Galilean Reeh--Schlieder Obstruction

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This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Two Different Rules for Reality

Imagine the universe has two different sets of rulebooks for how particles behave:

The Relativistic Rulebook: This is Einstein's world. It's fast, rigid, and everything is connected in a specific way.
The Galilean Rulebook: This is the "everyday" world of Isaac Newton. It's slower, and time flows the same way for everyone, regardless of where they are.

For a long time, physicists thought these two rulebooks were just different versions of the same game. They believed that if you built a quantum theory (the math of tiny particles) using the Newtonian rules, it would eventually look like the Einsteinian one if you just added a few extra conditions.

This paper says: "No, they are fundamentally different."

The author, Leonardo A. Pachón, proves that you cannot build a Newtonian (Galilean) quantum theory that satisfies a specific, powerful mathematical property called the Reeh–Schlieder property. If you try to force this property into the Newtonian rulebook, the whole system collapses.

The Key Concept: The "Perfect Vacuum"

To understand the proof, we need to understand the Reeh–Schlieder property.

Imagine a room (a region of space) and a "Vacuum" (a state of absolute nothingness, or zero energy).

In Einstein's World (Relativistic): The vacuum is incredibly powerful. Even though the room is empty, the vacuum contains "seeds" of everything. If you have a magic wand (a local field operator) and you wave it inside this empty room, you can create any possible state of the universe. You can conjure a particle, a star, or a galaxy just by acting on the empty space in that one room. The vacuum is "cyclic" (it can generate everything) and "separating" (it's so unique that if your magic wand does nothing to it, the wand must be broken/empty).
In Newton's World (Galilean): The paper proves that in a Newtonian universe, the vacuum is not this powerful. It cannot generate everything just by acting on a small patch of space.

The "Obstruction": Why Newton's Rules Break the Magic Wand

The paper identifies a specific structural reason why the Newtonian vacuum fails to be "perfect" like the Einsteinian one. It's a clash between two ingredients:

1. The "Mass Charge" (The Bargmann Superselection)
In Newtonian physics, particles have a "mass charge." Think of this like a specific color or a unique ID tag.

A particle has a mass ID of $+1$ .
An "anti-particle" (or the hole left behind) has a mass ID of $-1$.
The Rule: You cannot mix these IDs. You cannot have a single object that is half $+1$ and half $-1$ at the same time. They live in separate "sectors" of reality.

2. The "Hermitian Combination" (The Magic Trick)
In Einstein's world, the math allows you to mix the particle and the anti-particle into a single, neutral object (a "Hermitian combination"). This neutral object is the one that lives in the local room and has the power to create anything.

Analogy: Imagine you have a red ball and a blue ball. In Einstein's world, you can glue them together to make a purple ball. This purple ball is the "local" object that can do magic.

The Problem:
In Newton's world, the "Mass Charge" rule forbids you from gluing the red and blue balls together. You can only hold the red ball alone or the blue ball alone.

The paper shows that in Newtonian physics, the "local" objects in your room are the red balls and blue balls separately.
But here is the catch: Red balls and blue balls (the basic fields) always kill the vacuum. If you wave a red ball at the empty vacuum, the vacuum stays empty (or rather, the red ball annihilates it).
Because the red ball kills the vacuum, and the red ball is the only thing allowed to be in the room (you can't make the purple ball), the vacuum cannot be "separating." If your tool kills the vacuum, the tool isn't necessarily broken; it's just that the vacuum is too "weak" to distinguish it.

The "No-Go" Conclusion

The paper proves a "No-Go Theorem." It says:

"You cannot have a Newtonian quantum theory that follows the standard rules of local fields AND has a vacuum that can generate the entire universe from a small room."

If you try to force the "Perfect Vacuum" (Reeh–Schlieder) into a Newtonian theory, the math forces the fields to become zero. The theory collapses into nothingness.

Why This Matters (According to the Paper)

The author argues that this difference is the structural divider between the two types of physics:

Relativistic Physics (Einstein): The Reeh–Schlieder property is a natural theorem. It works automatically. This is why modular theory (a complex math tool used to study time and entropy) works so well in Einstein's universe.
Galilean Physics (Newton): The Reeh–Schlieder property is impossible. Therefore, the fancy math tools that rely on it (like Tomita–Takesaki modular flow) do not exist in Newtonian quantum theories.

