5. Path Integral: Quantum Dynamics That Sums “All” Paths

In Chapter 4, we learned about the “principle of least action” in classical mechanics (\(\delta S = 0\)). This was an elegant and powerful principle stating that nature selects only a single “classical path” among all possible paths when going from \(t_1\) to \(t_2\).

In the 1940s, Richard Feynman extended this concept to quantum mechanics by posing a fundamentally different question: “If particles can interfere like waves (see Chapter 7), why must they follow only a single path?”

The Path Integral formalism is a revolutionary answer to this question. Quantum particles do not follow a single path, but instead simultaneously pass through all possible paths from \(x_i\) at \(t_1\) to \(x_f\) at \(t_f\). This perspective, mathematically equivalent to the Hamiltonian dynamics in Chapter 3 (\(U(t)=e^{-iHt/\hbar}\)), is a powerful tool that most intuitively demonstrates quantum interference and the emergence of the classical world.

1. Fundamental Concepts

Path Integral Postulate The total transition amplitude (Kernel) \(K\) for a particle moving from \((x_i, t_i)\) to \((x_f, t_f)\) is the sum (integral) of the amplitudes of all paths.
- Amplitude of each path: Each path has a complex “phase” given by \(e^{iS[x]/\hbar}\). Here, \(S[x]\) is the classical action of that path, as learned in Chapter 4.
- Sum over all paths: The notation for “integrating over all paths” is written as \(\int \mathcal{D}x(t)\) (functional integral).
\[K(x_f, t_f; x_i, t_i) = \int_{x(t_i)=x_i}^{x(t_f)=x_f} \mathcal{D}x(t) \, \exp\left(\frac{i}{\hbar} S[x(t)]\right)\]
Meaning of the Functional Integral \(\mathcal{D}x\) (Time Slicing) How is “summing over all paths” mathematically defined? There are infinitely many paths, and each is continuous.

💡 Detailed Explanation: Infinite-Dimensional Integral, “Time Slicing”

\(\int \mathcal{D}x\) is not a general integral. Feynman defined this infinite-dimensional integral by “slicing” time into small pieces.
1. Divide time into \(N\) small pieces (\(\Delta t\)) \((t_0, t_1, \dots, t_N)\).
2. Approximate “all continuous paths” as polyline paths passing through points \(x_k\) at each time \(t_k\).
3. “Integrating over all paths” \(\int \mathcal{D}x\) becomes a multiple integral \(\int dx_1 \int dx_2 \dots \int dx_{N-1}\) over all intermediate points (\(x_1, \dots, x_{N-1}\)).
4. The action \(S = \int L dt\) also becomes the sum (\(\sum\)) of the Lagrangian \(L_k \Delta t\) for each piece.
In other words, the path integral defines an impossible infinite-dimensional integral as the limit of a finite-dimensional integral (\(N \to \infty\)) that we can compute. \[K \approx \lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar \Delta t}\right)^{N/2} \int \prod_{k=1}^{N-1} dx_k \, \exp\left(\frac{i}{\hbar} \sum_{j=0}^{N-1} S_j\right)\]
Classical Limit (Connection to Chapter 4)
If quantum particles traverse all paths, why do we only observe one classical path (with \(\delta S = 0\)) in the macroscopic world (as learned in Chapter 4)?
- \(\hbar\) (Planck’s constant) is an extremely small value, \(10^{-34}\) J·s. Thus, \(S[x]/\hbar\) becomes an astronomically large number.
- Non-classical path: Paths near this one experience rapid changes in \(S\). Consequently, the phase of \(e^{iS/\hbar}\) rotates unpredictably, causing amplitudes from different paths to destructively interfere and cancel each other out.

[Image of destructive wave interference]

* **Classical path:** This is the path with $\delta S = 0$ from Chapter 4. Near this path, $S$ changes very little. Thus, the phase of $e^{iS/\hbar}$ remains nearly constant, allowing amplitudes from all neighboring paths to **constructively interfere**, surviving strongly.

