When I was an undergrad taking quantum mechanics, I had a secret weapon. It was that the work-study student at the front desk of the science library was a gorgeously beautiful coed, so of course I had to spend much time at the science library. Late at night, she and I would often be the only two people in the place. And it seemed rude not to talk.
Then, as long as I was in the library, anyway, doing my QM homework (which took forever, but I wasn’t in any hurry), one day I decided to look to see what was there on the library shelf. And I found a book on the mathematics of quantum mechanics, which was a game-changer. Often, in working a QM problem, we can get to a certain place in doing the math where we are stymied because we can tell that we want to end up “over there” in the math, but we can’t quite figure out how to get there. We can discern that there ought to be a math trick allowing us to go from here (where we are) to over there (where we want to be), but we don’t know the trick.
And here was a book full of the tricks.
Suddenly, I was a whiz kid at QM. I’m not claiming that I understood most of the book, but it definitely helped me with the elementary problems that I was working.
Yet there was an even greater point to my story; in fact, there were several corollaries. One was that our QM professor described for the class how the original discoverers of QM had undergone a similar problem to what I had experienced. They would get so far in the math, and then they’d realized that they’d needed a trick to get mathematically from here to there. But they, too, had been stymied, having no such tricks available to them. Yet then what had happened to them was this: Some math friend of theirs had told them that hey, there was a looney mathematician who thought that he had discovered a way to contact the spirit world by using math. And his math seemed to be just what the QM founders needed to do their tricks.
And yes, that was an interesting “interpretation” of QM. Was it a way to contact the spirit world? What did this mathematician mean by that, anyway? (We have to recall that Spiritualism was a big movement back in the 1800’s, séances and all of that).
And I didn’t know about the spirit world, but let me tell you. When it came to solving problems, this spiritualism stuff really worked.
Today, I am pretty sure that the looney mathematician was Hamilton—he was definitely a loopy enough guy—and there are Hamiltonians throughout QM. But I am too lazy to track that down for sure. (And Einstein had encountered a similar problem with his relativity theory, but he had discovered his own solution in Riemannian geometry, which being non-Euclidean was hardly feet-on-the-ground stuff, either). For that matter, Newton had invented calculus to do his physics.
There seemed to be a pattern here: mathematical breakthroughs as part of physics breakthroughs. (Also, Boltzmann had succeeded after the discovery of probability math).
Was there some comparable new math for QM?
I am coming to describe an interesting historical controversy of the late 1940’s, regarding the appropriateness of science adopting this “spiritualism ” approach. But I am not finished setting it up because, back then, as a student, I hadn’t discovered it yet. (How I had found out about it is a part of my story; yes, it had to do with the sumptuously gorgeous coed). I hadn’t yet happened upon two additional corollaries to my good fortune in finding this book. The next corollary (besides learning that these math tricks might have begun as Spiritualism) was this:
I had never heard of the author of this book that I had found—I had assumed that he was some obscure professor from the University of Somewhere—but even so, I had considered him to be my obscure professor (since no one else I knew was reading him). I’d left the book on the shelf so that I would have to read it while in the library, for obvious reasons.
And no, the author wasn’t Hamilton. But it turned out to be John von Neumann, and this was a famous book. At the time, as a student, I had been familiar with most of the big names in QM—the founding scientists—but I hadn’t gotten to any of the ones who came a half generation later. If I had told the spectacularly beautiful coed of my false earlier assumptions about “my” obscure professor, I would have been profoundly embarrassed, but fortunately, I hadn’t.
Yet maybe you can tell where I am going with this. The book left me hungry to understand more about, not just QM, but the math of QM. And it was in further researching that subject that I then fortuitously came across Feynman, which is what I want to talk about here. What was interesting was that, back in the 1940’s, Feynman was considered a whacko nutcase for even proposing certain math explanations. It seemed to be related to this Spiritualism stuff.
But, of course, I hadn’t known who Feynman was at the time, either. All I’d known was that he was writing on the subject that I wanted to read about (how to make sense of QM). As far as I’d known, he was another one of those obscure professors from the University of Somewhere Else.
But I had liked him—I’d liked how his mind worked and how I could absorb his thinking—so obviously I’d thought of him as another one of my obscure professors. He was my other obscure professor who was cluing me in on QM.
