Chapter 3: Path Integrals

The path integral formulation, due to Feynman, provides an alternative to canonical quantization that is manifestly Lorentz-covariant and naturally extends to gauge theories. Instead of operators and commutation relations, we sum over all possible field configurations weighted by $e^{iS/\hbar}$. This approach makes the connection between quantum field theory and statistical mechanics transparent and leads directly to Feynman diagrams.

Derivation 1: The Feynman Path Integral

The path integral expresses quantum mechanical amplitudes as a sum over all possible histories of the system, weighted by the phase $e^{iS}$. We derive it from first principles using time-slicing.

Starting Point: The Transition Amplitude

We want to compute the probability amplitude for a particle to go from position $x_i$ at time $t_i$ to $x_f$ at $t_f$:

$K(x_f, t_f; x_i, t_i) = \langle x_f | e^{-iH(t_f - t_i)} | x_i \rangle$

Time-Slicing Derivation

Divide the time interval into $N$ small steps of size $\epsilon = (t_f - t_i)/N$:

Step 1: Insert complete sets of position states at each intermediate time:

$K = \int dx_1 \cdots dx_{N-1} \prod_{j=0}^{N-1} \langle x_{j+1} | e^{-iH\epsilon} | x_j \rangle$

Step 2: For small $\epsilon$, with $H = p^2/(2m) + V(x)$:

$\langle x_{j+1} | e^{-iH\epsilon} | x_j \rangle \approx \left(\frac{m}{2\pi i \epsilon}\right)^{1/2} \exp\left(i\epsilon \left[\frac{m}{2}\left(\frac{x_{j+1}-x_j}{\epsilon}\right)^2 - V(x_j)\right]\right)$

Step 3: Recognize $(x_{j+1}-x_j)/\epsilon \to \dot{x}$ and $\epsilon \sum_j \to \int dt$:

$\epsilon \sum_{j=0}^{N-1} \left[\frac{m}{2}\dot{x}_j^2 - V(x_j)\right] \to \int_{t_i}^{t_f} L(x, \dot{x}) \, dt = S[x(t)]$

Taking $N \to \infty$, the multiple integral becomes a functional integral:

$\boxed{K(x_f, t_f; x_i, t_i) = \int \mathcal{D}x(t) \, e^{iS[x(t)]/\hbar}}$

$\mathcal{D}x = \lim_{N\to\infty} \left(\frac{m}{2\pi i\epsilon}\right)^{N/2} dx_1 \cdots dx_{N-1}$

Physical Interpretation

The particle takes all paths simultaneously. Each path contributes a phase $e^{iS}$. In the classical limit $\hbar \to 0$, the rapidly oscillating phases cancel except near the stationary phase path where $\delta S = 0$ — recovering the classical equations of motion. Quantum mechanics arises from the constructive interference of paths near the classical trajectory.

Derivation 2: The Generating Functional

For field theory, the path integral becomes a functional integral over all field configurations. The generating functional is the master object from which all physics can be extracted.

Definition

The generating functional $Z[J]$ is defined by adding a source term $J(x)$ coupled to the field:

$\boxed{Z[J] = \int \mathcal{D}\phi \, \exp\left(i \int d^4x \left[\mathcal{L}(\phi) + J(x)\phi(x)\right]\right)}$

The source $J(x)$ is an arbitrary external function — a mathematical tool that allows us to generate correlation functions by differentiation.

