Chapter 31: Path Integrals — Feynman's Formulation

In which we discover that a quantum particle takes every possible path simultaneously — and that classical mechanics is the universe's way of selecting the one path that matters most

Opening: A Radical Democracy of Trajectories

Everything we have done in this book so far can be traced back to a single equation: the Schrödinger equation. We wrote it down in Chapter 1, solved it in increasingly sophisticated contexts through Chapters 2–5, reformulated it in the elegant language of Dirac notation in Chapter 8, and used it to derive perturbation theory, scattering amplitudes, and the behavior of identical particles. The Schrödinger equation has never failed us.

And yet.

In 1948, a twenty-nine-year-old Richard Feynman published a paper that reformulated the entirety of quantum mechanics without the Schrödinger equation. No wave functions. No operators. No eigenvalue problems. Instead, Feynman proposed something that sounded, at first hearing, completely mad: to find the probability amplitude for a particle to go from point $A$ to point $B$, you should sum up contributions from every conceivable path connecting the two points — straight lines, wild spirals, paths that loop backward in time, paths that visit the Andromeda galaxy and return. Every path contributes. No path is forbidden. The contribution of each path is a complex exponential whose phase is the classical action of that path, divided by $\hbar$.

This is the path integral formulation of quantum mechanics. It is exactly equivalent to the Schrödinger equation — every prediction is identical — yet it offers something the Schrödinger equation does not: a direct, intuitive picture of why classical mechanics works. In the classical limit, the wild paths cancel each other out through destructive interference, and only the path of stationary action — the classical trajectory — survives. Classical mechanics is not a separate theory. It is the constructive interference pattern of quantum mechanics.

Feynman's formulation also provides something else the Schrödinger equation handles poorly: a natural framework for quantum field theory. Feynman diagrams — those iconic doodles of particle interactions that decorate the walls of every physics department — are not merely mnemonics. They are terms in a perturbative expansion of a path integral over field configurations. Without the path integral, much of modern theoretical physics would be unthinkable.

In this chapter, we will derive the path integral from first principles, starting from the time-evolution operator you already know from Chapter 7. We will compute it exactly for two systems: the free particle and the quantum harmonic oscillator (our anchor example, now in its most sophisticated incarnation). We will show that the stationary phase approximation recovers classical mechanics — the Euler-Lagrange equations and Hamilton's principle emerge automatically. And we will glimpse the connection to statistical mechanics and Feynman diagrams that makes the path integral the central tool of modern theoretical physics.

Learning paths: - 🏃 Streamlined path: Sections 31.1–31.4 give the core derivation and the classical limit. If time is limited, stop after 31.4. - 🔬 Deep dive path: All seven sections. The QHO path integral (31.5) is beautiful and connects directly to quantum field theory (Chapter 37). The statistical mechanics connection (31.6) opens the door to an entire field.

31.1 From the Time-Evolution Operator to the Path Integral

The Propagator: What We Want to Calculate

Recall from Chapter 7 that the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ takes a quantum state at $t = 0$ to its state at time $t$:

$$|\Psi(t)\rangle = \hat{U}(t)|\Psi(0)\rangle$$

In the position representation, this becomes:

$$\Psi(x_f, t) = \int_{-\infty}^{\infty} K(x_f, t; x_i, 0)\, \Psi(x_i, 0)\, dx_i$$

where the propagator (or kernel) is the matrix element of $\hat{U}$ in the position basis:

$$K(x_f, t; x_i, 0) = \langle x_f | \hat{U}(t) | x_i \rangle = \langle x_f | e^{-i\hat{H}t/\hbar} | x_i \rangle$$

The propagator is the amplitude for the particle to go from position $x_i$ at time $0$ to position $x_f$ at time $t$. If you know the propagator, you know everything — you can evolve any initial state forward in time. It is the Green's function of the time-dependent Schrödinger equation.

The propagator also has a spectral decomposition in terms of energy eigenstates. If $\hat{H}|\psi_n\rangle = E_n|\psi_n\rangle$, then:

$$K(x_f, t; x_i, 0) = \sum_n \psi_n(x_f)\psi_n^*(x_i)\, e^{-iE_n t/\hbar}$$

This is perfectly correct and perfectly useless for most practical purposes — it requires solving the eigenvalue problem first, which is the very thing we are trying to avoid. What Feynman realized is that there is another representation of $K$ that does not require knowing the eigenstates at all.

Our goal in this section is to write $K(x_f, t; x_i, 0)$ as a sum over paths. We will do this by a trick so simple it almost seems like cheating: we will repeatedly insert the identity operator.

Slicing Time: The Key Step

Divide the time interval $[0, t]$ into $N$ equal slices of duration $\epsilon = t/N$. The time-evolution operator factors as:

$$\hat{U}(t) = \hat{U}(\epsilon)^N = \left(e^{-i\hat{H}\epsilon/\hbar}\right)^N$$

This is exact — it uses only the composition property $\hat{U}(t_1 + t_2) = \hat{U}(t_1)\hat{U}(t_2)$ and the fact that $\hat{H}$ is time-independent.

Now insert the completeness relation $\hat{I} = \int dx\,|x\rangle\langle x|$ between each pair of adjacent $\hat{U}(\epsilon)$ factors. With $N-1$ insertions:

$$K(x_f, t; x_i, 0) = \int dx_1 \int dx_2 \cdots \int dx_{N-1} \prod_{j=0}^{N-1} \langle x_{j+1}| e^{-i\hat{H}\epsilon/\hbar} |x_j\rangle$$

where $x_0 \equiv x_i$ and $x_N \equiv x_f$.

Each factor $\langle x_{j+1}|e^{-i\hat{H}\epsilon/\hbar}|x_j\rangle$ is the amplitude for a short hop: the particle moves from $x_j$ to $x_{j+1}$ during a tiny time interval $\epsilon$. The full propagator is the integral over all possible intermediate positions $x_1, x_2, \ldots, x_{N-1}$.

💡 Key Insight: This is not an approximation. For any finite $N$, the formula is exact. But we will find that the integrand simplifies dramatically when $\epsilon$ is small (i.e., $N$ is large), and the limit $N \to \infty$ is where the path integral lives.

Evaluating the Short-Time Propagator

For the single-step propagator, we use the Trotter decomposition. For a Hamiltonian of the standard form $\hat{H} = \hat{p}^2/(2m) + V(\hat{x})$:

$$e^{-i\hat{H}\epsilon/\hbar} = e^{-i[\hat{p}^2/(2m) + V(\hat{x})]\epsilon/\hbar} \approx e^{-iV(\hat{x})\epsilon/\hbar}\, e^{-i\hat{p}^2\epsilon/(2m\hbar)} + O(\epsilon^2)$$

The error is of order $\epsilon^2$ because $[\hat{x}, \hat{p}] = i\hbar$ generates commutator corrections. When we take $N \to \infty$ (so $\epsilon \to 0$), these corrections vanish.

⚠️ Mathematical subtlety: The Trotter product formula guarantees that $\lim_{N\to\infty} (e^{-iA\epsilon}e^{-iB\epsilon})^N = e^{-i(A+B)t}$ under appropriate conditions. We are using this in the exponent, separating kinetic and potential energy.