Summary of the Verification

The author checked five famous, real-world examples of Newtonian quantum theories (like the Lee model and others).

Result: None of them have the "Perfect Vacuum" property.
Why? In all these models, the basic particles annihilate the vacuum. Because they annihilate it, they cannot be "separating" in the way required by the Reeh–Schlieder property.

The Takeaway

The paper concludes that the "rigidity" of Einstein's universe (where everything is tightly connected) is not just a result of speed limits or time dilation. It is a fundamental algebraic feature that cannot exist in a Newtonian universe. The Newtonian universe is "less constrained" in some ways, but it is also "less connected" in a very specific, mathematical way: its empty space cannot conjure the whole universe from a single room.

1. Problem Statement

The paper addresses a fundamental structural distinction between Relativistic Algebraic Quantum Field Theory (AQFT) and Galilean (non-relativistic) AQFT.

The Context: In relativistic AQFT, the Reeh–Schlieder property (the vacuum state $\Omega$ being both cyclic and separating for every local field algebra $\mathcal{F}(\mathcal{O})$ ) is a proven theorem derived from the axioms of microcausality, the spectrum condition, and Lorentz covariance. This property is crucial for the existence of Tomita–Takesaki modular theory, which underpins many deep results like the Bisognano–Wightman theorem and the thermal time hypothesis.
The Gap: While it is known that Galilean QFTs satisfy a set of axioms (Haag–Kastler adapted for Galilean symmetry) and admit interacting models, the status of the Reeh–Schlieder property in the Galilean context has been unclear. Previous literature suggested Galilean theories are "less constrained" but lacked a rigorous proof that they cannot satisfy the Reeh–Schlieder property alongside standard axioms.
The Core Question: Can a Galilean Haag–Kastler net, satisfying standard axioms (including Bargmann mass superselection), possess a vacuum that is cyclic and separating for all local algebras?

2. Methodology

The author employs a rigorous axiomatic approach, constructing a no-go theorem by demonstrating an internal contradiction within a specific set of axioms.

Axiomatic Framework: The paper defines a set of Galilean Haag–Kastler axioms (G1–G7):
- (G1) Isotony.
- (G2) Equal-time locality (microcausality).
- (G3) Galilean covariance with Bargmann central extension (mass superselection).
- (G4) Canonical fields satisfying equal-time Canonical Commutation Relations (CCR).
- (G5) Positive energy spectrum.
- (G6) Unique translation-invariant vacuum.
- (G7) Fock representation (initially).
The Contradiction Mechanism: The proof relies on the interplay between two specific features of Galilean QFT that do not exist (or exist differently) in Relativistic QFT:
1. Annihilation of the Vacuum: In Galilean Fock space, the canonical field operators $\hat{\psi}(f)$ (annihilation operators) annihilate the vacuum: $\hat{\psi}(f)\Omega = 0$ .
2. Bargmann Mass Superselection: The central charge $\hat{M}$ (mass) forbids the formation of Hermitian combinations of creation and annihilation operators as local generators. Unlike the relativistic case where the local algebra is generated by Hermitian fields $\hat{\phi} = \hat{a} + \hat{a}^\dagger$ (which do not annihilate the vacuum), the Galilean local algebra $\mathcal{F}(\mathcal{O})$ must be generated by the individual charged fields $\hat{\psi}$ and $\hat{\psi}^\dagger$ separately.
Logical Flow:
1. Assume the Reeh–Schlieder property holds (vacuum is separating).
2. Use the axioms to show $\hat{\psi}(f)\Omega = 0$ for any test function $f$ .
3. Use the locality axiom (G4) to show $\hat{\psi}(f) \in \mathcal{F}(\mathcal{O})$ .
4. Apply the separating property: If $A \in \mathcal{F}(\mathcal{O})$ and $A\Omega = 0$ , then $A=0$ .
5. This implies $\hat{\psi}(f) = 0$ as an operator.
6. This contradicts the CCR axiom (G4), which requires $[\hat{\psi}, \hat{\psi}^\dagger] \neq 0$ .

