sum-over-paths approach, transition amplitudes are computed by summing over all possible trajectories a particle can take. Although it is most extensively applied to nonrelativistic quantum mechanics,[@KleinertQM] for which it was originally designed, the worldline approach also found applications in the context of relativistic systems.[@Feynman50; @proptime] (For overviews and references to a host of applications in high energy physics, see Refs. \[\].) In these notes, we explore its use in the theory of critical phenomena.

The possibility to adapt Feynman’s sum-over-paths approach to describe phase transitions derives from particle-field duality. Featuring the particle content of the system, the spacetime approach provides a completely equivalent alternative to the quantum field theoretic description. Although by no means the only successful method,[@FisherRMPII] field theory is frequently used to describe critical phenomena,[@Amit; @Dimo; @KlSF] whether the fluctuations are thermal as in phase transitions taking place at finite temperature, or quantum as in zero-temperature transitions. Particle-field duality then opens the possibility for a spacetime approach to critical phenomena. In the context of classical critical phenomena, the line objects described by the field theory are current lines.

Though not formulated as such, the high-temperature expansion of spin models[@Stanley; @Polyakov] probably provides the first example of the spacetime 