Correlation Functions from Functional Derivatives

The $n$-point correlation function (Green's function) is obtained by taking functional derivatives with respect to the source:

One-point function (field expectation value):

$\langle \phi(x) \rangle_J = \frac{1}{Z[J]} \frac{\delta Z[J]}{i \, \delta J(x)}$

Two-point function (propagator):

$\langle \phi(x_1)\phi(x_2) \rangle = \frac{1}{Z[0]} \left.\frac{\delta^2 Z[J]}{(i)^2 \, \delta J(x_1) \delta J(x_2)}\right|_{J=0}$

General $n$-point function:

$\langle \phi(x_1) \cdots \phi(x_n) \rangle = \frac{1}{Z[0]} \left.\frac{\delta^n Z[J]}{(i)^n \, \delta J(x_1) \cdots \delta J(x_n)}\right|_{J=0}$

Connected Generating Functional

The connected correlation functions (which cannot be factored into products of lower-order functions) are generated by $W[J] = -i \ln Z[J]$:

$\langle \phi(x_1)\phi(x_2) \rangle_c = \frac{\delta^2 W[J]}{\delta J(x_1) \delta J(x_2)}\bigg|_{J=0}$

Connected functions are what appear in actual physical processes. The factored (disconnected) parts correspond to vacuum-to-vacuum processes that cancel in cross sections.

Derivation 3: The Feynman Propagator

The propagator is the two-point correlation function of the free field — the probability amplitude for a particle to travel from one spacetime point to another.

Computing the Free Propagator

For the free Klein-Gordon field with $\mathcal{L} = \frac{1}{2}(\partial\phi)^2 - \frac{1}{2}m^2\phi^2$, the equation of motion in the presence of a source is:

$(\Box + m^2)\phi(x) = J(x)$

The propagator $\Delta_F(x-y)$ is the Green's function of this equation:

$(\Box_x + m^2)\Delta_F(x-y) = -i\delta^4(x-y)$

Fourier transforming to momentum space:

Substituting $\Delta_F(x-y) = \int \frac{d^4k}{(2\pi)^4} \tilde{\Delta}_F(k) \, e^{ik(x-y)}$:

$(-k^2 + m^2)\tilde{\Delta}_F(k) = -i$

$\boxed{\Delta_F(x-y) = \int \frac{d^4k}{(2\pi)^4} \frac{i}{k^2 - m^2 + i\varepsilon} \, e^{-ik \cdot (x-y)}}$

The Feynman $i\varepsilon$ Prescription

The integral has poles at $k^0 = \pm\omega_\mathbf{k} = \pm\sqrt{|\mathbf{k}|^2 + m^2}$. The $i\varepsilon$ prescription (with $\varepsilon > 0$ infinitesimal) tells us how to handle these poles:

$\frac{i}{k^2 - m^2 + i\varepsilon} = \frac{i}{(k^0)^2 - \omega_\mathbf{k}^2 + i\varepsilon}$

The poles are shifted to $k^0 = \omega_\mathbf{k} - i\varepsilon'$ and $k^0 = -\omega_\mathbf{k} + i\varepsilon'$.

Closing the contour in the complex $k^0$ plane:

• For $x^0 > y^0$: close in the lower half-plane, picking up the pole at $k^0 = +\omega_\mathbf{k}$. This propagates positive energy forward in time.

• For $x^0 < y^0$: close in the upper half-plane, picking up the pole at $k^0 = -\omega_\mathbf{k}$. This propagates positive energy backward in time (= antiparticles forward).

The result is the time-ordered product:

$\Delta_F(x-y) = \langle 0 | T\{\hat{\phi}(x)\hat{\phi}(y)\} | 0 \rangle$

where $T$ denotes time ordering. The Feynman propagator automatically includes both particle propagation ($x^0 > y^0$) and antiparticle propagation ($x^0 < y^0$).

Derivation 4: Gaussian Integrals in Field Theory

The free field path integral is Gaussian and can be evaluated exactly. This is the foundation for perturbation theory, where interactions are treated as corrections to the Gaussian integral.

