Theories of Everything with Curt Jaimungal - Debunking the “All Possible Paths” Myth: What Feynman Really Showed

Starting point is 00:00:00 Ooh, nice! Air Canada has a worldwide sale. Wow, look at this deal to the Philippines. Nice, let's book it. But wait, Naples is also a steal. Saving seafood and sun? You want sun? There's a hot deal to Mexico. And even hotter to Yellowknife. Nice, but I thought you wanted tropical hot.

Starting point is 00:00:16 You're all over the map. Well, yeah, we've got over 180 destinations to choose from. Saving on every single destination. Nice. Hurry, book at aircanada.com or contact your travel agent. Conditions apply. Air Canada, nice travels. You've probably been seeing viral videos claiming that particles take all possible paths.

Starting point is 00:00:35 Everything is actually exploring all possible paths. When we don't let all possible paths, we can describe all possible paths. All possible paths. Over the space of all possible paths. Consider this all possible paths. the space of all possible paths. This is based on a significant misunderstanding of path integrals. Let's clarify what the path integral is about and why Feynman's tool isn't a literal map

Starting point is 00:00:54 of reality. Firstly, we need to stop saying that electrons go through both slits. That's not what quantum mechanics like textbook quantum mechanics says. For more on this, you can read this sub this substack article or watch this interview with Jacob Barandas on the Dirac von Neumann axioms. This whole particle goes through both slits as a hangover from misinterpreting wave functions in 3D space when they actually exist in something else called configuration space. This confuses a calculational trick with physical ontology.

Starting point is 00:01:25 It's popular because it seems many people want their quantum mechanics magical. But let's demystify some of this confusion with some rigor. People love to say that particles explore every possible path simultaneously, even going back in time or to the moon or what have you. Firstly, what does this word possible mean? Possible isn't a physics word. So do you mean to say every continuous path in R4? Do you mean every once differentiable path in R3? Like what is it? What's the domain here? Also, when someone says possible, it just makes you stop and think that, okay, well, if quantum mechanically you can tunnel and plenty else that I thought wasn't possible is actually possible, then how informative

Starting point is 00:02:10 is it to say that the particle takes all possible paths? What is the rigorous domain of possible? Furthermore, we're already including many of these impossible paths classically, like going backward in time and not being differentiable. So why can we not go through the blocked parts of the slit? Why is that not possible? There's a host of questions that occur to a student when this word, quote unquote, possible, is brought up.

Starting point is 00:02:34 This isn't modal logic. It's best to just drop this. All possible paths seems to be echoed due to doctrinal inheritance without thinking, just like the word equal footing. Time and space are relative and treated on equal footing. Time and space are supposed to enter on equal footing. We should think of space and time on the same footing. That past, present, and future seem to be on equal footing.

Starting point is 00:02:51 Space and time on equal footing. When you hear someone repeat a certain word or a lexical bundle that they don't repeat in any other place except at this one specific circumstance, then it's likely they've just inherited it from hearing other people say it. What is equal footing, for instance? Doesn't mean that they're the same? Well, no, because there's the opposite sign on the time part. So are they on equal footing? Have you seen a mathematical definition of equal footing? We're supposed to be rigorous. Does it mean that you can just add and subtract

Starting point is 00:03:21 spatial and temporal degrees of freedom? Well, derivative operators can be added even if the order of the operators is mixed. So I talk about that here with Tim Maudlin. Anyhow, path integrals are a computational tool or shortcut for combining unitary evolution and the Born rule for a specific measurement basis. The cool diffraction grading experiment doesn't actually prove that particles take all possible paths or all paths or what have you. It demonstrates wave optics, which can be calculated with path integrals, but it doesn't necessitate the all paths ontology. Wave phenomena explain it just fine. Now, dealing with classical electromagnetic waves propagating in our familiar 3D space is a fundamentally different beast than the abstract wave function of an electron, which

Starting point is 00:04:10 lives in configuration space. So let's take a look at the origin story. Path integrals weren't invented by Feynman out of thin air to describe particles taking these scenic routes. Paul Dirac introduced the core idea behind them in 1932. Dirac's goal? To understand the quantum role of the Lagrangian, which felt sidelined by the Hamiltonian focused formulations of early quantum mechanics by Schrödinger and Heisenberg, for instance. Dirac figured out how to express the transition amplitude by dividing time into these tiny intervals