[Image of constructive wave interference]

* **Conclusion:** Path integrals perfectly explain how quantum mechanics reduces to classical mechanics ($\delta S=0$) in the $\hbar \to 0$ limit, known as the "stationary phase approximation."

Composition Law
The kernel \(K\) acts as a propagator for wave functions. Thus, the amplitude for \(i \to f\) must equal the sum over all intermediate points \(m\) of the amplitude for \(i \to m\) followed by \(m \to f\):
\[\int K(x_f, t_f; x_m, t_m) \, K(x_m, t_m; x_i, t_i) \, dx_m = K(x_f, t_f; x_i, t_i)\]
This matches precisely with the property of the unitary operator (time evolution) from Chapter 3: \(U(t_f-t_m)U(t_m-t_i) = U(t_f-t_i)\), demonstrating that path integrals satisfy the Schrödinger equation.

2. Core Computational Tool: Gaussian Integrals and Euclidean Rotation

Path integrals are conceptually elegant but computationally tractable only for Gaussian cases (almost never otherwise). Fortunately, free particles and harmonic oscillators (the basis of all vibrations) fall into this category.

Gaussian Integral:
The time-sliced integral reduces to repeating \(N\) times an integral of the form \(e^{\text{i} \cdot (\text{quadratic})}\). The fundamental building block is:
\[\int_{-\infty}^{\infty} e^{iax^2} dx = \sqrt{\frac{i\pi}{a}}\]
Applying this formula \(N-1\) times in a chain allows exact computation of the kernel for certain models.
Euclidean Rotation (Wick Rotation):
The \(i\) in \(e^{iS/\hbar}\) causes the integral to oscillate, making convergence difficult to handle.
Idea: Apply the mathematical technique of changing time from \(t\) to \(t = -i\tau\) (imaginary time).
- Result: The Lagrangian \(L = T - V\) becomes the Euclidean Lagrangian \(L_E = T + V\) (energy sum), and the phase of the amplitude changes to real exponent.
  \[\frac{i}{\hbar}S = \frac{i}{\hbar}\int (T - V) dt \quad \xrightarrow{t \to -i\tau} \quad -\frac{1}{\hbar}\int (T + V) d\tau = -\frac{S_E}{\hbar}\]
- Meaning: The oscillating \(e^{iS/\hbar}\) becomes the real damping function \(e^{-S_E/\hbar}\).
- Connection to statistical mechanics: This \(e^{-S_E/\hbar}\) form exactly matches the Boltzmann factor (\(e^{-E/kT}\)) in statistical mechanics. That is, \(\hbar\) behaves like the inverse of temperature (\(1/T\)). The Euclidean path integral shows that quantum mechanics (\(t\)) and statistical mechanics (\(\tau = i t\)) are deeply mathematically connected.

3. Key Examples and Extension to Quantum Field Theory (QFT)

Example 1: Free particle (\(L = \frac{1}{2}m\dot{x}^2\))
Calculating the Gaussian integral in the \(N \to \infty\) limit yields an exact kernel using the classical path action \(S_{cl} = \frac{m(x_f - x_i)^2}{2t}\).
\[K_0(x_f, t; x_i, 0) = \sqrt{\frac{m}{2\pi i\hbar t}} \exp\left(\frac{i}{\hbar} S_{cl}\right)\]
Example 2: Harmonic oscillator (\(L = \frac{1}{2}m(\dot{x}^2 - \omega^2 x^2)\))
A more complex form of the Gaussian integral, but it can still be solved exactly. The result (kernel) is expressed as the product of the classical action \(S_{cl}\) and the quantum fluctuation term.
Example 3: Extension to Quantum Field Theory (QFT)
The true power of path integrals emerges when dealing with fields (Field, \(\phi(x, t)\)) rather than particles (\(x(t)\)).
- “All paths” \(\int \mathcal{D}x\) \(\to\) “All field configurations” \(\int \mathcal{D}\phi\)
- Generating functional \(Z[J]\): The central object in QFT containing everything, representing the total amplitude for the state with external ‘source’ \(J\).
  \[Z[J] = \int \mathcal{D}\phi \, \exp\left(iS[\phi] + i \int J(x)\phi(x) d^4x\right)\]
- If \(Z[J]\) can be calculated, differentiating it with respect to \(J\) allows computation of all desired physical quantities (particle interactions, scattering probabilities, etc.).

