Worldline Quantum Field Theory

• Worldline Quantum Field Theory is a reformulation of QFT that uses particle path integrals over proper time to represent propagators, determinants, and effective actions.
• It incorporates spin, color, and higher-spin dynamics via Grassmann variables and auxiliary fields, streamlining the treatment of gauge structures and boundary conditions.
• WQFT unifies scattering amplitudes, classical observables, and holographic mappings by reorganizing Feynman diagrams into efficient one-dimensional path integrals.

Worldline Quantum Field Theory (WQFT) denotes a first-quantized reformulation of perturbative quantum field theory in terms of particle path integrals over worldlines and proper time; in more recent work on classical scattering it also denotes a mixed worldline-plus-bulk path integral in which both mediating fields and worldline fluctuations are quantized around straight-line backgrounds. In this framework propagators, determinants, effective actions, amplitudes, heat kernels, and classical observables such as impulse, waveform, and radiated momentum are represented by one-dimensional sigma models or worldline correlators rather than exclusively by second-quantized field integrals (Edwards et al., 2019, Mogull et al., 2020, Wang, 2022).

1. Proper-time representation and first-quantized foundations

The basic worldline representation starts from the Schwinger proper-time identity and rewrites field-theoretic propagators and one-loop determinants as quantum-mechanical path integrals. For a scalar particle in an external Maxwell background, the propagator takes the form

$$D^{x'x}[A] = \int_{0}^{\infty} dT \, e^{-m^{2} T} \int_{x(0) = x}^{x(T) = x'} \mathscr{D}x \, e^{- \int_{0}^{T} d\tau\, \left[ \frac{\dot{x}^{2}}{4} + ie\dot{x} \cdot A(x(\tau)) \right]},$$

while the one-loop effective action is obtained by closing the worldline,

$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$

The interval describes propagation, the circle computes one-loop effective actions through traces, and the infinite line is used for tree amplitudes (Edwards et al., 2019, Corradini et al., 2015, Fecit, 20 Mar 2026).

In this first-quantized language, the worldline parameter is a one-dimensional modulus, and the proper time $T$ plays the role of a Schwinger parameter. Gauge fixing the einbein produces the proper-time integral, and the distinction between periodic, antiperiodic, Dirichlet, or open-line boundary conditions encodes whether one is computing a trace, a propagator, a local heat kernel, or an amplitude building block. The worldline formalism therefore reorganizes perturbation theory rather than changing its physical content; its advantage is that an entire one-loop diagram is often represented by a single path integral, all cyclic orderings of external insertions are naturally combined, and gauge structure is frequently easier to organize (Edwards et al., 2019, Corradini et al., 2015).

A central structural consequence is that effective actions, heat kernels, and amplitudes are all expressed in terms of one-dimensional Green functions and Wick contractions. In scalar theories this yields standard heat-kernel expansions; in modern developments it also supports resummed derivative expansions and strong-background calculations. The thesis literature further emphasizes that this top-down representation has a complementary bottom-up side: one may instead begin with a constrained worldline gauge system and quantize it to recover the target-space field theory (Fecit, 20 Mar 2026).

2. Worldline degrees of freedom, spin, colour, higher spin, and boundaries

Spin is incorporated by adjoining Grassmann worldline variables. For spin-$\tfrac12$ matter the standard spinning-particle action is

$$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$

and the worldline theory has a rigid worldline supersymmetry,

$$\delta x^{\mu} = -2\wp \psi^{\mu}, \qquad \delta\psi^{\mu} = \wp\dot{x}^{\mu}.$$

In second-order form, spinor QED is recovered by squaring the Dirac operator and representing the spin factor by a Grassmann path integral (Edwards et al., 2019, Corradini et al., 2015).

Non-Abelian colour can be encoded by auxiliary worldline colour fields rather than explicit matrix insertions. In the modern formulation one introduces colour variables $\bar c_f,c_f$ and gauges a worldline $U(F)$ symmetry so that the Hilbert space is projected onto a chosen irreducible representation. This makes path ordering emerge automatically and permits worldline descriptions of loops transforming in arbitrary irreducible $SU(N)$ representations (Edwards et al., 2019).

Higher spins arise from constrained worldline systems with extended local supersymmetry. The $O(N)$-extended spinning particle has constraints

$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$0

and quantization yields Bargmann–Wigner-type equations for massless spin $$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$1 fields. More recent work also uses Grassmann-even bosonic oscillators for integer-spin amplitudes and massive vector particles, and extends the same logic to massless and massive spin-2 worldline models (Corradini et al., 2015, Fecit, 20 Mar 2026).