I’m pretty sure that I’d found the first article of Feynman’s on my own. But then, are you ready for this?
Clearly, I’d had to tell the gorgeous woman at the front desk about my good fortune in finding this first Feynman article. Upon finding it, I was probably looking way too unusually happy for her not to notice how joyful I was, and she had asked me to explain (since that wasn’t the usual condition of someone staying up late at night doing long QM problems).
So, here we go. . . .
The spectacularly gorgeous woman at the front desk of the science library started finding me, on successive nights, articles by Feynman for my further reading. In those days, one didn’t just find articles online because there wasn’t any online. You had to find the printed-out articles in the library, which could often prove very troublesome.
But she was helping me.
And sure, like that might happen in a Hollywood movie or in my dreams or something, but here it was, really happening!
So now I had two obscure professors secretly cluing me in on the math of QM, and I’m almost ready to get to my point in this essay.
Feynman did prevail. And he prevailed so thoroughly that today the issue is not much considered a problem in interpretation anymore (to my knowledge). But that is what I want to talk about. I sort of became privy to this old controversy that nobody worries about anymore, just because this beautiful woman was giving me Feynman articles from back when he was controversial. But it affected my own thinking way beyond QM, since I then learned, on my own, more of what he was talking about. It was indeed another one of these different ways of doing math. It wasn’t exactly the same as connecting with the spirit world, but it maybe was about why Hamilton might have once thought it was.
I’ll finally cut to the bottom line (at which point my interest in all of this will become apparent). Feynman proposed that QM should be understood in terms of energy rather than just in terms of forces. Yet the very word “mechanics”—the study of motion—had seemed heretofore to be in the provenance of Newton and his description of one thing bumping into other things to make them move. What was this about describing motion in terms of energy (which doesn’t even have a direction to it) instead of using forces (which do have a direction so as to point things in the proper heading)?
In chemistry, we already think in terms of energy. When we write an equation such as oxygen and hydrogen combine to make water, part of that very equation includes the amount of energy being transferred in the process, to make it “go.” But even chemists hadn’t thought of that as “mechanics.” (Energy doesn’t need a direction to make chemical reactions proceed because a reaction is about zillions of molecules mixing randomly but imparting energy as they do so).
But there were three senses in which the physicists had learned that it didn’t work to use Newton for very small particles. (It was like how Einstein had showed that Newton didn’t work for very fast particles, either, although relativity turned back into Newtonian mechanics as velocities were reduced). The three problems were 1) there is a limit to how accurately things can be measured; 2) there is a limit to how many things can be measured simultaneously; and 3) the values of some variables are not continuous. Yet Newton assumes the opposite in all three scenarios.
So a different approach was needed, to address very small particles, and that different approach was QM.
And now the issue was: Was this new approach also about solving problems in terms of energy rather than with Newtonian forces? Could considerations of energy handle these three obstacles?
Appealing to energy for a solution—instead of Newtonian forces—could indeed feel like appealing to the spirits.
What Feynman proposed, back in the 1940s, was that QM could be understood in terms of what was then called “least action,” which was a well-known phenomenon to the physicists, yet I had never heard of it. And back then, it had carried connotations, to many physicists, of being a strongly malodorous meta-magical heresy. It was frowned upon as the antithesis of Newtonian thinking, which in many ways of looking at it, it was.
So, me being me, I had plainly had to learn all about this stuff. But with Feynman providing my introduction, I had not necessarily started out with the notion that it was whacko. (But I’m not here claiming that Feynman would agree with everything I am about to say about it—although I would hope that he would—I’m merely saying that I found out about it through him).
It turned out, I discovered, that least action, under various names, had a long history (centuries) of dealing with motion in terms of energy. Least action itself was the precursor of Lagrangian mechanics—and Hamilton was basically an updating of Lagrange—so altogether these methods constituted that long history. And I, having just finished an intensive course in thermodynamics, whereupon I pretty much thoroughly bought into its way of seeing the world in terms of energy, became an eager learner of the Lagrangian approach. Now I was learning that even mechanics, the science of motion, could be explained in terms of energy?
