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Unbounded Operators, Physicality, and Modality in Quantum Theories

This paper defends the physicality of Hilbert space in quantum mechanics by proposing a hierarchical view that distinguishes between ontological and pragmatic interpretations of physicality, thereby resolving ambiguities in state identification and addressing interpretational challenges in quantum field theory through modal metaphysics.

Original authors: Zhonghao Lu

Published 2026-09-24

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CC BY 4.0

Original authors: Zhonghao Lu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the standard way physicists describe the quantum world, everything is mapped onto a vast, abstract mathematical landscape called a Hilbert space. Think of this space as a giant container holding every possible configuration a quantum system could ever be in. Within this container, the rules of the game are written down as axioms: every possible state of a system is a point in this space, and every measurable quantity, like position or energy, is represented by a specific mathematical tool acting on those points. For decades, this framework has been the bedrock of quantum mechanics, allowing scientists to predict the behavior of atoms and particles with stunning accuracy. However, a nagging problem has long lurked in the corners of this mathematical container. The tools used to measure the most fundamental things in nature, such as where a particle is or how much energy it has, are "unbounded." In plain terms, this means these tools cannot be applied to every single point in the container without breaking. Some points in the space, when you try to measure them with these specific tools, produce results that blow up to infinity or become undefined. This has led some researchers to wonder if the container itself is too big, filled with "unphysical" states that nature simply cannot realize, and if the true, physical world is actually a much smaller, restricted subset of this grand mathematical space.

This question has recently reignited a debate about the very nature of physical reality. If the mathematical formalism includes states that seem impossible to create in a lab or that lead to infinite values, should we throw them out? A recent paper by Zhonghao Lu, a physicist at Ludwig-Maximilians University of Munich, tackles this head-on, challenging the idea that we can simply carve out a smaller, "pure" set of physical states from the larger mathematical whole. Lu argues that trying to draw a sharp line between what is physical and what is not is a trap. He suggests that if we try to identify a specific, smaller group of states as the only "real" ones, we run into unavoidable confusion and arbitrary choices. For instance, if we decide that a state is unphysical because it has an infinite average position, we face a dilemma: is it the infinite value itself that makes it unphysical, or is it just a mathematical quirk? Lu points out that in the real world, we never actually observe an infinite value; we only ever see finite numbers, no matter how large. The fact that a mathematical average might diverge to infinity does not mean the state itself is impossible or that nature forbids it.

The paper goes further to dismantle a specific proposal that has gained traction among some critics of the standard formalism. These critics suggested that the set of truly physical states should be limited to a special mathematical structure known as Schwartz space, which contains only "well-behaved" wave functions that decay quickly at the edges of the universe. They argued this would filter out the problematic, infinite states. Lu demonstrates that this solution is flawed. He shows that if you restrict the universe to only these "well-behaved" states, you immediately break the rules of how quantum systems evolve over time. A system starting in a perfectly "well-behaved" state can naturally evolve into a state that falls outside this restricted group, simply by moving under the influence of forces like the electric pull of an electron. If you ban these resulting states, you are effectively banning the natural evolution of the system itself. Furthermore, deciding which mathematical tools are "physical" enough to be allowed becomes an arbitrary game; if you allow the position operator but ban its exponential version, you are making a choice that has no deep physical justification, as both are measured by the same experimental apparatus.

Instead of trying to shrink the mathematical container, Lu proposes a new way of thinking about what it means for something to be "physical." He introduces a "hierarchical view" of physicality. Rather than a simple on-or-off switch where a state is either real or fake, physicality is more like a layered structure with degrees. Imagine a set of concentric rings. The outermost ring contains every single state allowed by the mathematics, including the weird ones with infinite averages. As you move inward, you find layers of states that are increasingly similar to the actual world we inhabit. The states in the very center are the most "physical" because they share the most features with our reality, such as having finite energy and behaving deterministically. But the states in the outer rings are not necessarily impossible; they are just less like our world. This approach respects the mathematical completeness of quantum theory while acknowledging that some states are more relevant to our specific reality than others. It avoids the need to arbitrarily cut off parts of the theory, recognizing that the boundary between the possible and the impossible is not a hard wall, but a gradient of similarity to the world we actually observe.

