Entangled Fields: Proving Bell’s Theorem in the Realm of Quantum Field Theory

Author: Denis Avetisyan

Researchers have demonstrated a clear violation of Bell’s inequality within the framework of relativistic quantum field theory, solidifying the foundations of non-local quantum mechanics.

This work provides an explicit realization of Summers-Werner’s theorems by utilizing cat states and Weyl operators to showcase Bell-CHSH inequality violation in a relativistic setting.

The foundations of relativistic quantum mechanics pose a challenge to our understanding of locality and entanglement. This is addressed in ‘Cat states and violation of the Bell-CHSH inequality in relativistic Quantum Field Theory’, which investigates the emergence of non-classical correlations within a relativistic framework. By employing cat states and analyzing the Bell-CHSH inequality, the authors demonstrate an explicit realization of Summers-Werner’s theorems, showcasing a concrete mechanism for entanglement in quantum field theory. Could this approach provide new insights into the interplay between quantum information and the structure of spacetime?

The Erosion of Classical Intuition: A Foundation for Non-Locality

For centuries, the understanding of the physical world rested upon the principle of local realism, a concept deeply ingrained in classical physics. This principle asserts that every physical system possesses definite properties at all times, irrespective of whether or not those properties are being measured. Furthermore, local realism dictates that an object can only be influenced by its immediate surroundings – no instantaneous action at a distance is permitted. This intuitive notion suggests a universe comprised of well-defined entities with predictable behaviors, where cause and effect are localized and readily discernible. It provided a solid foundation for scientific inquiry, allowing for precise predictions and a deterministic view of reality – a worldview that, however, would later face profound challenges from the burgeoning field of quantum mechanics.

Quantum mechanics introduces a radical departure from classical intuition by predicting correlations between distant particles that defy explanation through local realism. These correlations are mathematically formalized through Bell-CHSH inequalities, which establish limits on the strength of correlations achievable if properties are predetermined and influenced only by local factors. Experiments, and now theoretical calculations within the framework of relativistic quantum field theory, demonstrate a violation of these inequalities, meaning the observed correlations are stronger than any classically permissible limit. This violation doesn’t imply a breakdown of quantum mechanics, but rather signals that the fundamental assumptions of local realism – that objects possess definite properties independent of measurement and that influences cannot travel faster than light – must be reconsidered, revealing a distinctly non-classical nature to reality.

Establishing a violation of local realism within the rigorous framework of Relativistic Quantum Field Theory is paramount to defining the boundaries of how accurately physical properties can be said to exist independently of observation. Recent calculations demonstrate a clear departure from classical expectations, yielding a violation value of approximately 2.012 when assessing the Bell-CHSH inequality. This result not only confirms the non-classical nature of quantum correlations but also solidifies the theoretical foundation for understanding the limits of realism in a universe governed by both quantum mechanics and special relativity; a value significantly exceeding the classical limit of 2 indicates a robust and measurable departure from locally realistic models, offering compelling evidence that quantum entanglement represents a fundamentally different mode of physical connection.

Constructing a Quantum Witness: Cat States and Causal Regions

A Cat State, utilized as a sensitive probe for non-classical correlations, is a quantum state created by the superposition of two coherent states. These coherent states, $\ket{\alpha}$ and $\ket{-\alpha}$ , represent minimum uncertainty wave packets analogous to classical oscillations, but exist in a superposition described as $\frac{1}{\sqrt{2}}(\ket{\alpha} + \ket{-\alpha})$ . This superposition results in a state exhibiting increased sensitivity to quantum fluctuations and entanglement, allowing for the detection of non-classical correlations that would be imperceptible to classical systems. The state’s properties, specifically its large amplitude and phase uncertainty, amplify the effects of these correlations, making it an effective tool for quantum information processing and the investigation of fundamental quantum phenomena.

A Wedge Region in quantum field theory defines a specific spacetime volume, typically characterized by angles $\{x \in \mathbb{R}^{1+1} : x^0 > 0, x^1 > |x^0| \}$ , which is fundamental for establishing the causal structure of the field. Localization of a quantum state, such as a ‘Cat State’, within this Wedge Region allows for precise investigation of correlations that respect this causality. The boundaries of the Wedge Region define the limits of information propagation, ensuring that any observable effects are constrained by the light cone and preventing acausal influences. This localization is critical for experiments designed to test the boundaries between classical and quantum behavior in curved spacetime and to explore the emergence of spacetime itself from quantum entanglement.

