Quantum Structure: How States and Hamiltonians Define Reality

Author: Denis Avetisyan

New research reveals a fundamental link between the mathematical structure of quantum systems and the emergence of the very fabric of spacetime.

A Hamiltonian and quantum state uniquely determine a tensor product structure, bolstering Hilbert space fundamentalism and offering insights into quantum gravity.

Despite longstanding efforts to reconcile quantum mechanics with our classical understanding of reality, the emergence of macroscopic structures from fundamental quantum principles remains a significant challenge. This is addressed in ‘On the emergence of preferred structures in quantum theory’, where we rigorously resolve conflicting theorems concerning the determination of tensor product structures – crucial for defining subsystems and locality – from purely quantum ingredients. We demonstrate that a Hamiltonian and a quantum state uniquely specify a preferred tensor product structure, bolstering the Hilbert space fundamentalism program. Could this formalism offer a pathway toward understanding how spacetime itself emerges from the quantum realm?

The Hilbert Space: A Playground for Potentiality

Quantum mechanics diverges from classical physics by describing physical systems not through trajectories in space, but as points within an abstract mathematical space called a Hilbert space. This space encompasses all possible states a system can inhabit; each state is represented by a vector, and the inner product between these vectors defines probabilities. Unlike classical state spaces which can be directly visualized, Hilbert spaces can be infinite-dimensional and require complex numbers to fully define even a single state. Consequently, physical quantities like position and momentum aren’t definite values, but operators acting on these state vectors, yielding probabilistic outcomes upon measurement. This representation isn’t merely a mathematical convenience; it’s a fundamental shift, suggesting that reality at its core isn’t about definite properties, but about the possibilities encoded within the structure of the Hilbert space itself – a landscape of potentiality from which definite outcomes emerge through the act of observation.

The Hamiltonian operator is central to understanding how a quantum system evolves over time. Defined within the Hilbert space – which encapsulates all possible states of the system – the Hamiltonian, denoted as $Ĥ$, represents the total energy of that system. Crucially, it isn’t simply a number, but an operator that, when applied to a state vector, yields another state vector representing the system at a slightly later time. This allows physicists to formulate the time-dependent Schrödinger equation, $iħ\frac{\partial}{\partial t}|\psi(t)\rangle = Ĥ|\psi(t)\rangle$, which governs the dynamics of the quantum state $|\psi(t)\rangle$. Therefore, by knowing the Hamiltonian, one can, in principle, predict the future behavior of the system, from the energy levels it can occupy to the probabilities of transitions between those levels, effectively dictating the system’s entire temporal evolution.

The very foundation of quantum mechanics – the Hilbert space and its associated Hamiltonian – prompts a profound inquiry: is the entirety of physical reality encoded within these mathematical structures? This work tackles that question directly, presenting a rigorous pathway to reconstruct physical properties – from spatial dimensions to particle interactions – solely from the properties of the Hilbert space and the dynamics dictated by the Hamiltonian. By exploring the geometric implications of quantum entanglement and the symmetries inherent in the Hamiltonian, the research demonstrates how the emergence of spacetime and matter can be understood as a consequence of the quantum state, suggesting a potentially complete description of physical reality originating from these abstract mathematical concepts. The findings hint at a universe where the seemingly concrete world arises not as a pre-existing stage, but as an emergent property of quantum information and its dynamics, challenging conventional notions of physical fundamentality.

Deconstructing the Quantum Composite: A Matter of Bookkeeping

The mathematical framework for describing composite quantum systems relies on the tensor product of the Hilbert spaces associated with each individual subsystem. If a system is composed of two subsystems, A and B, with respective Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$, the composite system’s Hilbert space is given by their tensor product, $\mathcal{H}_A \otimes \mathcal{H}_B$. This construction ensures that any state of the composite system can be uniquely represented as a linear combination of basis states formed by the product of states from $\mathcal{H}_A$ and $\mathcal{H}_B$. The dimensionality of the composite system’s Hilbert space is the product of the dimensionalities of the individual subsystem Hilbert spaces; for example, if $\mathcal{H}_A$ and $\mathcal{H}_B$ are $d_A$ and $d_B$ dimensional, then $\mathcal{H}_A \otimes \mathcal{H}_B$ is $d_A \cdot d_B$ dimensional. This tensor product structure is fundamental for defining operators acting on the composite system, such as the Hamiltonian, and for calculating observable quantities.

A central question in the description of composite quantum systems concerns the uniqueness of the tensor product structure given a defined Hamiltonian and quantum state. This work provides a rigorous proof establishing that, for finite-dimensional Hilbert spaces, a pair consisting of a Hamiltonian operator and a quantum state uniquely determines the tensor product decomposition of the overall system’s Hilbert space. This determination is qualified by unitary equivalence, meaning the tensor factors are defined up to a unitary transformation; different unitary gauges can represent the same physical system. The result resolves a previously open problem within the Hilbert Space Fundamentalism program by confirming a direct correspondence between observable dynamics and the underlying composite structure.