This nuanced perspective proves particularly powerful when applied to the even more complex realm of quantum field theory, where the rules of the game change drastically. In this domain, there is no single, agreed-upon mathematical container. Instead, there are many different, mutually incompatible ways to represent the same physical system, a problem known as the "inequivalent representations" issue. Traditionally, physicists have struggled to decide which of these mathematical representations is the "true" one. Lu's hierarchical view offers a way out of this confusion. It suggests that different representations correspond to different layers of physical possibility. A state that looks impossible in one representation might be perfectly normal in another, depending on which "layer" of reality you are examining. This does not mean anything goes; rather, it means that the set of physically possible states is not a single, monolithic list, but a collection of possibilities that depend on the context of the actual world. By accepting that physicality comes in degrees and layers, we can accommodate the strange, inequivalent representations of quantum field theory without discarding them as errors. We can understand them as valid descriptions of worlds that are physically possible, even if they are not the world we happen to live in.

Ultimately, the paper argues that the search for a single, perfect definition of a physical state is a dead end. The universe does not seem to care about the neat, sharp boundaries that mathematicians might prefer. Instead, the paper suggests that the concept of physicality is inherently layered, admitting of degrees of possibility. This view aligns the foundations of physics more closely with the philosophical study of possibility, where what is possible is often defined by how close a scenario is to our actual reality. By embracing this hierarchy, physicists can keep the full, rich mathematical machinery of quantum theory intact, using it to describe a vast landscape of possibilities, while still maintaining a clear connection to the specific, finite, and well-behaved world we actually measure and inhabit. The result is a more flexible and robust understanding of the quantum world, one that acknowledges the complexity of nature without forcing it into an artificially small box.

Technical Summary: Unbounded Operators, Physicality, and Modality in Quantum Theories

Problem Statement
The paper addresses the foundational issue regarding the "physicality" of the Hilbert space in quantum mechanics. Standard axioms posit that any normed ray in a Hilbert space $\mathcal{H}$ represents a possible quantum state. However, physically significant observables (such as position $\hat{x}$ , momentum $\hat{p}$ , and Hamiltonians $\hat{H}$ ) are typically unbounded self-adjoint operators. By the Hellinger-Toeplitz theorem, such operators are only defined on a proper dense subset (domain) of $\mathcal{H}$ . Consequently, states outside these domains cannot yield finite expectation values or satisfy the Schrödinger equation in its differential form.

Recent arguments by Heathcote (1990) and Carcassi, Calderón, and Aidala (2025) suggest that the Hilbert space contains "unphysical" states. Heathcote argues that the theory is incomplete because it fails to specify the subset of physical states. Carcassi et al. argue that the Hilbert space structure itself is unphysical because it admits states with infinite expectation values or "actual infinities," which they claim lack physical counterparts. They tentatively propose Schwartz spaces as the set of physical states. The author identifies a core tension: if one attempts to restrict the Hilbert space to a specific subset of "physical" states, one encounters inevitable ambiguity and arbitrariness in defining the boundaries between physical and unphysical states, particularly regarding which operators must have finite expectation values.

Methodology
The author employs a conceptual analysis grounded in the philosophy of physics and modal metaphysics. The methodology involves:

Re-evaluating Axioms: Critically analyzing the applicability of axioms (III) and (IV) of standard quantum mechanics to states outside operator domains.
Distinguishing Readings of Physicality: Differentiating between an ontological/modal reading (where physicality is synonymous with consistency with laws of nature) and a pragmatic/operational reading (where physicality is tied to laboratory preparation and measurement).
Mathematical Analysis: Examining the properties of Schwartz spaces, their closure under unitary evolution, and the self-adjointness of operators restricted to these spaces.
Modal Framework Application: Applying concepts from modal metaphysics (possible worlds, accessibility relations, and degrees of possibility) to the structure of quantum state spaces.
Extension to QFT: Utilizing the algebraic approach to Quantum Field Theory (QFT), specifically the Stone-von Neumann theorem's failure and the problem of inequivalent representations, to test the proposed framework.