Well, I sort of became an eager learner. I must admit that, at first, I did raise a few eyebrows as I was learning it, wondering if this could be really happening. It was so different from Newton.
I didn’t learn that Feynman was an extra big big-cheese until years later when I saw his book “Surely, You’re Joking, Mr. Feynman” at the bookstore, and I thought, “Hey, I know that guy.” It turned out that he had won the Nobel for applying this very thing to quantum electrodynamics. (His book consists of anecdotes, such as how, while working on the atomic bomb at Las Alamos, he cracked the top secret safe as a practical joke).
Here, in part one of this essay, I’m going to briefly sketch the scientific history of least action—what it is and why it was controversial for Feynman to advocate it—as well as look deeper into what it means to see mechanics in terms of energy rather than in terms of forces.
The name “least action” is not even used anymore—the word “least” implies using mathematical minimums when some applications don’t utilize minimums—so we can think of my subject here more accurately as being Lagrangian dynamics and how Lagrange developed his mechanics from a study of least action.
Then, in part two, I will compare how Lagrange and Newton entertained different “philosophies of motion,” as well as view them both in contrast with still other theories of how things change, all of the theories claiming to be “scientific.” I’ve been calling my website “Souls of Lagrange,” but I haven’t been writing any posts on that subject . . . , until now.
My overall project, to repeat yet again, is to show how we can start with two qualities of energy, that it animates particles to move randomly via their internal energy, and that it enables them to stick together into arrangements, and from that we can derive all the characteristics of the world including those traditionally attributed to factors residing outside the physical Universe (such as Platonic Forms or external physical Laws). And now, to further that objective, we can add how even mechanics itself can be understood in terms of energy. (We might think that mechanics would be an exception, but it is not).
Again, the difference is that to see the world just in terms of forces is to see things only being pushed around. But to see it in terms of energy is to likewise see motion but also to recognize that there is a role for the arrangement of a situation in making what can happen. And that role of the arrangement turns out to be well-illustrated in Lagrangian mechanics, as well.
There is also more to my story of the hauntingly beautiful woman in the science library, but I am not going to go into that here. She didn’t know QM, but she did know about finding papers in the science library, which is how I learned all of this stuff. (I repeat that, by the time I read these articles, decades after they were written, the issues had all been resolved in Feynman’s favor).
The usual story of Lagrange starts with Fermat (1662) and his description of how light travels by taking the path that requires the least time. But then the question becomes: How does light know, at the outset of its journey, which direction is correct?
Maupertuis (1744) generalized Fermat’s principle by saying it applied, not just to light, but to everything. He formulated mathematically an entity that he called “action,” which today, from looking at the formula, we can recognize as being basically kinetic energy, the energy of motion. But that was the start of seeing movement in terms of energy rather than forces. He was saying that light—and everything—takes the path that requires the least action (the least energy) since the Universe, he claimed, is basically lazy and always taking the laziest route. We might think of it as the path of least resistance (although it’s not really about resistance); it takes the path that requires the least energy (the “least action”) to get there.
And that is the principle of least action in a nutshell: The Universe is lazy and always takin the easiest path.
And then Lagrange (1750’s and 1780’s) added in potential energy, to go with the kinetic energy, and he explicitly used those terms. In Lagrange’s famous formulation, the “action”—the animus to move—was actually the integral over time of the difference of kinetic energy minus potential energy. Why that should be the case remained pretty much a mystery, but it worked. It generated an equation that could be used to predict the path of the light.
Finally, Hamilton did the same, but with KE + PE, rather than KE – PE, because in Hamilton’s formulation, it is about momenta and how that has properties equivalent to the total energy of the system. (And the total energy is less mysterious than the “action”).
Of course, it is more complicated than that because it is formulated in Lagrange’s calculus of variations, about minimums and maximums. (And today, it’s often re-envisioned in terms of Feynman’s path integral analysis, but I’m not going into that).
Yet we can see how all of that is different from Newton’s formulation wherein motion happens via a succession of things pushing on other things (or by physical laws pushing things along).
Indeed, Lagrangians and Hamiltonians were mostly rejected until the 20th Century because they were recognized as anathema to Newton’s approach. That was why Feynman met with so much resistance when he first started using them in quantum electrodynamics. (The idea of doing so, in the abstract, was first proposed by Dirac, but Feynman did it in practice with actual equations).
