Reverse-Engineering Quantum Mechanics, III.

An Option for the ‘Ground’ State

In previous posts we have encountered the three main equivalent re-formulations of classical (i.e., Newtonian) mechanics—Lagrangian, Hamiltonian, and Hamilton-Jacobi—as well as their quantum mechanical counterparts, Feynman path integrals, Heisenberg’s operator mechanics, and Schrödinger’s wave mechanics, respectively. (A few more are possible, cf. Styer et al. 2002, but these are the main ones of relevance here). We are now ready to think about what the final step ‘down’ might be to the quantum side of the bottom level of the ladder of abstraction, the quantum equivalent of the ‘ground’ from which classical mechanics arose.

But before we go there, we need to be aware of the many ways in which the mathematics of quantum mechanics (or quantum theory) can be interpreted. You might think that this should be fairly straightforward. After all, quantum theory has made some of the most spectacularly accurate predictions in the history of science. For example, the atomic energy levels in the hydrogen spectrum are accurate to $\sim1$ part in $10^{12}$ (Maisenbacher et al. 2026). Unfortunately, it is not at all straightforward.

As I mentioned in the first post in this series, there are a number of competing what are often called “interpretations” of quantum mechanics (e.g., Jammer 1974; Jaeger 2009). These days the broad topic of thinking about this also goes by the somewhat more neutral term foundations of quantum mechanics (e.g., Auletta 2019; Norsen 2017), possibly because the term “interpretation” carries too much of a post-modern ring to it, and connotes the idea that you can pretty much think what you like about what quantum mechanics is saying (and some do, cf. Bricmont 2017). A bibliography from 2004 lists over 10,300 sources on the topic of quantum foundations, which the author states is “incomplete” (Cabello 2000), so this is a live-wire topic indeed!

The 100th anniversary of the development of quantum mechanics recently ticked over, so naturally people were thinking about how it has fared over the course of that century. Sean Carroll (2019a, 2025), no doubt channelling Richard Feynman’s famous remark from 1965 that “no one understands quantum mechanics”, claims that physicists still don’t understand it, now even 60 years later. A recent survey in Nature of working physicists asked about their views of it (Gibney 2025), and the findings were stark, reflecting Steven Weinberg’s (2017) lament from the previous post: there is not only no consensus but, in the words of the Nature article, “wild disagreement”. Therefore, we can expect expert views to be contradictory, but that need not bother us if we are staying true to the Feynman approach of ignoring what the experts say and just working it out for ourselves.

The interpretation-in-use that was most commonly given by the survey respondents was the Copenhagen Interpretation (so named because it was basically developed and championed by Niels Bohr, who was based in Copenhagen, and by many of his students or associates, such as Heisenberg and Pauli). But, and here is the interesting data point, while 36% listed it as their preferred interpretation, the degree of confidence in it among them as the best interpretation was somewhat less. It is the Copenhagen Interpretation that was dominant during the history of quantum physics, and which represents the ‘received view’ that physics students are exposed to and expected to learn to use when they study quantum mechanics. But there is, as we will see, something not quite right about the Copenhagen interpretation.

The ‘Received View’

As I noted on the ‘landing page’ for this series, I found when I read Carroll’s (2019b) book in full that the term “reverse-engineering” was mentioned three times (pp. 268, 282 & 284). The fact that I bought it after the title of this series occurred to me suggests that the general idea is hanging around in the æther. Over the years I’ve read several of his books, as well as watched a couple of his Great Courses courses. In fact, we might even have exchanged a couple of emails way back in the very early 1990s when we were both graduate students. So, perhaps the resonance-at-a-distance is not so spooky. I’ve always been drawn to contrarian views in physics, probably installed by reading Feynman as an undergrad (Feynman 1986, 1989, 1965), which triggered my almost-allergic visceral reaction to the “everybody knows Einstein’s unified field theory doesn’t work” conventional “wisdom”. That led to my doing a PhD on perhaps the most unfashionable classical field theory ever, or at least in the 20th Century. (Fortunately, I had a similarly contrarian supervisor, who as it happens was also happy to supervise research on quantum foundations, as one or two of my cohort fellows did.) And (ahem!), by the way, showing that that conventional view was in fact misplaced and based on an analysis that was actually inconclusive as a test of the viability of the theory (Voros 1995, 2002). But that’s another story.

