Euclidean Path Integrals and Applications
Imaginary time, statistical mechanics, instantons, and the road to quantum field theory
Wick Rotation to Imaginary Time
The oscillatory phase $e^{iS/\hbar}$ in the Minkowski path integral makes rigorous mathematical treatment and numerical computation difficult. A powerful technique is the Wick rotation: the analytic continuation to imaginary time$t \to -i\tau$ where $\tau$ is real and positive.
Under this substitution, the kinetic and potential energies transform as:
The Minkowski action $iS = i\int(T-V)\,dt$ becomes $-\int(T_E + V)\,d\tau = -S_E$.
The Euclidean Path Integral
After Wick rotation, the path integral becomes:
where the Euclidean action is $S_E[x] = \int_{\tau_a}^{\tau_b}\!\left[\frac{m}{2}\!\left(\frac{dx}{d\tau}\right)^{\!2} + V(x)\right]d\tau$.
The crucial change is that $e^{iS/\hbar} \to e^{-S_E/\hbar}$: the oscillatory integrand becomes a real, damped exponential. This has profound consequences:
- The path integral is now mathematically well-defined (Wiener measure)
- Paths with large Euclidean action are exponentially suppressed
- Monte Carlo numerical methods converge (importance sampling works)
- The Euclidean propagator corresponds to $\langle x_b| e^{-\hat{H}(\tau_b-\tau_a)/\hbar}|x_a\rangle$
Connection to Statistical Mechanics
The Euclidean propagator at imaginary time $\tau_b - \tau_a = \beta\hbar$ is the thermal density matrix element:
where $\beta = 1/k_BT$ is the inverse temperature.
The partition function of quantum statistical mechanics is obtained by tracing over position states — setting $x_b = x_a$ and integrating:
The circle on $\oint$ indicates periodic boundary conditions:$x(0) = x(\beta\hbar)$. The particle returns to its starting point in imaginary time.
This establishes a deep correspondence:
- Quantum mechanics in $d$ dimensions $\longleftrightarrow$ Classical statistical mechanics in $d$ dimensions (with imaginary time as an extra dimension)
- Temperature $\longleftrightarrow$ imaginary time period: $T = \hbar/k_B\beta$
- Ground state energy: $E_0 = -\lim_{\beta\to\infty}\frac{1}{\beta}\ln Z$
- Correlation functions in imaginary time $\longleftrightarrow$ thermodynamic response functions
Tunneling via Instantons
The Euclidean path integral provides an elegant treatment of quantum tunneling. Consider a symmetric double-well potential with minima at $x = \pm a$:
In the Euclidean theory, the potential is inverted: $V_E = -V$ in the equation of motion. Classical solutions that interpolate between the two minima — impossible in real time — now exist. These are called instantons:
where $\omega = 2a\sqrt{2\lambda/m}$ and $\tau_0$ is the center of the instanton (a collective coordinate). The solution is localized in imaginary time, hence the name "instanton."
The instanton has finite Euclidean action:
The tunneling amplitude (energy splitting between the two lowest states) is dominated by the instanton contribution:
This non-perturbative result ($\sim e^{-1/\hbar}$) is invisible to any finite order of perturbation theory — a triumph of the path integral approach.
Phase Space Path Integral
The path integral can also be written in phase space, integrating over both positions and momenta:
The exponent is $\frac{i}{\hbar}\int(p\dot{x} - H)\,dt$, the phase-space action in Hamiltonian form.
For the standard Hamiltonian $H = p^2/2m + V(x)$, integrating out the momenta (completing the square in $p$) recovers the configuration-space path integral:
The phase-space form is essential for:
- Systems with non-standard kinetic terms (e.g., particles on curved spaces)
- Constrained systems and gauge theories (Faddeev-Popov procedure)
- Spin systems where no natural configuration space exists
- Coherent-state path integrals for bosons and fermions
Advantages and the Road to Quantum Field Theory
The path integral generalizes naturally from quantum mechanics to quantum field theory by replacing the particle trajectory $x(t)$ with a field configuration $\phi(x,t)$:
Now the sum is over all field configurations in spacetime, weighted by $e^{iS[\phi]/\hbar}$.
Key advantages of the path integral formulation:
- Manifest symmetries: Lorentz invariance, gauge invariance, and other symmetries are preserved throughout
- No operator ordering: Classical expressions are used directly; no ambiguity in quantization
- Non-perturbative effects: Instantons, solitons, and topological terms arise naturally
- Generating functionals: Adding source terms $J\phi$ generates all correlation functions
- Lattice formulation: Discretized Euclidean path integral enables ab initio numerical computation (lattice QCD)
Three Formulations Compared
| Aspect | Schrodinger | Heisenberg | Path Integral |
|---|---|---|---|
| Central object | Wave function $\psi(x,t)$ | Operators $\hat{A}(t)$ | Propagator $K(b;a)$ |
| Starting point | Hamiltonian $\hat{H}$ | Hamiltonian $\hat{H}$ | Lagrangian $L$ |
| Time evolution | $i\hbar\partial_t\psi = \hat{H}\psi$ | $i\hbar\dot{\hat{A}} = [\hat{A},\hat{H}]$ | Sum over all paths |
| Classical limit | WKB ansatz | $[\hat{x},\hat{p}]\to 0$ | Stationary phase |
| Best suited for | Bound states, spectra | Symmetries, selection rules | QFT, gauge theories, tunneling |
| Difficulty | PDEs in many variables | Operator algebra | Functional integrals |
Key Concepts Summary
- Path integral: $K = \int\mathcal{D}x\,e^{iS[x]/\hbar}$ — sum over all paths weighted by complex phase
- Functional measure: $\mathcal{D}x = \lim_{N\to\infty}(m/2\pi i\hbar\varepsilon)^{N/2}\prod dx_j$
- Classical limit: Stationary phase ($\delta S = 0$) selects classical paths as $\hbar\to 0$
- Free particle: $K = \sqrt{m/2\pi i\hbar T}\;\exp[im(x_b-x_a)^2/2\hbar T]$
- Harmonic oscillator: Mehler kernel with $\sqrt{m\omega/2\pi i\hbar\sin\omega T}$ prefactor
- Semiclassical: Van Vleck formula $K \approx |\partial^2 S_{\text{cl}}/\partial x_b\partial x_a|^{1/2}\,e^{iS_{\text{cl}}/\hbar}$
- Euclidean path integral: Wick rotation $t\to -i\tau$ gives $K_E = \int\mathcal{D}x\,e^{-S_E/\hbar}$
- Partition function: $Z = \oint\mathcal{D}x\,e^{-S_E/\hbar}$ with periodic boundary conditions
- Instantons: Euclidean classical solutions that mediate tunneling ($\Delta E \sim e^{-S_{\text{inst}}/\hbar}$)
- Phase space form: $K = \int\mathcal{D}x\,\mathcal{D}p\;\exp[\frac{i}{\hbar}\int(p\dot{x}-H)\,dt]$
Looking Ahead: The path integral is the foundation of modern quantum field theory. In QFT, the sum over particle paths becomes a sum over field configurations. Feynman diagrams — the lingua franca of particle physics — emerge as the perturbative expansion of the path integral. Non-perturbative techniques like lattice gauge theory and instanton calculations are formulated entirely within the Euclidean path integral framework.