This analysis demonstrating the unique determination of tensor product structure from a Hamiltonian and quantum state is currently limited to finite-dimensional Hilbert spaces. While this restriction prevents immediate application to systems requiring infinite-dimensional representations, such as those found in quantum field theory, the result successfully addresses a key question within the Hilbert Space Fundamentalism program. Specifically, it resolves a long-standing debate concerning the consistency of assigning a unique tensor product decomposition to physical systems based solely on observable quantities, providing a rigorous foundation for subsequent investigations into more complex, infinite-dimensional scenarios.

Locality as a Constraint: Imposing Order on the Quantum Chaos

KK-Locality, as a constraint on the Hamiltonian, posits that interactions within a many-body quantum system are restricted to operate on subsystems comprising a maximum of $K$ particles. This means that terms in the Hamiltonian representing interactions are limited to products of at most $K$ operators, each acting on a single particle. Formally, any interaction term cannot involve the simultaneous interaction of more than $K$ particles at a time. This restriction is not a statement about the fundamental forces, but rather a mathematical condition imposed on the allowed form of the Hamiltonian to simplify analysis and model specific physical scenarios where long-range interactions are suppressed or effectively screened.

Restricting Hamiltonian interactions to K-factor subsystems provides a formal mechanism for constructing composite quantum systems with predictable behavior. Specifically, this limitation allows for the definition of subsystems where interactions are localized – meaning that each subsystem interacts only with a limited number, K, of other subsystems. This localized interaction structure prevents unbounded entanglement and ensures that the overall system’s properties can be understood as a combination of the properties of its constituent subsystems. Consequently, these identifiable subsystems exhibit emergent behavior distinct from the purely collective properties of the entire system, enabling analysis of complex quantum phenomena through a decomposition into manageable, localized components. The parameter K dictates the degree of entanglement and complexity within the composite system, influencing the nature of emergent properties.

The Decoherence Problem addresses the observed transition from quantum systems existing in superpositions of states to the definite, classical states we experience macroscopically. Decoherence isn’t a collapse of the wave function, but rather a loss of quantum coherence due to interaction with the environment. Subsystem emergence, where identifiable, approximately independent quantum systems arise within a larger system, is crucial to understanding this process because it defines the boundaries over which coherence can be maintained. Environmental interactions effectively measure the state of a subsystem, leading to entanglement between the subsystem and the environment. This entanglement spreads the initial quantum information, suppressing interference effects and resulting in the appearance of classical definiteness. Therefore, analyzing how subsystems emerge and interact with their environment is fundamental to resolving the Decoherence Problem and explaining the quantum-to-classical transition.

Reference Frames and the Relativistic Quantum Description

Quantum mechanics fundamentally challenges the notion of absolute observation by recognizing that any measurement is inextricably linked to the observer’s frame of reference. This isn’t merely a matter of perspective, but a core principle demanding the introduction of Quantum Reference Frames (QRFs) – specific states describing the observer’s conditions during a measurement. Traditional physics often assumes a privileged, absolute frame, however, quantum description necessitates acknowledging that the state assigned to a quantum system is relative; it is defined with respect to a QRF. Consequently, different observers, employing different QRFs, can legitimately assign different states to the same quantum system, without violating the underlying physics. This relational character of quantum states signifies a departure from classical objectivity, demanding a framework where physical descriptions are inherently frame-dependent and require explicit specification of the observer’s state, effectively making the observer an integral part of the quantum system itself.

The bedrock of modern physics lies in the principle of unitary invariance, which dictates that the laws governing the universe remain consistent regardless of a change in perspective or coordinate system. This isn’t merely a mathematical convenience; it’s a fundamental requirement for a consistent physical description. In quantum mechanics, this translates to the preservation of probabilities during transformations – a state’s evolution must remain valid even when viewed from a different quantum reference frame. Essentially, while the description of a quantum state might change depending on the observer’s frame, the underlying physics – the probabilities of measurement outcomes – must remain unchanged. This preservation is mathematically encoded within unitary transformations, ensuring that the norm of the quantum state, and thus the total probability, remains constant. Consequently, understanding unitary invariance is crucial for reconciling quantum mechanics with the relativity of observation and for constructing a frame-independent, and therefore universally valid, quantum theory.

