A skepticism on the concept of quantum state related to quantum field theory on curved spacetime

The paper argues that the absence of a distinguished vacuum state in quantum field theory on curved spacetime undermines the physical reality of quantum states, while further challenging pragmatic realism by suggesting that the concept of a quantum state may be dispensable even in non-relativistic quantum mechanics and flat spacetime field theory.

The Gist

Imagine you are trying to describe the weather. In a small, sunny town, you might say, "It's 75 degrees and sunny." That's a state. It's a specific description of the system at a specific time.

Now, imagine trying to describe the weather of the entire universe at once. Or imagine trying to describe the weather in a place where the rules of physics change depending on how you look at them (like near a black hole). Suddenly, that simple "75 degrees and sunny" description breaks down. You can't agree on what "sunny" even means.

This is the core puzzle that Hideyasu Yamashita tackles in his paper. He is skeptical about the idea that a "quantum state" (like a wave function) is a real, physical thing that exists out there in the world. Instead, he argues that "states" are just useful tools we use to calculate things, and in the most extreme environments of the universe, those tools might not even exist.

Here is a breakdown of his argument using simple analogies:

1. The "Map vs. Territory" Problem

In standard quantum physics (like in a lab on Earth), we usually assume there is a "ground floor" or a Vacuum State. Think of this like the "ocean level" on a map. Once you have the ocean level, you can measure everything else relative to it (mountains, valleys, waves).

• The Old View: We believe the "ocean level" (the vacuum) is a real, physical thing. All other "states" are just waves on top of this real ocean.
• Yamashita's View: In Curved Spacetime (like near a black hole or in an expanding universe), there is no single, universal "ocean level." One observer might see a calm ocean, while another observer (falling into a black hole) sees a storm.
• The Result: If you can't agree on the "ocean level," you can't agree on what the "waves" (the quantum states) are. If the foundation is shaky, the whole building of "physical reality" for quantum states crumbles.

2. The "Fictional Characters" Analogy

Yamashita makes a distinction between Real things and Fictional things in math.

• Real States: These are like characters in a movie that you can actually see on screen. You can prepare them in a lab.
• Fictional States: These are like characters that exist only in the script or the director's imagination. They are mathematically possible, but you can never actually create them in the real world.

In normal physics, we can tell the difference. We know which characters are "real" because we have a fixed camera (a fixed "Hilbert space").
But in Curved Spacetime, the camera keeps changing angles and zooming in and out. Suddenly, you can't tell if a character is real or just a trick of the lens. If you can't distinguish the real from the fictional, why do we insist the "real" ones exist?

3. The "Pragmatic Realist" Trap

Some scientists argue: "Well, even if we can't define the state perfectly, we need the concept of a 'state' to do our calculations. Since it's indispensable, it must be real."
Yamashita compares this to Vector Potentials in electricity.

• The Analogy: Imagine you are driving. You need a map (Vector Potential) to get to the store. The map is incredibly useful. Does that mean the map is a physical object you can hold in your hand? No. It's just a tool.
• The Twist: Yamashita argues that even in quantum mechanics, we don't actually need the "state" to describe what happens. We can describe the universe using Operations (what we do) and Probabilities (what happens) without ever mentioning a "state."
• The Metaphor: Imagine describing a game of chess.
  • The "State" way: "The board is in State X." (This requires a complex, invisible snapshot of the whole board).
  • Yamashita's "Operation" way: "If you move the Knight here, and then I move the Pawn there, the probability of me winning is 60%."
  • He argues the second way is enough. We don't need to believe in the "State of the Board" as a physical object; we just need to know the rules of movement.

4. The "Scope" of Information

Yamashita introduces a new idea: Scope-Dependence.
Instead of saying "The state depends on the observer" (which sounds like "it's all in your head"), he says "The state depends on the scope of your information."

• Analogy: Think of a giant library.
  • If you are looking at just one book (a small "scope"), you can describe it perfectly.
  • If you try to describe the entire library at once (a "global state"), the description becomes impossible because the library is too big and the books interact in ways you can't see from one spot.
  • Yamashita suggests we should stop trying to describe the "Whole Universe State" and focus only on the "Scope" we can actually measure.

