Thread by @TOEwithCurt

“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. (1/13)

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.) (2/13)

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: . (3/13)curtjaimungal.substack.com/p/the-interpre…

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. (4/13)

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. (5/13)

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 () is that it turns out there's a mathematical correspondence. (6/13)arxiv.org/pdf/2302.10778

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). (7/13)

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 (youtu.be/7oWip00iXbo), 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. (8/13)

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

By the way, those of you who don't deal with math often may mistake the ρ for a p. This is dangerous but forgivable mistake. It's okay. I feel you. (9/13)

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. (10/13)

Recall (and you can take a look at my write-up mentioned in my third tweet above) 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) = Γ_ij(t) = |Θ_ij(t)|². (11/13)

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" but "non-Markovian." Mathematicians love to confuse you with non-descriptive terminology, but the translation is that Markovian can be read as "memory-less," which means non-Markovian is non-memoryless, thus memory-full. We get transition probabilities given by the equation Γ_ij(t) = tr(Θ†(t) P_i Θ(t) P_j), where Θ(t) is the time-evolution operator, and P_i, P_j are projection operators. (12/13)

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: youtu.be/7oWip00iXbo and let me know in follow-up tweets. (13/13)