Scattering Amplitudes and the Navier-Stokes Equation (2010.15970v1)

Abstract: We explore the scattering amplitudes of fluid quanta described by the Navier-Stokes equation and its non-Abelian generalization. These amplitudes exhibit universal infrared structures analogous to the Weinberg soft theorem and the Adler zero. Furthermore, they satisfy on-shell recursion relations which together with the three-point scattering amplitude furnish a pure S-matrix formulation of incompressible fluid mechanics. Remarkably, the amplitudes of the non-Abelian Navier-Stokes equation also exhibit color-kinematics duality as an off-shell symmetry, for which the associated kinematic algebra is literally the algebra of spatial diffeomorphisms. Applying the double copy prescription, we then arrive at a new theory of a tensor bi-fluid. Finally, we present monopole solutions of the non-Abelian and tensor Navier-Stokes equations and observe a classical double copy structure.

• The paper introduces a novel framework applying scattering amplitude methods to classical fluid dynamics, unveiling universal infrared structures similar to gauge theories.
• The paper demonstrates on-shell constructibility of the NSE and NNSE via recursion relations, simplifying the study of complex fluid interaction dynamics.
• The paper reveals color-kinematics duality and extends the double copy construction to a tensor bi-fluid theory, opening new avenues for unified analysis of fluid and gauge interactions.

Scattering Amplitudes and the Navier-Stokes Equation: A Treatise

The paper "Scattering Amplitudes and the Navier-Stokes Equation" authored by Clifford Cheung and James Mangan, presents a meticulous theoretical exploration of the scattering amplitudes for fluid dynamics as governed by the Navier-Stokes equation (NSE) and its non-Abelian extension, which they denominate as the non-Abelian Navier-Stokes equation (NNSE). This work is characterized by the novel application of quantum field theory techniques to classical fluid dynamics, framing the problem within the broader context of modern scattering amplitude methods.

Overview of Theoretical Development

The authors address the intricate problem of turbulence inherent to fluid dynamics, acknowledging its analogy to strong coupling problems in quantum field theories like quantum chromodynamics (QCD). Without resorting to an action-based formulation, they articulate how scattering amplitudes can be constructed using the S-matrix method. Notably, this approach has successfully resolved complex dynamics in gauge theories and general relativity attributable to on-shell recursion relations, exploiting symmetries like Poincare invariance.

In their analysis, Cheung and Mangan rigorously derive scattering amplitudes using the equations of motion from the NSE and NNSE. The NNSE is a particularly striking development in this text, achieved by incorporating both structure constants from a gauge theory and projecting these onto the framework of Navier-Stokes dynamics. This progression from classical fluid mechanics to a colored fluid model is both subtle and ambitious, providing insights that extend beyond classical Navier-Stokes theory.

Key Results

1. Universal Infrared Structures: The paper identifies that the scattering amplitudes for the NSE exhibit universal infrared properties analogous to quantum field theoretic concepts such as the Weinberg soft theorem and Adler zeros. These features are critical, as they influence the understanding of gauge symmetries within fluid dynamics.
2. On-Shell Constructibility: The article demonstrates the on-shell constructibility of the NSE and NNSE through recursion relations. This asserts that all higher-point amplitudes can be recursively generated from lower-point ones, simplifying the paper of complex interaction scenarios and offering parallels to techniques used in calculating tree-level amplitudes in gauge theories.
3. Color-Kinematics Duality: The NNSE surprisingly exhibits color-kinematics duality off-shell, a profound symmetry usually observed in gauge theories, suggesting the kinematic algebra mimics spatial diffeomorphisms.
4. Double Copy and Tensor Bi-Fluid Theory: By applying the double copy construction—a method utilized to relate gauge and gravity theories—the authors extend the NNSE to a tensor Navier-Stokes equation (TNSE), conceptualizing dynamics of a bi-fluid system, offering a broader theoretical structure to paper such interactions.
5. Monopole Solutions: They further explore classical solutions within these frameworks, finding intriguing monopole-like structures and highlighting a classical double copy similarity.

Theoretical and Practical Implications

Theoretical implications of this paper are profound. By applying techniques traditionally reserved for particle physics to fluid dynamics, this work bridges an interdisciplinary gap. The realization of color-kinematics duality in classical non-Abelian fluids suggests potential unified vistas in the paper of fluid dynamics, chromodynamics, and perhaps gravitational systems.

Practically, insights drawn from such abstract theoretical analysis could eventually influence computational fluid dynamics approaches, especially in turbulent regimes where classical solvers struggle. Though speculative, future developments in AI-assisted fluid modeling might benefit from this refined theoretical foundation.

Future Directions

Future research could delve into understanding turbulence through the lens of high-multiplicity scattering processes using the outlined formalism. It may also explore supersymmetric or extended-string theories within this context, seeking connections to other double copy constructs. Moreover, integrating classical and amplitude-based double copies could reveal new aspects of the classical double copy concept, potentially impacting the comprehension of known fluid structures and solutions.

In conclusion, Cheung and Mangan's work invites both appreciation and further inquiry into the intersection of classical fluid mechanics and modern quantum field theory, paving the way for future explorative studies within this fertile domain.