5. The "Stateless" Universe

The paper's ultimate goal is to show that we can rewrite the laws of physics without ever using the word "State."

• Current Physics: "If the universe is in State A, then Event B happens."
• Yamashita's Physics: "If we perform Operation A, then the probability of Event B is X."

He uses complex math (like "Causal Nets" and "Sorkin densities") to prove that you can calculate everything you need to know about particles and fields just by looking at how measurements connect to each other, without ever needing to assume a "ghostly" quantum state is floating around.

Summary

Yamashita is saying: "Stop worrying about what the 'Quantum State' really is. It might not be a real thing at all. It's just a mathematical crutch we use because we're used to it. In the wild, curved universe, that crutch breaks. We should learn to walk using 'Operations' and 'Probabilities' instead."

He isn't saying physics is broken; he's saying our language for physics is outdated. We are trying to describe a fluid, shifting universe with a rigid, static concept of "states," and it's time to switch to a language that flows with the universe.

Technical Summary

1. Problem Statement

The paper addresses the ontological status of the quantum state in modern physics. While standard interpretations (like Copenhagen) often treat the quantum state as a physical reality, alternative views (Relational Quantum Mechanics, QBism) treat it as observer-dependent or epistemic.

The author identifies a specific gap in the "pragmatic realist" argument, which claims quantum states must be real because they are indispensable for physical theory. The paper argues that:

1. In Quantum Field Theory on Curved Spacetime (QFTCS): There is no distinguished vacuum state or physical Hilbert space. Consequently, the algebraic formalism (AQFTCS) renders the distinction between "physically real" states and "fictional" states impossible.
2. Dispensability: The concept of the quantum state is not empirically indispensable. The author proposes that physical laws can be formulated entirely in terms of operations (measurements) and conditional probabilities, rendering the state vector or density matrix superfluous.

2. Methodology

The author employs a rigorous algebraic approach to quantum theory, utilizing the framework of Causal Nets of Algebras.

• Algebraic Framework: The paper unifies Non-Relativistic QM, QFT in Minkowski Spacetime (QFTM), and QFTCS under a single mathematical structure: a causal net $\{ \mathcal{A}_\alpha \}_{\alpha \in I}$ of local observable algebras.
  • $I$ is a directed set representing "scopes" (regions of information or observation).
  • $\alpha \leq \beta$ implies $\mathcal{A}_\alpha \subseteq \mathcal{A}_\beta$.
  • $\alpha \perp \beta$ (orthogonality) implies commutativity $[\mathcal{A}_\alpha, \mathcal{A}_\beta] = 0$.
• Hypotheses on Local Algebras: The author introduces working hypotheses assuming local algebras $\mathcal{A}_\alpha$ are Type I factors (isomorphic to $B(\mathcal{H}_\alpha)$) or atomic W*-algebras. This allows for the definition of a canonical trace, which is crucial for defining probabilities without global states.
• Stateless Formalism: Instead of using state vectors $|\psi\rangle$ or density matrices $\rho$, the author defines prior conditional probabilities directly via the trace of products of projection operators (or POVM elements).
  • Formula: $P(B|A) = \frac{\text{Tr}((AB)^* AB)}{\text{Tr}(A^* A)}$.
• Geometric Analysis: For systems with symplectic/Kähler structures (like spin systems), the author utilizes holonomy and Sorkin density functions to reconstruct interference patterns and probabilities without invoking a state.

3. Key Contributions

A. The Skepticism of Global States in QFTCS

• The Vacuum Problem: In QFTM, the vacuum state provides a privileged representation (GNS representation), distinguishing physical states (density matrices on the vacuum Hilbert space) from non-physical ones.
• The Curved Spacetime Issue: In QFTCS, no global vacuum exists (due to the lack of time-translation symmetry and the Unruh effect). Without a distinguished vacuum, there is no unique GNS representation.
• Conclusion: In QFTCS, one cannot mathematically distinguish between "realizable" states and "fictional" states. Therefore, the concept of a global quantum state loses its physical reality and becomes a mathematical fiction.