Finite-Dimensional Gaussian Integral

Recall the basic Gaussian integral in $n$ dimensions:

$\int d^n x \, \exp\left(-\frac{1}{2} x^T A x + J^T x\right) = \frac{(2\pi)^{n/2}}{\sqrt{\det A}} \exp\left(\frac{1}{2} J^T A^{-1} J\right)$

where $A$ is a positive-definite symmetric matrix. This is proved by completing the square:

Shift $x \to x + A^{-1}J$:

$-\frac{1}{2}x^T A x + J^T x = -\frac{1}{2}(x - A^{-1}J)^T A (x - A^{-1}J) + \frac{1}{2}J^T A^{-1}J$

The Field Theory Generalization

For the free scalar field, the action is quadratic in $\phi$:

$S[\phi] + \int J\phi = -\frac{1}{2}\int d^4x \, \phi(x)(-\Box - m^2)\phi(x) + \int d^4x \, J(x)\phi(x)$

The operator $A = -\Box - m^2 + i\varepsilon$ plays the role of the matrix, and its inverse is the Feynman propagator: $A^{-1}(x,y) = \Delta_F(x-y)$. The functional integral gives:

$\boxed{Z_0[J] = Z_0[0] \exp\left(-\frac{1}{2}\int d^4x \, d^4y \, J(x) \Delta_F(x-y) J(y)\right)}$

The prefactor $Z_0[0] \propto (\det A)^{-1/2}$ is an (infinite) constant that cancels in normalized correlation functions. The entire physics of the free theory is encoded in the propagator $\Delta_F(x-y)$.

The Two-Point Function

Taking two functional derivatives and setting $J = 0$:

$\langle \phi(x)\phi(y) \rangle = \frac{1}{i^2} \frac{\delta^2 \ln Z_0}{\delta J(x)\delta J(y)}\bigg|_{J=0} = \frac{1}{i}\Delta_F(x-y)$

This confirms that the propagator is the two-point function of the free field, derived entirely from the path integral without ever using operators or commutation relations.

Derivation 5: Wick's Theorem

Wick's theorem is the fundamental combinatorial identity that reduces any free-field correlation function to products of propagators. It is the bridge from the abstract path integral to concrete Feynman diagram calculations.

Statement of Wick's Theorem

For the free field, the $2n$-point function is:

$\boxed{\langle \phi(x_1) \cdots \phi(x_{2n}) \rangle = \sum_{\text{pairings}} \prod_{\text{pairs}} \Delta_F(x_i - x_j)}$

The sum runs over all possible ways to pair up the $2n$ fields into $n$ pairs. Odd-point functions vanish: $\langle \phi(x_1) \cdots \phi(x_{2n+1}) \rangle = 0$.

Proof from the Generating Functional

Since $Z_0[J] = Z_0[0] \exp(\frac{1}{2}\int J \Delta_F J)$, the Taylor expansion of the exponential gives:

$Z_0[J] = Z_0[0] \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{1}{2}\int d^4x \, d^4y \, J(x)\Delta_F(x-y)J(y)\right)^n$

Taking $2n$ derivatives with respect to $J$, only the $n$-th term in the sum survives (lower terms have too few $J$'s, higher terms still contain$J$'s that vanish when $J = 0$). The $n!$ from the expansion cancels against the$n!$ from distributing the derivatives, and the remaining combinatorial factor gives exactly the sum over all pairings.

Examples

Two-point function (1 pairing):

$\langle \phi(x_1)\phi(x_2) \rangle = \Delta_F(x_1 - x_2)$

Four-point function (3 pairings):

$\langle \phi_1 \phi_2 \phi_3 \phi_4 \rangle = \Delta_F(x_1-x_2)\Delta_F(x_3-x_4) + \Delta_F(x_1-x_3)\Delta_F(x_2-x_4) + \Delta_F(x_1-x_4)\Delta_F(x_2-x_3)$

Number of pairings for $2n$ fields:

$(2n-1)!! = (2n-1)(2n-3) \cdots 3 \cdot 1$

For 4 fields: 3, for 6 fields: 15, for 8 fields: 105, ...