Well, here’s another spooky “resonance at a distance”. As I read Carroll’s book, I noted a reference in the Further Reading to a book about the historical development of quantum mechanics by Adam Becker (2018). Between finishing the previous post and working on this one, I read the Becker book right through (as well as listening to an audio version of it at times). And, what do I find on page 270? A reference to none other than Wile E. Coyote! So this is definitely getting spooky. Of course, the broadly similar cultural background we share most probably primed this, so it is much more likely to be due to the “hidden variables” of watching Warner Bros cartoons when growing up, rather than any truly non-local spooky action at a distance. And the idea of “both feet planted firmly in mid-air” is a trope from the futurist Donald Michael (1985), which naturally lends itself to cartoon physics. Of course, reading the story of how people who were interested in quantum foundations were sidelined, gaslit, ridiculed, persecuted or just plain ignored for daring to question the received view was slightly triggering, given the three years in the wilderness I went through before my doctoral thesis was finally accepted. I can definitely attest to feeling a certain kinship with them.¹

So, reading the story of how the Copenhagen Interpretation (i.e., “shut up and calculate!”) came to dominate 20th Century quantum physics and how its proponents shut down or sidelined any and all discussion of possible alternatives, usually with scorn or ridicule, was both triggering and cathartic.

Becker’s book basically describes the many, many ways that the ‘received view’ of the Copenhagen Interpretation is not only vague, but so full of inconsistencies and sloppy thinking as to be largely incoherent. Quantum physics has a reputation for being weird anyway, but in the words of David Albert (who himself was essentially punished by his university physics department for daring to work on quantum foundations as a graduate student): “there’s a huge difference between being weird and being incoherent or unintelligible. … The Copenhagen Interpretation is not weird, it’s gibberish, it’s unintelligible!” (Albert quoted in Becker 2018, p.283). All those years of feeling that there was somehow something wrong at the heart of the conventional view of quantum physics now click into place. But the existence of the book also means that I can point you towards it to get a sense of why the topic of quantum foundations is worth pursuing despite the widespread (mistaken) view that everything has been settled for nearly a century. It most certainly has not. A shorter version of the same basic history was also published recently in Nature (Baggott 2024).

One of the main criticisms of the Copenhagen view is the need for what is sometimes called the “Heisenberg cut” (Carroll 2019b, p.35)—a (usually un- or poorly-defined) separation between the (macroscopic classical) observer and the (microscopic quantum) observed system, necessary to cause the (required) “collapse” of the wave function. John Bell famously called this the “shifty split” (e.g., Bell 2004a). It is here that a lot of the metaphysical weirdness (or rather, nonsense, cf. Bricmont 2017) comes into play in the conventional Copenhagen view, so that some physicists, in all seriousness, make nonsensical claims, such as “we now know that the moon is demonstrably not there when nobody looks” (Mermin 1981, p.397; 1985).²

If the Copenhagen Interpretation is incoherent, as so many researchers on quantum foundations suggest (and my own experience of it concurs), then what other interpretation(s) do we have as viable alternatives? Becker goes into several of these in a pretty fair and even-handed way. But I am following the Feynman-Seneca path of starting out from scratch to see where this leads without favour, and branching off wherever it seems the right thing to do. Hopefully the first two posts have made this clear. It is how we crossed over from classical mechanics to the quantum side and came back down to the path integral formulation, which has many advantages for attempting to visualise what is going on.

That leaves the question of what the final step down might be to the ‘ground state’ after all the ‘excited’ states that have been going on at higher levels of abstraction (sorry, in the end I couldn’t resist that terrible physics pun). And the answer is pretty obvious (at least, to begin with).

Particles-and-waves: The de Broglie-Bohm option

The simplest possible conceptual step down to the Newtonian level on the ladder of abstraction is to go right back to particles again, albeit now being influenced by, or at least associated with, wave motions. This was essentially de Broglie’s original 1924 thesis proposal which Schrödinger took up and used to develop his equation and which de Broglie also developed a bit more into a theory of ‘pilot waves’ in 1927. In fact, there is a sense in which this interpretation may actually have been the first major one, even before the Copenhagen view had been formulated and allegedly “won” the day. This latter claim seems to be a bit of rewriting of history by the Copenhagen crew. For example, at the famed Fifth Solvay Conference of 1927 (Bacciagaluppi and Valentini 2009, p.xviii):

