Worldline Formalism in Phase Space

• Worldline formalism in phase space is a first-quantized approach that maps a one-dimensional worldline to a symplectic target, encoding both position and momentum dynamics.
• It employs noncanonical coordinate transformations to cubicize Feynman rules, naturally enforce LSZ reduction, and reveal manifest gauge invariance.
• The method unifies amplitude computations across QED, Yang–Mills, and gravity by leveraging symplectic geometry to derive universal propagators and interaction vertices.

The worldline formalism in phase space is a first-quantized approach to quantum dynamics and quantum field theory, implemented as a worldline sigma model mapping a one-dimensional manifold (the worldline) into a symplectic target which encodes the phase space. This framework generalizes conventional worldline path integrals by formulating dynamics directly in phase space, i.e., on a manifold equipped with a symplectic structure given by a closed two-form $\omega = d\theta$. The action takes the generic form

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

where $z^i$ combines positions and momenta (or other phase space coordinates), $\theta$ is a potential for the symplectic form, and $H$ is the Hamiltonian. The formalism supplies universal Feynman rules dictated by the symplectic geometry, admits computational advantages via noncanonical coordinates, and can natively reproduce on-shell amplitude structures through appropriate worldline topologies and boundary conditions. Below, the main structural and methodological aspects are outlined.

1. Universal Structure of Feynman Rules and Symplectic Geometry

The phase space worldline formalism derives Feynman rules by expanding the action $S[z]$ about a free saddle (with $(\theta^0, \omega^0, H^0)$) and treating all interactions — from both $H$ and $\theta$ — perturbatively.

• Propagators: The quadratic part of $S[z]$ determines the free propagator, universally given by the inverse of the free symplectic form,

$$\langle z^i(\tau_1) z^j(\tau_2)\rangle = (\omega^{0\,-1})^{ij} \cdot \Theta(\tau_1, \tau_2),$$

where $\Theta$ is a Green’s function reflecting the topology (interval, half-line, full line).

• Vertices: Interaction vertices split into:
  • Hamiltonian vertices: Arise from Taylor expanding $H(z)$ about the background, coupling to fluctuation fields via derivatives $H_{,i_1\ldots i_n}$;
  • Symplectic vertices: Arise from expanding $\theta(z)$ and hence $\omega(z)$, with the "valence one" (linear) symplectic vertex

  $$V_{\text{symp}}^{(1)} \sim (\omega'_{ij}) \dot{\delta z}^j,$$

  where $\omega' = \omega - \omega^0$. - The symplectic geometry, via the Poisson bracket defined by $(\omega^0)^{-1}$,

  $$\{f,g\}^0 = (\omega^{0\,-1})^{ij} \partial_i f\, \partial_j g,$$

  is thus intrinsic to both the propagation and interaction structure.

• Universality: All diagrams, regardless of interaction details, are constructed from these building blocks governed by $(\theta, \omega, H)$ in direct analogy with underlying symplectic geometry of phase space.

2. Worldline Topology and LSZ Reduction

A salient feature of the phase-space worldline approach is its treatment of external states and the natural embedding of LSZ (Lehmann–Symanzik–Zimmermann) reduction:

• Interval ($[0,1]$): Path integration with fixed endpoints computes bulk-to-bulk propagators (off-shell Green functions).

• Half-line ($[0,\infty)$): One end lies at a finite position state, the other at momentum $\tau\to\infty$ (asymptotic), yielding a partition function

$$Z(p_2, x_1) = \mathcal{K}(p_2, \tau\to\infty \mid x_1, \tau=0),$$

automatically enforcing on-shell boundary conditions at $\tau\to\infty$ and thus automating LSZ reduction for scattering amplitudes.

• Full line ($(-\infty,\infty)$): Both ends represent asymptotic momentum eigenstates; translation invariance introduces a trivial infinite moduli (volume) factor, accounted for by dividing by $\bar{\delta}(0)$.

• Moduli space: On noncompact topologies (half-/full-line), the moduli space for the worldline “metric” (einbein) is trivial after gauge fixing — no integration over proper time remains.

This ties boundary data (position/momentum) and moduli space structure directly to external state choice, making amplitude extractions natural.

3. Noncanonical Coordinates: Cubicization and Manifest Gauge Invariance

The formalism achieves significant simplification and manifest gauge invariance by performing a field redefinition:

• Noncanonical transformation: For massive QED with minimal coupling, transition from canonical momentum $P$ to “kinetic momentum” $p = P - qA(x)$ yields a new action,

$$S[x,p] = \int d\tau [p_\mu \dot{x}^\mu + qA_\mu(x)\dot{x}^\mu - \tfrac{1}{2}(p^2 + m^2)],$$

where now the Hamiltonian remains quadratic (free), and all background interactions reside in $\theta$ (via $A$).