Yet, for all the controversy, it still was recognized (in the 1940’s and all through the centuries) that Lagrangian and Hamiltonian methods work very well—and not just for QM—and that often they are much easier to use than is Newton. One reason for the ease-of-use is that it frequently is simpler to calculate the energies compared to summing up all of what could be zillions of forces working on a moving body. Yet both ways yield the same answer. And a second reason is that Lagrangians work independently of the coordinate system being used, unlike Newton’s absolute system, and that can make possible solutions that would otherwise be too complicated to solve at all (which was part of how Feynman won out).
But the aspect of all of this which most caught my attention—and still does—is all of this sudden talk about the potential energy, the significance of which is often glossed over.
Suddenly, we are describing motion, not just with kinetic energy (the energy of motion), but with potential energy mixed into the account.
But where does the potential come from? And what is it doing there?
And of course I think of the potential as deriving from the arrangement of the situation. By our factoring in the potential to our description of the actual movement, we are thereby incorporating the role of arrangement into our accounts of how bodies move.
The first step to solving a problem with Lagrangian dynamics is to write an equation describing the potential energy. Then that is inserted into Lagrange’s equation, and the result is an equation that can be used to find specific answers regarding an object in motion in that situation.
So, what does it mean to first find the potential?
The classic example (from Newtonian mechanics) is a flowerpot sitting on a window ledge. It is because of the arrangement of the situation (being on a window ledge) that it has the potential to fall off and thereby have its potential energy converted to kinetic energy. (If it was already on the ground, then it wouldn’t have any PE). We can calculate the potential energy by measuring the height of the window ledge off the ground, thereby also setting the coordinate system with that measurement.
And likewise, we can write other equations about how other situations are fitting together, to find their PE. There can be a lot of variety and complication to what exactly constitutes a potential. But usually it entails that in some way the position of an individual object is described relative to the other objects that are present. The potential exists because the object is in some way residing within a greater setup.
So, to use the Lagrangian approach is to see an action as occurring as per how it is enabled to act given its setup. If it is falling off a window ledge, that is because it started out by being on the ledge, so its action is to move accordingly. The action arises from how a lot of things fit together around the object in question. And we can see how that is different from the Newtonian approach where the motion is strictly about one object impinging on another object, to push it along as if the overall setup is irrelevant and as if the only important factor is the immediately prior contact that is doing the pushing.
I actually believe that that last sentence is unfair to Newton because the first step in working a Newtonian problem is to diagram the situation at hand, and that enables us to visualize the forces that are in play given the arrangement of the setup. That also helps us to realize which equations are germane to a solution. So in practice, with Newton we are still mentally factoring in the role of the setup—we are picturing ourselves as dealing with a pot on a ledge—the difference is that with Lagrange the role of the setup is factored right into the actual math. With both approaches, we get the same answer.
There was a time when I had thought that it was difficult to translate the role of arrangement into mathematics. But Lagrange is showing us how to do that. (And it’s the same way with causality. We can begin by describing the potential—the arrangement—and show how that setup enables motion. The Enlightenment folks saw causality as they saw Newton—as just about chains of motion, as if the world is only kinetic energy and not a total energy including potential—but the arrangement must be factored in).
I have a lot more to say on this difference between describing motion in terms of energy or with forces, but I am saving that for part two of this essay.
At this juncture, I’ll just repeat that I view the two approaches as more compatible than we might think at first. We use the method that is easiest to use for a given application.
In one case, we are looking to the overarching horizon above us, to see our action as per how it is fitting in with others. And in the second case, we are looking down at where our feet are planted firmly on the ground. But there is no reason to have to choose one approach as automatically better than the other.
Arguably, it is beneficial to be able to use either method (although looking at each might result in jumping to different philosophical conclusions).
We shouldn’t let ourselves be told that the only choice is between a clockwork Universe or the spirits.
My own understanding of Lagrangian mechanics comes from Schaum’s Outline of Lagrangian Dynamics, by Dare Wells, which is full of practical problems.
IMAGE: unretouched photo by my son Andrew Rasmussen; the colors really are that rich in the desert because of all the dust in the air.