Now evaluate:

$$\langle x_{j+1}| e^{-i\hat{H}\epsilon/\hbar} |x_j\rangle \approx e^{-iV(x_j)\epsilon/\hbar}\, \langle x_{j+1}| e^{-i\hat{p}^2\epsilon/(2m\hbar)} |x_j\rangle$$

The remaining matrix element is a free-particle propagator for time $\epsilon$. Insert the momentum completeness relation $\hat{I} = \int dp/(2\pi\hbar)\,|p\rangle\langle p|$:

$$\langle x_{j+1}| e^{-i\hat{p}^2\epsilon/(2m\hbar)} |x_j\rangle = \int \frac{dp}{2\pi\hbar}\, e^{-ip^2\epsilon/(2m\hbar)}\, e^{ip(x_{j+1} - x_j)/\hbar}$$

This is a Gaussian integral in $p$. Using $\int_{-\infty}^{\infty} dp\, e^{-\alpha p^2 + \beta p} = \sqrt{\pi/\alpha}\, e^{\beta^2/(4\alpha)}$ with $\alpha = i\epsilon/(2m\hbar)$ and $\beta = i(x_{j+1}-x_j)/\hbar$:

$$\langle x_{j+1}| e^{-i\hat{p}^2\epsilon/(2m\hbar)} |x_j\rangle = \sqrt{\frac{m}{2\pi i\hbar\epsilon}} \exp\left(\frac{im(x_{j+1} - x_j)^2}{2\hbar\epsilon}\right)$$

Combining with the potential energy factor:

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

Recognize what is in the exponential: $\frac{m}{2}\left(\frac{\Delta x}{\Delta t}\right)^2 - V(x)$ is the Lagrangian $L(\dot{x}, x)$, evaluated with $\dot{x} \approx (x_{j+1} - x_j)/\epsilon$. The argument of the exponential is $(i/\hbar) \cdot L \cdot \epsilon$, which is $(i/\hbar)$ times the action for the $j$-th time step.

This is the moment where Dirac's "analogy" becomes Feynman's equality. The short-time propagator is not merely analogous to $e^{iL\epsilon/\hbar}$ — it is $e^{iL\epsilon/\hbar}$, up to a normalization factor and corrections that vanish in the continuum limit.

✅ What we have so far: The propagator for one small time step $\epsilon$ is a known function of the endpoints $x_j$ and $x_{j+1}$. The propagator for $N$ steps is the product of $N$ such factors, integrated over all intermediate positions. This is already a "sum over paths" — each choice of $(x_1, x_2, \ldots, x_{N-1})$ defines a piecewise-linear path through spacetime, and the integral sums over all such paths.

Assembling the Path Integral

Substituting back into the multi-step propagator and taking $N \to \infty$:

$$\boxed{K(x_f, t; x_i, 0) = \int \mathcal{D}[x(t')]\, \exp\left(\frac{i}{\hbar} S[x(t')]\right)}$$

where the classical action is:

$$S[x(t')] = \int_0^t L\left(\dot{x}(t'), x(t')\right) dt' = \int_0^t \left[\frac{1}{2}m\dot{x}^2 - V(x)\right] dt'$$

and the path integral measure is the formal limit:

$$\int \mathcal{D}[x(t')] \equiv \lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int_{-\infty}^{\infty} dx_1 \int_{-\infty}^{\infty} dx_2 \cdots \int_{-\infty}^{\infty} dx_{N-1}$$

The integral is over all paths $x(t')$ satisfying the boundary conditions $x(0) = x_i$ and $x(t) = x_f$. There are no constraints on what the path does in between. It can be smooth, discontinuous, fractal — everything contributes.

📊 Dimensional check: The action $S$ has dimensions of $[E \cdot T] = [J \cdot s]$, which is the same as $\hbar$. So $S/\hbar$ is dimensionless, as it must be for the exponent to make sense. The normalization prefactor $\left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2}$ ensures the correct dimensions for the propagator.

This is Feynman's path integral. Let us now understand what it means.

31.2 The Sum Over All Paths

What "Sum Over All Paths" Really Means

The path integral $\int \mathcal{D}[x]\, e^{iS[x]/\hbar}$ is a sum over the uncountably infinite set of all continuous paths from $(x_i, 0)$ to $(x_f, t)$. Each path $x(t')$ contributes a complex number $e^{iS[x]/\hbar}$ whose magnitude is 1 (every path contributes equally in magnitude) and whose phase is $S[x]/\hbar$ (the classical action in units of $\hbar$).

The propagator is obtained by adding up all these unit-magnitude, varying-phase complex numbers. Paths with similar phases add constructively; paths with very different phases cancel through destructive interference.

This is the democratic principle of the path integral: no path is more important than any other a priori. The classical path has no special status in the formulation itself. What makes it special is that it is the path of stationary phase, as we will see in Section 31.4.

Visualizing the Sum

Imagine a particle that starts at $x_i = 0$ at $t = 0$ and arrives at $x_f$ at time $t$. Consider three representative paths:

1. The straight-line path (constant velocity $v = x_f/t$): This has action $S_{\text{straight}} = m x_f^2/(2t)$ for a free particle.

2. A path that detours slightly: Perhaps the particle moves a little faster at first, then slows down. Its action will differ from the straight-line action by a small amount $\delta S$.

3. A wildly looping path: The particle speeds off to $x = 1000 x_f$, loops back, and arrives at $x_f$. Its action is enormously different from $S_{\text{straight}}$.

For the straight and slightly-detoured paths, the phase difference $\delta S/\hbar$ is small, so the contributions add nearly constructively. For the wild path, the phase $S/\hbar$ is huge (because $S \gg \hbar$ for macroscopic motions), and nearby paths in "path space" have very different phases — they cancel through destructive interference.

This is why classical mechanics works: for macroscopic systems, $S \gg \hbar$, and only paths near the classical trajectory (where $\delta S = 0$) survive the interference. For microscopic systems, $S \sim \hbar$, and many paths contribute — this is quantum mechanics.

💡 Key Insight: The path integral explains the transition from quantum to classical physics as a crossover from a regime where many paths contribute (quantum) to a regime where only one path dominates (classical). There is no sharp boundary — just a smooth transition governed by the ratio $S/\hbar$.

The Double-Slit Experiment Revisited

The path integral provides perhaps the most intuitive explanation of the double-slit experiment (Chapter 1). A particle going from the source to a point on the screen has multiple classes of paths:

• Paths that go through slit 1
• Paths that go through slit 2
• Paths that go through neither slit (but around the edges, backward, etc.)

All contribute. But the paths going through neither slit generally have very large actions relative to $\hbar$ and cancel each other out. The dominant contributions come from paths near the two straight lines through the two slits. The interference between these two families of paths produces the fringe pattern.

When you place a detector at one slit, you modify the Hamiltonian in the path integral (adding an interaction with the detector). The paths through that slit now have different actions, the interference is destroyed, and the pattern disappears.

The Path Integral and the Uncertainty Principle

The path integral provides a vivid visualization of the uncertainty principle. Consider a particle localized at $x_i$ at $t = 0$, propagating freely to time $t$. In the path integral, the particle "explores" all intermediate positions. But the paths that contribute most are those near the classical trajectory, within a "tube" of width:

$$\Delta x_{\text{tube}} \sim \sqrt{\frac{\hbar t}{m}}$$

This is the quantum uncertainty in position after time $t$ — exactly the wave-packet spreading width from Chapter 7. The path integral tells us why wave packets spread: it is because paths that deviate by $\Delta x_{\text{tube}}$ from the classical trajectory have action differences of order $\hbar$, so they contribute constructively rather than canceling.