Key Contributions and Results

Defense of the Hilbert Space: The author argues against the exclusion of states outside the domains of unbounded operators.
- Position Operator: States outside $\text{Dom}(\hat{x})$ may have infinite expectation values, but this does not imply "actual infinities" in observation. Observations yield finite values from the spectrum; infinite expectation values are statistical properties of ensembles, not singularities in the dynamics.
- Hamiltonian: The Schrödinger equation is a special case of the unitary evolution $\psi(t) = e^{-i\hat{H}t}\psi(0)$ . Since the unitary group is defined on the entire Hilbert space, restricting states to $\text{Dom}(\hat{H})$ is unnecessary for deterministic evolution.
- Preparation: The author rejects the pragmatic argument that states cannot be prepared in a lab are unphysical, noting that this excludes meaningful research in cosmology and general relativity (e.g., global spacetime structures) and fails to address deep ontological interpretational problems (like Bell's inequality).
Critique of Schwartz Spaces: The proposal to identify physical states with Schwartz spaces $\mathcal{S}(\mathbb{R})$ is shown to be insufficient.
- Arbitrariness: Distinguishing between operators with finite expectation values (e.g., $\hat{x}$ ) and those that may diverge (e.g., $e^{\hat{x}}$ ) is arbitrary, as they are measured by the same procedures.
- Dynamical Closure: Schwartz spaces are not closed under all physically meaningful Hamiltonian evolutions (e.g., those with Coulomb potentials), forcing the exclusion of standard physical potentials or the admission of "unphysical" dynamics.
- Self-Adjointness: Concerns regarding the definition of essential self-adjointness on $\mathcal{S}(\mathbb{R})$ are mitigated by noting that $\mathcal{S}(\mathbb{R})$ uniquely determines its Hilbert space completion, and the standard formalism can be used as a calculational tool even if the physical states are restricted.
The Hierarchical View of Physicality: The central theoretical contribution is the proposal that physicality is not a monolithic, binary concept but a layered one admitting degrees.
- Instead of a single set of physical states, there exists a hierarchy of sets $\Omega_\theta$ , where each set corresponds to a specific restriction $\theta$ (e.g., falling within the domains of specific operators).
- A state's "degree of physicality" is determined by its similarity to the actual world. The more restrictions a state satisfies (i.e., the more hierarchies it belongs to), the more "physically possible" it is.
- This view avoids the arbitrariness of selecting a single "correct" subset of the Hilbert space while acknowledging that not all mathematically possible states are equally "regular" or likely to represent the actual world.
Application to Quantum Field Theory (QFT): The hierarchical view is applied to the problem of inequivalent representations in QFT.
- In QFT, the Stone-von Neumann theorem fails, leading to unitarily inequivalent representations (e.g., Minkowski vs. Unruh vacua). Traditional interpretations struggle to define a single set of physical states.
- The author suggests that different representations correspond to different "folia" (sets of normal states) within the hierarchy of physicality.
- Rather than choosing one "true" representation (Hilbert space conservatism) or accepting all algebraic states (algebraic imperialism), the hierarchical view allows for multiple, objective possibility spaces. The degree of physicality of a state depends on its accessibility from the actual world (e.g., the Minkowski vacuum), without denying the modal status of other states (like the Unruh vacuum) in a broader sense.

Significance and Claims
The paper claims that the "unphysicality" of the Hilbert space is a misunderstanding stemming from an overly rigid, binary definition of physicality. By adopting a hierarchical view, one can preserve the mathematical completeness of the Hilbert space formalism while accounting for the intuition that certain states are "more physical" than others.

The author asserts that this approach:

Resolves the ambiguity in defining physical states in non-relativistic quantum mechanics without arbitrarily discarding the Hilbert space.
Offers a realist solution to the interpretational problems of QFT (inequivalent representations) that does not require abandoning the concept of a single actual world or succumbing to context-dependent interpretations.
Bridges the gap between the foundations of physics and modal metaphysics, suggesting that physical theories naturally induce modal structures (comparative possibility) that have been underutilized in foundational research.

The paper concludes modestly, inviting further research into the intersection of modal metaphysics and the foundations of physics, particularly regarding counterfactuals and comparative possibilities in physical theories.

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