Have you heard the good news about quantum mechanics?

Something interesting happened! I swear!

Let’s start with what’s weird about quantum mechanics (QM). This is leaning heavily on my own personal take, and others may have issues with how I’ve built it. I have some issues with how I’ve built it! I mean how do you define weirdness? How can you rank-order weirdness? So anyway, here is a hierarchical ranking of quantum weirdness:

• Entanglement (superposition)

  • Wavefunction description (probability / Born rule)

    • Matter interference (complex numbers)

      • Wave-particle duality (basis vectors / representation)

        • Uncertainty principle (commutators)

First, the non-parentheticals. The next item down in the list “explains” the one above it, i.e. if we use wavefunctions, entanglement is pretty straightforward; if we have interference for matter systems then wavefunctions are a straightforward way to obtain it; given the wave-particle duality, interference becomes obvious; given the uncertainty principle, a wave-like nature is pretty much a given. Also as you get to the bottom of the list, the “weirdness” can be achieved even in classical systems. Light carrying energy-momentum like a particle can be classically shown in gravity — even in Newtonian gravity, but the actual lensing is off by a factor of 2 compared to General Relativity (GR). The uncertainty principle exists for purely classical waves. It’s basically not weird.

In parentheses, I added “mathematical” components of QM related to the weirdness. The top half falls loosely under the concept of “measurement” and the bottom half falls loosely under the concept of “complementarity”. At the top we have EPR and Schrodinger’s cat (weird!) while at the bottom we have Heisenberg’s microscope (extremely intuitive, but also wrong and completely unnecessary because we have a classical wave analog for the uncertainty principle which is not weird).

From this hierarchy, you get to my “three big questions”¹ about QM (some of which are more of a big comment):

• *muttering under my breath* I think the cat would know if it was alive or dead.

• *points to Born rule* I know what “axiom” means but it looks like you just made that up.

• *raises hand in class* Where does the √−1 come from?

The “traditional” answers to these are something like:

• Look, I know the cat suddenly becoming alive or dead when you open the box (Copenhagen) or you being in a different universe (many worlds) doesn’t make sense but that’s how the math works (shut up and calculate) and if you try to add stuff so someone could know it will be wrong (Bell).

• We have all this lovely theory and it would be a shame if we couldn’t use it.

• Right here → 𝒾 ℏ (∂/∂t) ψ(x, t) = H ψ(x, t). “You put that there.” Ok, here → 𝒾 ℏ = [x, p]. “Gah!”

Before going on to my main point, a discussion of Jacob Barandes’ recent papers (yes that’s what this is about), let me note that that last bullet is kind of showing the two “pictures” of quantum mechanics — the Schrodinger picture where states have time dependence and the Heisenberg picture where operators have time dependence.

“The Office” meme about the “same” picture, except the pictures have been replaced with the Schrodinger and Heisenberg pictures and the text changed to say “equivalent” picture.

If we have an operator A acting on a state |ψ⟩, then those two different “pictures” are represented mathematically as A |ψ(t)⟩ versus A(t) |ψ⟩. Moving the time parameter around has no effect on the observables, only the math — and sometimes makes some stuff easier like how polar versus cartesian coordinates can make some calculus problems easier. I’ve also put in a little bit of Dirac notation. If you’re wondering how you get to ψ(x, t), the wavefunction in the Schrodinger equation, from that notation it looks like this:

ψ(x, t) = ⟨x|ψ(t)⟩

If you noticed I called time a parameter earlier and if you also noticed this new “unequal” footing of x and t manifest in the Dirac notation in the Schrodinger picture, you are most of the way to “the problem of time” — a big conceptual incompatibility between General Relativity and Quantum Mechanics.

Now a wavefunction ψ(x, t) is also, unfortunately, sometimes used in quantum field theory (QFT) but it's, um, different:

ψ(x, t) = ⟨0|ϕ(x, t)|ψ⟩

Here ϕ is an operator at every point in space time — a quantum field. And we’re back to the Heisenberg picture, too, because that t goes with the operator. The Heisenberg picture is great for combining QM with Special Relativity. And then there are the grassmann fields ψ in the functional integral in the path integral for the case of Dirac spinor fields:

\(Z[\bar{\psi}, \psi] = \int D \bar{\psi} D \psi \; \exp \left(i S[\bar{\psi}, \psi] \right)\)

which are also different. Once you get to quantum field theory, the Fock states (states of a particular particle number) become more useful than wavefunctions because in quantum field theory you can create particles. The raising and lowering operators (“1st quantization”) that create or destroy particles are “inside” that ϕ(x, t) operator (“2nd quantization”) above. And that |ψ⟩ in QM is an N = 1 single particle Fock state …

… because the wavefunction most people know and love (read: have a fuzzy non-technical familiarity with and are otherwise indifferent to) is a single particle wavefunction which by a weird coincidence (well, 3 × 1 = 3) also can seem like a wave in 3D space (since 3 dof = 3 dim) instead of an abstract construct in configuration space (since 1 state = 1 particle). When N = 2 in 3D space 3 × 2 = 6 and we have (back in QM rather than QFT)

\(\psi \left( \textbf{x}_{1} , \textbf{x}_{2} , t\right) = \langle \textbf{x}_{1} , \textbf{x}_{2} | \psi (t) \rangle\)

where x₁ and x₂ are 3D vectors. Just try doing intuitive reasoning about waves in 7D = 6D (space) plus 1 (time)!