For shorter times, the tube is narrower (less uncertainty). For heavier particles, the tube is narrower (more classical). For $\hbar \to 0$, the tube collapses to a single line — the classical trajectory. These are all manifestations of the correspondence principle, made geometrically transparent by the path integral.

Why the Lagrangian, Not the Hamiltonian?

It is striking that the path integral uses the Lagrangian $L = T - V$ rather than the Hamiltonian $H = T + V$. This is not arbitrary. The Lagrangian formulation has two key advantages for the path integral:

1. It treats position and velocity on equal footing. The Lagrangian $L(\dot{x}, x)$ depends on the path $x(t')$ and its derivative — precisely the objects over which we integrate. The Hamiltonian depends on $x$ and $p$, but the path integral over positions naturally generates the Lagrangian, not the Hamiltonian.

2. It is manifestly Lorentz-invariant. In relativistic physics, the Lagrangian density $\mathcal{L}$ is a Lorentz scalar, while the Hamiltonian density is the time-time component of the stress-energy tensor — inherently frame-dependent. For quantum field theory, where Lorentz invariance is essential, the Lagrangian path integral is the natural starting point.

There is also a phase-space path integral that uses the Hamiltonian directly (see Problem 31.5). Integrating out the momentum variables in the phase-space path integral recovers the Lagrangian path integral. The two are equivalent, but the Lagrangian version is more commonly used.

Mathematical Subtleties

The path integral as we have written it raises serious mathematical questions:

Problem 1: The measure $\mathcal{D}[x]$ is not a well-defined mathematical measure. The limit $N \to \infty$ of the discrete product of Lebesgue integrals does not converge in any standard sense. The oscillatory factor $e^{iS/\hbar}$ has magnitude 1 everywhere, so there is no convergence factor.

Problem 2: The "typical" path in the integral is nowhere differentiable. The dominant paths in a path integral are continuous but not smooth — they are fractal-like, resembling Brownian motion. The velocity $\dot{x}$ does not exist as a classical derivative. This can be shown by examining the scaling of the discrete differences $(x_{j+1} - x_j) \sim \sqrt{\epsilon}$ rather than $\sim \epsilon$.

Resolution: In practice, the path integral is defined by the discretized version: you always compute with finite $N$ and take the limit at the end. For many important cases (free particle, QHO, perturbation theory around these), the limit can be taken explicitly and gives well-defined, finite answers. The Wick rotation to imaginary time (Section 31.6) converts the oscillatory integral into a well-defined probability measure — the Wiener measure of Brownian motion.

These are not deficiencies that undermine the theory. The path integral has been verified against the Schrödinger equation in every case where both can be computed. Its mathematical foundations are an active area of research in mathematical physics, but for the working physicist, the discrete definition is sufficient.

31.3 The Free-Particle Propagator

Setting Up the Calculation

The simplest nontrivial path integral is the free particle ($V = 0$). The action is purely kinetic:

$$S[x] = \int_0^t \frac{1}{2}m\dot{x}^2\, dt'$$

In the discretized form:

$$K_{\text{free}}(x_f, t; x_i, 0) = \lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int dx_1 \cdots dx_{N-1} \exp\left[\frac{im}{2\hbar\epsilon}\sum_{j=0}^{N-1}(x_{j+1} - x_j)^2\right]$$

This is a product of $N$ coupled Gaussian integrals. The coupling arises because each $x_j$ appears in two terms (the $j$-th and $(j-1)$-th steps). However, Gaussian integrals are tractable — we can evaluate them sequentially, starting from $x_1$.

Sequential Gaussian Integration

Consider the integral over $x_1$. The terms in the exponent that involve $x_1$ are:

$$\frac{im}{2\hbar\epsilon}\left[(x_1 - x_0)^2 + (x_2 - x_1)^2\right]$$

Expanding and completing the square in $x_1$:

$$(x_1 - x_0)^2 + (x_2 - x_1)^2 = 2\left(x_1 - \frac{x_0 + x_2}{2}\right)^2 + \frac{(x_2 - x_0)^2}{2}$$

The Gaussian integral over $x_1$ gives:

$$\int dx_1 \exp\left[\frac{im}{2\hbar\epsilon}\cdot 2\left(x_1 - \frac{x_0+x_2}{2}\right)^2\right] = \sqrt{\frac{2\pi i\hbar\epsilon}{2m}} = \sqrt{\frac{\pi i\hbar\epsilon}{m}}$$

The remainder is $\exp\left[\frac{im}{2\hbar\epsilon}\cdot\frac{(x_2 - x_0)^2}{2}\right] = \exp\left[\frac{im(x_2 - x_0)^2}{2\hbar \cdot 2\epsilon}\right]$.

Notice the pattern: after integrating over $x_1$, the result looks like a propagator from $x_0$ to $x_2$ over time $2\epsilon$. Continuing this procedure for $x_2, x_3, \ldots, x_{N-1}$, we find by induction that after integrating out all intermediate positions:

$$K_{\text{free}}(x_f, t; x_i, 0) = \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \cdot \left(\frac{\pi i\hbar\epsilon}{m}\right)^{(N-1)/2} \cdot \frac{1}{\sqrt{N}} \cdot \exp\left[\frac{im(x_f - x_i)^2}{2\hbar N\epsilon}\right]$$

Since $N\epsilon = t$, this simplifies to:

$$\boxed{K_{\text{free}}(x_f, t; x_i, 0) = \sqrt{\frac{m}{2\pi i\hbar t}} \exp\left(\frac{im(x_f - x_i)^2}{2\hbar t}\right)}$$

Verification

We can verify this result in several ways:

Check 1: Schrödinger equation. The propagator must satisfy $i\hbar\,\partial K/\partial t = -(\hbar^2/2m)\,\partial^2 K/\partial x_f^2$. Taking derivatives:

$$\frac{\partial K}{\partial t} = K\left[-\frac{1}{2t} + \frac{im(x_f - x_i)^2}{2\hbar t^2} \cdot (-1)\right] = K\left[-\frac{1}{2t} - \frac{im(x_f - x_i)^2}{2\hbar t^2}\right]$$

$$\frac{\partial^2 K}{\partial x_f^2} = K\left[\frac{im}{\hbar t} + \left(\frac{im(x_f - x_i)}{\hbar t}\right)^2\right]$$

Substituting confirms $i\hbar\,\partial_t K = -(\hbar^2/2m)\,\partial_{x_f}^2 K$. ✅

Check 2: Initial condition. As $t \to 0^+$, the Gaussian $\exp[im(x_f - x_i)^2/(2\hbar t)]$ becomes a rapidly oscillating function that vanishes everywhere except at $x_f = x_i$, where it diverges. The prefactor $1/\sqrt{t}$ ensures the correct normalization. In the distributional sense:

$$\lim_{t\to 0^+} K_{\text{free}}(x_f, t; x_i, 0) = \delta(x_f - x_i)$$

This is exactly the identity operator in the position basis: $\hat{U}(0) = \hat{I}$. ✅