Wick's Theorem with Interactions

When interactions are present (e.g., $\mathcal{L}_\text{int} = -\frac{\lambda}{4!}\phi^4$), we expand the interaction exponential perturbatively:

$Z[J] = \int \mathcal{D}\phi \, e^{i\int(\mathcal{L}_0 + \mathcal{L}_\text{int} + J\phi)} = e^{i\int \mathcal{L}_\text{int}[\frac{1}{i}\frac{\delta}{\delta J}]} Z_0[J]$

Each interaction vertex brings down factors of $\phi$, and Wick's theorem pairs them with propagators — this is the origin of Feynman diagrams!

Simulation: Path Integral Monte Carlo

This simulation implements the path integral for the quantum harmonic oscillator using Monte Carlo sampling in Euclidean time. The Wick rotation $t \to -i\tau$ transforms$e^{iS}$ into the real weight $e^{-S_E}$, enabling statistical sampling. The simulation computes the ground-state energy, two-point correlation function, effective mass (energy gap), and the ground-state wave function.

Path Integral Monte Carlo: Quantum Harmonic Oscillator

Python

script.py240 lines

#!/usr/bin/env python3
"""
path_integrals.py - 1D Path Integral Monte Carlo for Quantum Harmonic Oscillator
Computes ground state energy, correlation functions, and wave function using lattice methods
"""
import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import base64
from io import BytesIO

# ============================================================
# Path Integral Monte Carlo for Quantum Harmonic Oscillator
# S_E = integral dt [m/2 (dx/dt)^2 + 1/2 m omega^2 x^2]
# Discretized: S = sum_i [m/(2*a) (x_{i+1}-x_i)^2 + a/2 m omega^2 x_i^2]
# where a = epsilon is the lattice spacing
# ============================================================

# Physical parameters
m = 1.0           # mass
omega = 1.0       # oscillator frequency
N_sites = 64      # number of time slices
a = 0.5           # lattice spacing (Euclidean time step)
beta = N_sites * a  # total Euclidean time

# Monte Carlo parameters
N_therm = 2000     # thermalization sweeps
N_meas = 5000      # measurement sweeps
N_skip = 5         # sweeps between measurements
delta = 1.0        # Metropolis step size

# Initialize lattice with random positions
np.random.seed(42)
x = np.random.randn(N_sites) * 0.5

def euclidean_action(x, m, omega, a):
    """Compute the Euclidean action for the path"""
    N = len(x)
    S = 0.0
    for i in range(N):
        ip1 = (i + 1) % N  # periodic boundary
        kinetic = m / (2.0 * a) * (x[ip1] - x[i])**2
        potential = a * 0.5 * m * omega**2 * x[i]**2
        S += kinetic + potential
    return S

def local_action(x, i, m, omega, a):
    """Action contribution from site i"""
    N = len(x)
    ip1 = (i + 1) % N
    im1 = (i - 1) % N
    kinetic = m / (2.0 * a) * ((x[ip1] - x[i])**2 + (x[i] - x[im1])**2)
    potential = a * 0.5 * m * omega**2 * x[i]**2
    return kinetic + potential

# Metropolis algorithm
def metropolis_sweep(x, m, omega, a, delta):
    """One sweep of Metropolis updates"""
    N = len(x)
    accepted = 0
    for i in range(N):
        S_old = local_action(x, i, m, omega, a)
        x_old = x[i]
        x[i] += delta * (2.0 * np.random.random() - 1.0)
        S_new = local_action(x, i, m, omega, a)
        dS = S_new - S_old
        if dS < 0 or np.random.random() < np.exp(-dS):
            accepted += 1
        else:
            x[i] = x_old
    return accepted / N

# Thermalization
print("=" * 72)
print("  PATH INTEGRAL MONTE CARLO: QUANTUM HARMONIC OSCILLATOR")
print("=" * 72)
print()
print(f"Parameters: m = {m}, omega = {omega}, N_sites = {N_sites}, a = {a}")
print(f"Total Euclidean time: beta = {beta}")
print(f"Exact ground state energy: E_0 = omega/2 = {omega/2:.6f}")
print()
print("Thermalizing...")

