insight - Computational Complexity - # Learning the Modulus-Phase Relation in Elastic 2→2 Scattering

Core Concepts

Neural Operators can learn the non-trivial relationship between the modulus and phase of scattering amplitudes in elastic 2→2 scattering, without directly using the unitarity constraint.

Abstract

The authors use Fourier Neural Operators (FNOs) to study the relation between the modulus and phase of amplitudes in 2→2 elastic scattering at fixed energies. Unlike previous approaches, they do not employ the integral relation imposed by unitarity, but instead train FNOs to discover it from many samples of amplitudes with finite partial wave expansions.
When trained only on true samples, the FNO correctly predicts (unique or ambiguous) phases of amplitudes with infinite partial wave expansions. When also trained on false samples, it can rate the quality of its prediction by producing a "fidelity index". The authors observe that the value of this index is strongly correlated with the violation of the unitarity constraint for the predicted phase, and present examples where it delineates the boundary between allowed and disallowed profiles of the modulus.
The authors' application of FNOs is unconventional, involving a simultaneous regression-classification task, and emphasizes the role of statistics in ensembles of Neural Operators. They comment on the merits and limitations of the approach and its potential as a new methodology in Theoretical Physics.

Stats

The modulus B(z) and sine of the phase sin φ(z) are used as the input and output of the Neural Operators, respectively.

Quotes

"Neural Operators can generalize non-trivially beyond their training set, learning important properties about the structure of the system."
"The value of the fidelity index is strongly correlated with the violation of the unitarity constraint for the predicted phase."

Key Insights Distilled From

by V. Niarchos,... at **arxiv.org** 04-24-2024

Deeper Inquiries

The insights gained from using Neural Operators to study the relation between the modulus and phase of amplitudes in 2→2 elastic scattering can be extended to more complex quantum field theories by adapting the methodology to handle higher-order scattering processes and more intricate interactions.
Higher-order Scattering Processes: The approach can be applied to study amplitudes in processes involving more particles, such as 2→3 or 3→3 scattering. By training Neural Operators on samples of amplitudes with different particle configurations, the model can learn the underlying patterns and constraints present in the scattering amplitudes.
Non-Elastic Scattering: Extending the analysis to non-elastic scattering processes, where particles can change their identities during the interaction, would provide valuable insights into the dynamics of quantum field theories beyond elastic scattering.
Incorporating Quantum Field Theory: By incorporating principles from quantum field theory, such as gauge symmetries and renormalization, the Neural Operators can be trained to capture the intricate structures present in quantum field theories and their scattering amplitudes.
Non-Perturbative Effects: Exploring the application of Neural Operators to study non-perturbative effects in quantum field theories, such as confinement in quantum chromodynamics, can provide new perspectives on the behavior of particles at strong coupling regimes.
By adapting the Neural Operator framework to handle these more complex scenarios, researchers can gain deeper insights into the behavior of quantum field theories beyond simple elastic scattering processes.

The fidelity index can indeed be used to constrain the space of allowed scattering amplitudes in a systematic way by providing a measure of confidence in the predictions made by the Neural Operators. Here's how the fidelity index can be utilized for constraining scattering amplitudes:
Validity Assessment: The fidelity index can serve as a metric to assess the validity of predicted scattering amplitudes. By setting a threshold for the fidelity index, one can distinguish between reliable predictions that adhere to the underlying constraints and less reliable predictions that may violate physical principles.
Constraint Enforcement: Researchers can use the fidelity index to enforce constraints derived from physical principles, such as unitarity, causality, and analyticity. Predictions with high fidelity indices can be considered as satisfying these constraints, while those with low fidelity indices may indicate potential violations.
Model Selection: When exploring different theoretical models or parameter spaces, the fidelity index can guide the selection of models that produce predictions with high confidence levels. This can help in narrowing down the space of allowed scattering amplitudes based on the fidelity of the predictions.
Quantifying Uncertainty: The fidelity index can also be used to quantify the uncertainty in the predictions, providing a measure of how confident one can be in the validity of the results. This can aid in robust decision-making and interpretation of the predicted scattering amplitudes.
By incorporating the fidelity index into the analysis, researchers can systematically constrain the space of allowed scattering amplitudes and enhance the reliability of the predictions made by the Neural Operators.

The unconventional application of Neural Operators that combines regression and classification tasks can benefit various physical systems beyond 2→2 elastic scattering. Here are some other physical systems that could benefit from this approach:
Particle Physics: Studying particle interactions in high-energy physics experiments, such as collider experiments, where complex scattering processes involving multiple particles need to be analyzed and understood.
Condensed Matter Physics: Investigating the behavior of condensed matter systems, such as quantum materials and superconductors, where the relationship between different phases and excitations can be studied using Neural Operators.
Astrophysics and Cosmology: Analyzing cosmological data and astrophysical phenomena, such as gravitational wave signals and cosmic microwave background radiation, to extract information about the universe's evolution and structure.
Biophysics: Understanding biological systems at the molecular level, including protein folding, molecular interactions, and genetic processes, where the application of Neural Operators can aid in modeling complex biological phenomena.
Materials Science: Predicting material properties, such as electronic structures, band gaps, and phase transitions, by training Neural Operators on experimental and computational data to accelerate materials discovery and design processes.
By applying Neural Operators to these diverse physical systems, researchers can gain new insights, make accurate predictions, and uncover hidden structures in complex phenomena across different branches of physics.

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