De Broglie’s pilot-wave theory was the subject of extensive and varied discussions. This is rather startling in view of the claim – in Max Jammer’s classic historical study The Philosophy of Quantum Mechanics – that de Broglie’s theory ‘was hardly discussed at all’ and that ‘the only serious reaction came from Pauli’ (Jammer 1974, pp.110–11). Jammer’s view is typical even today. But in the published proceedings, at the end of de Broglie’s report there are 9 pages of discussion devoted to de Broglie’s theory; and of the 42 pages of general discussion, 15 contain discussion of de Broglie’s theory, with serious reactions and comments coming not only from Pauli but also from Born, Brillouin, Einstein, Kramers, Lorentz, Schrödinger and others. Even the well-known exchange between Pauli and de Broglie has been widely misunderstood.

However, the subsequent silencing of this alternative was not merely a matter of scientific consensus emerging naturally, it rather seems more like it was actively suppressed, as were later interpretations, such as Hugh Everett’s (1957, 1973) “relative state” interpretation, which is nowadays better known as “many-worlds” and is the favoured interpretation of Sean Carroll (2019b). While Becker’s book provides a highly-readable (I did it in essentially a ‘gulp’) popular history of this suppression, the somewhat damning contention is corroborated by more rigorous scholarly and technical accounts. John Cushing (1994) argues that the dominance of the Copenhagen interpretation was not inevitable but rather the result of specific historical contingencies and institutional power dynamics. Mara Beller (1999) gives a rhetorical analysis showing how the “Copenhagen interpretation” was constructed through strategic argumentation that often obscured genuine disagreements among its proponents, so as to provide a united front against any and all detractors. And Olival Freire (2015) documents the specific, often punitive, professional consequences faced by physicists like David Bohm, Hugh Everett, David Albert and John Clauser (see below), who dared to challenge the orthodoxy. These works collectively reveal that the “received view” was maintained not just by its explanatory power, but by an active, sometimes hostile, discouragement of alternatives; a hegemony that marginalised the pilot-wave approach, and any other, for decades and perhaps still influences the field today, as the results of the survey mentioned above conceivably show.

But when David Bohm essentially re-discovered and extended the pilot-wave approach in 1952, de Broglie returned to it and spent the rest of his life working on it. His Paris Institute was one of the few places that Copenhagen dissent was possible (Freire 2015, p.42). In the literature one therefore finds reference to de Broglie-Bohm theory, pilot-wave theory, sometimes just ‘Bohmian mechanics’, and occasionally “the causal interpretation”; it had something of a resurgence in the latter part of the 20th Century (Dewdney 2023).

So, in terms of the ladder of abstraction, this is (classical) Newtonian particle mechanics turned into (quantum) de Broglie-Bohm pilot-wave particle-and-wave mechanics. The predictions of this theory are exactly the same as for conventional “shut-up-and-calculate” quantum mechanics, the Copenhagen interpretation. There are some interesting dynamics that come out of the mathematics, such as non-local interactions—essentially instantaneous correlations between distant particles—but this is consistent with the Copenhagen view, which tends to hide this “spooky action-at-a-distance” (Einstein’s famous expression) in the form of the “collapse of the wave function”. And it is also consistent with experimental results. This will be discussed in the next post on Bell’s Theorem, but suffice it so say that the 2022 Nobel Prize in Physics was awarded to experimentalists, including Clauser, who showed that quantum mechanics really does do “spooky action-at-a-distance”, so the fact that the de Broglie-Bohm theory implies this (as well as showing an actual mechanism for it) is a feature, not a bug.

For decades, de Broglie-Bohm theory has been the “obvious” candidate for the ground state of quantum reality, by restoring the intuitive picture of particles moving in 3D space, resolving the measurement problem by eliminating the need for wave function “collapse”, and providing a clear mechanism for the strange correlations found in experimental results. It was the interpretation that John Stewart Bell, perhaps the most influential voice in quantum foundations, following the 1964 theorem that bears his name, championed for much of his career. For Bell, this was the theory that finally allowed us to take quantum mechanics seriously as a description of a real world, rather than just as an unthinking calculator for experimental probabilities. He witheringly criticised the Copenhagen view for what he considered its incoherent vagueness and ambiguity: “I think that conventional formulations of quantum theory […] are unprofessionally vague and ambiguous. Professional theoretical physicists ought to be able to do better. Bohm has shown us a way” (Bell 2004b, p.173).

