Core Concepts

This paper presents a theoretical framework for incorporating quantum field theory (QFT) corrections into calculations of quantum tunnelling amplitudes, addressing the limitations of traditional single-particle quantum mechanics in describing many-particle effects.

Abstract

Zielinski, R., Simenel, C., & McGlynn, P. (2024). Quantum corrections to tunnelling amplitudes of neutral scalar fields. arXiv:2406.18086v1 [hep-th].

This research paper aims to develop a formalism for calculating quantum tunnelling amplitudes that incorporates quantum field theory (QFT) corrections, going beyond the limitations of single-particle relativistic quantum mechanics.

The authors utilize a simplified model of a massive neutral scalar field interacting with both an external scalar field (acting as a potential barrier) and a quantized scalar field. They derive an all-order recursive expression for the loop-corrected scalar propagator, focusing on vertex-corrected Feynman diagrams. A perturbative coupling approximation is then introduced to make the calculations more tractable.

- The authors successfully formulate an integral equation that captures the QFT corrections to tunnelling amplitudes, providing a framework for future quantitative analysis.
- They demonstrate a perturbative approach, treating the coupling to the external field non-perturbatively while handling the quantized field perturbatively, similar to the Furry expansion in QED.

This work lays the theoretical foundation for integrating QFT corrections into quantum tunnelling calculations. While a closed-form solution remains elusive, the derived integral equations and perturbative approximations offer a pathway for future numerical investigations.

This research holds significant implications for understanding tunnelling phenomena in a QFT framework, potentially impacting fields where many-particle effects are crucial, such as high-energy physics and condensed matter physics.

The complexity of the derived integral equations poses challenges for obtaining analytical solutions. Future work should explore numerical methods to solve these equations and quantify the impact of QFT corrections on tunnelling probabilities for various potentials. Additionally, extending this formalism to more realistic and complex QFT Lagrangians, such as QED, would be a valuable next step.

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by Rosemary Zie... at **arxiv.org** 10-10-2024

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Including self-energy terms in the dressed propagator would indeed refine the calculation of tunnelling amplitudes, capturing more intricate quantum effects. Here's how:
Renormalized Mass: Self-energy diagrams directly contribute to the renormalization of the particle's mass. While the paper argues that this doesn't affect the interaction with the external field for transmitted particles, it does impact the overall propagation and, consequently, the tunnelling process. A more accurate mass for the tunnelling particle, accounting for its interactions with the quantum field Φ, would lead to a more precise tunnelling amplitude.
Virtual Particle Effects: Self-energy terms represent the particle's interaction with virtual particles from the quantum field. These virtual particles can briefly modify the properties of the tunnelling particle (like its effective mass or coupling to the external field) within the barrier region. Even though these effects are transient, they can influence the overall tunnelling probability.
Higher-Order Corrections: The paper focuses on a specific class of "vertex-corrected" diagrams. Including self-energy terms would introduce a broader range of higher-order corrections, capturing more complex interactions between the tunnelling particle and the quantum field. These higher-order terms, while typically smaller in magnitude, can become significant in certain regimes or for precise calculations.
However, incorporating self-energy terms introduces significant computational challenges:
Divergences: Self-energy diagrams often involve loop integrals that are ultraviolet divergent, requiring renormalization procedures to extract physically meaningful results. This adds complexity to the calculation.
Integral Equations: The inclusion of self-energy terms would likely turn the already complex integral equation (Eq. 13) into a system of coupled integral equations, significantly increasing the difficulty of finding a solution.
Therefore, while including self-energy terms would lead to a more complete and accurate description of quantum tunnelling, it comes at the cost of increased mathematical complexity.

Yes, the limitations of perturbative methods in describing tunnelling, an inherently non-perturbative phenomenon, could potentially be addressed by employing non-perturbative techniques from other areas of theoretical physics. Here are some promising avenues:
Lattice Field Theory: Lattice field theory provides a non-perturbative way to study quantum field theories by discretizing spacetime onto a lattice. This method is well-suited for strongly coupled systems where perturbative expansions break down. Applying lattice field theory to the tunnelling problem could offer insights into the non-perturbative regime of the interaction with the external field.
Semi-Classical Methods (WKB Approximation): The Wentzel-Kramers-Brillouin (WKB) approximation is a semi-classical method widely used to study tunnelling in quantum mechanics. Extending the WKB approach to incorporate quantum field-theoretic effects could provide a more accurate description of tunnelling in the presence of particle creation and annihilation.
Functional Renormalization Group: The functional renormalization group (FRG) is a powerful non-perturbative technique that allows for the systematic study of quantum field theories at different energy scales. Applying FRG to the tunnelling problem could help to resum important classes of diagrams and capture non-perturbative effects.
Holographic Methods (AdS/CFT Correspondence): In certain cases, the AdS/CFT correspondence, which relates strongly coupled quantum field theories to weakly coupled gravitational theories in higher dimensions, could provide a dual description of tunnelling phenomena. This approach has been successful in studying other non-perturbative aspects of quantum field theories.
It's important to note that applying these non-perturbative techniques to the specific problem of quantum tunnelling in the presence of external fields would require careful adaptation and development of new theoretical tools. However, the potential rewards are significant, as they could lead to a deeper understanding of this fundamental quantum phenomenon.

Incorporating QFT corrections to tunnelling could have profound implications in fields like cosmology and quantum computing, where tunnelling plays a crucial role:
Cosmology:
False Vacuum Decay: The current state of our universe might be metastable, susceptible to decay via quantum tunnelling to a lower energy state (false vacuum decay). QFT corrections to tunnelling rates could significantly alter our understanding of the stability of the universe and the potential for such catastrophic events.
Early Universe Cosmology: Tunnelling processes in quantum field theory are believed to have played a crucial role in the very early universe, potentially driving inflationary periods or phase transitions. Accurate calculations of tunnelling rates, incorporating QFT corrections, are essential for building precise models of the early universe.
Black Hole Physics: Hawking radiation, the process by which black holes evaporate, can be understood as a form of quantum tunnelling near the event horizon. Incorporating QFT corrections could refine our understanding of black hole evaporation rates and their eventual fate.
Quantum Computing:
Qubit Stability: Quantum bits (qubits), the building blocks of quantum computers, are extremely fragile and susceptible to decoherence due to interactions with their environment. Tunnelling processes can contribute to qubit decoherence. Understanding QFT corrections to tunnelling could help in designing more stable qubits and improving the coherence times of quantum computers.
Quantum Annealing: Quantum annealing is a computational technique that leverages quantum tunnelling to find the global minimum of complex energy landscapes. Incorporating QFT corrections could lead to more accurate simulations of quantum annealing processes and potentially enhance their efficiency.
Topological Quantum Computing: Some proposals for topological quantum computers rely on the manipulation of quasiparticles that exhibit exotic tunnelling properties. QFT corrections to tunnelling could be crucial for understanding and controlling these quasiparticles for robust quantum computation.
In summary, incorporating QFT corrections to tunnelling could lead to significant advancements in our understanding of fundamental physics and potentially revolutionize fields like cosmology and quantum computing. However, it also poses significant theoretical and computational challenges that need to be addressed.

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