Comment on "On the emergence of preferred structures in quantum theory" by Soulas, Franzmann, and Di Biagio

Here is an explanation of the paper in simple, everyday language, using analogies to make the complex physics concepts accessible.

The Big Picture: The "Magic Box" vs. The "Real World"

Imagine you have a Magic Box (the Quantum Universe). Inside this box, there are only two things:

A Rulebook (the Hamiltonian, $H$ ), which tells you how things change over time.
A Snapshot (the State Vector, $|\psi\rangle$ ), which shows you the current arrangement of everything inside.

The Big Question: Can you look only at the Rulebook and the Snapshot, and magically figure out what the world looks like? Can you deduce where the "particles" are, how they are grouped, and what "space" is, just from those two ingredients?

This idea is called Hilbert Space Fundamentalism (HSF). It's the dream that the universe is just pure math, and everything else (like space, time, and particles) "emerges" from it automatically.

The Author's Argument:
Cristi Stoica (the author) says: "No, you can't."

He argues that if you try to build a world only from the Rulebook and the Snapshot, you run into a logical trap. You either end up with a world that never changes (boring!), or you have to secretly sneak in extra information to make it work (cheating!).

The New Contender: Soulas et al.'s "Magic Recipe"

Recently, a group of physicists (Soulas, Franzmann, and Di Biagio) tried to prove Stoica wrong. They said, "We found a way to build a unique world from just the Rulebook and the Snapshot!"

They created a specific recipe to define Tensor Product Structure (TPS).

What is a TPS? Think of it as the "cutting board" for the universe. It decides how to slice the big Magic Box into smaller pieces (subsystems/particles). Without a TPS, you don't know if you have one giant blob or a billion tiny atoms.

Soulas et al. claimed their recipe cuts the box in exactly one unique way, solving the problem.

Stoica's Reply:
Stoica says, "Actually, your recipe doesn't prove you can build a world from nothing. Instead, your recipe perfectly illustrates why you can't."

He calls this a "Trilemma" (a three-way trap). To make your recipe work, you have to choose one of three bad options:

Option 1: The Frozen World (No Change)

If you make the recipe strictly follow the rules of the Magic Box without any extra help, the "cutting board" (TPS) becomes locked in place.

The Analogy: Imagine you are watching a movie, but the camera angle is glued to the wall. No matter how the actors move, the camera never pans, zooms, or cuts.
The Problem: In this scenario, particles can never interact, and "entanglement" (the spooky connection between particles) can never change. But in our real world, things do change and interact. So, this version of the universe is dead.

Option 2: The Cheat Code (Fixing a Moment)

To fix the problem, you might say, "Okay, let's freeze the camera at one specific moment in time (say, $t=0$ ) and use that to define the cutting board forever."

The Analogy: You take a photo of a messy room, decide that this is the only way the room can be organized, and then ignore the fact that the room changes every second.
The Problem: This breaks the laws of physics because it treats time as special. It's like saying, "The universe only makes sense if we look at it at exactly 12:00 PM." This isn't a true "emergence" from the rules; it's just manually setting the rules.

Option 3: The Moving Target (Changing Rules)

Or, you could say, "The cutting board changes every second to match the Snapshot."

The Analogy: Imagine a puzzle where the picture on the box changes every second, so you have to constantly reshuffle the pieces to keep the picture looking right.
The Problem: To make this work, you have to secretly program the puzzle pieces to move in a very specific, pre-determined way to compensate for the changes. You aren't discovering the structure; you are forcing it to happen. It's like saying, "I can predict the weather," but you are just secretly controlling the wind.

The Conclusion on the Recipe:
Soulas et al.'s recipe works, but only because they secretly added a huge list of "invariants" (fixed numbers) to make it fit.

Stoica's Analogy: It's like trying to prove that the mass of an electron emerges naturally from the laws of physics. But instead of calculating it, you just say, "Let's fix the mass to be 9.109 × 10⁻³¹ kg."
If you just "fix" the numbers by hand, you haven't explained anything. You've just stated the answer. Soulas et al. did the same thing with the structure of space; they fixed the "entanglement numbers" by hand to get the result they wanted.

The "Relational" Misunderstanding

Soulas et al. argue that they are being "relational." They say, "It doesn't matter if we rotate the whole universe; the relationships stay the same, so the physics is the same."

Stoica agrees that relationships matter, but he says they are applying the wrong kind of relationship.