The de Broglie-Bohm Theory in brief

So what exactly is de Broglie-Bohm theory? The basic idea is very simple: particles have definite positions at all times, and their motion is “guided” by the wave function (hence “pilot wave”). That’s it. The mathematics makes this precise, and the structure is familiar to anyone who knows the classical mechanics we have been discussing in these posts. I’ll use a capital $\Psi$ for the wave function in what follows, which is the notation Bohm (1952a, 1952b) used.

Start with the standard Schrödinger equation for the wave function, $\Psi(\mathbf{x}, t)$:
$$
i\hbar \frac{\partial \Psi}{\partial t} = \left( -\frac{\hbar^2}{2m} \nabla^2 + V \right) \Psi \; .
$$
Now, write $\Psi$ in “polar” form, separating it into a real amplitude $R$ and a real phase $S$:
$$
\Psi = R \, e^{iS/\hbar} \;.
$$
Substitute this into the Schrödinger equation and separate the “real” and “imaginary” (i.e., the terms multiplied by the unit imaginary $i$) parts, yielding two coupled real equations.

The first is a continuity equation, which ensures conservation of the wave “intensity”: $\rho=R^2=|\Psi|^2$. In conventional quantum mechanics this is a separate postulate introduced by Max Born, now called the “Born Rule”, and interpreted as the “conservation of probability”:
$$
\frac{\partial\rho}{\partial t} + \sum_{k=1}^{N} \nabla_{k}\cdot (\rho\mathbf{v}_k) = 0
$$
where the sum on $k$ is over all $N$ particles. It defines a velocity field for the particles:
$$
\mathbf{v}_k = \frac{1}{m_{k}} \nabla_k S \; .
$$
This is the guidance equation: the $k$th particle’s velocity is determined by the gradient of the phase $S$ at its location. This is exactly how particle velocities are defined in classical mechanics, albeit usually in terms of momenta, for the case of Hamiltonian dynamics (recall that particle trajectories are orthogonal to the lines of constant $S$):
$$
\mathbf{p}_k = m_k\mathbf{v}_k = \nabla_k S \; .
$$
The second equation looks almost exactly like the classical Hamilton-Jacobi equation mentioned (but not written down) in the first post. It describes how the phase $S$ evolves over time:
$$
\frac{\partial S}{\partial t} + \frac{(\nabla S)^2}{2m} + V + V_Q = 0 \;,
$$
here with an “extra” term at the end: $V_Q$. This term is what is called the quantum potential, defined by
$$
V_Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R} \; ,
$$
which depends on the amplitude $R$.

This is the key part of the theory. In classical mechanics, the Hamilton-Jacobi equation has no $V_Q$ term because there is no “amplitude” associated with the particles (they are idealised zero-width trajectories, governed solely by the action function $S$), so particles are guided purely by the classical potential $V$. In de Broglie-Bohm theory, however, particles are guided by $V$ plus this new quantum potential $V_Q$. In the classical limit as $\hbar\rightarrow0$, it vanishes entirely, so we recover the standard Hamilton-Jacobi equation of classical mechanics again.

The quantum potential is where all the “quantum magic” happens that physicists find so strange (cf. Feynman 1967, p.128, who called this behaviour “screwy”). Because it depends on the amplitude $R$ (which is derived from the wave function $\Psi$), it encodes the interference patterns and the non-local correlations of the system. If $R$ has a complicated structure (like the interference fringes in a double-slit experiment), then $V_Q$ will have a correspondingly complicated structure that will steer the particles away from zones of destructive interference (the dark bands) and toward zones of constructive interference (the bright ones).

Interestingly, the Born Rule ($P = |\Psi|^2$) is not a separate postulate here. Instead, it emerges rather naturally: if the particles start distributed according to $R^2$, the continuity equation guarantees they stay that way. This is the “quantum equilibrium” hypothesis, analogous to thermal equilibrium in classical statistical mechanics.

So, in summary: de Broglie-Bohm theory postulates real particles moving in 3D space, guided by a wave that evolves via the Schrödinger equation. The wave function acts as a “pilot,” determining the particles’ velocities through the gradient of its phase $S$ and exerting a force on the particles that also includes the quantum potential $V_Q$, which is the mathematical signature of the wave’s influence on the particles’ motions. The theory is deterministic, non-local (because $V_Q$ depends on the positions of all particles instantaneously), reproduces all standard quantum predictions, and, unlike Copenhagen, requires no wave function “collapse” and no undefined (“shifty split”) “cut” between the observer and the system.