Starting point is 00:04:42 and inserting complete sets of states. Feynman, years later, as a PhD student, turned Dirac's formal insight into this powerful computational recipe, the path integral that we know. But notice the motivation. It was about a mathematical representation and calculation, not primarily painting a literal picture of particle trajectories. Actually, Feynman himself in his 1948 review stated that his path integral couldn't, at least at the time, calculate anything that standard methods couldn't. Now let's get back to the electron goes through both slits. This visual relies on thinking that the electron's wave function is like a ripple in 3D space or in the 3D space of the experiment.

Starting point is 00:05:24 Now that works okay if you have one particle but if you add a second one or more then you don't have a wave function you have a six dimensional configuration space over here and then you can obviously extend this to n particles of three n dimensional space. This psi here doesn't assign a value technically known as an amplitude in this case at each point in our 3D space instead it assign assign a value, technically known as an amplitude in this case, at each point in our 3D space. Instead, it assigns a value to each possible arrangement of all n particles.

Starting point is 00:05:52 It's fundamentally not a wave in physical space. So this intuitive picture of waves splitting and going through two slits simultaneously is already a misleading simplification from the n equals 1 case, incorrectly extrapolated. The all paths story inherits this spatial misconception. The paths being summed over, so here on screen, are trajectories in configuration space, not simply paths in a 3D space that you can easily draw. The standard quantum mechanics, so as I mentioned the Dirac von Neumann axioms, which you can see on screen here linked to the substack, doesn't actually describe what

Starting point is 00:06:30 happens between measurements. If you want to know what the particle is doing when it's not being measured, then you need a theory that describes what's going on in between, like Bohm for instance, or many worlds, or even indivisible stochastic processes by Jacob Barandas. I'll put a link to both my conversation with Sean Carroll on screen and my conversation with Jacob Barandas in the description. Now let's talk about the math itself. To make these path integrals mathematically well-defined and convergent, you have to employ some tricks. And these tricks are not for kids.

Starting point is 00:07:00 So a common one is giving time a small imaginary component. Or you can perform a full wick rotation. This essentially is doing a calculation in Euclidean space-time, imaginary time, and then rotating back at the end. But ask yourself, look, if the path integral literally depicts reality, then does reality fundamentally occur in complex or imaginary time? These are mathematical regularizations needed to make the calculational tool work. The formalism isn't a direct ontological description. Path integrals are math and not a literal movie of nature.

Starting point is 00:07:38 Plus, there are other approaches to quantizing like the one you'll first learn called canonical quantization. There's an even more advanced one based on symplectic manifold or symplectic geometry called geometric quantization and here there isn't an obvious particle takes all paths interpretation and so if we're just going by whatever the math gets you that's correct well other quantization methods are shown to be equivalent in most cases. So how do you know which math determines your ontological commitments? Okay, this brings us back to the Veritasium video and the Looking Glass Universe response.

Starting point is 00:08:14 The experiment with the laser, the mirror, and diffraction grading is cool. However, does it prove that particles take all possible paths? No. As Mathuna from Looking Glass Universe eventually shows, the phenomenon is perfectly explained by standard wave optics, so Huygens' principle, and diffraction. Light waves do spread out, even laser beams aren't localized. Thus, they do hit the whole grating. And the grating does show diffraction patterns according to well-understood physics. Now, by the way, even defining what light is isn't trivial.

Starting point is 00:08:48 I'll put a link on screen. And then also same with defining what energy is. I'll put a link on screen and in the description. Now, the path integral can calculate this experiment, this outcome, because wave equations can often be derived from action principles. But, the wave explanation is sufficient, and arguably it's more direct. Claiming that particles must be taking these odd paths relies on the premise that the wave – so the laser beam – is perfectly localized and doesn't hit parts of the mirror away

Starting point is 00:09:17 from the primary reflection point. But as Mathuna from the Looking Glass video showed, it demonstrably does. In other words, this experiment confirms wave behavior, which can be modeled mathematically via path integrals. But it doesn't prove that all paths chronicle is physically real. As an aside, I've spoken to Mathuna a few years ago of the Looking Glass universe on this podcast here on quantum immortality and many worlds. Mathuna makes fantastic approachable and informative videos.

