Wick rotation

Wick rotation
Name	Wick rotation
Field	Mathematical physics, Quantum field theory, Statistical mechanics
Namedafter	Gian-Carlo Wick
Relatedconcepts	Analytic continuation, Imaginary time, Path integral formulation, Euclidean field theory

Wick rotation. In theoretical physics, particularly quantum field theory and statistical mechanics, it is a method of transforming a Minkowski spacetime description into a Euclidean space formulation by substituting an imaginary time coordinate. This technique, named after physicist Gian-Carlo Wick, converts expressions involving the Minkowski metric into mathematically more tractable expressions involving the Euclidean metric, often turning oscillatory path integrals into exponentially damped ones. It serves as a crucial bridge connecting the formalism of relativistic quantum mechanics with that of statistical field theory.

Definition and mathematical formulation

The core operation substitutes the real time coordinate \( t \) with an imaginary time coordinate \( \tau = i t \), where \( i \) is the imaginary unit. This transforms the Minkowski metric \( ds^2 = -c^2 dt^2 + d\mathbf{x}^2 \) into the Euclidean metric \( ds^2 = c^2 d\tau^2 + d\mathbf{x}^2 \). Consequently, the Lorentzian manifold structure of spacetime is replaced by a Riemannian manifold. In the context of action (physics), the Minkowski space action \( S_M \) is related to the Euclidean action \( S_E \) by \( i S_M = -S_E \). This transformation directly affects key quantities like the Feynman propagator, turning it into the Schwinger function.

Physical interpretation and motivation

The primary motivation is to circumvent the mathematical difficulties posed by the oscillatory nature of the path integral formulation in Minkowski space, as famously developed by Richard Feynman. By performing this substitution, the complex phase factor \( e^{iS/\hbar} \) becomes a real, decaying exponential \( e^{-S_E/\hbar} \), which is far more amenable to rigorous mathematical techniques like the method of steepest descent. This imaginary time \( \tau \) is often interpreted as an inverse temperature in statistical mechanics, providing a profound link between quantum field theory at zero temperature and thermal field theory.

Applications in quantum field theory

It is a foundational tool for defining and calculating path integrals in a mathematically controlled setting, leading to the development of Euclidean field theory. This approach is essential for constructing rigorous formulations of Yang-Mills theory and for applications in lattice gauge theory, where fields are discretized on a Euclidean spacetime lattice, as pioneered by Kenneth G. Wilson. Calculations of correlation functions, vacuum expectation values, and anomaly (physics) often rely on this technique, and it underpins the OS positivity axioms in axiomatic quantum field theory.

Applications in statistical mechanics

The transformation establishes a direct equivalence between the partition function (statistical mechanics) of a quantum statistical system and the path integral of a Euclidean quantum field theory. Specifically, the partition function \( Z = \operatorname{Tr} e^{-\beta \hat{H}} \) is identified with a path integral over a Euclidean time interval of length \( \hbar \beta \), where \( \beta = 1/(k_B T) \) is the inverse temperature. This connection allows techniques from quantum field theory, such as the renormalization group developed by Leo P. Kadanoff and Kenneth G. Wilson, to be applied to critical phenomena in condensed matter physics, including studies of the Ising model and superfluidity.

Relation to analytic continuation

The procedure is a specific instance of analytic continuation in the complex time plane. The expectation is that physical observables calculated in the Euclidean domain can be analytically continued back to real Minkowski time to yield physical predictions. This relies on the assumption of certain analyticity properties, such as those encoded in the Schwinger-Dyson equations. The Wightman axioms and the Osterwalder-Schrader theorem provide a rigorous framework for this back-and-forth continuation, ensuring the recovery of a unitary quantum theory from its Euclidean counterpart.

Mathematical rigor and limitations

While immensely powerful, the process is not universally applicable. Its rigorous justification depends on the specific quantum field theory in question and often requires the theory to satisfy the Osterwalder-Schrader axioms. Theories containing chiral fermions or suffering from a sign problem, such as quantum chromodynamics at finite baryon chemical potential, pose significant challenges. Furthermore, the analytic continuation back to real time can be ill-defined or non-unique for certain classes of observables, like real-time correlation functions, limiting the technique's scope in studying non-equilibrium thermodynamics and real-time dynamics.

Category:Quantum field theory Category:Theoretical physics Category:Mathematical physics