Have we landed? Maybe

Let’s take another look at the dual ladder of abstraction, now with the last cell at bottom right filled in. I’ll consider this a first approximation (i.e., provisional for now).

Table 1: First Approximation of Full ‘Dual Ladder’ of Abstraction

Classical Mechanics	Quantum Mechanics
Hamilton-Jacobi Equation Field $S(\vec{q}, t)$ on configuration space.	Schrödinger (Wave) Equation Field $\Psi(\vec{q}, t)\sim e^{iS/\hbar}$ on configuration space.
Hamiltonian Formalism Phase space $(q, p)$. Poisson brackets $\{A, B\}$.	Heisenberg (Operator Mechanics) Operators $\hat{A}(t)$ on Hilbert space. Commutators $\frac{1}{i\hbar}[\hat{A}, \hat{B}]$.
Lagrangian Formalism Action $S = \int L \, dt$. Principle: $\delta S = 0$ selects a single path.	Feynman (Path Integral) Sum over histories $\int_{\cal{P}} e^{iS/\hbar} \mathcal{D}[q]$. Principle: $\sum e^{iS/\hbar}$ sums all paths. Classical path emerges as stationary phase limit.
Newtonian Mechanics Particles in 3D space. Forces: $F = ma$.	de Broglie-Bohm (Pilot Wave) Particles guided by $\Psi$ in 3D space. Quantum potential $V_Q$ .

There is certainly a strong resonance between the two Level 1 pictures of particle mechanics. One wonders what might have happened to quantum theory if de Broglie’s view had not been rejected, sidelined and ignored. We might have foregone and avoided a whole century of quantum nonsense (Bricmont 2017).

So, de Broglie-Bohm theory stands as the most viable (and obvious) “Level 1” landing we have found (at least, so far; although you can probably guess where this might be going, right?). But before we accept it as a potentially final answer, we need to consider the empirical constraints that define what any quantum theory must obey. The next post will therefore examine Bell’s famous Theorem, the implications it has for quantum physics, the classes of theories these implications allow and exclude, and the subsequent experimental tests that have ultimately forced us to have to accept that nature is, actually (yes, really), fundamentally “spooky action-at-a-distance” non-local at least at the quantum level…

Next: Part IV – For Whom the Bell (Theorem) Tolls

Notes

1. But these are just a few data points on what seems to be an unfortunately pervasive aspect of science: there are ‘received views’ which are so entrenched that even raising a legitimate question as to how strongly we can rely on them is greeted by some people as almost literally heretical. Science is done by scientists, who are people, and so subject to all of the wonderful attributes and maddening foibles that people may have. The heat that can be generated by the simple desire to shed some more light on a particular topic is astounding. And it matters a lot whether you are a ‘known name’, too. I can remember receiving a desk rejection letter from Physical Review for a paper on the electrodynamics of Einstein’s unified field theory while still a graduate student; ‘desk rejection’ means that the editorial staff has nixed it before even sending it out for review (ouch!). The letter stated that it was (and I remember this quite vividly) “theoretical speculation whose inclusion in the journal cannot be justified”. Because “everybody knows” Einstein’s UFT doesn’t work, right? On the same day as this letter arrived in my mail slot at the university (grad students could get mail delivered to general pigeon holes sorted by first letter of surname), a paper was published in Phys Rev using the same mathematics as EUFT, but instead it was examining the question of whether there might be as-yet unknown fields implied by the mathematics that might have novel couplings to matter. Not even electromagnetism, mind you. But that was not theoretical speculation whose inclusion could not be justified, no. It turns out the author was a well-known North American theoretician at a “name” university who had been publishing along roughly similar lines for a couple of decades by then. Therefore, who was proposing the idea seemed far more important than the idea itself. That was the beginning of my disillusionment as a grad student. It is fairly common, I’m told, to have one’s naïve view, that simply doing good work is enough, rudely popped in some way, but that was only the precursor to what followed…↩︎

2. This is a reference to a story told by Abraham Pais in his scientific biography of Einstein, Subtle is the Lord (Pais 1982, p.5), which I obviously read in great detail while working on my doctorate. During one of their walks to Einstein’s home from the Institute for Advanced Study in Princeton, Pais relates, “Einstein […] suddenly stopped, turned to me, and asked me if I really believed that the moon exists only if I look at it.”↩︎

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