The generation of quantum operators required for constructing the system utilizes Weyl operators, which provide a direct link between classical phase space functions and quantum operators. Specifically, a Weyl operator $W(f)$ acting on a quantum state maps it to a new state based on the classical function $f$ . This is achieved through a specific integral transform, effectively translating classical descriptions of field behavior into corresponding quantum operators. The application of these Weyl operators allows for the creation of displacement and squeezing operators, crucial for preparing the coherent states that form the basis of the ‘Cat State’ within the defined ‘Wedge Region’. The formalism ensures that the resulting quantum operators satisfy the canonical commutation relations of the quantum field theory.

Formalizing Non-Locality: Operators and the Bell-CHSH Correlator

The Bell-CHSH correlator, denoted as $S = E(A,B) + E(A,B') + E(A',B) - E(A',B')$ , is a central observable in tests of local realism. It quantifies the degree of correlation between measurements performed on two entangled particles. Values of S exceeding 2 violate Bell’s inequalities, demonstrating a contradiction with the assumptions of locality and realism. The magnitude of the violation directly indicates the strength of the quantum correlations; larger deviations from the classical bound of 2 signify stronger entanglement and non-classical behavior. This correlator is constructed from expectation values $E(A,B)$ of measurement outcomes A and B, where A and A’ represent two distinct measurement settings on one particle, and B and B’ represent settings on the other particle.

The Bell-CHSH correlator is constructed using $\hat{O}_A$ and $\hat{O}_B$ , which are bounded Hermitian operators representing measurable observables. These operators act on a ‘smeared scalar field’, denoted as $\phi(x)$ , where ‘smearing’ involves integrating the field with a suitable test function to ensure finite variance and well-defined observables. The Hermitian property of these operators, $\hat{O} = \hat{O}^\dagger$ , guarantees that the measured quantities are real-valued. The specific form of the smearing function determines the spatial extent and characteristics of the observable being measured, impacting the overall value of the Bell-CHSH correlator and its ability to demonstrate non-local correlations.

The Tomita-Takesaki (TT) theory establishes a rigorous framework for constructing and validating the Hermitian operators used in calculating the Bell-CHSH correlator. Specifically, TT theory links the algebraic properties of a quantum system – described by a von Neumann algebra – to its dynamical evolution and observable quantities. It defines a ‘modular operator’ Δ and ‘modular conjugation’ $J$ that characterize the time-reversal and spatial reflection symmetries of the system. These elements are crucial for defining a consistent time evolution and ensuring that the constructed operators representing physical observables are self-adjoint and bounded, which is essential for obtaining physically meaningful correlations and verifying violations of local realism. The theory guarantees the existence of a unique, strongly continuous one-parameter group of automorphisms representing the time evolution, thereby ensuring the mathematical consistency of the operator construction.

A Definitive Departure: Closed-Form Solution and Violation of Local Realism

The calculation of the Bell-CHSH correlator, a crucial step in testing Bell’s theorem, often involves complex mathematical manipulations. To streamline this process, researchers introduced a ‘Sign Operator’ – a specific type of bounded Hermitian operator – which effectively simplifies the correlator’s formulation. This operator, acting as a mathematical shorthand, allows for a more direct and efficient calculation of the correlations between entangled particles. By leveraging the properties of this operator, the typically cumbersome calculations become significantly more tractable, enabling a precise determination of whether quantum correlations violate classical bounds and confirming the non-classical nature of quantum mechanics. The use of this operator isn’t merely a computational trick; it represents a focused approach to isolating the essential features of entanglement relevant to Bell’s inequality.

The adoption of a specific ‘Sign Operator’ facilitated a remarkably direct analytical result – a closed-form solution for calculating the Bell-CHSH correlator. This solution leverages the properties of the $Imaginary Error Function$ , a special function within complex analysis, to express the correlations with unprecedented precision. Rather than relying on numerical approximations or complex simulations, this analytical form allows for an exact evaluation of the correlator’s value for any given set of measurement settings. The resulting clarity is crucial, as it not only confirms the quantum mechanical predictions but also provides a solid foundation for rigorously demonstrating the violation of local realism, bypassing the uncertainties inherent in other computational methods.

Calculations reveal a definitive breach of the Bell-CHSH inequality, firmly establishing the non-classical characteristics of the quantum state under investigation. The computed value of approximately 2.012 significantly surpasses the classical limit of 2, a threshold that local realism predicts cannot be overcome. This result provides compelling evidence against the possibility of explaining quantum correlations through any theory reliant on local hidden variables. The observed disparity isn’t merely a statistical fluctuation; it represents a fundamental divergence between quantum predictions and the constraints imposed by classical physics, reinforcing the inherently non-local nature of quantum entanglement and its incompatibility with intuitive, everyday notions of causality.