Worldline methods on manifolds with boundary require an additional construction, because one must restrict the path integration domain to worldlines compatible with the boundary condition. For a scalar field confined to the $$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$2-dimensional ball, the boundary problem is converted into a worldline problem on a doubled space with a singular metric, and the heat trace is reconstructed from a direct/indirect image decomposition. The same strategy has been extended to a spinor field in a two-dimensional curved half-plane with MIT bag boundary conditions, where the doubled Hamiltonian contains a projected $$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$3 interaction and the reflected contribution carries a spinorial reflector $$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$4. In both cases the first few Seeley–DeWitt coefficients are recovered, verifying that curved boundaries and mixed boundary conditions can be handled within WQFT (Corradini et al., 2019, Manzo, 2024).

3. Relation to ordinary quantum field theory and scattering amplitudes

A major modern development is the precise matching between worldline correlators and ordinary QFT quantities. In classical black-hole scattering, a worldline path-integral representation of the graviton-dressed scalar propagator can be inserted into the standard time-ordered correlator definition of the S-matrix, producing a direct relation between on-shell scalar-graviton form factors and WQFT expectation values. In that setting the WQFT partition function is identified with the classical eikonal phase,

$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$5

and the deflection follows from

$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$6

This establishes that classical eikonal exponentiation is the exponentiation of connected worldline diagrams (Mogull et al., 2020).

For spinning matter, the natural object matched between WQFT and ordinary QFT is not directly the usual Dirac propagator but the worldline kernel or second-order form of the dressed propagator. In spin-$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$7 QED this requires a symbol map that identifies worldline Grassmann bilinears with gamma-matrix Lorentz generators,

$$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$8

The thesis literature shows that WQFT correlators match amputated dressed kernels, and from there standard QED amplitudes are recovered by the kernel-to-propagator relation. This is why spinning WQFT matches the kernel/second-order formulation more naturally than first-order QED (Kopp, 2023).

This matching also clarifies a recurrent misconception. WQFT is not merely a classical bookkeeping device: tree-level WQFT gives the leading classical contribution because $$\Gamma[A] = \int_{0}^{\infty} \frac{dT}{T} e^{-m^{2}T} \oint \mathscr{D}x \, e^{-\int_{0}^{T} d\tau \left[ \frac{\dot{x}^{2}}{4} + ie \dot{x}\cdot A(x(\tau)) \right]}.$$9-counting is tied directly to the number of internal loops in WQFT diagrams, but if one does not truncate the worldline-loop expansion then quantum contributions are retained. In particular, summing infinite towers of worldline loops reconstructs the internal scalar propagator familiar from the kernel or second-order formulation (Kopp, 2023).

4. Classical scattering, radiation, and post-Minkowskian observables

In the post-Minkowskian program, WQFT treats compact objects as worldlines expanded around straight trajectories,

$T$0

and computes classical observables directly from tree-level one-point functions. The impulse is extracted from the worldline fluctuation through

$T$1

while the radiative field is obtained from $T$2 or its analogues in other theories (Mogull et al., 2020, Jakobsen et al., 2022).

In scalar QED, this logic leads to a classical generating function $T$3 defined by the WQFT path integral,

$T$4

whose connected-diagram expansion directly encodes binary dynamics and from which the scattering angle follows by

$T$5

The notable claim is that, unlike standard eikonal treatments, this connected generating function requires no subtraction of iteration terms generated by exponentiating lower-order contributions. The formalism was worked out through 3PM order in scalar QED, including conservative and radiative pieces (Wang, 2022).

For gravitational scattering, the in-in or Schwinger–Keldysh formulation gives a causal initial-value version of WQFT. In the Keldysh basis, tree-level one-point functions for the classical observables of interest reduce to a simple rule: use the same tree-level WQFT diagrams as in the in-out formalism, but replace all propagators by retarded propagators pointing toward the measured outgoing leg. This yields the complete radiation-reacted impulse and radiated four-momentum at 3PM, as well as the leading 2PM far-field gravitational waveform, including tidal effects for neutron stars (Jakobsen et al., 2022).

The waveform problem illustrates the economy of the method. In leading post-Minkowskian gravitational bremsstrahlung from two spinless bodies, the far-zone time-domain waveform is obtained from only three WQFT diagrams at leading order, and the resulting expression reproduces the classic Kovacs–Thorne result. The same waveform yields the total radiated angular momentum and energy by standard null-infinity flux formulas (Jakobsen et al., 2021).

The same one-point-function machinery has recently been extended from impulse and radiated momentum to total angular-momentum flux. The new observable is the impact-parameter kick $T$6, extracted from the subleading soft behavior of the worldline fluctuation via

$T$7

Its computation is in one-to-one correspondence with the impulse problem in the dynamical region, but zero-frequency gravitons generate a distinct static region. These static contributions are organized by $T$8-point functions called static correlators, which reduce to a simple one-loop integral family. The method produces the $T$9 total flux of angular momentum in gravity and the analogous $\tfrac12$0 result in electromagnetism (Jakobsen et al., 12 May 2026).