Confused?

Good!

Because I’m about to talk about Barandes’ papers and you really need that background of darkness and bafflement to appreciate them.

First, I should note that they are a reformulation of (1-D time) QM and appear AFAICT to be largely mathematically equivalent. I say largely only because I am a flawed human and do not know everything. I have not even the slightest hint of evidence there is anything wrong here.

Now the “detractors” (not really and definitely read Aaronson’s blog on the subject) have basically only come up with “what new do we learn?” My QFT² / HEP professor Stephen Ellis asked me after I did a class paper on reformulating gauge theory as fiber bundles³ “what is it good for?” — advice couched as a question that I have taken with me to this day.

Equivalent reformations have a few uses:

1. Creativity: Giving people a new perspective to think about

2. Practicality: Can sometimes be easier to solve problems with

3. Novelty: Can result in new stuff unto itself / be interesting in its own right

Barandes’ papers at least represent 1 & 3. So what happened?

In a few papers plus several hours of video interviews / talks Jacob Barendes has presented a reformulation of quantum mechanics in terms of stochastic processes. There is a long history of this general tack — the Schrodinger equation is a diffusion equation in imaginary time, after all. But the new part is not just abandoning the Markov property, but salting the Earth where it once lay.

I’m turning the arrows around compared to the standard notation in Barandes’ paper because this is my blog. Anyway, a Markov process can be thought of as a “thing” Γ that propagates a “state” from a time 0 to a time t, represented as Γ(0 → t) where

Γ(0 → t) = Γ(0 → δt) Γ(δt → 2 δt) Γ(2 δt → 3 δt) … Γ((n-1) δt → t)

But if Γ(k δt → (k+1) δt) is the same for all k we have

Γ(0 → t) = Γ(δt)ⁿ

i.e. the same memoryless Γ(δt) applied over and over. You can of course ask all kinds of questions of these definitions, which creates a hierarchy of different kinds of non-Markov processes (see e.g. Fig 1 here). But the key one for us, and one that seems to have not been really asked until the mid aughts, is what if:

Γ(0 → t) ≠ Γ(0 → t’) Γ(t’ → t)

What if it is impossible to take a process and a time t’ between 0 and t such that you can create a “propagator” Γ to move your state from 0 to some intermediate time t’ to some final time t? This has been called (for not even 20 years at this point⁴) an indivisible stochastic process.

Barandes works through the math (of course, standing on the shoulders of others as we all do) and finds that this kind of non-Markov process is basically equivalent to quantum mechanics. This has so many interesting ramifications.

One, as is shown in the papers, trying to “invent” times between 0 and t (“division events”) necessarily results in interference terms.

Γ(0 → t) − Γ(0 → t’) Γ(t’ → t) = interference ≠ 0

We can see the various other formulations of quantum mechanics, in attempting to overcome this “temporal nonlocality”, this indivisible dynamics where no times “exist” between 0 and t, have to pull in complex numbers and wave properties to fill in the intermediate times. As Barandes puts it:

One sees from this analysis that interference is a direct consequence of stochastic dynamics not generally being divisible. More precisely, interference is nothing more than a generic discrepancy between actual indivisible stochastic dynamics and hypothetically divisible stochastic dynamics

This means, for example, the “path integral” formulation is a mathematical fiction to help solve a problem. No times (well, “division events” but let me get to that in a minute) t’ in a sense “exist” between 0 and t, so no paths exist.

This is analogous to the fact that the “light takes every path” derivation of Snell’s law is purely a mathematical contrivance. Several derivations exist where there’s no such thing. The path integral formulation of QM was always just another approach to the physics the Schrodinger equation and Heisenberg’s matrix mechanics were trying to capture. But it doesn’t mean that’s what’s “really happening” (ontology) — and Barandes’ approach makes that far more manifest. It also says Schrodinger wavefunctions (defined for 0 < t’ < t) aren’t what is “really happening” either.

You might be asking why I’ve drunk the kool aid on this. I mean, I haven’t. I remain as skeptical of the ultimate utility of yet another reformulation as Scott Aaronson in the video linked above. But!

I have my own crackpot take that the “uncertainty” in QM is basically the same lack of knowledge of events outside our causal region we have in special relativity (possibly separated by a Rindler horizon) — the “elsewhere” interpretation of quantum mechanics. Or, modifying a controversial claim from Leonard Susskind, SR = QM. I’ve talked about that elsewhere (lol) on this blog (here, here) and even mentioned it in a sci fi short story.

A particle undergoing an acceleration (blue line that curves to the right) and the distance to the Rindler horizon (brown). The light rays forming the caustic representing the horizon are in red and an observer is shown with a yellow past and future light cone.

