"There is no wavefunction"

My write up, as I prepare for another talk with Harvard's Jacob Barandes

“There is no wave function…”

This claim by Jacob Barandes sounds outlandish, but allow me to justify it with a blend of intuition regarding physics and rigor regarding math.

We’ll dispel some quantum woo myths along the way.

Is the Wave Function a Rudimental Element of Reality?

Most people think of quantum mechanics as being about wave functions. What if Ψ isn’t fundamental? What if it’s just a mathematical convenience?

What happens to the associated devices like Hilbert spaces and state vectors? To Jacob, sure, they’re useful, but they’re not “real.”

(I understand that you should technically be dealing with the endomorphisms of Hilbert spaces rather than direct members of them, but this is something unnecessary to get into currently.)

There are five axioms of quantum mechanics. I’ve spelled them out here with both their math and their meaning in case you’re interested.

Jacob’s version is below. Or more specifically, my version of Jacob’s version.

EDIT (Jan 18, 2025): My previous version was less compact, and had redundancies, so now I've had approval from Jacob himself. This above is technically Jacob's version of my version of Jacob's version.

I would put the above in LaTeX here on Substack but currently it’s unwieldy to do so via their editing software.

The New Paradigm

Instead, Jacob suggests you start with a more general notion: indivisible stochastic processes. “Indivisible” means atomic (in the sense that you can’t break it down further), and “stochastic” is the mathematician’s word for random.

“Processes” means something that starts off initially in some way and gets transformed into something else. Extremely general.

The difference with these ISPs is that they’re unlike a coin flip or a Markov chain; the probabilities for these systems can’t be divided into smaller time steps.

Also, they don’t monitor your shady internet usage you disgusting pig.

The laws are indivisible. They don’t tell you what happens from moment to moment but over finite chunks of time.

The Correspondence to Quantum-ness

You’re likely thinking “Curt, this in some way sounds like QM, but also it sounds so vague. Where the heck is superposition? The interference? The complex numbers?”

What Jacob found in his 2023 paper here is that it turns out there’s a mathematical correspondence.

You can take any of these indivisible stochastic systems and represent them in a Hilbert space. The Hilbert space is a representation in this model, not a fundamentality!

(“Representations” in math mean something specific so I do have to be careful saying this.) The probabilities become mod-squares of complex amplitudes, just like in vanilla QM.

Superposition in the Hilbert space picture is actually just a classical probability distribution over configurations in the indivisible stochastic picture. No cats that are both alive and dead.

There’s just a system that will be in one state or another (with certain probabilities).

Now for measurement, it’s not some undefined collapse caused by large systems or conscious observers. All it is is an interaction that creates a new “division event.”

I specifically asked Jacob if a division event was just pushing the measurement problem back a step here, and he said that when a “measuring device” interacts with a system under observation, their combined evolution is still governed by indivisible stochastic laws.

What we perceive as “collapse” is the probabilistic evolution of the composite system into a configuration where the measuring device displays a definite outcome.

Importantly (and interestingly), this is NOT instantaneous. However, in practice, it does happen considerably quickly for macroscopic devices.

The probabilities for each outcome are determined by the underlying stochastic dynamics, and they precisely match the predictions of the Born rule, given by pₖ(t) = Tr(Pₖ 𝝆(t)), where pₖ(t) is the probability of outcome k at time t, Pₖ is the projection operator for that outcome, and 𝝆(t) is the density matrix.

Now, note that the conditioning times t₀ are precisely what Jacob calls “division events.” They're the times at which the indivisible evolution divides and can start anew.

By the way, those of you who don’t deal with math often may mistake the 𝝆 for a p. This is a dangerous but forgivable mistake.

It’s okay. I feel you.

Greta Thunberg Asks: “What About the Environment?”

Decoherence is often invoked to explain the emergence of classicality, but Jacob has a different interpretation of it.

When a system interacts with a large environment, the indivisible stochastic process describing their joint evolution leads to a rapid suppression of interference terms in the Hilbert space representation.

This is because the environment effectively carries away information about the system’s configuration, making it practically impossible to observe interference effects.

Mathematically, this is captured by the decay of off-diagonal elements in the density matrix when expressed in the configuration basis.

The density matrix is defined as 𝝆(t) = Θ(t) 𝝆(0) Θ†(t), where Θ(t) is the time-evolution operator. So, 𝝆(0) is the initial density matrix.

This suppression arises directly from the indivisible stochastic dynamics and doesn’t require any new postulates or interpretations.

Recall that in the standard formulation, the Born rule states that the probability of a measurement outcome is given by the squared magnitude of the corresponding amplitude. It’s simply postulated.

Here, though, it’s induced as a consequence of the correspondence between indivisible stochastic processes and their Hilbert space representations.

The probabilities in the stochastic picture, which are fundamental, are mapped to the mod-squares of amplitudes in the Hilbert space picture, so p(i, t | j, 0) = Γᵢⱼ(t) = |Θᵢⱼ(t)|².

Furthermore, in the stochastic picture, probabilities are real and non-negative, as they should be. However, the mapping to the Hilbert space actually introduces complex amplitudes.

This can be understood as a consequence of representing indivisible dynamics in a divisible formalism. The complex phases, which are responsible for interference effects, encode the MEMORY of the indivisible process in the divisible language of the Hilbert space.

This is the key that no one else saw.

These are not “Markovian dynamics.” Mathematicians love to confuse you with non-descriptive terminology, but the translation is that Markovian can be read as “memory-less.”

But they don’t quite match the usual “non-Markov” story either. The future depends on more than just the system’s present state, yet the theory doesn’t define those multi-time probabilities p(i,t | j_1,t_1; j_2,t_2; ...). That means the system has memory, but it’s a different breed of memory than you get in standard non-Markov processes.

\(\Gamma_{ij}(t) = \text{Tr}(\Theta^\dagger(t) P_i \Theta(t) P_j)\)

We get transition probabilities given by the equation above, where Θ(t) is the time-evolution operator, and Pᵢ, Pⱼ are projection operators.

Jacob’s approach is interesting to me because of its implications for quantum field theory and (specifically) the standard model, and thus for unifying the SM with GR (a so-called “Theory of Everything”).

I’ll be speaking with him in just a day or two, so if you have any questions, watch his interview here and let me know in the comments.

- Curt Jaimungal