Worldline formalism in phase space (2509.06058v1)

Abstract: We implement the worldline formalism in phase space to compute scattering amplitudes. First, the Feynman rules exhibit several useful universal features, reflecting elements of the symplectic geometry of the phase space target. Next, noncompact worldline topologies automate LSZ reductions in accordance with the boundary conditions available in phase space, provided correct identifications of the associated moduli spaces. Further, employing noncanonical coordinates cubicizes the Feynman rules and manifests gauge invariance. As a result, our phase space implementation could provide a framework optimized for computing scattering amplitudes while retaining the nice features of the original formalism. For an explicit demonstration, we compute the multi-photon Compton amplitudes up to six points in the classical limit. Compton amplitudes in Yang-Mills theory and gravity are also computed in a uniform fashion by supposing the backgrounds of nonlinearly superposed plane waves.

• The paper introduces a phase space reformulation of the worldline formalism that leverages symplectic geometry to derive universal Feynman rules.
• It distinguishes between Hamiltonian and symplectic perturbation theories, enabling manifest gauge invariance and automated LSZ reductions.
• The approach efficiently computes multi-photon Compton amplitudes in QED, Yang-Mills, and gravity, with implications for gravitational wave physics.

Worldline Formalism in Phase Space: A Technical Analysis

The paper "Worldline formalism in phase space" (2509.06058) presents a comprehensive reformulation of the worldline approach to quantum field theory (QFT) amplitudes, recasting it in the language of phase space and symplectic geometry. The work systematically develops the Feynman rules for perturbation theory in phase space, clarifies the role of worldline topologies in automating LSZ reductions, and demonstrates the practical computation of multi-photon Compton amplitudes in QED, Yang-Mills, and gravity. The formalism is shown to be both conceptually and computationally advantageous, especially for classical and eikonalized limits relevant to gravitational wave physics.

Symplectic Sigma Models and Phase Space Actions

The foundation of the approach is the identification of the worldline path integral as a sigma model into a symplectic manifold $(P, \omega)$, with the action

$$S[\zeta] = \int d\tau \left( \theta_i(\zeta) \dot{\zeta}^i - H(\zeta) \right),$$

where $\theta$ is the symplectic potential and $H$ the Hamiltonian. The choice of polarization (i.e., the boundary conditions and representation) is encoded in $\theta$, and the symplectic structure $\omega = d\theta$ governs the Poisson brackets and propagators.

A key technical distinction is made between Hamiltonian perturbation theory (canonical coordinates, interactions in $H$) and symplectic perturbation theory (noncanonical coordinates, interactions in $\omega$). The latter, especially when using kinetic momenta, leads to manifestly gauge-invariant Feynman rules and cubicizes the worldline vertices, eliminating spurious quadratic couplings present in canonical formulations.

Universal Feynman Rules in Phase Space

The paper derives all-order Feynman rules for perturbation theory in phase space, with the following universal features:

• Propagator: Encodes the free theory Poisson bracket, with the Green's function structure determined by the symplectic form.
• Vertices: Two classes—Hamiltonian (from $H'$) and symplectic (from $\omega'$), with the latter further split into regular and pinched types.
• Valence-One Vertices: Directly correspond to the leading corrections to the classical equations of motion, with the symplectic vertex yielding a Lorentz-force-like structure.
• Pinched Vertices: Implement contact interactions by canceling propagators, crucial for resumming the Poisson structure in the interacting theory.

Worldline Topologies and LSZ Reduction

A central technical advance is the explicit connection between worldline topology and the automation of LSZ reductions:

• Interval ($[0,1]$): Yields off-shell propagators; both endpoints are in the bulk.
• Half-Line ($[0,\infty)$): One endpoint at infinity, corresponding to a single LSZ reduction; directly computes bulk-to-boundary amplitudes.
• Full Line ($(-\infty, \infty)$): Both endpoints at infinity, corresponding to fully on-shell amplitudes; the path integral is modded out by the infinite volume of residual translations.

This correspondence is made precise by analyzing the moduli spaces of one-dimensional metrics (einbeins) and their residual gauge redundancies. The half-line topology is shown to be optimal for practical amplitude computations, as it requires only a single LSZ reduction and directly yields amplitudes in the momentum basis.

Figure 1: Various worldline topologies for particle propagation in a sandwich geometry, illustrating the correspondence between interval, half-line, and full-line worldlines and the degree of LSZ reduction.

Efficient Computation of Scattering Amplitudes

The formalism is applied to the computation of multi-photon Compton amplitudes in QED, with explicit Feynman rules derived for the phase space worldline action. The key features include:

• Direct Momentum Representation: The path integral is performed with mixed boundary conditions (position at the initial point, momentum at infinity), eliminating the need for post-hoc Fourier transforms.
• Manifest Gauge Invariance: All vertices are constructed from field strengths (bivector polarizations), and gauge invariance is preserved at every step.
• Automated LSZ Reduction: The worldline topology ensures that only a single amputation is required, and the remaining on-shell limit is straightforward.
• Scalability: The approach is demonstrated up to six-point amplitudes, with the combinatorics and tensor structures handled efficiently via the symplectic Feynman rules.

The explicit expressions for the $n$-photon Compton amplitudes are given in terms of Lorentz-force-like contractions of field strengths and momenta, with the eikonalized propagators appearing naturally from the worldline integrals.

Extension to Yang-Mills and Gravity

The formalism is extended to nonabelian gauge theory and gravity by identifying an isomorphism at the level of symplectic structures:

• Yang-Mills: The color charge is implemented as a composite variable in phase space, and the symplectic perturbation theory yields cubic worldline couplings to the gauge field.
• Gravity: The kinetic momentum is identified via the tetrad, and the symplectic structure is perturbed by the torsion two-form, paralleling the gauge theory case.
• Color-Kinematics Duality: The structural similarity between the Feynman rules for Yang-Mills and gravity is made explicit, with the double copy structure manifest at the level of worldline vertices.

The computation of two-point Compton amplitudes in both theories is shown to follow from the same master formulae, with the only difference being the replacement of color factors by kinematic ones.

Implications and Future Directions

The phase space worldline formalism developed in this work has several significant implications:

• Computational Efficiency: The approach streamlines the calculation of tree-level amplitudes, especially in the classical and eikonal limits, by minimizing the need for post-processing and manifesting gauge invariance.
• Theoretical Clarity: The explicit connection between worldline topology, boundary conditions, and LSZ reduction provides a first-quantized understanding of amplitude construction, with clear geometric and algebraic underpinnings.
• Versatility: The formalism is applicable to in-in and in-out frameworks, quantum corrections, and can be extended to spinning particles and more general backgrounds.
• Double Copy and Classical Limit: The symplectic perturbation theory provides a natural language for exploring color-kinematics duality and the double copy at the level of worldline actions, with potential applications to gravitational wave physics and the effective field theory of compact objects.

Conclusion

The paper establishes a robust and general framework for the computation of scattering amplitudes via the worldline formalism in phase space. By leveraging symplectic geometry, noncanonical coordinates, and a careful analysis of worldline topologies, the approach achieves both conceptual clarity and practical efficiency. The results are directly relevant for high-multiplicity classical amplitudes in gauge theory and gravity, and the formalism is poised for further development in quantum corrections, double copy structures, and the treatment of spinning and extended objects.