The thing is that if there’s a system you’re observing that passes into the “elsewhere” region of a space-time diagram (and for quantum phenomena, some back of the envelope calculations, it typically does⁵), you can’t in good conscience talk about what happens at the times t’ between when it left your past light cone (call it 0) and when it entered your future light cone at t. It should be an indivisible process, at least in terms of accessible information to the observer — and Barandes’ papers show that basically results in quantum mechanics. To paraphrase Wittgenstein:

Whereof one cannot speak (i.e. elsewhere), thereof one must be silent (i.e. live with fundamental quantum uncertainty)

Regardless of my own crackpottery, this is an interesting new take on a “hidden variable” theory. I mean, just one step up in abstraction from Markov processes are hidden Markov processes — this is like 5 or 6 steps up where we’re basically at the most general stochastic process possible and you essentially know nothing about the time between the start and the end.

Barandes presents his formulation purely as such. A new formulation that might help you think in a new way. I mean, it doesn’t appear to make solving any quantum mechanics homework problems any easier! He presents helpful intuition. What it needs is a connection to physics, which something like the elsewhere interpretation and its connection to Bousso’s bound could provide. But there could easily be better (read: not wrong) ways of doing it.

To get back to my three big questions above (and as I understand Barandes):

• The cat being in a superposition of alive + dead (suitably normalized) is a fiction based on hypothetical divisible dynamics — as well as scaling up such an absurd amount that it would likely be impossible for actual divisible dynamics (which are shown as a limit) to arise (via e.g. interaction with the environment).

• It’s a stochastic process so probability is just there the whole time and the Born rule is due to the mathematical trick Barandes did on purpose to show the equivalence to what Schrodinger did by accident.

• The √−1 is there because you didn’t know you were trying to understand an indivisible stochastic process as a divisible one, but also helps the math work out easier — like changing the t → 𝒾 t (Wick rotation) to make your path integrals easier.

However, and we’re back to my crackpot speculation now, if the process is indivisible because it’s elsewhere⁶ there’s some alternative intuition:

• Is the cat alive + dead? If you could create a big enough pocket of elsewhere to put a cat in (or a person), then this could be true to the best knowledge of the outside observer — but the interactions between not only the apparatus (the box, the air in it, the photons due to non-zero temperature) but within the cat itself (the cat’s toenail is not a critical life component) render this statistically impossible⁷ plus the cat would know.

• Why is it probabilistic? It’s elsewhere. It is causally disconnected from you, and perhaps even on the other side of an event horizon. There are places in spacetime⁸ where you just don’t have complete information so probability is all you got.

• Where does the √−1 come from? The √−1 is not necessary! You technically only need special relativity and a bit of humility. But if you like to pretend you can know about stuff you cannot possibly know about, then plain old QM is an option. (Think of QM as “male answer syndrome⁹” in response to the question of what happens between the times you observe a very small particle.)

Don’t take my crackpottery as detracting from Barandes’s papers. They’re genuinely interesting. There are some somewhat accessible videos on YouTube if you want to hear more details. Even the “click-baity” thumbnails are not so “click-baity” (I mean the vloggers have to get the views). Spent last weekend with several of these on in the background (at least at first … until I took out my notebook) because I am super cool.

1. There is another question: “Where does ℏ come from?” but we can just take that as the scale at which QM starts to matter for our purposes here. I think it might come from here. J/K. Here.

2. I technically did not have a “QFT professor”. There was no professor available to teach it the year my grad student class was to take it, so as students we self-organized a class where we all wrote up a lecture to give, and then chose someone at random from our class to give it. We had occasional guest lectures, and Prof. Ellis was the sponsor / occasional guest lecturer of this experiment. It was literally the hardest class I’ve ever had (imagine preparing 3 lectures a week to give to people who had also prepared a lecture on the subject while you did not actually know the material two days before). The first quarter had about a dozen or so of us grad students in it, the second quarter had three. I am the slacker of that trio given they are both professors now.

3. This is not my useless grad student paper, but shows the ideas (better than I did).

4. It should be noted the possibility of this non-equality was mentioned in a review from Aaronson in 2004 citing Gillespie in 1994 (gated), but more as a “this doesn’t happen” than “this defines a more general kind of stochastic process”. But I should also note here, this is what I meant about point #3 in my list: taking on this kind of indivisible stochastic process is novel unto itself even if it doesn’t lead to revolutionizing QM.

5. The “acceleration” at the 1-sigma edge of a diffusing Gaussian wavepacket is such that c²/a ~ 1481 fm or about 3.8 Compton wavelengths where c²/a is the distance to the horizon in Rindler coordinates.

6. Which incidentally would make sense why it is easy to add QM plus special relativity — there is still the aforementioned “problem of time” since the indivisible stochastic process approach also privileges time.

7. I am curious about what happens to the argument I made here.

8. The spacetime a nanosecond in the future more than a foot (30 cm) away from you is elsewhere.

9. Another piece of advice from another professor, Alan Cline.