Check 3: Composition. The free-particle propagator should satisfy $\int K(x_f, t_2; x, t_1)\, K(x, t_1; x_i, 0)\, dx = K(x_f, t_2; x_i, 0)$. This reduces to a Gaussian integral over $x$ that can be evaluated explicitly — and it works. ✅

Check 4: Classical limit. The phase of the propagator is $S_{\text{cl}}/\hbar = m(x_f - x_i)^2/(2\hbar t)$, which is the classical action for a free particle traveling from $x_i$ to $x_f$ in time $t$ with constant velocity $v = (x_f - x_i)/t$. Indeed, $S_{\text{cl}} = \frac{1}{2}mv^2 t = m(x_f - x_i)^2/(2t)$. ✅

📊 Numerical example: An electron ($m = 9.1 \times 10^{-31}$ kg) travels 1 nm in 1 fs. The classical action is $S_{\text{cl}} = m\Delta x^2/(2\Delta t) = 9.1 \times 10^{-31} \times (10^{-9})^2/(2 \times 10^{-15}) = 4.6 \times 10^{-34}$ J·s $\approx 0.43\hbar$. The phase is less than one radian — many paths contribute, and quantum effects dominate. For a baseball ($m = 0.145$ kg, $\Delta x = 18.4$ m, $\Delta t = 0.5$ s), $S_{\text{cl}} \approx 4.9$ J·s $\approx 4.6 \times 10^{33}\hbar$. The phase is enormous, and only the classical path survives.

Propagation of a Gaussian Wave Packet

As a consistency check, let us verify that the free-particle propagator correctly evolves a Gaussian wave packet. Take the initial state:

$$\Psi(x, 0) = \left(\frac{1}{2\pi\sigma_0^2}\right)^{1/4} \exp\left(-\frac{x^2}{4\sigma_0^2} + ik_0 x\right)$$

a Gaussian centered at the origin with initial momentum $\hbar k_0$ and width $\sigma_0$. Apply the propagator:

$$\Psi(x_f, t) = \int_{-\infty}^{\infty} K_{\text{free}}(x_f, t; x_i, 0)\, \Psi(x_i, 0)\, dx_i$$

This is a Gaussian integral in $x_i$ (the product of two Gaussians is a Gaussian). After completing the square and evaluating, the result is exactly the spreading wave packet of Chapter 7:

$$|\Psi(x_f, t)|^2 = \frac{1}{\sqrt{2\pi}\,\sigma(t)} \exp\left[-\frac{(x_f - v_g t)^2}{2\sigma(t)^2}\right]$$

where $v_g = \hbar k_0/m$ is the group velocity and $\sigma(t) = \sigma_0\sqrt{1 + \hbar^2 t^2/(4m^2\sigma_0^4)}$ is the time-dependent width. The propagator approach reproduces the Schrödinger equation result, as it must — but notice how naturally the calculation flows. There are no differential equations to solve, no eigenfunction expansions to sum. You simply multiply two functions, complete the square, and evaluate the Gaussian integral.

The Free Propagator in Higher Dimensions

The free-particle propagator generalizes straightforwardly to $d$ dimensions:

$$K_{\text{free}}(\mathbf{r}_f, t; \mathbf{r}_i, 0) = \left(\frac{m}{2\pi i\hbar t}\right)^{d/2} \exp\left(\frac{im|\mathbf{r}_f - \mathbf{r}_i|^2}{2\hbar t}\right)$$

This is simply the product of $d$ independent one-dimensional propagators — a consequence of the fact that the free-particle Hamiltonian $\hat{H} = \hat{\mathbf{p}}^2/(2m)$ separates in Cartesian coordinates. For $d = 3$, the prefactor is $(m/(2\pi i\hbar t))^{3/2}$, and the propagator is the retarded Green's function of the three-dimensional free Schrödinger equation.

31.4 Stationary Phase and the Classical Limit

The Stationary Phase Approximation

The path integral $K = \int \mathcal{D}[x]\, e^{iS[x]/\hbar}$ sums contributions of magnitude 1 with rapidly varying phases when $S \gg \hbar$. The dominant contribution comes from the neighborhood of paths where the phase $S[x]/\hbar$ is stationary — i.e., where the first-order variation of $S$ vanishes:

$$\delta S[x_{\text{cl}}] = 0$$

This is the principle of stationary action (often called the principle of least action, though "stationary" is more accurate). It is the defining equation of classical mechanics.

Recovering the Euler-Lagrange Equations

Let $x_{\text{cl}}(t')$ be the classical path satisfying $\delta S = 0$ with boundary conditions $x_{\text{cl}}(0) = x_i$ and $x_{\text{cl}}(t) = x_f$. Write a general path as $x(t') = x_{\text{cl}}(t') + y(t')$, where $y(t')$ is the deviation from the classical path, satisfying $y(0) = y(t) = 0$.

The action expands as:

$$S[x_{\text{cl}} + y] = S[x_{\text{cl}}] + \underbrace{\delta S}_{\text{= 0 on classical path}} + \frac{1}{2}\delta^2 S + \cdots$$

where the first-order variation vanishes by the definition of $x_{\text{cl}}$, and the second-order variation is:

$$\delta^2 S = \int_0^t \left[m\dot{y}^2 - V''(x_{\text{cl}})y^2\right] dt'$$

The condition $\delta S = 0$ for arbitrary variations $y(t')$ gives the Euler-Lagrange equation:

$$m\ddot{x}_{\text{cl}} = -V'(x_{\text{cl}})$$

This is Newton's second law. We have derived classical mechanics from the path integral.

💡 Key Insight: In the Schrödinger formulation, the classical limit requires Ehrenfest's theorem and a discussion of when expectation values track classical trajectories (Chapter 7). In the path integral formulation, the classical limit is immediate and transparent: when $S_{\text{cl}} \gg \hbar$, the stationary phase approximation selects the classical path automatically. No additional assumptions are needed.

The Semiclassical (WKB) Approximation

Keeping the second-order term in the expansion:

$$K(x_f, t; x_i, 0) \approx e^{iS_{\text{cl}}/\hbar} \int \mathcal{D}[y]\, \exp\left(\frac{i}{2\hbar} \delta^2 S[y]\right)$$

The integral over fluctuations $y(t')$ is Gaussian (since $\delta^2 S$ is quadratic in $y$) and can in principle be evaluated exactly. The result is:

$$K(x_f, t; x_i, 0) \approx A(x_f, x_i, t)\, e^{iS_{\text{cl}}(x_f, x_i, t)/\hbar}$$

where $A$ is a slowly varying amplitude determined by the second variation of the action. This is the semiclassical approximation to the propagator — the path integral version of the WKB method.

For the free particle, $V = 0$ so $V'' = 0$, and the second variation is $\delta^2 S = \int_0^t m\dot{y}^2\,dt'$. There are no higher-order terms — the path integral is exactly Gaussian — and the semiclassical approximation is exact. This is why we were able to compute the free-particle propagator exactly.

