Osterwalder-Schrader and Euclidean Spinor Fields

Key Takeaways

• OS method doubles spinor degrees of freedom for Euclidean rotation
• ψ and ψ̄ become independent fields (ψ1, ψ2) Euclidean
• Reconstruction yields Minkowski Wightman functions despite extra structure
• Approach fails for chiral Weyl fermions in Euclidean QFT
• Weyl fields would require quadrupling degrees, complicating simulations

Summary

The 1972 Osterwalder‑Schrader framework tackles the long‑standing problem of Wick rotating spinor fields by introducing a pair of independent fermionic variables, effectively doubling the degrees of freedom when moving from Minkowski to Euclidean space. Their construction preserves the Dirac adjoint relation only in Minkowski signature, using a reflection operator that swaps the two fields and inserts a gamma‑matrix factor. While the method successfully reconstructs Wightman functions and the physical state space, it breaks down for chiral Weyl fermions, which would require a further doubling of fields. This limitation raises questions for Euclidean formulations of the Standard Model.

Pulse Analysis

Wick rotation is a cornerstone of quantum field theory, allowing calculations in a mathematically tractable Euclidean regime before analytically continuing back to physical Minkowski space. For scalar fields the Osterwalder‑Schrader (OS) axioms provide a rigorous bridge, but extending this to spinors introduces subtleties. In 1972 OS proposed treating the Dirac spinor and its adjoint as two separate Grassmann fields, ψ₁ and ψ₂, thereby doubling the fermionic degrees of freedom. This maneuver restores reflection positivity and enables a reconstruction theorem that reproduces the original Minkowski correlators, albeit with an added γ₀ factor that swaps left‑ and right‑handed components.

The OS construction hinges on a modified reflection operator that not only reverses Euclidean time but also interchanges ψ₁ and ψ₂. By abandoning the traditional Dirac adjoint relation in Euclidean space, the framework sidesteps the incompatibility of complex conjugation with Lorentzian spinor representations. The resulting Euclidean action remains local and amenable to lattice discretization, making it attractive for non‑perturbative studies of gauge theories with fermions. However, the price paid is a proliferation of field components, which can inflate computational costs and obscure the physical interpretation of intermediate steps.

A more pressing challenge emerges when the theory involves chiral Weyl fermions, the building blocks of the Standard Model. Because a single Weyl spinor lacks a Dirac partner, the OS doubling is insufficient; a second duplication would be required to preserve chirality, effectively quadrupling the field content. This complication undermines straightforward Euclidean simulations of neutrinos or other massless chiral particles and motivates alternative approaches, such as the right‑handed spacetime formulation recently proposed by the author. Resolving these issues is critical for advancing lattice QCD, electroweak precision studies, and beyond‑Standard‑Model explorations that rely on accurate Euclidean treatments of spinor dynamics.