3. Key Contributions

A. The Main Theorem (The Obstruction)

The paper proves Theorem 1: The standard Galilean Haag–Kastler axioms (G1–G7) are inconsistent with the Reeh–Schlieder property. No such net exists.

Mechanism: The obstruction arises because Bargmann mass superselection forces the local algebra to contain the annihilation operator $\hat{\psi}$ as a distinct element. Since $\hat{\psi}$ annihilates the vacuum, the separating property forces the operator to be zero, violating the CCR.
Contrast with Relativity: In relativistic AQFT, the local algebra is generated by Hermitian fields $\hat{\phi}$ . While $\hat{\phi}$ is local, its decomposition into $\hat{a}$ and $\hat{a}^\dagger$ is non-local (due to the complex structure of the one-particle space). Thus, the fact that $\hat{a}\Omega=0$ does not force $\hat{a}=0$ because $\hat{a} \notin \mathcal{R}(\mathcal{O})$ . This "evasion mechanism" is structurally impossible in Galilean QFT.

B. Generalization Beyond Fock Space

The author extends the result beyond the initial Fock representation hypothesis (G7) through Theorem 2:

The result holds for any Galilean net where:
1. Canonical fields carry definite Bargmann mass charges.
2. Time-zero restrictions of fields exist on a common dense domain.
Derived Consequences: The paper demonstrates that the "bounded-below mass spectrum" and "vacuum at spectral minimum" (often assumed as axioms) are actually derived consequences of the Galilean boost identity and positive energy spectrum (Lemma 4). This removes the need to assume them explicitly, making the obstruction theorem robust and structural.

C. Verification Against Literature

The paper systematically verifies the theorem against five major published interacting Galilean QFT constructions (Lévy-Leblond, Schrader, Hepp, Eckmann, and Lampart et al.).

Result: All these models satisfy the axioms (G1–G7) but fail the Reeh–Schlieder property.
Reason: In all cases, the dynamical ground state coincides with the bare Fock vacuum, which is annihilated by the bare fields. This confirms the theorem is non-vacuous and that existing models naturally evade the obstruction by failing the separating condition.

4. Key Results

Structural Divider: The Reeh–Schlieder property is identified as the precise structural ingredient distinguishing relativistic from Galilean AQFT. Relativistic theories have it as a theorem; Galilean theories cannot have it.
Failure of Modular Theory: As a direct corollary (Corollary 1 & 2), the Tomita–Takesaki modular flow is undefined for any Galilean local field algebra satisfying the standard axioms. The modular operator $\Delta$ and conjugation $J$ require a separating vector, which does not exist in this context.
No-Go Theorem: It is impossible to construct a Galilean QFT that simultaneously satisfies:
- Standard Galilean covariance (Bargmann extension).
- Canonical commutation relations.
- Positive energy.
- The Reeh–Schlieder property.

5. Significance

Clarification of "Rigidity": The paper resolves a long-standing ambiguity about why Galilean QFTs evade relativistic "rigidity" theorems (like CPT, Spin-Statistics, and Haag's Theorem). It clarifies that they do not evade them by violating microcausality (which holds in equal-time form) but by lacking the Reeh–Schlieder property.
Implications for Quantum Gravity: Many approaches to quantum gravity (e.g., Connes–Rovelli thermal time hypothesis, modular localization) rely heavily on the existence of a cyclic-and-separating vacuum to define time evolution and thermodynamic properties. This result implies that these approaches are intrinsically relativistic. They cannot be directly applied to Galilean limits or non-relativistic regimes without fundamental modification, as the necessary modular structure collapses in the Newton–Cartan limit.
Algebraic Kinematics: The work provides a precise algebraic characterization of why Special Relativity is "selected" over Galilean kinematics in the context of local quantum field theory. It suggests that the Reeh–Schlieder property should be promoted to an explicit axiom in any characterization of relativistic kinematics.
Foundational Rigor: By proving a precise no-go theorem on an explicit axiom set, the paper moves beyond heuristic arguments or analytic limit arguments (which show suppression of non-local effects) to a structural impossibility proof.

In summary, Pachón demonstrates that the Reeh–Schlieder property is incompatible with Galilean mass superselection, thereby establishing a fundamental algebraic barrier between relativistic and non-relativistic quantum field theories and explaining the absence of modular structures in Galilean QFT.