Corrections to Classical Mechanics

The path integral makes the structure of quantum corrections to classical mechanics explicit:

$$K = e^{iS_{\text{cl}}/\hbar} \times \underbrace{\text{(Gaussian fluctuations)}}_{\text{one-loop}} \times \underbrace{\text{(anharmonic corrections)}}_{\text{higher-loop}}$$

The "one-loop" correction comes from the Gaussian integral over $y$ and involves the determinant of the differential operator $-m\partial_t^2 - V''(x_{\text{cl}})$. Higher-order corrections involve the cubic, quartic, etc. terms in the Taylor expansion of $V(x)$ around the classical path and are organized by powers of $\hbar$.

This expansion in powers of $\hbar$ is the loop expansion of the path integral. The classical limit ($\hbar \to 0$) retains only the leading term $e^{iS_{\text{cl}}/\hbar}$. The first quantum correction is the one-loop determinant. Each subsequent order brings a factor of $\hbar$ and corresponds to increasingly quantum behavior. The terminology "loop" comes from the diagrammatic representation of these corrections, which we will encounter in Section 31.7.

A Worked Example: The Stationary Phase for a Falling Particle

To make the stationary phase approximation concrete, consider a particle of mass $m$ falling under gravity: $V(x) = mgx$. The classical path from $(x_i, 0)$ to $(x_f, t)$ is:

$$x_{\text{cl}}(t') = x_i + \left(\frac{x_f - x_i}{t} - \frac{1}{2}gt\right)t' + \frac{1}{2}gt'^2$$

The classical action is:

$$S_{\text{cl}} = \frac{m(x_f - x_i)^2}{2t} - \frac{mg(x_f + x_i)t}{2} - \frac{m g^2 t^3}{24}$$

The first term is the free-particle action. The second is the work done by gravity. The third is a purely time-dependent correction. Notice that $\partial S_{\text{cl}}/\partial x_f = m(x_f - x_i)/t - mgt/2 = mv_f$ gives the final momentum, and $-\partial S_{\text{cl}}/\partial x_i = m(x_f - x_i)/t + mgt/2 = mv_i$ gives the initial momentum. These are Hamilton-Jacobi relations.

Since $V(x) = mgx$ is at most linear in $x$, we have $V'' = 0$, and the path integral is again exactly Gaussian. The semiclassical propagator is exact, just as for the free particle. This is a general rule: the path integral is exactly solvable whenever the Lagrangian is at most quadratic in $x$ and $\dot{x}$.

Connection to Hamilton-Jacobi Theory

The phase of the semiclassical propagator is the classical action $S_{\text{cl}}(x_f, x_i, t)$, viewed as a function of the endpoints. This is precisely Hamilton's principal function from classical mechanics. The Hamilton-Jacobi equation:

$$\frac{\partial S_{\text{cl}}}{\partial t} + H\left(x_f, \frac{\partial S_{\text{cl}}}{\partial x_f}\right) = 0$$

is the classical limit of the Schrödinger equation. Specifically, if you write $\Psi = A\, e^{iS/\hbar}$ and take $\hbar \to 0$, the Schrödinger equation reduces to the Hamilton-Jacobi equation at leading order.

The path integral thus provides a seamless bridge: at one end is quantum mechanics (sum over all paths), at the other end is classical mechanics (one path dominates), and the Hamilton-Jacobi equation sits precisely at the junction.

31.5 The Path Integral for the Quantum Harmonic Oscillator

Setting Up the QHO Path Integral

The quantum harmonic oscillator is the second system (after the free particle) for which the path integral can be evaluated exactly. This is because the QHO Lagrangian:

$$L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2$$

is quadratic in both $x$ and $\dot{x}$, making the action a quadratic functional — and the path integral therefore Gaussian.

The classical equation of motion is $\ddot{x}_{\text{cl}} + \omega^2 x_{\text{cl}} = 0$, with the boundary conditions $x_{\text{cl}}(0) = x_i$ and $x_{\text{cl}}(t) = x_f$. The solution is:

$$x_{\text{cl}}(t') = \frac{x_i \sin[\omega(t - t')] + x_f \sin(\omega t')}{\sin(\omega t)}$$

(This requires $\sin(\omega t) \neq 0$, i.e., $t$ is not an integer multiple of $\pi/\omega$. At these special times, called focal points or caustics, the classical path is not unique, and the propagator has singularities.)

Computing the Classical Action

The classical action is:

$$S_{\text{cl}} = \int_0^t \left[\frac{1}{2}m\dot{x}_{\text{cl}}^2 - \frac{1}{2}m\omega^2 x_{\text{cl}}^2\right] dt'$$

Substituting the classical trajectory and evaluating (using the identity $\int_0^t \sin^2(\omega t')\, dt' = t/2 - \sin(2\omega t)/(4\omega)$ and similar):

$$S_{\text{cl}} = \frac{m\omega}{2\sin(\omega t)}\left[(x_i^2 + x_f^2)\cos(\omega t) - 2x_i x_f\right]$$

This is the QHO analogue of the free-particle classical action $m(x_f - x_i)^2/(2t)$. In fact, in the limit $\omega \to 0$, $\sin(\omega t) \to \omega t$ and $\cos(\omega t) \to 1$, and the QHO action reduces to the free-particle action (verify this as an exercise).

The Fluctuation Determinant

Since the action is exactly quadratic, the path integral factorizes:

$$K_{\text{QHO}} = e^{iS_{\text{cl}}/\hbar} \int \mathcal{D}[y]\, \exp\left(\frac{i}{2\hbar}\int_0^t [m\dot{y}^2 - m\omega^2 y^2]\,dt'\right)$$

where $y(t') = x(t') - x_{\text{cl}}(t')$ satisfies $y(0) = y(t) = 0$. The integral over fluctuations is a Gaussian functional integral that depends on the operator $\hat{O} = -m(\partial_t^2 + \omega^2)$ acting on functions vanishing at both endpoints.

The eigenfunctions of this operator with Dirichlet boundary conditions are $\phi_n(t') = \sin(n\pi t'/t)$ for $n = 1, 2, 3, \ldots$, with eigenvalues:

$$\lambda_n = m\left(\frac{n^2\pi^2}{t^2} - \omega^2\right)$$

The functional determinant is:

$$\det\hat{O} = \prod_{n=1}^{\infty} \lambda_n = \prod_{n=1}^{\infty} m\left(\frac{n^2\pi^2}{t^2} - \omega^2\right)$$

This infinite product can be regularized by dividing by the free-particle determinant (which we already know how to handle):

$$\frac{\det\hat{O}_{\text{QHO}}}{\det\hat{O}_{\text{free}}} = \prod_{n=1}^{\infty}\left(1 - \frac{\omega^2 t^2}{n^2\pi^2}\right) = \frac{\sin(\omega t)}{\omega t}$$

The last step uses the celebrated infinite product formula for the sine function: $\sin(z)/z = \prod_{n=1}^{\infty}(1 - z^2/(n^2\pi^2))$. Since the free-particle prefactor is $\sqrt{m/(2\pi i\hbar t)}$, the QHO prefactor is:

$$A_{\text{QHO}} = \sqrt{\frac{m}{2\pi i\hbar t}} \cdot \sqrt{\frac{\omega t}{\sin(\omega t)}} = \sqrt{\frac{m\omega}{2\pi i\hbar \sin(\omega t)}}$$

The Complete QHO Propagator

$$\boxed{K_{\text{QHO}}(x_f, t; x_i, 0) = \sqrt{\frac{m\omega}{2\pi i\hbar\sin(\omega t)}} \exp\left\{\frac{im\omega}{2\hbar\sin(\omega t)}\left[(x_i^2 + x_f^2)\cos(\omega t) - 2x_i x_f\right]\right\}}$$

This is one of the most beautiful exact results in quantum mechanics. Let us verify it.

