Topic

What is Wick Rotation and why does it work?

QFT Path Integrals Mathematics

Short Answer

Wick rotation is the analytic continuation from real time $t$ to imaginary time $\tau = it$ . It transforms oscillatory path integrals in Minkowski space into convergent integrals in Euclidean space. It works because of the analytic properties of quantum field theories.

The Problem: Oscillatory Integrals

Path Integrals in Minkowski Space

The quantum mechanical amplitude is:

$Z = \int \mathcal{D}\phi \, e^{iS[\phi]/\hbar}$

The integrand $e^{iS}$ is a phase—it oscillates wildly and doesn’t converge in any naive sense.

Example: Free Particle

For a free particle, the path integral involves:

$\int_{-\infty}^{\infty} dx \, e^{i\alpha x^2}$

This doesn’t converge absolutely. We need a prescription to make sense of it.

The Solution: Wick Rotation

The Idea

Analytically continue time to imaginary values:

$t \to -i\tau$

(The sign convention varies in the literature.)

What Changes

Minkowski	Euclidean
Time $t$	Imaginary time $\tau = it$
Metric $ds^2 = -dt^2 + d\vec{x}^2$	Metric $ds^2 = d\tau^2 + d\vec{x}^2$
Signature $(-,+,+,+)$	Signature $(+,+,+,+)$
Lorentzian	Riemannian

The Action Transforms

For a scalar field with action:

$S = \int dt \, d^3x \left[ \frac{1}{2}(\partial_t \phi)^2 - \frac{1}{2}(\nabla\phi)^2 - V(\phi) \right]$

Under $t = -i\tau$ :

$iS \to -S_E$

where the Euclidean action is:

$S_E = \int d\tau \, d^3x \left[ \frac{1}{2}(\partial_\tau \phi)^2 + \frac{1}{2}(\nabla\phi)^2 + V(\phi) \right]$

The Path Integral Converges

$Z_E = \int \mathcal{D}\phi \, e^{-S_E[\phi]}$

Now the integrand is $e^{-S_E}$ , which is exponentially suppressed for large field configurations. The integral converges!

Why It Works: Analytic Continuation

Analyticity of Correlation Functions

In QFT, correlation functions like:

$\langle \phi(x_1) \phi(x_2) \cdots \phi(x_n) \rangle$

are analytic functions of the time coordinates (in suitable regions).

The Key Theorem

If a function is analytic, knowing it on one region determines it everywhere (by analytic continuation).

Strategy:

Compute in Euclidean space (where integrals converge)
Analytically continue back to Minkowski space

Osterwalder-Schrader Axioms

These axioms specify conditions under which a Euclidean QFT defines a valid Lorentzian theory:

Euclidean covariance: Rotation invariance in $\mathbb{R}^4$
Reflection positivity: A positivity condition involving time reflection
Regularity: Suitable growth conditions

If satisfied, the Euclidean theory can be continued to a unitary Lorentzian QFT.

Physical Interpretations

Thermal Field Theory

Euclidean time with period $\beta$ corresponds to temperature $T = 1/\beta$ :

$Z(\beta) = \text{Tr}(e^{-\beta H}) = \int_{\phi(\tau+\beta) = \phi(\tau)} \mathcal{D}\phi \, e^{-S_E}$

The path integral with periodic boundary conditions computes the thermal partition function!

Tunneling and Instantons

In Euclidean space, classical solutions called instantons mediate tunneling between different vacua.

Example: Double-well potential

Minkowski: Particle oscillates in one well
Euclidean: Instanton solution connects the two wells

The tunneling amplitude is $\sim e^{-S_E[\text{instanton}]}$ .

Quantum Mechanics as Euclidean Statistics

The Euclidean path integral:

$Z_E = \int \mathcal{D}\phi \, e^{-S_E}$

looks like a statistical mechanics partition function:

$Z = \sum_{\text{configs}} e^{-E/kT}$

Quantum fluctuations ↔ Thermal fluctuations

Practical Applications

Lattice QFT

Discretize Euclidean spacetime on a lattice:

Time direction: $N_t$ points
Space directions: $N_s^3$ points

The path integral becomes a finite sum, computable by Monte Carlo.

This is how we compute:

Hadron masses
QCD phase diagram
Non-perturbative quantities

Perturbation Theory

Feynman rules are often derived in Euclidean space, then continued:

$\frac{1}{k^2 + m^2} \xrightarrow{\text{continue}} \frac{1}{k^2 - m^2 + i\epsilon}$

The $i\epsilon$ prescription handles the poles correctly.

Subtleties and Limitations

When Wick Rotation Fails

Not all theories can be Wick-rotated:

Theories with complex actions
Chern-Simons theory (topological)
Real-time dynamics (scattering, decay)

The Sign Problem

For some theories (finite density QCD), the Euclidean action is complex:

$e^{-S_E} \text{ is not real positive}$

Monte Carlo fails. This is a major unsolved problem.

Real-Time Physics

Some questions require real time:

Scattering amplitudes
Time-dependent phenomena
Non-equilibrium systems

These require direct Minkowski calculations or Schwinger-Keldysh formalism.

Summary

Aspect	Minkowski	Euclidean
Time	Real $t$	Imaginary $\tau = it$
Signature	$(-,+,+,+)$	$(+,+,+,+)$
Geometry	Lorentzian	Riemannian
Path integral	$e^{iS}$ (oscillatory)	$e^{-S_E}$ (convergent)
Symmetry	Lorentz $SO(1,3)$	Rotation $SO(4)$
Use	Physical amplitudes	Computation, lattice

References

G. C. Wick, “Properties of Bethe-Salpeter Wave Functions,” Physical Review 96, 1124 (1954) — Original paper
K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions,” Communications in Mathematical Physics 31, 83–112 (1973)
J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd ed. (Springer, 1987)
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995), Section 9.3
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Oxford University Press, 2002)
M. Creutz, Quarks, Gluons and Lattices (Cambridge University Press, 1983) — Lattice QFT introduction