for sweep in range(N_therm):
    acc = metropolis_sweep(x, m, omega, a, delta)

print(f"Thermalization complete. Final acceptance rate: {acc:.3f}")
print()

# Measurements
x2_samples = []
corr_data = np.zeros(N_sites)
n_corr = 0
path_samples = []

print("Measuring...")
for sweep in range(N_meas * N_skip):
    acc = metropolis_sweep(x, m, omega, a, delta)
    if sweep % N_skip == 0:
        x2_samples.append(np.mean(x**2))
        # Two-point correlation function <x(t) x(0)>
        for dt in range(N_sites):
            for i in range(N_sites):
                j = (i + dt) % N_sites
                corr_data[dt] += x[i] * x[j]
        n_corr += N_sites
        if len(path_samples) < 50:
            path_samples.append(x.copy())

corr_data /= n_corr

# Extract ground state energy from correlator
# <x(t)x(0)> ~ exp(-E_1 * t) for large t, where E_1 = omega (first excitation)
# More precisely: <x(t)x(0)> = 1/(2m*omega) * exp(-omega*t) for harmonic oscillator
t_lattice = np.arange(N_sites) * a
# Use effective mass: E_eff(t) = -log(C(t+a)/C(t)) / a
E_eff = np.zeros(N_sites - 1)
for i in range(N_sites - 1):
    if corr_data[i+1] > 0 and corr_data[i] > 0:
        E_eff[i] = -np.log(corr_data[i+1] / corr_data[i]) / a
    else:
        E_eff[i] = 0

# Ground state energy from <x^2> = 1/(2*m*omega)
x2_mean = np.mean(x2_samples)
x2_err = np.std(x2_samples) / np.sqrt(len(x2_samples))
E0_from_x2 = 0.5 * m * omega**2 * x2_mean + 0.5 * omega * 0  # virial theorem: <T> = <V> = E/2
# Better: E_0 = omega/2 and <x^2> = 1/(2*m*omega)
omega_measured = 1.0 / (2.0 * m * x2_mean)

print(f"<x^2> = {x2_mean:.6f} +/- {x2_err:.6f}")
print(f"Exact <x^2> = {1/(2*m*omega):.6f}")
print(f"Measured omega from <x^2>: {omega_measured:.4f} (exact: {omega:.4f})")
print()

# Effective energy from plateau in E_eff
plateau_region = E_eff[3:N_sites//3]
E_gap = np.mean(plateau_region[plateau_region > 0])
print(f"Energy gap (first excitation) from correlator: {E_gap:.4f}")
print(f"Exact energy gap: {omega:.4f}")
print()

# ============================================================
# Plotting
# ============================================================
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0f0a1a')

# --- Plot 1: Sample paths ---
ax1 = axes[0, 0]
ax1.set_facecolor('#1a1025')
for spine in ax1.spines.values():
    spine.set_color('#4c1d95')
ax1.tick_params(colors='#c4b5fd')
ax1.grid(True, color='#4c1d95', alpha=0.3)

for i in range(min(15, len(path_samples))):
    ax1.plot(t_lattice, path_samples[i], alpha=0.3, linewidth=0.8, color='#a78bfa')

ax1.set_xlabel('Euclidean time tau', color='#a78bfa', fontsize=11)
ax1.set_ylabel('x(tau)', color='#a78bfa', fontsize=11)
ax1.set_title('Monte Carlo Path Samples', color='#e9d5ff', fontsize=13, fontweight='bold')