The Geometry Analogy:
- Affine Geometry: Imagine a world where you can stretch and squish triangles. A triangle with sides 3-4-5 is the "same" as a triangle with sides 6-8-10 because they are just stretched versions of each other.
- Euclidean Geometry: In our real world, a 3-4-5 triangle is different from a 6-8-10 triangle. Distance matters.
- The Mistake: Soulas et al. are using "Affine Geometry" (stretching is fine) to describe a world that clearly needs "Euclidean Geometry" (distance matters).
- The Result: If you use the wrong geometry, you can't tell the difference between a star and a speck of dust, or between "now" and "yesterday." Nature clearly distinguishes between them, so the "Affine" (purely relational) view is wrong for our universe.

What Can We Save? (The "Law" vs. The "World")

Stoica isn't saying the math is useless. He suggests we shrink the goal.

The "World" Dream (Failed): Trying to build the entire universe (space, time, particles, history) only from the Rulebook and Snapshot. Verdict: Impossible. You need extra info (like "QT3" in the paper) to tell the math what it represents.
The "Law" Dream (Possible): Using the Rulebook to understand the laws of interaction.
- Analogy: Think of a musical score. The score (the Hamiltonian) tells you the relationships between notes. You can study the score to understand the structure of the music (the law). But the score alone doesn't tell you which instrument is playing, or where the concert hall is. You need that extra info to hear the actual music.

Summary in One Sentence

You cannot build a unique, changing, physical world out of thin air (just a rulebook and a snapshot); if you try, you either get a frozen universe, or you have to secretly cheat by hand-picking the answer, which isn't true "emergence."

The Takeaway: The universe is more than just abstract math; it requires specific, physical "labels" (like position and momentum) to make sense of the changes we observe.

Here is a detailed technical summary of the paper "On the emergence of preferred structures in quantum theory" by Cristi Stoica.

1. Problem Statement

The paper addresses the Hilbert Space Fundamentalism (HSF) program, which posits that the complete physical reality of the universe (including space, fields, particles, and the Tensor Product Structure (TPS) defining subsystems) can be uniquely derived from only two fundamental ingredients:

The Hamiltonian operator ( $H$ ).
The unit state vector ( $|\psi\rangle$ ).

The specific problem under scrutiny is the emergence of a unique TPS. A TPS is required to define subsystems, local observables, and entanglement. Stoica argues that previous attempts to derive a unique TPS from $(H, |\psi\rangle)$ fail because they either contradict physical observations (specifically the time-evolution of entanglement) or rely on arbitrary, non-emergent parameters.

The paper serves as a direct rebuttal to Soulas, Franzmann, and Di Biagio (2025), who claimed to have constructed a unique TPS from $(H, |\psi\rangle)$ , thereby refuting Stoica's earlier "no-go" theorems (Stoica, 2022a, 2024b).

2. Methodology

Stoica employs a combination of group-theoretic analysis, dimensional counting, and dynamical consistency checks:

Dimensional Analysis: He compares the dimension of the stabilizer group of the Hamiltonian ( $Stab(H)$ ) against the stabilizer group of a TPS ( $Stab(T)$ ).
Invariance and Symmetry: He analyzes the conditions under which a construction of a structure $S$ from $(H, |\psi\rangle)$ is invariant under unitary transformations. He distinguishes between "essential uniqueness" (unique up to physically unobservable differences) and strict uniqueness.
Dynamical Consistency: He tests proposed constructions against the time evolution of the state vector $|\psi(t)\rangle = e^{-iHt}|\psi(0)\rangle$ . He examines whether the emergent structure allows for the physical observation that entanglement entropy changes over time.
Case Study: He uses the specific construction proposed by Soulas et al. (2025) as a "toy model" to illustrate the general obstructions. He dissects their method of fixing von Neumann entropies via invariants to see if it holds up under time evolution.

3. Key Contributions and Arguments

A. Refutation of Soulas et al.'s Counterexample

Soulas et al. constructed a unique TPS by fixing a set of invariants (von Neumann entropies $s_{R,k}$ ) derived from the state vector and Hamiltonian. Stoica demonstrates that this construction fails to be a valid counterexample due to a trilemma:

Time-Dependence of Invariants: If the invariants $s_{R,k}$ are fixed constants, the resulting TPS is time-independent. However, for a TPS to be physically relevant, it must allow entanglement to change over time. A fixed TPS with fixed entropies implies no interactions and no change in entanglement, which contradicts empirical reality.
Absolute Time: To allow entanglement to change while keeping the TPS unique, one must fix the construction to a specific absolute time $t_0$ (using $|\psi(t_0)\rangle$ ). This violates the principle of time-translation invariance inherent in HSF.
State-Dependent Invariants: If one allows the invariants $s_{R,k}$ to depend on time to compensate for the state evolution, the construction is no longer "emergent" from the fundamental triple alone; it requires an external, time-dependent rule. This effectively amounts to manually fixing the TPS, similar to specifying particle masses by hand.