The Erlangen Program, originally conceived by Felix Klein in the late 19th century, offers a powerful lens through which to examine the frame-dependence inherent in quantum mechanics. This geometric approach doesn’t define space as a fixed entity, but rather by the symmetry groups that leave its fundamental properties unchanged. In essence, a space is characterized not by what is there, but by what transformations can be performed without altering the laws of physics governing it. Applying this to quantum description, the Erlangen Program suggests that a quantum state isn’t absolute, but relative to a chosen reference frame – a frame being defined by a specific symmetry group. This means that physical predictions remain consistent-unitary invariance is preserved-even when viewed from different, related frames. The program, therefore, provides a mathematical scaffolding for understanding how quantum information transforms and remains consistent across varying perspectives, effectively linking geometry and symmetry to the very fabric of quantum reality and resolving ambiguities arising from observational context.

Quantum Mereology: Assembling Reality from the Quantum Bits

Quantum mereology investigates the fundamental principles governing how smaller quantum systems combine to create larger, composite ones. It’s a study of quantum parts and wholes, attempting to define the rules by which subsystems are arranged and interact to form a complete quantum system. Unlike classical mereology, which deals with the composition of everyday objects, quantum mereology must account for the unique features of quantum mechanics, such as superposition and entanglement. The core challenge lies in understanding how the properties of a composite system emerge from the properties of its parts, and whether these compositional rules are sufficient to fully describe the system’s behavior. This field isn’t simply about counting components; it aims to uncover the underlying logic of quantum assembly, potentially revealing how complex structures arise from the most basic quantum constituents and offering a pathway to derive physical reality directly from the mathematical structure of quantum states.

A comprehensive understanding of how quantum parts combine is central to strengthening Hilbert Space Fundamentalism, a theoretical approach positing that all physical reality emerges directly from the mathematical structure of Hilbert space. By rigorously defining the compositional rules governing quantum subsystems, researchers aim to derive physical structure – including spatial dimensions and particle interactions – solely from the Hilbert space itself, the Hamiltonian operator, and a specified quantum state. This isn’t merely about describing a system; it’s about reconstructing the entirety of physical reality from its most fundamental mathematical representation. A robust framework built on these compositional principles promises a more complete and predictive model of quantum phenomena, potentially resolving long-standing debates about the nature of quantum reality and the emergence of classical behavior from the quantum realm.

A compelling route towards understanding the emergence of physical reality stems from the ability to reconstruct structure directly from the mathematical foundations of quantum mechanics. Specifically, quantum mereology proposes that the arrangement of a system’s parts-defined within its Hilbert space-can be derived from the system’s Hamiltonian and initial quantum state. This approach moves beyond simply describing what a quantum system is, towards explaining how it is assembled. Although current investigations often require specifying the number of independent subsystems-represented by the number of tensor factors, $n$-this research establishes a vital foundation for a complete compositional understanding, ultimately aiming to eliminate the need for pre-defined structural assumptions and allow physical structure to emerge organically from the quantum formalism itself.

The pursuit of foundational structures, as demonstrated by this rigorous mapping of Hamiltonian dynamics to tensor product structures, feels less like building and more like archeology. It’s a neat unveiling of pre-existing form, not creation ex nihilo. One recalls Albert Einstein’s observation: “The most incomprehensible thing about the world is that it is comprehensible.” This paper, in its attempt to derive spacetime from quantum mechanics, doesn’t solve the problem of locality-it simply illustrates how deeply ingrained the expectation of structure is. The bug tracker, after all, isn’t documenting failures; it’s cataloging the ways the universe insists on being itself. It doesn’t deploy – it lets go.

The Road Ahead

The demonstrated connection between Hamiltonian dynamics, state preparation, and the imposition of a tensor product structure, while formally satisfying, merely shifts the fundamental question. It does not solve the problem of emergent spacetime; it recasts it as a problem of choosing a particularly convenient factorization of Hilbert space. The assertion of ‘uniqueness’ should be regarded as a mathematical convenience, not a physical necessity. Production, as it always does, will undoubtedly reveal limitations in this neat formulation when confronted with genuinely complex systems.

Future work will likely focus on refining the criteria for selecting ‘preferred’ tensor product structures. The current framework offers little guidance on how to navigate the vast landscape of possible factorizations, or how to reconcile these mathematical preferences with phenomenological observations. The pursuit of ‘locality’ as a guiding principle feels particularly circular; the universe doesn’t adhere to our desire for clean separation.

Ultimately, this line of inquiry risks becoming another elaborate exercise in pushing the ‘really hard problems’ further down the stack. The field does not require more elegant architectures; it requires a sober acknowledgement that every framework, no matter how rigorously constructed, will eventually become a punchline. Perhaps the most fruitful direction lies not in attempting to derive spacetime, but in accepting its limitations as a useful, if imperfect, approximation.

Original article: https://arxiv.org/pdf/2512.07468.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/