5. Worldline holography and emergent higher-dimensional geometry

A distinct line of development shows that WQFT can reorganize an ordinary four-dimensional source generating functional into a five-dimensional field theory. In the worldline representation of a one-loop determinant,

$\tfrac12$1

the proper-time measure is exactly that of $\tfrac12$2, with metric

$\tfrac12$3

The central conceptual move is therefore that Schwinger proper time becomes the fifth holographic coordinate (Dietrich, 2015).

The result is “all-orders worldline holography”: the sources of a quantum field theory over $\tfrac12$4 naturally form a field theory over $\tfrac12$5, and this holds to all orders in the elementary fields. The derivation is performed entirely within quantum field theory and does not select a subset of diagrams. For higher-spin sources of a free scalar theory, the worldline representation yields a Fronsdal-like action on $\tfrac12$6, while the mass of the underlying four-dimensional matter appears as a tachyon profile $\tfrac12$7 (Dietrich, 2015).

The holographic dictionary is implemented by worldline smearing. Translation operators exponentiate to Gaussian smearing operators, so a source is dressed into a bulk profile

$\tfrac12$8

which solves a linearized gradient or Wilson flow equation. In this sense the bulk profile is not imposed externally: boundary sources are extended into the fifth dimension by worldline-induced Wilson or gradient flow. The paper further argues that varying the five-dimensional action with respect to this profile reproduces the standard holographic computation (Dietrich, 2015).

This construction does not rely on large $\tfrac12$9, is not postulated from string theory, and is not a full nonperturbative string dual. Its explicit derivations are carried out mainly for a free scalar matter theory with external vector and higher-spin sources on flat four-dimensional spacetime, and the interacting higher-spin bulk dynamics is not fully developed. A plausible implication is that the worldline formalism exposes a hidden higher-dimensional geometry already present in ordinary QFT source functionals, but in a scope that remains more specific than canonical AdS/CFT (Dietrich, 2015).

6. Double copy, scope, and limitations

WQFT also admits a classical double-copy organization. For worldline bi-adjoint scalar theory, worldline Yang–Mills theory, and worldline dilaton-gravity, the eikonal/free energies can be written schematically as

$$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$0

where the bi-adjoint scalar theory fixes the classical locality kernel $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$1. In this formulation the double copy acts directly on classical observables such as deflection, radiation, and the eikonal/free energy, and the equivalence to the double copy of the classical limit of quantum amplitudes has been shown explicitly up to next-to-leading order (Shi et al., 2021).

For spinning particles the double copy becomes more restrictive. Using supersymmetric worldline models, the gauge-theory side is the $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$2 spinning particle in a Yang–Mills background, while enforcing worldline supersymmetry and $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$3 R-symmetry on the double-copied integrands identifies the gravity-side theory as the $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$4 particle coupled to dilaton-gravity. The spin map is

$$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$5

with the Grassmann nature of spin playing a decisive role. If the double copy is performed without preserving SUSY and R-symmetry, the antisymmetric $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$6-field also couples to the worldline (Comberiati et al., 2022).

The breadth of WQFT should not obscure its domain restrictions. Several constructions are explicitly limited: the spinning $$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$7 gauge-theory worldline captures only linear order in spin per copy; the amplitude matching for gravity with spin-$$S[x, \psi | A] = \int_{0}^{T}d\tau\left[ \frac{\dot{x}^{2}}{4} + \frac{1}{2}\psi \cdot \dot{\psi} + ie\dot{x} \cdot A(x(\tau)) - ie \psi^{\mu} F_{\mu\nu}(x(\tau))\psi^{\nu} \right],$$8 matter is verified explicitly only at one-graviton order; the holographic construction is developed mainly for free scalar matter with external vector and higher-spin sources; the confined-scalar and bounded-spinor boundary constructions are verified in particularly tractable geometries; and the massive spin-2 worldline model in curved space is consistent on Ricci-flat backgrounds only (Kopp, 2023, Dietrich, 2015, Corradini et al., 2019, Manzo, 2024, Fecit, 20 Mar 2026).

Taken together, these developments show that WQFT is simultaneously a reformulation of perturbative QFT, a computational framework for classical PM observables, and a source of structural relations connecting first quantization, amplitudes, double copy, heat kernels, boundary-value problems, and holography. This suggests a unifying interpretation: the worldline description does not merely replace Feynman graphs by a different notation; it reorganizes QFT in a way that makes proper-time geometry, constraint algebras, causal observables, and source-theoretic structure manifest (Edwards et al., 2019, Mogull et al., 2020).