Lorentz versus Euclidean Symmetry

Key Takeaways

• Wick rotation maps SO(3,1) symmetry to SO(4) symmetry.
• Euclidean OS reconstruction retains only SO(3) subgroup.
• Analytic continuation via extended tube yields single-valued holomorphic functions.
• Recovering Lorentz representations from Euclidean data is nontrivial.
• Spinor continuation remains an unresolved challenge.

Summary

The article explains how Wick rotation swaps the Lorentz symmetry SO(3,1) of Minkowski quantum field theory for the Euclidean rotation group SO(4), and how the reverse process is more subtle. It shows that Osterwalder‑Schrader (OS) reconstruction in Euclidean space breaks SO(4) down to an SO(3) subgroup because an imaginary‑time direction must be chosen. Analytic continuation of Wightman and Schwinger functions via the extended tube and the Bargmann‑Hall‑Wightman theorem yields a single‑valued holomorphic result linking the two symmetry regimes. While scalar fields are well understood, extending these ideas to spinors remains an open problem.

Pulse Analysis

Wick rotation is the mathematical bridge that connects the Lorentzian world of real‑time quantum field theory with its Euclidean counterpart used in many non‑perturbative calculations. By analytically continuing Wightman functions into complex spacetime, the original SO(3,1) invariance is transformed into an SO(4) rotation symmetry on the Euclidean slice. The Bargmann‑Hall‑Wightman theorem guarantees that this continuation is single‑valued across an extended tube region, allowing physicists to treat Euclidean correlators as legitimate representations of the underlying Minkowski theory.

The Osterwalder‑Schrader reconstruction formalism, however, introduces a crucial asymmetry. To recover a physical Hilbert space from Euclidean Schwinger functions one must select an imaginary‑time direction, which reduces the full SO(4) invariance to its SO(3) stabilizer. This symmetry breaking is not merely a technicality; it shapes how lattice simulations interpret correlation functions and how they project back onto Lorentz‑invariant observables. Representation‑theoretic analyses by Seiler, Klein‑Landau, and Frohlich‑Osterwalder‑Seiler detail the intricate steps required to lift the residual SO(3) action to a full SO(3,1) representation on the reconstructed state space.

While scalar fields fit neatly into this framework, spinor fields pose deeper challenges. The analytic continuation of fermionic degrees of freedom must respect both spinor representations of the Lorentz group and the doubled SU(2) structure of Spin(4). Ongoing research investigates whether one of the Euclidean SU(2) factors can survive as an internal symmetry after Wick rotation, a prospect that could reshape our understanding of symmetry emergence in quantum field theory. Addressing these spinor issues will be pivotal for extending rigorous Euclidean methods to realistic gauge theories and for ensuring that lattice results faithfully capture Minkowski physics.