Starting point is 00:09:48 I recommend you check her out. Links on screen and in the description for everything. So when you see path integrals being used in physics, like advanced QFT topics, it's primarily because path integrals handle field, theoretic expansions, and perturbation series in a more streamlined manner, especially when dealing with sums over topologies, which string theorists love. See my video on string theory here dealing with the graduate level math. If you don't know what the word perturbation series is here, you can just think of it as a certain approximation that you can make sometimes that allows

Starting point is 00:10:20 calculations to be both computationally and conceptually easier. Now none of this means that there's a consensus among physicists to accept the path integral as the ontologically correct portrayal of reality. It often is used simply because it's more convenient for a litany of tools like gauge fixing and Feynman diagrams and something called large-scale expansions. Gauge fixing in case you're interested, just means selecting convenient representative field configurations, which amounts to removing overcounting so that you can properly count.

Starting point is 00:10:54 And Feynman diagrams are those diagrammatic expansions which depict these particle interactions, those ones that you've seen before, each of which is actually a visual shorthand for an integral. So, some fancy math. To summarize, none of the Veritasium video proves that a particle actually takes all these paths simultaneously in reality. The video is filled with a mixture of interpretation-dependent claims and some statements that are just incorrect. Also on this channel and on my sub stack, I go into detail about these unexamined slogans that even brilliantly

Starting point is 00:11:27 informed people sometimes perpetuate like Decoherence solves the measurement problem. It doesn't or that Heisenberg uncertainty is caused by the physical act of measurement Disturbing the system or that distances below the Planck length are meaningless. They state these without rigorous justification or acknowledging the underlying philosophical assumptions. So none of these are necessarily true. So for instance, the Heisenberg uncertainty relation is right there in the algebra. And the last one about going smaller than the Planck length assumes operationalism, for instance. Now scrutinizing these requires asking for axioms and justification. Another falsehood is that the idea of erasing one bit must dissipate this amount of heat, also known as Landauer's Principle.

Starting point is 00:12:12 And this underpins the it from bit, or the slogan that information is physical. But John Norton has longstanding objections to its universal validity. Soon I'll be doing a write-up and a video about how John Bell's theorems depend explicitly on specific causation theories. So locality, counterfactual definiteness, and statistical independence aren't the only assumption. PS, notice that realism isn't actually there despite it being there in PopSci accounts. Bell in 1964 relied on an interventionalist causation account, but that has some problems for microphysics. And then Bell in 1964 relied on an interventionist causation account, but that has some problems for microphysics.

Starting point is 00:12:47 And then Bell in 1975 and 1990 relied on something called Reichenbach's principle or Reichenbachian factorization or related criteria, which Bell himself doubted and faced objections, like quantum interactions aren't conditional upon beables. It's a John Bell term. When you ignore the dependence on these debatable metaphysical causation accounts, it actually neuters the theorem's purported power to prove non-locality is inherent to quantum mechanics. Actually, I think in Bell's 1990 paper, he modified the assumptions again, trying to get away from that strict rock and bike, rock and bike, rock, sorry, reichenbach factorization.

Starting point is 00:13:26 But this created other problems by requiring beables to be on a complete spatial slice and maybe that conflicts with statistical independence. This is detailed by Joanna Luke. I'll also be making a video on how quantum expectation values aren't averages

Starting point is 00:13:42 over stuff happening. Like, the spoiler, by the way, is that this over here, this expectation value is defined as a statistical average of measurement outcomes weighted by the born probabilities. But when you mistake this for the average value of an observable between measurements when nothing is being measured, that's a category error. And this is pervasive in discussions of the classical limits like Ehrenfest's theorem and semi-classical gravity. You can subscribe for an upcoming podcast with Eric Kuriel on this topic. And if you're just generally interested in the interpretations of

Starting point is 00:14:16 quantum mechanics and theories of everything, unifying theories in physics. Now despite these comments, Veritasium videos are a useful go-to resource for an initial overview. Just know that it's an initial overview and don't wholesale buy what someone is saying at the introductory level. In fact, you shouldn't wholesale buy what anyone says, including myself.

Theories of Everything with Curt Jaimungal - Debunking the “All Possible Paths” Myth: What Feynman Really Showed

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