# --- Plot 2: Two-point correlation function ---
ax2 = axes[0, 1]
ax2.set_facecolor('#1a1025')
for spine in ax2.spines.values():
    spine.set_color('#4c1d95')
ax2.tick_params(colors='#c4b5fd')
ax2.grid(True, color='#4c1d95', alpha=0.3)

t_half = N_sites // 2
t_plot = t_lattice[:t_half]
corr_plot = corr_data[:t_half]
# Exact result
corr_exact = 1.0/(2*m*omega) * (np.exp(-omega*t_plot) + np.exp(-omega*(beta - t_plot))) / (1 + np.exp(-omega*beta))

ax2.plot(t_plot, corr_plot, 'o', color='#a78bfa', markersize=4, label='MC data')
ax2.plot(t_plot, corr_exact, '-', color='#22d3ee', linewidth=2, label='Exact')
ax2.set_xlabel('Euclidean time separation tau', color='#a78bfa', fontsize=11)
ax2.set_ylabel('<x(tau) x(0)>', color='#a78bfa', fontsize=11)
ax2.set_title('Two-Point Correlation Function', color='#e9d5ff', fontsize=13, fontweight='bold')
ax2.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax2.set_yscale('log')

# --- Plot 3: Effective energy ---
ax3 = axes[1, 0]
ax3.set_facecolor('#1a1025')
for spine in ax3.spines.values():
    spine.set_color('#4c1d95')
ax3.tick_params(colors='#c4b5fd')
ax3.grid(True, color='#4c1d95', alpha=0.3)

t_eff = t_lattice[:N_sites//3]
E_eff_plot = E_eff[:N_sites//3]
valid = E_eff_plot > 0
ax3.plot(t_eff[valid], E_eff_plot[valid], 'o-', color='#f472b6', markersize=5, linewidth=1.5, label='Effective energy')
ax3.axhline(y=omega, color='#22d3ee', linewidth=2, linestyle='--', label=f'Exact Delta E = {omega:.2f}')
ax3.set_xlabel('Euclidean time tau', color='#a78bfa', fontsize=11)
ax3.set_ylabel('E_eff(tau)', color='#a78bfa', fontsize=11)
ax3.set_title('Effective Energy (plateau -> energy gap)', color='#e9d5ff', fontsize=13, fontweight='bold')
ax3.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)
ax3.set_ylim(0, 3)

# --- Plot 4: Probability distribution (wave function squared) ---
ax4 = axes[1, 1]
ax4.set_facecolor('#1a1025')
for spine in ax4.spines.values():
    spine.set_color('#4c1d95')
ax4.tick_params(colors='#c4b5fd')
ax4.grid(True, color='#4c1d95', alpha=0.3)

all_x = np.concatenate(path_samples)
x_bins = np.linspace(-3, 3, 60)
ax4.hist(all_x, bins=x_bins, density=True, color='#a78bfa', alpha=0.6, edgecolor='#7c3aed', label='MC histogram')

# Exact ground state |psi_0(x)|^2 = sqrt(m*omega/pi) * exp(-m*omega*x^2)
x_exact = np.linspace(-3, 3, 300)
psi2_exact = np.sqrt(m * omega / np.pi) * np.exp(-m * omega * x_exact**2)
ax4.plot(x_exact, psi2_exact, color='#22d3ee', linewidth=2.5, label='Exact |psi_0|^2')

ax4.set_xlabel('x', color='#a78bfa', fontsize=11)
ax4.set_ylabel('Probability density', color='#a78bfa', fontsize=11)
ax4.set_title('Ground State Wave Function from Path Integral', color='#e9d5ff', fontsize=13, fontweight='bold')
ax4.legend(facecolor='#1a1025', edgecolor='#4c1d95', labelcolor='#c4b5fd', fontsize=10)

plt.tight_layout(pad=2.0)

buf = BytesIO()
plt.savefig(buf, format='png', dpi=150, bbox_inches='tight', facecolor='#0f0a1a')
buf.seek(0)
img_b64 = base64.b64encode(buf.read()).decode()
print(f'<img src="data:image/png;base64,{img_b64}" alt="Path Integral Monte Carlo" />')
buf.close()

print()
print("The path integral approach recovers exact quantum mechanics results")
print("from a purely classical statistical mechanics computation in Euclidean time!")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Euclidean Path Integral and Statistical Mechanics

The oscillatory Minkowski path integral $e^{iS}$ is difficult to define rigorously and impractical for numerical computation. The Wick rotation$t \to -i\tau$ transforms it into a well-defined Euclidean integral.