Conclusion on Soulas et al.: Their construction does not refute the no-go theorem; it confirms it. It either fails to describe physical change or ceases to be an invariant emergence from $(H, |\psi\rangle)$ .

B. The "No-Go" Theorem Clarified

Stoica clarifies his 2022 theorem, addressing misconceptions that it relies on "absolute uniqueness" or ignores relational perspectives.

Theorem: If a structure $S$ is physically relevant (i.e., its relation to $|\psi(t)\rangle$ changes in time, allowing for observable phenomena like entanglement evolution), it cannot emerge uniquely (even up to physically unobservable differences) from $(H, |\psi\rangle)$ in an invariant way.
Reasoning: If the construction is invariant under unitary transformations, and the time evolution operator $U_t = e^{-iHt}$ is a unitary symmetry, then the structure $S$ at time $t$ must be unitarily equivalent to $S$ at time $0 $. If$ S $is unique,$ S(t) $must be the same as$ S(0)$. This implies the structure cannot change, contradicting the requirement that it describes a changing world.

C. Critique of "Orbit Selection" as Emergence

Stoica argues that Soulas et al.'s method is not true emergence but merely orbit selection.

By fixing the invariants $s_{R,k}$ , one is effectively selecting a specific orbit of TPSs.
He draws an analogy to the mass of elementary particles: One could claim the electron's mass "emerges" by fixing the Casimir invariants of the Poincaré group. However, since these invariants are the mass, this is circular. Similarly, fixing the entanglement invariants to select a TPS is equivalent to manually specifying the TPS, not deriving it from the dynamics.

D. Relational Philosophy vs. Physical Evidence

The paper critiques the "relational philosophy" advocated by Soulas et al., which suggests that any global unitary transformation of the entire system depicts the same physical reality.

Stoica argues that while global unitaries map isomorphic structures, they do not map physically distinct states if those states have different time-evolution histories.
He uses an analogy with geometry: Just as affine geometry cannot distinguish between triangles of different sizes (it treats them as equivalent), HSF (with full unitary symmetry) cannot distinguish between different times or different entanglement configurations.
Physical Reality: Nature distinguishes these states (entanglement changes, time passes). Therefore, the symmetry group of the physical world is smaller than the full unitary group $U(H)$ . QT2 (TPS) and QT3 (observables) are necessary to break this symmetry and define the physical reality.

4. Results

Impossibility of Unique Emergence: It is impossible to construct a unique, physically relevant TPS (or space, or pointer basis) solely from $(H, |\psi\rangle)$ that is invariant and allows for time-dependent changes in entanglement.
Validation of No-Go Theorems: The Soulas et al. construction, intended as a counterexample, actually illustrates the trilemma (Invariance vs. Uniqueness vs. Physical Relevance) predicted by Stoica's earlier work.
HSF is Incomplete: Hilbert Space Fundamentalism cannot provide a complete description of reality (ontology) because the triple $(H, H, |\psi\rangle)$ is underdetermined. It can describe infinitely many different physical worlds (different TPSs, different laws) that are isomorphic as mathematical triples.

5. Significance and Proposed Path Forward

Pedagogical Value: The paper establishes that the Soulas et al. construction is an excellent pedagogical tool for understanding why emergent structures fail in HSF, despite the authors' best efforts to avoid the pitfalls.
Refinement of HSF: Stoica proposes a weaker, potentially viable version of HSF called "HSF about laws" (or "nomic HSF").
- Instead of claiming to derive the ontology (the specific world configuration, space, and particles), HSF should be restricted to deriving the laws (the form of the Hamiltonian and its local structure).
- While one cannot derive the specific TPS from the spectrum alone, one might derive constraints on the locality of the Hamiltonian (as attempted by Cotler et al., 2019).
Conclusion: The paper concludes that the "strong" HSF (deriving the world from the triple) is untenable. The "weak" HSF (deriving laws from the triple) has limited scope and cannot fully replace the need for postulates regarding subsystems and observables (QT2 and QT3) in standard quantum theory.

In summary, Stoica demonstrates that the attempt to derive preferred structures (like space and subsystems) purely from the Hamiltonian and state vector is fundamentally blocked by the conflict between unitary invariance and physical time-evolution. Any construction that succeeds in being unique and invariant fails to describe a changing world, and any construction that describes a changing world must sacrifice uniqueness or invariance, effectively requiring external input.