Verification and Limits

Free-particle limit ($\omega \to 0$): Using $\sin(\omega t) \to \omega t$ and $\cos(\omega t) \to 1 - \omega^2 t^2/2$:

$$K_{\text{QHO}} \to \sqrt{\frac{m}{2\pi i\hbar t}} \exp\left[\frac{im}{2\hbar t}(x_i^2 + x_f^2 - 2x_i x_f)\right] = K_{\text{free}}$$

✅

Short-time limit ($t \to 0$): Same as the free-particle limit, since the potential is negligible for infinitesimal times: $K \to \delta(x_f - x_i)$. ✅

Periodicity: At $t = 2\pi/\omega$ (one full period), $\sin(\omega t) = 0$ and $\cos(\omega t) = 1$. The propagator becomes singular — it reduces to $\delta(x_f - x_i)$. After one full period, the QHO returns to its initial state. This is the quantum analogue of the classical fact that a harmonic oscillator is periodic.

At $t = \pi/\omega$ (half period), $\sin(\omega t) = 0$ and $\cos(\omega t) = -1$. The propagator becomes $\delta(x_f + x_i)$. The wave function is reflected: $\Psi(x, \pi/\omega) = e^{i\phi}\Psi(-x, 0)$. This is a remarkable quantum effect with no simple classical analogue.

Extracting the Energy Spectrum

The propagator contains the complete energy spectrum. Using the spectral decomposition:

$$K(x_f, t; x_i, 0) = \sum_n \psi_n(x_f)\psi_n^*(x_i)\, e^{-iE_n t/\hbar}$$

we can extract the energies by taking the trace (setting $x_f = x_i$ and integrating):

$$G(t) \equiv \int_{-\infty}^{\infty} K(x, t; x, 0)\, dx = \sum_n e^{-iE_n t/\hbar}$$

For the QHO propagator, setting $x_f = x_i = x$ and integrating the Gaussian over $x$:

$$G(t) = \int_{-\infty}^{\infty} \sqrt{\frac{m\omega}{2\pi i\hbar\sin(\omega t)}} \exp\left[\frac{im\omega x^2}{2\hbar\sin(\omega t)}\left(2\cos(\omega t) - 2\right)\right] dx$$

Evaluating the Gaussian integral and simplifying using $2\cos\theta - 2 = -4\sin^2(\theta/2)$ and $\sin\theta = 2\sin(\theta/2)\cos(\theta/2)$:

$$G(t) = \frac{1}{2\sin(\omega t/2)}\, e^{-i\pi/4 \cdot \text{sgn}(\sin\omega t)}$$

More carefully, using the correct branch cuts:

$$G(t) = \frac{e^{-i\omega t/2}}{1 - e^{-i\omega t}} = \sum_{n=0}^{\infty} e^{-i(n + 1/2)\omega t}$$

Reading off the coefficients: $E_n = \hbar\omega(n + 1/2)$. The path integral reproduces the QHO energy spectrum, including the zero-point energy $\hbar\omega/2$ that we first encountered in Chapter 4 through the algebraic method.

💡 Key Insight: The path integral derives the QHO spectrum without solving a differential equation, without finding eigenfunctions, and without using ladder operators. The energy levels emerge from the analytic structure of a single integral — specifically, from the poles of $1/\sin(\omega t)$ in the propagator.

The Ground-State Wave Function from the Path Integral

The spectral decomposition $K = \sum_n \psi_n(x_f)\psi_n^*(x_i)\,e^{-iE_n t/\hbar}$ offers another powerful application. In the limit of large imaginary time ($t \to -i\infty$), the sum is dominated by the ground state ($n = 0$), since $e^{-iE_n t/\hbar} = e^{-E_n \tau/\hbar}$ for $t = -i\tau$, and the lowest energy dominates:

$$K(x_f, -i\tau; x_i, 0) \xrightarrow{\tau\to\infty} \psi_0(x_f)\psi_0^*(x_i)\, e^{-E_0\tau/\hbar}$$

For the QHO, performing the Wick rotation $t \to -i\tau$ in the exact propagator and taking $\tau \to \infty$, we can read off:

$$\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right)$$

This is the Gaussian ground state we found in Chapter 4 — but now derived from the propagator without ever solving the Schrödinger equation. The excited-state wave functions can be extracted systematically by subtracting the ground-state contribution and repeating the procedure, though this is more involved.

Why the QHO Is So Special

The QHO occupies a unique position in the path integral formalism. It is the simplest interacting system (the free particle is "trivially" solvable), yet it is exactly solvable because of the quadratic action. Every other exactly solvable path integral — the particle in a uniform field, the particle in a magnetic field (Landau levels), the Morse potential — either reduces to a Gaussian integral or can be mapped onto one through a coordinate transformation.

In quantum field theory, the QHO becomes the fundamental building block: each mode of a free quantum field is an independent QHO, and the field theory propagator is built from the QHO propagator. Interactions in field theory correspond to anharmonic perturbations ($\lambda x^3$, $\lambda x^4$, etc.), which are treated perturbatively using the QHO path integral as the starting point. This is the deep reason why the QHO is, as Sidney Coleman said, "the most important problem in all of physics."

31.6 Connection to Statistical Mechanics

The Wick Rotation: Imaginary Time

One of the most powerful aspects of the path integral is its deep connection to statistical mechanics. The bridge is an elegant mathematical trick: replace real time $t$ with imaginary time $\tau = it$ (or equivalently, $t \to -i\tau$). This is called the Wick rotation, after the Italian physicist Gian-Carlo Wick.

Under the Wick rotation: - The time-evolution operator $e^{-i\hat{H}t/\hbar}$ becomes $e^{-\hat{H}\tau/\hbar}$ - The oscillatory phase $e^{iS/\hbar}$ becomes a real exponential $e^{-S_E/\hbar}$

where $S_E$ is the Euclidean action:

$$S_E[x(\tau)] = \int_0^{\beta\hbar} \left[\frac{1}{2}m\left(\frac{dx}{d\tau}\right)^2 + V(x)\right] d\tau$$

Notice the crucial sign change: both the kinetic and potential energy appear with the same (positive) sign. The Euclidean action is bounded below, and the integrand $e^{-S_E/\hbar}$ is a genuine, convergent probability weight rather than an oscillatory phase. This solves the mathematical convergence problem of the real-time path integral.

The Partition Function

The quantum partition function in statistical mechanics is:

$$Z = \text{Tr}\left(e^{-\beta\hat{H}}\right) = \sum_n e^{-\beta E_n}$$

where $\beta = 1/(k_B T)$ is the inverse temperature. Comparing with the Wick-rotated propagator at $\tau = \beta\hbar$:

$$Z = \int dx\, K(x, -i\beta\hbar; x, 0) = \int dx \int_{x(0) = x}^{x(\beta\hbar) = x} \mathcal{D}[x(\tau)]\, e^{-S_E[x]/\hbar}$$

The trace enforces $x_f = x_i$, which means the paths are periodic: $x(0) = x(\beta\hbar)$. The quantum partition function is a path integral over all closed paths of period $\beta\hbar = \hbar/(k_B T)$.

$$\boxed{Z = \oint \mathcal{D}[x(\tau)]\, e^{-S_E[x]/\hbar}}$$

where $\oint$ denotes the integral over periodic paths.