The Wick Rotation

Under $t \to -i\tau$, the Minkowski action becomes the Euclidean action:

$iS_M \to -S_E, \quad \text{where} \quad S_E = \int d^4x_E \left[\frac{1}{2}(\partial_E\phi)^2 + \frac{1}{2}m^2\phi^2\right]$

The Euclidean generating functional is:

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

This is identical in form to the partition function of classical statistical mechanics with $S_E$ playing the role of the energy and the functional integral playing the role of the phase space sum. This correspondence provides:

• Numerical methods: Monte Carlo sampling of $e^{-S_E}$ converges because it is a real, positive weight (unlike $e^{iS}$).
• Lattice QFT: Discretize spacetime and compute the Euclidean path integral numerically — the only non-perturbative approach to QCD.
• Thermal field theory: Compactifying the Euclidean time direction with period $\beta = 1/T$ gives the thermal partition function at temperature $T$.
• Instantons: Classical solutions of the Euclidean equations of motion describe quantum tunneling between different vacua.

Connection to the Harmonic Oscillator

For the quantum harmonic oscillator, the Euclidean path integral at finite temperature$\beta = 1/T$ gives:

$Z = \text{Tr}(e^{-\beta H}) = \int_{\text{periodic}} \mathcal{D}x \, e^{-S_E[x]}$

The path integral is over periodic paths $x(0) = x(\beta)$. In the$\beta \to \infty$ (zero temperature) limit, only the ground state survives:$Z \to e^{-\beta E_0}$. This is exactly what our Monte Carlo simulation exploits to extract the ground state energy and wave function.

The correlation function $\langle x(\tau)x(0)\rangle$ decays as $e^{-\Delta E \cdot \tau}$for large $\tau$, where $\Delta E = E_1 - E_0$ is the energy gap. This gives a clean method to extract excitation energies, used extensively in lattice QCD to compute hadron masses.

The Effective Action and 1PI Diagrams

The effective action $\Gamma[\phi_c]$ is the Legendre transform of $W[J]$ and generates one-particle-irreducible (1PI) diagrams:

$\Gamma[\phi_c] = W[J] - \int d^4x \, J(x)\phi_c(x), \quad \text{where} \quad \phi_c(x) = \frac{\delta W}{\delta J(x)}$

The equation of motion for the classical field $\phi_c$ in the presence of the source is:

$\frac{\delta \Gamma}{\delta \phi_c(x)} = -J(x)$

Setting $J = 0$, the stationary points of $\Gamma$ give the quantum-corrected equations of motion. The effective action includes all quantum corrections and is the generating functional for 1PI vertex functions — the basic building blocks from which all scattering amplitudes are constructed.

Hierarchy of generating functionals: $Z[J]$ generates all correlation functions; $W[J] = -i\ln Z[J]$ generates connected correlators;$\Gamma[\phi_c]$ generates 1PI vertex functions. Each level removes more redundancy, and $\Gamma$ is the most economical description of the quantum theory.

Summary: The Path Integral Framework

Sum Over Histories

Quantum amplitudes are computed by summing $e^{iS}$ over all field configurations. The classical path emerges as the stationary phase in the limit $\hbar \to 0$.

Generating Functional

$Z[J]$ encodes all correlation functions via functional derivatives. Connected correlators come from $W[J] = -i\ln Z[J]$.

Free Theory is Gaussian

The free field path integral evaluates to a Gaussian, with Wick's theorem reducing all correlators to products of propagators $\Delta_F(x-y)$.

Gateway to Feynman Diagrams

Perturbative expansion of interactions combined with Wick's theorem generates Feynman diagrams systematically. Each vertex and propagator has a direct path-integral origin.

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