Quarks and Leptons, M.Y. Han, World Scientific Publishing, 2004, ISBN 981-256-034-3. Requires advanced undergraduate level physics and math. Only 106 pages. Very succinct.

    From Photons to Higgs - A Story of Light (2nd edition), M.Y. Han, World Scientific Publishing, 2014, ISBN 978-981-4583-86-2(pbk). 127 pages. New coverage of Higgs mechanism, etc

  35. @ piein skee,

    I think, most people agree that calling it Pr. of least time or least action is wrong. See for example Feynman's lectures. There are many examples for which the action is maximum. For example, in the case of reflection from a concave mirror, the time is maximum. It is minimum for reflection from plane and convex mirrors.Please google for it. I have difficulty in copying the picture here.

  36. What if there are multiple paths that satisfy the principle of minimum action then which path will be preferable? Is there other criteria the system will have to engage or it will simply engage all paths simultaneously?

  37. Stuart,

    If the (classical) paths reconverge to the same point you get what's called a 'caustic' - interesting phenomenon, best known maybe from optics.

  38. Thanks Bee. Okay what about a particle on top of a Mexican hat potential in unstable equilibrium,is it stuck there because of indecision since all directions satisfy the principle of minimum action? If yes then does this prove that if our universe is in a state of unstable equilibrium then therefore there are no external universes to interact with to break the symmetry?

  39. Thanks, good diagram. I can see why your editor wanted it out! No doubt I'll enjoy your book, but how many others? The aphorism "every equation loses half your readers" might be applicable. Anyway, good luck with it.

    In modern physics "a theory" means: a Lagrangian. Any new idea must be put in this form, subject to all the rules of QFT. Then you're guaranteed to satisfy covariance, conservation laws, etc. The problem is, it becomes impossible to break the rules. Physicists have built this wonderful hammer. But what if nature uses a screw somewhere? It won't be noticed by people trained (indoctrinated) in this approach.

    You say "the universe we find ourselves in is the one for which the action ...". This statement needs the phrase "as far as we know at this time". For example, Milgrom MOND matches actual behavior of galaxies very well. But it can't fit this format. It's not a conservative field, isn't covariant. Therefore, according to modern mind-set, it can't be right. But in fact we don't know that those rules hold at galactic scale. I don't particularly care about MOND. The point is NOT that it's right; probably isn't. The point is, after getting PhD mastering the rigid strait-jacket of QFT it becomes impossible to investigate such an idea. You're locked in a box, can't think outside.

    You figure "it's good to be locked in that box because everything outside it is definitely wrong". Maybe so. Modern physics is betting the farm on this hope. If it turns out wrong, as may well happen, new physics will have to be discovered by non-physicists.

    Thanks again, great diagram!

  40. George,

    My editor said nothing about it. I decided myself to remove it.

  41. >> My editor said nothing about it. I decided myself to remove it.

    Tough decision but probably right. Some famous author, I forget who, was asked the secret of his great prose. He said "Whenever I see a passage I'm very proud of, I edit it out." If it's too clever, the reader won't take the time to understand it.

  42. @Stuart "...a particle on top of a Mexican hat potential in unstable it stuck there because of indecision since all directions satisfy the principle of minimum action?"

    If all directions are locally identically unstable (e.g., slope), there is no "indecision." They are all equally valid. The particle cannot locally know the globally deepest potential path.

    Momentum and position are conjugate, ΔxΔp ≥ ħ. Angular momentum and angular position are conjugate ΔLΔΘ ≥ ħ. Conjugate observables cannot simultaneously both be zero re Heisenberg. The particle falls down the potential. We can classically derive over ~15 lines that the integral of "stability" is divergent. It cannot be better than a saddle point, if that.

  43. "[The principle of least action is] the most important idea in physics, and it underlies everything."

    This is debatable. What led Einstein to the theory of general relativity was the principle of general covariance. The technical step of finding the Riemann-Hilbert action came later, and was not crucial.

    As a practitioner of the (conformal) bootstrap method, I would argue that actions are not very important. There are even quantum field theories for which it is known that no action exists.

    The most important idea in physics is certainly symmetry. If you have enough symmetry, you can solve your theory using symmetry and consistency, without any extra dynamical information such as an action. This bootstrap method used to work in two-dimensional conformal field theory only, but nowadays it works in higher-dimensional conformal field theories too, and there are forays into more general quantum field theories.

  44. Sylvian,

    I didn't say that it was historically the most important principle, just that it's now. And no, it's not symmetry. You try to calculate anything from symmetry without using the principle of least action. Indeed that so many people think this was the major reason I had this in the book.

  45. There is a(n) (in my opinion) funny footnote about this in the book "The theoretical minimum: classical mechanics" by Leonard Susskind. It goes:

    "... to my knowledge, there is no verb to express making a quantity stationary. I tried stationaryizing, stationizing, and a few others, but I eventually gave up and took the path of least action..."



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