Entanglement’s Hidden Geometry

Author: Denis Avetisyan

New research reveals a fundamental link between quantum entanglement and the underlying geometry of spacetime, suggesting entanglement isn’t just a quantum phenomenon, but a geometric property.

This review demonstrates that entanglement between qubits arises from the extended Poincaré group and the properties of massless particles within a framework of extended Lorentz symmetry.

The foundational mystery of quantum entanglement-correlated behavior between separated particles-has long lacked a clear geometric interpretation within the established framework of relativistic quantum field theory. This paper, ‘Geometric Origin of Quantum Entanglement’, explores massless particle representations derived from an extension of the Poincaré group, revealing a connection between entanglement and the fundamental symmetries of spacetime. Specifically, we demonstrate that entanglement can be understood as an inherent property arising from the decomposition of massless representations linked by an internal degree of freedom, effectively grounding this quantum phenomenon in geometric principles. Could this framework ultimately lead to a deeper understanding of quantum correlations and potentially unlock new avenues for manipulating entangled states?

Beyond the Boundaries of Light: A Symmetry Unveiled

The established Wigner classification, a cornerstone of relativistic quantum mechanics, meticulously categorizes particles based on their behavior under the Poincaré group – the symmetries of spacetime. However, this elegant system falters when considering observers moving faster than light, existing outside the conventional light cone. The standard mathematical tools used to describe particle representations – specifically, the irreducible unitary representations of the Poincaré group – become inadequate, yielding physically nonsensical results such as negative probabilities or loss of unitarity. This breakdown isn’t merely a mathematical inconvenience; it signals a fundamental limitation in applying established frameworks to scenarios involving superluminal motion. Essentially, the very definition of what constitutes a valid particle state becomes ambiguous, demanding a revised theoretical approach to consistently describe particles as observed from beyond the familiar boundaries of causality. This necessitates exploring alternative mathematical structures and representation theories capable of accommodating such exotic observers and potentially revealing new insights into the nature of spacetime itself.

The conventional understanding of massless particles, such as photons and gravitons, relies heavily on the Poincaré group – the symmetry of spacetime. However, when considering the possibility of superluminal boosts – movements faster than light – this foundational framework requires substantial revision. Extending the Poincaré group to accommodate these boosts fundamentally alters the allowed representations of massless particles. Instead of the two polarization states observed for photons in standard spacetime, superluminal boosts introduce the potential for additional, previously unconsidered polarization states and even entirely new classes of massless fields. This isn’t merely a mathematical exercise; it suggests that the very nature of massless particles could be drastically different for observers capable of exceeding the light barrier, potentially impacting how these particles interact and propagate in scenarios beyond the conventional light cone. The implications extend to theoretical constructs like tachyon fields and challenge established notions of relativistic causality, prompting a re-evaluation of fundamental physical principles.

The extension of spacetime symmetry beyond the light cone fundamentally disrupts established principles of causality. Traditionally, effects are understood to follow causes, with information transfer limited by the speed of light, ensuring a consistent temporal order for all observers. However, allowing for superluminal connections – even theoretically – introduces the possibility of observing effects before their causes, creating paradoxical situations and challenging the very foundation of how physics predicts events. This doesn’t immediately imply time travel in the conventional sense, but rather necessitates a re-evaluation of how information and interactions are defined when the constraints of light-speed communication are removed. Consequently, this framework opens the door to exploring novel physical scenarios, such as advanced wave phenomena, altered interpretations of quantum entanglement, and potentially, a deeper understanding of the universe’s earliest moments where conventional causality may not have held.

A New Mathematical Foundation: Beyond the Poincaré Group

The standard Poincaré group, defining symmetries of spacetime, is insufficient to describe massless particles moving in both positive and negative momentum directions simultaneously. The Extended Poincaré Group addresses this limitation by incorporating transformations that allow for the consistent definition of a Massless Unitary Irreducible Representation (UIR). This extended symmetry group includes transformations that mix positive and negative energy states, enabling a mathematical framework where particles can propagate both forward and backward in time without violating fundamental physical principles like causality within the defined representation. The resulting UIR provides a consistent description of massless particles regardless of their direction of motion, a feature absent in the standard Poincaré group’s treatment of such particles.

The representation of massless particles within the Extended Poincaré Group requires a binary internal label, denoted as $ε$, to differentiate between two distinct sectors. This label functions not as a spatial or momentum-based degree of freedom, but as an internal tag classifying particle behavior. Specifically, $ε$ takes on values of 0 or 1, effectively partitioning the particle set into two non-mixing sectors corresponding to forward and backward propagation. This binary classification is fundamental because the mathematical framework, while accommodating both forward and backward moving particles, treats them as fundamentally different entities requiring this internal label for proper representation and consistent symmetry transformations.