This is a profound result: quantum statistical mechanics in $d$ spatial dimensions is equivalent to classical statistical mechanics in $d+1$ dimensions (the extra dimension being imaginary time with period $\beta\hbar$). The quantum fluctuations of the original system map to thermal fluctuations in the $(d+1)$-dimensional classical system.

QHO Partition Function

For the QHO, $S_E = \int_0^{\beta\hbar} [\frac{1}{2}m\dot{x}^2 + \frac{1}{2}m\omega^2 x^2]\, d\tau$. Setting $t \to -i\beta\hbar$ in the QHO propagator:

$$\sin(\omega t) \to \sin(-i\omega\beta\hbar) = -i\sinh(\omega\beta\hbar)$$

The diagonal propagator becomes:

$$K(x, -i\beta\hbar; x, 0) = \sqrt{\frac{m\omega}{2\pi\hbar\sinh(\omega\beta\hbar)}} \exp\left[-\frac{m\omega x^2}{\hbar}\cdot\frac{\cosh(\omega\beta\hbar) - 1}{\sinh(\omega\beta\hbar)}\right]$$

Integrating over $x$ (a Gaussian):

$$Z_{\text{QHO}} = \frac{1}{2\sinh(\beta\hbar\omega/2)} = \sum_{n=0}^{\infty} e^{-\beta\hbar\omega(n+1/2)}$$

The last equality is the geometric series, which confirms the energy spectrum $E_n = \hbar\omega(n + 1/2)$.

📊 Physical consequences:

The high-temperature limit $Z \approx 1/(\beta\hbar\omega)$ reproduces the classical partition function of a harmonic oscillator (energy $k_B T$ by equipartition). The low-temperature limit shows that the ground state energy $\hbar\omega/2$ controls the thermodynamics — the zero-point energy is not just a theoretical curiosity but has observable thermal consequences.

Path Integrals and Brownian Motion

The Euclidean path integral for a free particle:

$$K_E(x_f, \tau; x_i, 0) = \sqrt{\frac{m}{2\pi\hbar\tau}} \exp\left[-\frac{m(x_f - x_i)^2}{2\hbar\tau}\right]$$

is a Gaussian distribution in $x_f - x_i$ with variance $\sigma^2 = \hbar\tau/m$. This is identical to the probability distribution for a Brownian particle diffusing for time $\tau$ with diffusion constant $D = \hbar/(2m)$.

The Euclidean path integral measure, with its exponential suppression $e^{-S_E/\hbar}$, is mathematically equivalent to the Wiener measure on the space of Brownian paths — one of the best-studied objects in probability theory. This connection places the Euclidean path integral on rigorous mathematical footing and is the basis for lattice gauge theory, one of the primary computational methods in quantum field theory.

⚠️ Caution: The correspondence is between quantum mechanics and a classical stochastic system (Brownian motion), not a classical deterministic system. The imaginary-time path integral describes a random walk, not a classical trajectory. The Wick rotation trades quantum coherence (interference, superposition) for classical randomness (diffusion, fluctuations). Both have the same mathematical structure, but the physical interpretation is completely different.

The Feynman-Kac Formula

The connection between quantum mechanics and stochastic processes is made rigorous by the Feynman-Kac formula. For a particle in a potential $V(x)$, the Euclidean propagator can be written as an expectation value over Brownian motion paths:

$$K_E(x_f, \tau; x_i, 0) = \mathbb{E}_{x_i \to x_f}\left[\exp\left(-\frac{1}{\hbar}\int_0^\tau V(B(s))\, ds\right)\right]$$

where $B(s)$ is a Brownian bridge (Brownian motion conditioned to start at $x_i$ and end at $x_f$), and the expectation $\mathbb{E}$ is over all realizations of the Brownian path. The kinetic energy term is already built into the Wiener measure (the probability distribution of Brownian paths), and the potential energy appears as an exponential weight.

The Feynman-Kac formula is a rigorous mathematical theorem — proved by Mark Kac in 1949, shortly after Feynman's paper. It places the Euclidean path integral on firm mathematical ground and provides the foundation for path integral Monte Carlo methods (see Case Study 31.2).

Applications: Instantons and Tunneling

The Euclidean path integral provides an elegant framework for quantum tunneling. Consider a particle in a double-well potential $V(x) = \lambda(x^2 - a^2)^2$. In the Euclidean path integral, tunneling between the two minima $x = \pm a$ is mediated by classical solutions of the Euclidean equations of motion:

$$m\frac{d^2 x}{d\tau^2} = V'(x) \quad \text{(note: + sign, not $-$)}$$

The sign flip (compared to the real-time equation) means the particle moves in the inverted potential $-V(x)$. A solution that interpolates from $x = -a$ at $\tau = -\infty$ to $x = +a$ at $\tau = +\infty$ is called an instanton. The tunneling amplitude is proportional to $e^{-S_E[\text{instanton}]/\hbar}$, providing a nonperturbative calculation of the tunnel splitting.

This instanton technique is indispensable in quantum field theory and condensed matter physics, where it describes phenomena ranging from vacuum decay to the formation of topological defects.

31.7 Feynman Diagrams: A Preview

Motivation: Beyond Exactly Solvable Systems

The free particle and QHO are exactly solvable because their actions are quadratic — the path integrals are Gaussian. But most physical systems of interest are not quadratic. Anharmonic oscillators, atoms in external fields, interacting quantum fields — all involve non-Gaussian path integrals that cannot be evaluated in closed form.

The standard strategy is perturbation theory: expand the non-Gaussian part of the action in a power series, and evaluate each term using the Gaussian (QHO or free-particle) path integral as a reference. This perturbative expansion has a beautiful diagrammatic representation — Feynman diagrams — that organizes the calculation and makes it tractable.

From Path Integrals to Perturbation Theory

Suppose the potential is $V(x) = \frac{1}{2}m\omega^2 x^2 + \lambda x^3 + \cdots$ — a harmonic oscillator plus anharmonic perturbations. The path integral is:

$$K = \int \mathcal{D}[x]\, \exp\left\{\frac{i}{\hbar}\int_0^t \left[\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2 - \lambda x^3 - \cdots\right] dt'\right\}$$

We can expand the non-Gaussian part of the exponential in a Taylor series:

$$e^{-(i/\hbar)\int \lambda x^3\, dt'} = 1 - \frac{i\lambda}{\hbar}\int x(t_1)^3\, dt_1 + \frac{1}{2}\left(\frac{-i\lambda}{\hbar}\right)^2 \int\!\!\int x(t_1)^3 x(t_2)^3\, dt_1\, dt_2 + \cdots$$

Each term is a Gaussian path integral (weighted by powers of $x$) that can be evaluated using Wick's theorem: the Gaussian integral of a product of $x$'s equals the sum over all pairings:

$$\langle x(t_1) x(t_2) \rangle_{\text{free}} = G(t_1, t_2) = \text{free propagator (Green's function)}$$

$$\langle x(t_1) x(t_2) x(t_3) x(t_4) \rangle_{\text{free}} = G(t_1,t_2)G(t_3,t_4) + G(t_1,t_3)G(t_2,t_4) + G(t_1,t_4)G(t_2,t_3)$$

Understanding Wick's Theorem

Wick's theorem is the computational engine behind Feynman diagrams. Let us state it more precisely. For a zero-mean Gaussian random variable $x$ with $\langle x^2 \rangle = \sigma^2$:

• $\langle x^2 \rangle = \sigma^2$ (one pairing)
• $\langle x^4 \rangle = 3\sigma^4$ (three pairings: 12-34, 13-24, 14-23)
• $\langle x^6 \rangle = 15\sigma^6$ (fifteen pairings)
• $\langle x^{2n} \rangle = (2n-1)!!\, \sigma^{2n}$ ($(2n-1)!!$ pairings)

The general rule is: the Gaussian expectation of a product of $2n$ variables equals the sum over all ways to pair them up, with each pair contributing a factor of the two-point function (the propagator). This is Wick's theorem.