Beyond Classical Limits: Theoretical Consistency and Broader Implications

The mathematical structure underpinning the ‘Smeared Scalar Field’ relies critically on the implementation of both Pauli-Jordan and Hadamard distributions, which serve to guarantee its internal consistency and uphold the principle of causality. These distributions aren’t merely technical tools; they actively prevent the propagation of information faster than light, a fundamental requirement for any physically plausible theory. The Pauli-Jordan representation, specifically, ensures that field operators commute at spacelike separation, preventing paradoxical situations where effects precede causes. Complementing this, Hadamard distributions provide a well-defined notion of ‘positive frequency,’ essential for constructing solutions that describe the evolution of quantum fields in a causal manner. By rigorously enforcing these mathematical constraints, the framework avoids the inconsistencies that plague many attempts to quantize fields and offers a robust foundation for exploring quantum phenomena.

The current findings demonstrate a compelling connection to the established Summers-Werner theorems, which delineate the boundaries of maximal non-classical correlations achievable in quantum systems. This work doesn’t merely confirm these theorems, but actively extends their applicability to the realm of smeared scalar fields, revealing that these maximal correlations aren’t limited to discrete systems. Specifically, the research shows that even with the inherent ‘smearing’ introduced to the field – a process designed to ensure causality – the highest degree of quantum entanglement, as predicted by the Summers-Werner criteria, can still be realized. This alignment strengthens the theoretical underpinnings of quantum information theory and provides a more complete picture of how non-classical correlations manifest in continuous variable systems, suggesting new avenues for exploring quantum communication and computation.

The developed framework, grounded in mathematically consistent quantum field theory, extends beyond simple models to offer a robust platform for investigating increasingly complex quantum systems. Researchers can now utilize this foundation to explore phenomena in regimes previously inaccessible to rigorous analysis, potentially revealing novel behaviors and correlations. Crucially, the approach facilitates a deeper examination of the boundaries between classical and quantum realities, allowing for precise tests of fundamental principles like locality and realism. By systematically probing these limits across diverse physical scenarios-from condensed matter physics to cosmology-the work promises to refine and potentially reshape our understanding of the very nature of physical reality and the constraints governing it, pushing the boundaries of what is considered classically possible.

The pursuit of demonstrable truth within relativistic quantum field theory, as evidenced by this work on Bell-CHSH inequality violations, echoes a fundamental principle of mathematical rigor. The construction of cat states, and the subsequent demonstration of non-locality through bounded Hermitian operators, isn’t merely a successful calculation – it’s a formalized proof. As Jean-Paul Sartre observed, “Existence precedes essence.” Similarly, here, the existence of a demonstrable violation-a concrete result-precedes any philosophical interpretation of non-locality. The paper doesn’t assume a violation; it proves one, establishing a firm foundation for understanding the limits of local realism and validating Summers-Werner’s theorems. If it feels like magic, one hasn’t revealed the invariant-in this case, the mathematical structure allowing for this specific violation.

What’s Next?

The explicit construction presented here, while satisfying in its mathematical rigor, merely scratches the surface of a deeper inquiry. Demonstrating a violation of the Bell-CHSH inequality within the established framework of relativistic quantum field theory is not, in itself, a resolution. Rather, it highlights the persistent tension between locality, causality, and the fundamentally non-classical nature of quantum correlations. The choice of cat states, while elegant, represents one particular instantiation; the question remains whether alternative states, or even entirely different operator constructions, might reveal a more general principle at play.

Further investigation should not be constrained by the pursuit of ‘realistic’ scenarios. The value lies not in mimicking physical experiments, but in pushing the boundaries of the theoretical framework. The Tomita-Takesaki theory, deployed as a tool to establish relativistic causality, demands further scrutiny. The modular operator, while mathematically precise, offers limited intuitive insight. A deeper understanding of its connection to the emergence of spacetime itself may be crucial.

Ultimately, the challenge is not simply to find violations of Bell inequalities – such demonstrations are now commonplace. The true endeavor lies in formulating a self-consistent theory that accommodates these violations without sacrificing the foundational principles of physics. A mathematically complete, logically sound framework-that is the only outcome worth striving for.

Original article: https://arxiv.org/pdf/2601.05216.pdf

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