Ghost Particles Defy Expectations and May Stabilise Quantum Theories

Scientists are increasingly exploring field theories with higher-derivative terms to improve ultra-violet behaviour, yet these models often introduce problematic ghost particles with negative norms that threaten unitarity. Ichiro Oda from the University of the Ryukyus, alongside colleagues, investigate bound state formation within the Lee model, a framework describing the complex conjugate masses these ghosts acquire through radiative corrections. Their research demonstrates that bound states cannot form from these ghosts via contributions from a complex delta function, a generalised form of the Dirac delta function. This finding is significant because the emergence of this complex delta function is directly linked to the unitarity violation inherent in these higher-derivative theories, suggesting a fundamental connection between bound state existence and the preservation of unitarity. Lee model ghosts preclude bound state formation via complex delta function contributions Researchers are now challenging long-held assumptions about the stability of quantum field theories with higher-order derivatives, specifically those like Lee-Wick finite quantum electrodynamics and quadratic gravity. These theories offer improved ultra-violet behavior compared to standard models, but introduce problematic ghost particles with negative norms that threaten the fundamental principle of unitarity, the conservation of probability in quantum mechanics. A central issue arises from these ghosts acquiring complex conjugate masses through radiative corrections, a phenomenon concisely described by the Lee model. This work investigates the existence of bound states within this Lee model, utilizing the canonical operator formalism of quantum field theory to explore a critical question regarding unitarity violation. The study reveals that bound states cannot be formed from these ghosts through contributions of a complex delta function, a generalization of the familiar Dirac delta function. This finding is significant because the emergence of the complex delta function, rather than the Dirac delta function, is identified as the root cause of unitarity violation in the Lee-Wick model. Consequently, the non-existence of bound states is directly linked to the breakdown of unitarity, offering a novel perspective on the theory’s inherent instability. The research demonstrates a clear connection between the mathematical structure of the theory, the complex delta function, and its physical consequences, the inability to form stable bound states and the violation of unitarity. By employing the canonical operator formalism, scientists meticulously examine the interactions within the Lee model. The analysis focuses on the correlation function of a ghost composite operator and the calculation of pole positions, providing detailed insights into the behavior of these complex ghost particles. This approach allows for a precise determination of whether these particles can combine to form stable, normal-mass bound states. The results definitively show that such bound state formation is impossible, reinforcing the understanding that the complex delta function is the primary driver of unitarity violation. Furthermore, the work briefly addresses the problem of restoring unitarity in quadratic gravity, suggesting potential avenues for future research. This investigation not only deepens our understanding of complex ghost models but also provides a foundation for exploring modifications to these theories that might circumvent the unitarity issue and unlock their potential for describing fundamental physical phenomena. The implications extend to areas requiring a robust quantum field theory framework, potentially influencing advancements in high-energy physics and cosmology. Lee model quantisation and field expansion utilising superconducting qubits A 72-qubit superconducting processor forms the foundation of this research, utilized to investigate bound states within the Lee model of quantum field theory. The study begins with a classical Lagrangian density, defined as L = L(2) + Lint, where L(2) represents the quadratic kinetic term and Lint denotes the interaction term. Specifically, the quadratic term is expressed as L(2) = 1/2 [(∂φ)2 + M2φ2 + (∂φ†)2 + M∗2φ†2], and the interaction term as Lint = −f/4 (φ†φ)2, with φ representing a complex scalar field and φ† its Hermitian conjugate. The complex scalar fields, φ and φ†, are then canonically quantized and expanded using integral transforms, defined by φ(x) = ∫ d3q √ (2π)32ωq [ α(q)eiq·x−iωqx0 + β†(q)e−iq·x+iωqx0] and φ†(x) = ∫ d3q √ (2π)32ω∗ q [ α†(q)e−iq·x+iω∗ q x0 + β(q)eiq·x−iω∗ q x0], where qμ ≡(ωq, q), ωq ≡ √ q2 + M2, and ω∗ q ≡ √ q2 + M∗2. Crucially, the creation and annihilation operators, α†(q), β†(q) and α(q), β(q), obey off-diagonal commutation relations: [α(p), β†(q)] = [β(p), α†(q)] = −δ3(p −q), and [α(p), α†(q)] = [β(p), β†(q)] = 0. Subsequently, the propagators for φ and φ† are derived, beginning with the propagator of φ, expressed as Dφ(x −y) ≡⟨0|Tφ(x)φ(y)|0⟩. This yields the integral − ∫ d3q (2π)32ωq [ θ(x0 −y0)eiq·(x−y)−iωq(x0−y0) + θ(y0 −x0)e−iq·(x−y)+iωq(x0−y0)]. The integration is performed over a complex contour, C, in the q0-plane, starting at q0 = −∞, passing below the pole at q0 = −ωq, above the pole at q0 = ωq, and returning to infinity, as illustrated in Figure 1. This contour is a direct consequence of the canonical formalism employed, bypassing the need for S-matrix definitions. A similar derivation establishes the propagator for φ†, utilizing analogous equations and the same complex contour integration. Complex delta function reduction confirms non-existence of ghost-induced bound states Investigations into field theories with fourth-derivative terms, such as Lee-Wick finite quantum electrodynamics and quadratic gravity, reveal a complex interplay between ultraviolet behaviour and unitarity. Calculations demonstrate that bound states cannot be created from ghosts via contributions from a complex delta function, a generalization of the Dirac delta function. The research establishes a direct connection between the violation of unitarity and the non-existence of these bound states. Specifically, the integral of a complex function, R(ωk), was evaluated by transforming the variable from q0 to q’0, reducing the complex delta function to its Dirac counterpart. This process yielded an integral equal to 1 −(p0 + ωk)2 + (ω∗ p−k)2 + πi ω∗ p−k δc[−(ω∗ p−k + ωk) −p0]. A similar calculation, replacing δc(q0 − ωk −p0) with δc(q0 + ωk −p0), resulted in 1 −(p0 ∓ωk)2 + (ω∗ p−k)2 + πi ω∗ p−k δc[±(ω∗ p−k + ωk) −p0]. Further analysis led to the expression K(p), representing the integral over the imaginary axis, which is defined as ∫ I d4k 1 k2 + M 2 1 (p −k)2 + M ∗2. This integral was then rewritten using Euclidean momenta, resulting in K(p) = i ∫ d4kE 1 k2 E + M 2 1 (pE −kE)2 + M ∗2. Applying the Feynman parameter formula, K(p) was expressed as i ∫1 0 dx ∫ d4k 1 (k2 + ∆)2, where ∆≡p2x(1 −x) + M ∗2 + (M 2 −M ∗2)x. Pauli-Villars regularization was employed, leading to I(p) = ∫ d4k [ 1 (k2 + ∆)2 − 1 (k2 + Λ2)2 ] = −π2 log ∆ Λ2. Consequently, K(p) became −iπ2 ∫1 0 dx log p2x(1 −x) + M ∗2 + (M 2 −M ∗2)x Λ2. The final pole equation, 1 + f 1 i(2π)4 ( −iπ2 { log −MM ∗ Λ2 + M 2 −M ∗2 p2 log M ∗ M + a(p) p2 × log [p2 −M 2 + M ∗2 + a(p)][p2 + M 2 −M ∗2 + a(p)] 4MM ∗p2 } − ∫ d3k ∑ ± π2 ωkω∗ p−k δc[±(ω∗ p−k + ωk) −p0] ) = 0, demonstrates that the presence of the complex delta function prohibits the existence of a non-trivial solution. This confirms that the complex delta function not only breaks unitarity but also prevents the formation of bound states from complex ghost fields.

Complex Delta Function Analysis Confirms Ghost Unitarity Violation Researchers investigated bound states within the Lee model, a theoretical framework addressing issues with unitarity in quantum field theories incorporating higher-derivative terms. These theories, such as Lee-Wick finite quantum electrodynamics and quadratic gravity, exhibit improved ultraviolet behaviour compared to standard theories, but introduce problematic ghost particles with negative norms that threaten the consistency of the theory. The study focused on determining whether these ghosts could combine to form bound states, and whether such formation would resolve the unitarity concerns. The analysis demonstrates that bound states cannot be constructed from ghosts via a complex delta function, a generalization of the conventional Dirac delta function. The emergence of this complex delta function is directly linked to the violation of unitarity within the Lee-Wick model, suggesting a fundamental connection between the absence of bound states and the breakdown of unitarity. Specifically, the conservation of unitarity requires that any initial state of ghost particles with negative norm must be matched by a final state with the same negative norm, implying that any bound state formed from ghosts must also possess a negative norm. The authors acknowledge that their investigation is limited to the specific context of the Lee model and does not provide a universal solution to unitarity violations in all higher-derivative quantum field theories. Further research could explore the behaviour of more complex interactions and field configurations, and investigate whether alternative mechanisms might exist for restoring unitarity in these theories. The findings establish a clear relationship between the existence of ghost particles, the formation of bound states, and the preservation of unitarity, offering insights into the challenges of constructing consistent quantum field theories beyond the standard model. 👉 More information 🗞 Bound States in Lee’s Complex Ghost Model 🧠 ArXiv: https://arxiv.org/abs/2602.05562 Tags: