Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Part III, Chapter 3

Path Integrals in QFT

Wick rotation, Euclidean field theory, and the connection to statistical mechanics

Page 4 of 4

3.11 Wick Rotation: Detailed Derivation

The Minkowski path integral $\int\mathcal{D}\phi\,e^{iS}$ has a purely oscillatory integrand that makes rigorous definition difficult. The Wick rotationremedies this by analytically continuing the time coordinate to imaginary values.

We rotate the time contour in the complex plane:

$$t = x^0 \;\longrightarrow\; -ix^0_E = -i\tau, \qquad x^0_E = \tau = it$$

Under this transformation the Minkowski metric changes to the Euclidean metric:

$$ds^2_M = -dt^2 + d\mathbf{x}^2 \;\longrightarrow\; ds^2_E = d\tau^2 + d\mathbf{x}^2$$

For the scalar field action, the time derivatives transform as:

$$\left(\frac{\partial\phi}{\partial t}\right)^2 = \left(\frac{\partial\phi}{\partial(-i\tau)}\right)^2 = -\left(\frac{\partial\phi}{\partial\tau}\right)^2$$

and the time integration measure gives $dt = -id\tau$. Combining these:

$$iS = i\int dt\,d^3x\left[\frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla\phi)^2 - \frac{1}{2}m^2\phi^2\right] \longrightarrow -\int d\tau\,d^3x\left[\frac{1}{2}\left(\frac{\partial\phi}{\partial\tau}\right)^2 + \frac{1}{2}(\nabla\phi)^2 + \frac{1}{2}m^2\phi^2\right]$$

We define the Euclidean action:

$$\boxed{S_E[\phi] = \int d^4x_E\left[\frac{1}{2}(\partial_\mu^E\phi)^2 + \frac{1}{2}m^2\phi^2 + V(\phi)\right]}$$

Note that all signs are positive in the Euclidean action: it is bounded below. The Euclidean path integral is therefore:

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

💡Convergence vs Oscillation

The Minkowski integrand $e^{iS}$ oscillates wildly and only converges through delicate cancellations. The Euclidean integrand $e^{-S_E}$ is exponentially suppressed for large field configurations, making the integral absolutely convergent. All practical non-perturbative calculations (lattice QFT, Monte Carlo simulations) work in Euclidean space.

3.12 Connection to Statistical Mechanics

The Euclidean path integral has a deep structural similarity to the partition function in classical statistical mechanics:

Euclidean QFT

  • $Z_E = \int\mathcal{D}\phi\,e^{-S_E[\phi]}$
  • $S_E$ = Euclidean action
  • $\phi(x_E)$ = field configuration
  • $\hbar$ = quantum parameter

Statistical Mechanics

  • $Z = \int\mathcal{D}\sigma\,e^{-\beta H[\sigma]}$
  • $\beta H$ = Hamiltonian / temperature
  • $\sigma(x)$ = spin/order parameter
  • $k_BT$ = thermal parameter

This correspondence is exact: a d-dimensional Euclidean quantum field theory is mathematically identical to a d-dimensional classical statistical mechanics system. The dictionary is $\hbar \leftrightarrow k_BT$ and$S_E \leftrightarrow \beta H$.

Correlation functions in QFT map to correlation functions in stat mech:

$$\langle\phi(x)\phi(y)\rangle_E = \frac{\int\mathcal{D}\phi\;\phi(x)\phi(y)\,e^{-S_E}}{\int\mathcal{D}\phi\,e^{-S_E}} \;\longleftrightarrow\; \langle\sigma(x)\sigma(y)\rangle = \frac{\int\mathcal{D}\sigma\;\sigma(x)\sigma(y)\,e^{-\beta H}}{\int\mathcal{D}\sigma\,e^{-\beta H}}$$

💡Phase Transitions and Quantum Field Theory

Critical phenomena in statistical mechanics (e.g., the Ising model near its critical temperature) are described by Euclidean QFT at long distances. The correlation length $\xi$ diverges at the critical point, and the system becomes scale-invariant - described by a conformal field theory. This is one of the deepest connections in theoretical physics.

3.13 Lattice Field Theory Motivation

The Euclidean path integral provides a natural framework for lattice field theory, where spacetime is discretized onto a hypercubic lattice with spacing a:

$$Z_{\text{lat}} = \prod_{n}\int d\phi_n\; \exp\left\{-a^4\sum_n\left[\sum_{\mu}\frac{(\phi_{n+\hat{\mu}}-\phi_n)^2}{2a^2} + \frac{m^2}{2}\phi_n^2 + \frac{\lambda}{4!}\phi_n^4\right]\right\}$$

On the lattice: derivatives become finite differences, the functional integral becomes an ordinary (high-dimensional) integral, and the theory has a built-in UV cutoff$\Lambda \sim \pi/a$. The continuum limit is recovered as $a \to 0$.

Lattice calculations can be performed numerically using Monte Carlo methods: field configurations are sampled with probability $\propto e^{-S_E}$ (importance sampling), and observables are computed as statistical averages. This is the primary non-perturbative tool for QCD.

3.14 Worked Example: Euclidean Propagator

Let us compute the Euclidean propagator for a free scalar field of mass m directly from the Euclidean path integral and verify it matches the Wick-rotated Minkowski result.

Step 1: Euclidean generating functional

The free Euclidean action is $S_E = \frac{1}{2}\int d^4x_E\,\phi(-\partial_E^2 + m^2)\phi$. With source J, completing the square as before:

$$Z_E[J] = Z_E[0]\,\exp\left[\frac{1}{2}\int d^4x_E\,d^4y_E\; J(x_E)\,\Delta_E(x_E - y_E)\,J(y_E)\right]$$

Step 2: Euclidean Green's function

The Euclidean propagator satisfies $(-\partial_E^2 + m^2)\Delta_E(x_E) = \delta^4(x_E)$. Fourier transforming:

$$\Delta_E(x_E) = \int\frac{d^4k_E}{(2\pi)^4}\frac{e^{ik_E \cdot x_E}}{k_E^2 + m^2}$$

Step 3: Verify via Wick rotation of Minkowski result

The Minkowski propagator is $D_F(k) = i/(k^2 - m^2 + i\epsilon)$ with$k^2 = -k_0^2 + \mathbf{k}^2$. Set $k_0 = ik_4$:

$$\frac{i}{k^2 - m^2 + i\epsilon} = \frac{i}{-k_0^2 + \mathbf{k}^2 - m^2} \;\xrightarrow{k_0 = ik_4}\; \frac{i}{k_4^2 + \mathbf{k}^2 - m^2}\cdot\frac{1}{i} = \frac{1}{k_E^2 + m^2}$$

This matches $\Delta_E(k_E)$ exactly.

Step 4: Position-space result

In position space the Euclidean propagator in d = 4 is:

$$\boxed{\Delta_E(x_E) = \frac{m}{4\pi^2|x_E|}\,K_1(m|x_E|)}$$

where $K_1$ is the modified Bessel function. For $|x_E| \gg 1/m$, this decays as $e^{-m|x_E|}$, confirming that correlations fall off exponentially with a correlation length $\xi = 1/m$.

This exponential decay is the Euclidean signature of the mass gap. A massless field ($m = 0$) gives power-law decay $\Delta_E \sim 1/|x_E|^2$, reflecting long-range correlations.

Key Concepts (This Page)

  • Wick rotation: $t \to -i\tau$ converts oscillatory $e^{iS}$ to damped $e^{-S_E}$
  • Euclidean action has all positive signs and is bounded below
  • Euclidean QFT in d dimensions = d-dimensional classical stat mech
  • Dictionary: $\hbar \leftrightarrow k_BT$, $S_E \leftrightarrow \beta H$
  • Lattice field theory discretizes the Euclidean path integral for numerical computation
  • Euclidean propagator: $\Delta_E(k_E) = 1/(k_E^2 + m^2)$, decays as $e^{-m|x|}$
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