3.4.1 Wick Rotation

3.4.1 Wick Rotation#

Prompts

  • What is the Wick rotation and why does transforming \(t \to -\mathrm{i}\tau\) turn oscillatory integrals into exponentially damped ones?

  • How does the real-time action transform under Wick rotation, and what is the resulting Euclidean action?

  • Why does the imaginary-time path integral converge when the real-time version oscillates wildly?

  • How is the partition function \(Z(\beta)\) related to the imaginary-time propagator?

  • Can you always analytically continue results from imaginary time back to real time, and are there subtleties?

Lecture Notes#

Overview#

The path integral oscillates as \(\mathrm{e}^{\mathrm{i}S/\hbar}\), making it difficult to evaluate directly. The Wick rotation \(t \to -\mathrm{i}\tau\) replaces oscillating phases with exponentially decaying weights \(\mathrm{e}^{-S_E/\hbar}\), where \(S_E\) is the Euclidean action. This transform reveals a deep connection: the imaginary-time propagator for interval \(\beta\hbar\) is precisely the thermal density matrix \(\mathrm{e}^{-\beta \hat{H}}\), linking quantum dynamics to statistical mechanics.

Wick Rotation

The Wick rotation from real time \(t\) to imaginary time \(\tau = it/\hbar\) transforms the oscillatory quantum path integral into an exponentially damped sum. This analytic continuation reveals that quantum mechanics and thermal statistical mechanics are intimately related—they are two faces of the same mathematical structure.

The Wick Rotation#

In quantum field theory and statistical mechanics, analytic continuation from real time \(t\) to imaginary time \(\tau = -it/\hbar\) simplifies calculations.

Definition#

The Wick rotation is the transformation

\[t \to -\mathrm{i}\tau, \quad \text{where} \quad \tau \in \mathbb{R}\]

Equivalently, \(\tau = it/\hbar\) is the imaginary time, with dimension of time. If thermal context applies, write \(\tau \in [0, \beta\hbar]\) where \(\beta = 1/(k_B T)\) is the inverse temperature.

Derivation: Wick Rotation Transform

The real-time action is

\[S[x] = \int_0^T \left(\frac{m}{2}\dot{x}^2 - V(x)\right) dt\]

Under \(t \to -\mathrm{i}\tau\), we have \(dt \to -i d\tau\) and \(\dot{x} = dx/dt \to i dx/d\tau\). Thus:

\[S[x] \to \int_0^{\beta\hbar} \left(\frac{m}{2}(i\dot{x})^2 - V(x)\right) (-i) d\tau = -i \int_0^{\beta\hbar} \left(-\frac{m}{2}\dot{x}^2 - V(x)\right) d\tau\]

The minus signs combine to give:

\[S[x] \to -\int_0^{\beta\hbar} \left(\frac{m}{2}\dot{x}^2 + V(x)\right) d\tau\]

Defining the Euclidean action as

\[S_E[x] = \int_0^{\beta\hbar} \left(\frac{m}{2}\dot{x}^2 + V(x)\right) d\tau\]

we have \(S \to -iS_E\), so

\[\mathrm{e}^{\mathrm{i}S/\hbar} \to \mathrm{e}^{-S_E/\hbar}\]

The rapidly oscillating phase \(\mathrm{e}^{\mathrm{i}S/\hbar}\) becomes a rapidly decaying exponential \(\mathrm{e}^{-S_E/\hbar}\). This convergence is crucial for numerical simulations and analytical approximations.

Why Wick Rotation Matters

Real-time path integrals with \(\mathrm{e}^{\mathrm{i}S/\hbar}\) are highly oscillatory—contributions from different paths interfere constructively and destructively, making direct integration intractable.

After Wick rotation, the Euclidean path integral with \(\mathrm{e}^{-S_E/\hbar}\) is exponentially damped—contributions from paths with large action are exponentially suppressed, enabling:

  • Numerical Monte Carlo simulations

  • Perturbative expansions

  • Semi-classical approximations

Euclidean Path Integral#

The Euclidean path integral is

\[Z(\beta) = \int_{\text{periodic}} \mathrm{e}^{-S_E[x]/\hbar} \, \mathcal{D}[x]\]

where the integral is over paths that are periodic in imaginary time: \(x(\tau + \beta\hbar) = x(\tau)\).

Key properties:

  1. Convergence: The exponential \(\mathrm{e}^{-S_E/\hbar}\) decays exponentially for large actions. Paths with large Euclidean action are suppressed. This makes numerical simulation feasible.

  2. Positive measure: Unlike the oscillating real-time path integral, the Euclidean integral has a well-defined probability measure (positive definite). This is essential for Monte Carlo simulations.

  3. Periodicity: The periodicity in imaginary time \(\tau\) reflects the cyclic nature of the thermal density matrix in quantum statistical mechanics.

Connection to the Partition Function#

In quantum statistical mechanics, the partition function at temperature \(T\) is

\[Z(T) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}}), \quad \beta = \frac{1}{k_B T}\]

where \(\hat{H}\) is the Hamiltonian and the trace sums over all quantum states.

Path Integral Representation of Partition Function

\[Z(\beta) = \int_{\text{periodic}} \mathrm{e}^{-S_E[x]/\hbar} \, \mathcal{D}[x]\]

where the sum is over all paths periodic in imaginary time with period \(\beta\hbar = 1/(k_B T)\).

The partition function \(Z = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\) equals the functional integral over periodic paths in Euclidean time. This is a profound connection:

  • Trace: sums over all eigenstates

  • Path integral: sums over all closed paths in imaginary time

Summary#

  • Wick rotation: \(t \to -\mathrm{i}\tau\) converts oscillating phases \(\mathrm{e}^{\mathrm{i}S/\hbar}\) to decaying exponentials \(\mathrm{e}^{-S_E/\hbar}\)

  • Euclidean action: \(S_E = \int(\frac{m}{2}\dot{x}^2 + V)\,\mathrm{d}\tau\) — same form as Hamiltonian but in imaginary time

  • Partition function: \(Z(\beta) = \int_{\text{periodic}}\mathrm{e}^{-S_E/\hbar}\mathcal{D}[x]\) — path integral over periodic paths of period \(\beta\hbar\)

  • Temperature duality: quantum mechanics in imaginary time \(\beta\hbar\) is thermal statistical mechanics at temperature \(T = 1/(k_B \beta)\)

See Also

Homework#

1. Wick rotation for the free particle. For a free particle (\(V = 0\)), the real-time action is

\[S[\boldsymbol{x}] = \int_0^T \frac{m}{2}\dot{\boldsymbol{x}}^2 \, \mathrm{d}t\]

Perform the Wick rotation \(t \to -\mathrm{i}\tau\) to find the Euclidean action \(S_E[\boldsymbol{x}]\). Show that the kinetic term changes sign: the Euclidean kinetic term is \(\frac{m}{2}\dot{\boldsymbol{x}}^2\) (positive). Why does this sign change make the Euclidean path integral convergent?

2. Phase vs. exponential in a toy model. Consider a one-dimensional free particle path \(x(t) = vt\) connecting \(x(0) = 0\) to \(x(T) = vT\). Compute:

  1. The real-time phase \(\mathrm{e}^{\mathrm{i}S/\hbar}\) where \(S = \frac{m}{2}\int_0^T v^2 \, \mathrm{d}t\).

  2. The Euclidean exponential \(\mathrm{e}^{-S_E/\hbar}\) after Wick rotation.

  3. Explain physically why the Euclidean form is easier for Monte Carlo integration than the oscillating phase.

3. Partition function from trace to path integral. The quantum partition function is \(Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\). For a single particle in a box of side length \(L\), expand the trace in position eigenstates:

\[Z(\beta) = \int_0^L dx \, \langle x | \mathrm{e}^{-\beta \hat{H}} | x \rangle\]

Explain how the path integral representation

\[Z(\beta) = \int_{\text{periodic}} \mathrm{e}^{-S_E[\boldsymbol{x}]/\hbar} \mathcal{D}[\boldsymbol{x}]\]

encodes the same information. Why must the paths be periodic in imaginary time?

4. Euclidean action for the harmonic oscillator. The real-time action for a harmonic oscillator (\(V = \frac{m\omega^2}{2}x^2\)) is

\[S[x] = \int_0^T \left(\frac{m}{2}\dot{x}^2 - \frac{m\omega^2}{2}x^2\right) dt\]

After Wick rotation, write the Euclidean action \(S_E[x]\). How do the kinetic and potential terms compare in magnitude in the Euclidean action? For very low temperature (\(\beta\hbar \gg 1/\omega\)), which term dominates?

5. Temperature and inverse period. Show that the Euclidean time period \(\tau_T = \beta\hbar = 1/(k_B T)\) correctly maps to physical temperature. If you increase \(T\):

  1. Does the period \(\beta\hbar\) increase or decrease?

  2. Physically, what does this mean for the density of imaginary-time paths in the partition function integral?

  3. Why does this behavior make intuitive sense in terms of thermal fluctuations?

6. Classical limit from high-temperature path integral. In the high-temperature limit (\(\beta \to 0\)), the Euclidean action is

\[S_E[\boldsymbol{x}] = \int_0^{\beta\hbar} \left(\frac{m}{2}\dot{\boldsymbol{x}}^2 + V(\boldsymbol{x})\right) d\tau\]

Show that the kinetic term \(\frac{m}{2}\dot{\boldsymbol{x}}^2\) becomes negligible compared to \(V(\boldsymbol{x})\) for small \(\beta\). Thus derive that the partition function reduces to

\[Z(T \to \infty) \approx \mathrm{e}^{-\beta V_0} \int \mathrm{e}^{-\beta V(\boldsymbol{x})} d^3\boldsymbol{x}\]

where \(V_0\) is a reference energy. What does this limit represent physically?

7. Ground state from zero-temperature path integral. In the limit \(T \to 0\) (or \(\beta \to \infty\)), the partition function becomes dominated by the ground state:

\[Z(\beta \to \infty) \approx \mathrm{e}^{-\beta E_0}\]

Show that at \(\tau = 0\) and \(\tau = \beta\hbar\), the paths in the path integral must return to the same point (periodicity). In this limit, argue that the weight in the path integral is dominated by paths that stay close to the ground-state configuration \(x_0(\tau)\). What is the physical meaning of this result for calculating ground-state properties?

8. Analytic continuation and Matsubara frequencies. Given a correlator in imaginary time,

\[C(\tau) = \langle x(\tau) x(0) \rangle_\beta\]

Expand this in Matsubara frequencies:

\[C(\tau) = \frac{1}{\beta\hbar} \sum_{n=-\infty}^{\infty} \tilde{C}(\mathrm{i}\omega_n) \mathrm{e}^{-\mathrm{i}\omega_n \tau}\]

where \(\omega_n = 2\pi n/(\beta\hbar)\) are the Matsubara frequencies.

  1. Show that for real frequency \(\omega\) on the axis, \(\tilde{C}(\omega)\) corresponds to the Fourier transform of the real-time correlator.

  2. Why is this approach more stable than direct Wick rotation \(\tau \to it\) for some observables?