B. The Dispensability of Quantum States

The author argues that the "pragmatic realism" argument (states are real because they are useful) fails because states are not empirically necessary.

• Reformulation of Laws: Physical laws are rephrased from "If the system is in state $\omega$, then..." to "If the system is prepared by operation $X$, then..."
• Local vs. Global: While global states are problematic, local states (states on finite-dimensional or Type I subalgebras) can be defined via preparation procedures (operations). However, even these can be replaced by the concept of operations themselves.

C. "Quantum Mechanics Without States"

The paper provides explicit mathematical constructions for a stateless formulation:

1. Spinless Particles: The author replaces the wavefunction and propagator with conditional probabilities derived from a product of Projection-Valued Measures (PVMs).
   • The standard propagator $K(x, t; x', t')$ is shown to contain "empirical redundancy" (gauge-like information).
   • A new function $\kappa(\vec{x}, \vec{t})$ is introduced, which is directly related to observable probabilities and possesses the correct symmetries (e.g., Galilei invariance) without the redundancy of the standard propagator.
2. POVM and Sorkin Additivity: For systems like spin, the author uses Positive-Operator-Valued Measures (POVMs).
   • Introduces Sorkin density functions ($\rho_1, \rho_2$) to calculate interference probabilities.
   • Demonstrates that interference (e.g., double-slit) can be described via the holonomy of paths in a Kähler manifold, where the probability is determined by the real part of the holonomy of a closed loop formed by two paths.
3. Free Fields (CCR/CAR Algebras):
   • Extends the stateless formalism to Free Scalar (CCR) and Free Fermion (CAR) fields in curved spacetime.
   • Shows that for free fields, the causal net of Type I factors allows the definition of physical laws purely through conditional probabilities of local operations, bypassing the need for a Fock space or global vacuum.

4. Results

• Mathematical Equivalence: The paper demonstrates that for Type I local algebras, the standard probabilistic predictions of quantum mechanics can be recovered entirely using trace formulas on products of operations, without referencing a state vector.
• Symmetry Restoration: The proposed "stateless" functions (like $\kappa$) are shown to be more "empirical" than standard propagators because they naturally satisfy covariance requirements (e.g., Galilei invariance) that standard propagators often violate due to gauge-fixing choices.
• Limitations: The author acknowledges that for interacting fields in 3+1 dimensions, the construction of a causal net of Type I factors is currently unknown (and likely impossible due to the nature of Type III factors in interacting QFT). Thus, the stateless formalism is rigorously proven for free fields and non-relativistic systems but remains a conjecture for interacting QFT.

5. Significance

• Ontological Shift: The paper provides a strong mathematical argument against the ontological reality of the quantum state, particularly in the context of General Relativity (QFTCS). It suggests that "state" is a convenient mathematical tool for specific scopes, not a fundamental physical entity.
• Operationalism: It advances the operationalist view of quantum mechanics, suggesting that physics should be formulated in terms of preparation procedures (operations) and measurement outcomes, rather than the evolution of an abstract state.
• Bridging QFT and Gravity: By highlighting the failure of the "vacuum state" concept in curved spacetime, the paper underscores the tension between standard QFT and General Relativity, suggesting that a future theory of quantum gravity may need to abandon the concept of a global quantum state entirely.
• New Mathematical Tools: The introduction of Sorkin density functions and the explicit construction of holonomy-based path integrals for probability calculation offers new tools for analyzing quantum systems without relying on the standard Hilbert space state formalism.

In summary, Yamashita argues that the concept of the quantum state is a "scope-dependent" fiction that becomes mathematically indistinguishable from non-physical states in curved spacetime. He proposes a rigorous alternative framework where physical laws are defined by the algebraic relations of local operations and their conditional probabilities, rendering the quantum state empirically dispensable.