• Feynman rule cubicization: In this “kinetic momentum” frame,

  • The photon–particle vertex is strictly linear in $A$, as all couplings arise from the symplectic term;
  • Higher-order photon couplings are combinatorically and algebraically simpler, facilitating higher-point computations.
• Gauge invariance: As $A \to A + d\lambda$ shifts $\theta$ by an exact form, the symplectic form $\omega$ — and thus all propagators and vertices — remains invariant:

$$\omega = dp_\mu \wedge dx^\mu + q dA(x) = dp_\mu \wedge dx^\mu + q F,$$

with $F = dA$ manifestly gauge-invariant.

• Extension to gravity/Yang–Mills: Analogous construction via noncanonical frames, with gravitational and Yang-Mills interactions encoded as deformations of $\theta$ and hence $\omega$.

4. Computation of Scattering Amplitudes

The phase-space worldline formalism yields a highly systematic and geometrically transparent construction of scattering amplitudes:

• Electromagnetic multiphoton Compton amplitudes: Using the half-line topology (bulk-to-boundary propagation) and expanding around the free worldline with fluctuations $\delta z$,
  • Vertices include both Hamiltonian derivatives (from $H$) and symplectic terms (from derivatives of $A$, i.e., $F$).
  • Amplitudes, such as the $n$-photon Compton amplitudes, appear as explicit sums over tree diagrams featuring propagators $(\omega^{0\,-1})^{ij}$ and the aforementioned vertices, with automized on-shell projection.
• Symplectic "pinching": Pinched symplectic vertices (arising when a propagator shrinks to zero) are crucial for obtaining correct on-shell amplitudes and correspond, algebraically, to enforcing physical-state conditions.
• Generalization to nonabelian and gravitational backgrounds:
  • For Yang-Mills: Couplings arise via $q_aA^a(x)$; color-ordered structures and additional vertices are handled diagrammatically, with analogous cubic simplifications.
  • For gravity: Adopting an orthonormal frame, the canonical-to-kinetic-momentum shift is implemented via $p$ and $e(x)$, with $de$ encoding gravitational field strengths in the symplectic sector.
  • Computing on pure plane-wave backgrounds (nonlinear superpositions) demonstrates that the phase-space methodology produces uniform results across QED, Yang-Mills, and gravity.

5. Comparison to Related Approaches and Structural Advantages

The phase-space worldline formalism, as implemented in this framework, exhibits several structural advances and contrasts with other phase-space and worldline approaches:

Feature	Phase-Space Worldline Formalism	Configuration-Space Path Integral/Worldline	Wigner–Moyal Formalism
Propagators	Encoded via $\omega^{-1}$	Green’s function for second-order operator	Not applicable
Vertices	Symplectic+Hamiltonian expansion	Potential expansions, path-ordered	Noncommutativity encoded in $\star$-product
Gauge invariance	Manifest with noncanonical coords	Varies (may require gauge fixing, ghosts)	Not manifest
External state handling	Boundary conditions: positions/momenta; topology automates LSZ	LSZ reduction required	Not applicable
Algebraic structure	Poisson brackets/symplectic geometry explicit	Operator algebra; commutators	Moyal bracket; pseudo-probabilities
Multiplicity scaling	Simple tree structure, cubic vertices	Combinatorically complex at high order	Not directly relevant

This structural approach tightly unifies algebraic, geometric, and physical aspects of amplitude computations.

6. Applications and Further Developments

• Efficient amplitude computation: Explicit computation provided for QED Compton amplitudes up to six external photons, as well as analogous amplitudes in Yang-Mills and gravitational theories, all using the same formalism and diagrammatics.
• Classical limit/eikonal regime: The simplification of denominators in the classical limit (e.g., to $1/K$ for multiphoton denominators) recovers eikonal-like amplitudes, with immediate connection to worldline effective field theory expansions of classical observables.
• Gauge and gravitational isomorphism: The formalism makes apparent isomorphisms between gauge and gravitational couplings at the level of symplectic data, facilitating unified treatments of amplitude calculations in different fundamental interactions.

7. Summary

The worldline formalism in phase space constructs a sigma model with action $$S[z] = \int d\tau\, (\theta_i(z)\,\dot{z}^i - H(z))$$ on a symplectic manifold, yielding universal Feynman rules governed by symplectic geometry, with propagators and vertices determined respectively by the inverse symplectic form and derivatives of $\theta$ and $H$. The choice of worldline topology (interval, half-line, full line) and associated boundary conditions provides direct access to physical on-shell amplitudes, natively implementing LSZ reductions. Noncanonical coordinates cubicize the Feynman rules and reveal manifest gauge invariance. Amplitudes in QED, Yang-Mills, and gravity can all be computed within a single, geometrically motivated framework. This approach is well-suited to both the analytic and numerical calculation of high-multiplicity and classical observables in quantum field theory (Kim, 7 Sep 2025).