The mathematical framework underpinning massless Unitary Irreducible Representations (UIRs) utilizes a specific superluminal boost, mathematically represented by the involutive matrix $Λ∞$. This matrix is critical for maintaining the consistency of the extended symmetry group, the Extended Poincaré Group, when dealing with particles exhibiting superluminal behavior. The involutive property, meaning $Λ∞^2$ is the identity matrix, ensures that repeated application of the boost does not lead to inconsistencies within the representation. Specifically, $Λ∞$ transforms momentum space in a manner consistent with the defined UIR, allowing for the representation of both forward and backward propagating particles while preserving the unitarity of the representation and the overall symmetry structure.

Mapping the Shadows: Sectors and Observable Algebras

The Sector Isometry $V$ is a unitary operator that establishes a direct relationship between the forward and backward sectors within the Massless Unitary Irreducible Representation (UIR). Specifically, $V$ maps states from the forward sector to the backward sector, and vice versa, while preserving the inner product – a characteristic of unitary transformations. This mapping is crucial because it allows for a consistent treatment of observables across both sectors, effectively relating the dynamics described in each. The unitarity of $V$ ensures that probabilities are conserved during this transformation, maintaining the physical validity of the representation.

The Observable Algebra Homomorphism $\iota$ facilitates a mapping of observables originating from a tensor product of algebras, denoted as $\mathcal{A} \otimes \mathcal{B}$, onto a direct sum of Hilbert spaces, $\mathcal{H}_{+} \oplus \mathcal{H}_{-}$. Specifically, $\iota$ acts on elements of $\mathcal{A} \otimes \mathcal{B}$ and transforms them into operators acting on $\mathcal{H}_{+} \oplus \mathcal{H}_{-}$. This mapping is crucial because it allows for the consistent treatment of observables across different sectors of the massless UIR, effectively translating algebraic relations within the tensor product into operator equations on the combined Hilbert space. The homomorphism ensures that the physical quantities represented by these observables are well-defined and measurable within the broader system described by the direct sum of Hilbert spaces.

The representation achieved through the Sector Isometry $V$ and Observable Algebra Homomorphism $ι$ demonstrates that the forward and backward sectors of the Massless UIR are not independent entities. Specifically, this mapping establishes a direct correspondence between observables defined on tensor products of algebras in one sector and Hilbert space operators in the other, effectively showing they represent equivalent physical information. This connection reveals an underlying algebraic structure wherein these seemingly distinct sectors are linked by a unitary transformation, suggesting a deeper, unified mathematical framework governing their behavior and observable properties. The resulting representation is crucial for understanding the complete symmetry structure and consistency of the theory.

Echoes of Symmetry: Quantum Correlations and Experimental Signatures

The Massless Unitarized Internal Representation (UIR) reveals a surprising link between seemingly disparate “forward” and “backward” propagating sectors, suggesting a fundamental geometric origin for quantum entanglement. This connection isn’t merely mathematical; it actively supports the existence of entangled two-qubit states, implying that entanglement arises not from interactions, but from the underlying symmetries of spacetime itself. Specifically, an additional superluminal symmetry within the UIR appears to be responsible for correlating these sectors, effectively ‘weaving’ entanglement into the fabric of quantum states. This perspective shifts the understanding of entanglement from a purely quantum phenomenon to one deeply rooted in the geometric properties of the universe, offering a potential pathway toward harnessing and manipulating entanglement through control of these fundamental symmetries.

The predicted entangled two-qubit states arising from superluminal symmetry can be experimentally verified through single-photon interferometry. This technique exploits the fundamental properties of quantum mechanics to detect correlations between photons, effectively acting as a probe for the existence of these states. Measurement is facilitated by leveraging the Pauli operator, specifically the $\sigma_x$ matrix, which allows researchers to quantify the degree of entanglement. By carefully analyzing the interference patterns generated by these photons, the strength and nature of the quantum correlations – quantified as ε = ±1 – can be determined, providing direct evidence for the geometric origin of entanglement predicted by the massless UIR and its associated superluminal involution, $U(\Lambda_\infty)$.