In the path integral context, $x$ is replaced by the field $x(t')$ at different times, and the two-point function $\langle x(t_1)x(t_2)\rangle$ is the free propagator $G(t_1, t_2)$. Each pairing connects two spacetime points with a propagator — a line in the diagram.

The Birth of Diagrams

Each pairing can be represented as a line connecting two time points. Each vertex (from the $\lambda x^3$ interaction) is a point where three lines meet. The resulting pictures are Feynman diagrams.

For the $\lambda x^3$ theory at second order in $\lambda$:

• First order: $\langle x^3 \rangle = 0$ (odd number of $x$'s in a Gaussian integral vanishes).

• Second order: $\langle x(t_1)^3 x(t_2)^3 \rangle$ has 15 pairings, which organize into diagrams:

• Connected diagrams: Two vertices connected by three propagators (the "sunset" or "basketball" diagram), and a vertex with a self-loop connected to the other vertex by a single propagator. These represent genuine interactions.
• Disconnected diagrams: Products of simpler diagrams. These cancel in the computation of connected correlation functions.

Feynman Rules (Preview)

In quantum mechanics (0+1 dimensions), the Feynman rules for computing the $n$-th order contribution to the propagator are:

1. Draw all topologically distinct diagrams with $n$ vertices and the appropriate number of external lines.
2. For each internal line connecting times $t_a$ and $t_b$, write a factor of $G(t_a, t_b)$ (the free propagator).
3. For each vertex, write a factor of $-i\lambda/\hbar$ (the coupling constant) and integrate over the vertex time.
4. Divide by the symmetry factor of the diagram (accounting for equivalent pairings).

These rules are a dramatic simplification: instead of computing a complicated functional integral, you draw pictures and evaluate ordinary integrals. This is why Feynman diagrams revolutionized physics.

From Quantum Mechanics to Quantum Field Theory

In quantum field theory (Chapter 37), the path integral becomes an integral over field configurations $\phi(\mathbf{x}, t)$ rather than particle trajectories $x(t)$:

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

The Feynman rules generalize naturally: - Lines become propagators in spacetime (functions of four-vectors rather than single times) - Vertices become interaction vertices determined by the field theory Lagrangian - Integrals over vertex times become integrals over all of spacetime (or, after Fourier transform, integrals over four-momenta)

The Feynman diagrams you see in particle physics textbooks — electron emitting a photon, virtual particle pairs appearing from the vacuum, quarks exchanging gluons — are all terms in this perturbative expansion. The path integral is their common ancestor.

An Analogy: From One Oscillator to Infinitely Many

The conceptual leap from quantum mechanics to quantum field theory is easier to understand through the path integral. In quantum mechanics, we integrate over all possible trajectories $x(t)$ of a single particle — a function of one variable (time). In quantum field theory, the "trajectory" is replaced by a field configuration $\phi(\mathbf{x}, t)$ — a function of four variables (space and time). The path integral becomes:

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

where $S[\phi] = \int d^4x\, \mathcal{L}(\phi, \partial_\mu\phi)$ is the field theory action, obtained by integrating the Lagrangian density $\mathcal{L}$ over all of spacetime.

The free field theory (no interactions) is a Gaussian path integral — each Fourier mode of the field is an independent QHO, and the field theory propagator is determined by the QHO propagator. Interactions add non-Gaussian terms (like $\lambda\phi^4$), which are treated perturbatively using Wick's theorem. The resulting Feynman diagrams now live in spacetime rather than just in time, with lines representing particle propagation and vertices representing particle interactions.

This is why the path integral approach to quantum mechanics is not merely an alternative formulation — it is the gateway to quantum field theory.

🔗 Connection forward: Chapter 37 (Quantum Mechanics to Quantum Field Theory) will develop this machinery in full, starting from the second-quantized path integral for scalar fields and working up to QED.

Limitations of Perturbation Theory

Not all path integrals can be evaluated perturbatively. When the coupling $\lambda$ is not small, or when the phenomenon of interest is inherently nonperturbative (tunneling, confinement, topological effects), the diagrammatic expansion fails. The instanton calculation of Section 31.6 is an example of a nonperturbative path integral technique — the tunneling amplitude goes as $e^{-\text{const}/\lambda}$, which is invisible to all orders of perturbation theory in $\lambda$.

Modern physics uses a variety of nonperturbative methods for path integrals: lattice Monte Carlo simulation (sampling the Euclidean path integral numerically), saddle-point expansions around nontrivial classical solutions, the renormalization group, and exact solutions using symmetry (supersymmetry, integrability). The path integral is the natural home for all of these.

31.8 Summary: The Path Integral at a Glance

The path integral reformulates quantum mechanics as follows:

The two formulations are exactly equivalent for all systems we consider. Each has situations where it is more natural:

• Use Schrödinger when you need eigenstates, when the system has discrete symmetries, or when you want bound-state wave functions.
• Use the path integral when you want the classical limit, when you are setting up quantum field theory, when you need nonperturbative results (instantons), or when you want to compute partition functions.

The deepest lesson of the path integral is perhaps philosophical: quantum mechanics is not about particles taking definite paths. It is about the coherent superposition of all possible histories. Classical mechanics is the story the universe tells when the phases align.

What Makes the Path Integral Indispensable

Despite being exactly equivalent to the Schrödinger equation, the path integral is not redundant. There are things you can do with the path integral that are extremely difficult (or impossible) in the operator formalism:

1. The classical limit is immediate. Stationary phase gives $\delta S = 0$ — Hamilton's principle — without any additional arguments.

2. Manifest Lorentz invariance. The action $S = \int \mathcal{L}\, d^4x$ is a Lorentz scalar, making relativistic quantum theories natural to formulate.

3. Nonperturbative physics. Instantons, tunneling amplitudes, topological effects, and vacuum structure are all naturally described by the path integral.

4. Numerical computation. The Euclidean path integral can be evaluated by Monte Carlo methods — the basis of lattice QCD and path integral molecular dynamics.

5. Quantum field theory. The path integral is the standard formulation of the Standard Model. Feynman rules, renormalization, anomalies, and effective field theories all flow from it.

If there is one formulation of quantum mechanics that a theoretical physicist must master above all others, it is the path integral.

Chapter 31 Notation Reference