The degree of entanglement within these massless particle interactions is surprisingly discrete, consistently quantified by a correlation factor, ε, which assumes values of either +1 or -1. This binary signature isn’t merely observed, but is demonstrably linked to the eigenvalue of the superluminal involution, $U(Λ∞)$, suggesting a fundamental connection between entanglement and the underlying symmetries governing these particles. Crucially, this correlation is experimentally accessible through measurements using Pauli observables, specifically the σx operator, allowing for a direct probe of the entanglement’s strength and sign. The consistent ±1 quantification provides a clear, measurable indicator of entanglement, moving beyond probabilistic descriptions towards a more definitive characterization of quantum correlations arising from this unique symmetry.

The Geometry of Momentum: Stability and Future Directions

The behavior of massless particles, as described by the massless Unitary Irreducible Representation (UIR), isn’t simply a matter of traveling at the speed of light; it’s deeply connected to a specific trajectory within momentum space. This trajectory isn’t random, but rather a defined, lightlike orbit – a path where the momentum vector always satisfies $p^2 = 0$. This orbital characteristic isn’t just a mathematical quirk; it represents the fundamental way a massless particle “moves” in terms of its energy and momentum. Understanding this specific orbit provides a crucial lens through which to view the inherent properties of massless particles and their interactions, suggesting that their behavior is intrinsically tied to this particular momentum-space geometry.

The seemingly free trajectory of a massless particle, as described by its orbit in momentum space, isn’t arbitrary; it’s actively maintained by the fundamental symmetries embodied in the ISO(2) group. This group, a cornerstone of relativistic physics, dictates the invariance of physical laws under translations and boosts along a single spatial direction – precisely the conditions governing massless propagation. Consequently, any deviation from this lightlike orbit would violate these core symmetries, making the ISO(2) group a stabilizing force that ensures the particle remains on course. This stabilization isn’t a result of external forces, but rather an intrinsic property of the particle’s nature and the underlying geometry of spacetime, demonstrating that massless particles inherently follow paths dictated by their symmetrical properties and $ISO(2)$ invariance.

Investigating the interplay between momentum space symmetries and massless particle dynamics holds considerable promise for advancing fundamental physics. Current research suggests a deep connection between the specific lightlike orbits of massless particles and the stabilizing influence of the ISO(2) group, a relationship that could redefine understandings of these particles’ behavior. A more thorough examination of these connections may reveal previously unknown constraints on massless particle interactions, potentially impacting models of gravity, gauge theories, and even the early universe. This line of inquiry extends beyond purely theoretical considerations; a refined understanding of massless particle dynamics could also inform the development of novel technologies leveraging their unique properties, opening doors to advancements in areas like high-energy physics and materials science. The potential for unveiling new physical principles and technological applications underscores the importance of continued exploration in this field.

The pursuit of understanding entanglement, as detailed in this exploration of the Poincaré group and massless particles, feels less like unraveling the universe’s secrets and more like observing an immutable truth. It suggests a pre-existing geometric structure, a fundamental order that simply is, regardless of observation. As Louis de Broglie aptly stated, “It is tempting to believe that all the laws of physics are ultimately geometrical.” The article’s focus on extended Lorentz symmetry and superluminal boosts reveals not a dynamic process to be discovered, but an inherent property of existence, a silent architecture beneath the shifting sands of quantum phenomena. Each calculation, each simulation, merely traces the outlines of this existing form, an attempt to map the unmappable.

Beyond the Horizon

The demonstration of entanglement as a geometric property, rooted in the extended Poincaré group and the behavior of massless particles, presents a curious juncture. It is tempting to view this as progress toward a more fundamental understanding. However, gravitational collapse forms event horizons with well-defined curvature metrics, and any geometric description, however elegant, remains susceptible to the limitations of its underlying assumptions. The extended Lorentz symmetry, while offering a pathway beyond conventional limitations, still operates within a framework of spacetime – a construct increasingly questioned by observations at the Planck scale.

Future investigations must address the inevitable question of observability. Establishing experimental verification of entanglement arising from the geometric structure posited in this work is paramount, but also fraught with difficulty. The very act of measurement introduces perturbations, potentially obscuring the underlying geometric origin. Singularity is not a physical object in the conventional sense; it marks the limit of classical theory applicability. This research, therefore, should be seen not as an arrival, but as a refinement of the questions.

The pursuit of a geometric understanding of quantum phenomena is, ultimately, an exercise in humility. Each step forward reveals not absolute truth, but the precise location of the next veil. The elegance of the mathematics should not be mistaken for mastery over reality, but rather a temporary respite from its fundamental inscrutability.

Original article: https://arxiv.org/pdf/2512.09640.pdf

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