4. Practice Problems

(Time Slicing): Use the action of a free particle \(S_j = \frac{m}{2} (\frac{x_{j+1}-x_j}{\Delta t})^2 \Delta t\) to express the kernel \(K\) for two time slices (\(N=2\)) as an integral over \(dx_1\).
(Verification of Composition Law): Directly verify using Gaussian integration that the free particle kernel \(K_0\) from Example 1 satisfies \(\int K_0(x_f, t_2; x, t_1) K_0(x, t_1; x_i, t_0) dx = K_0(x_f, t_2; x_i, t_0)\).
(Derivation of the Schrödinger Equation): Expand \(K(x_f, t+\epsilon; x_i, t)\) for very small time \(\epsilon\) and show that the kernel \(K\) satisfies \(i\hbar \frac{\partial K}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 K}{\partial x_f^2} + V(x_f)K\) (Schrödinger equation).
(Quasiclassical Approximation): Using the fact that when \(\hbar \to 0\), \(S[x] \approx S[x_{cl}] + \frac{1}{2}\int \delta^2 S \, \eta^2 dt\) (where \(x = x_{cl} + \eta\)), explain that the kernel takes the form \(K \approx (\text{Factor}) \times e^{iS_{cl}/\hbar}\).
(Harmonic Oscillator): Find the classical path \(x_{cl}(t)\) and classical action \(S_{cl}\) for the harmonic oscillator.
(Euclidean Path Integral): Obtain the Euclidean kernel \(K_E(x_f, \tau; x_i, 0)\) (with Euclidean rotation applied) for a free particle and show that it satisfies the diffusion equation \(\frac{\partial K_E}{\partial \tau} = \frac{\hbar}{2m}\frac{\partial^2 K_E}{\partial x_f^2}\).
(Generating Functional \(Z[J]\)): Conceptually explain why the two-point correlation function \(\langle T\{\phi(x)\phi(y)\} \rangle\) appears when \(Z[J]\) is twice functionally differentiated with respect to \(J\) (\(\frac{\delta^2 Z}{\delta J(x) \delta J(y)}\)).
(External Linear Potential): When \(L=\frac{1}{2}m\dot{x}^2 + Fx\) (constant force \(F\)), find the classical action \(S_{cl}\) and write the quasiclassical kernel (similar to Example 1).
(Perturbation Theory): When \(S = S_0 + \lambda S_I\) (\(S_0\) is free particle, \(S_I = -\int V(x) dt\)), expand \(e^{iS/\hbar}\) up to first order in \(\lambda\) and write the path integral form of first-order perturbation theory.
(Aharonov-Bohm Effect): When the term \(L \to L + q\vec{v}\cdot\vec{A}\) (vector potential) is added to the Lagrangian, explain how \(S\) changes and that this can affect the particle’s phase even in regions where the magnetic field is zero (interference).

5. Exercise Solutions

\(K \propto \int dx_1 \exp\left( \frac{im}{2\hbar\Delta t} \left[ (x_1-x_i)^2 + (x_f-x_1)^2 \right] \right)\).
The product of two Gaussian exponential functions becomes another Gaussian function. Integrating with respect to \(x\) (\(\int e^{-a(x-b)^2}dx = \sqrt{\pi/a}\)), the exact exponent and constant of \(K_0(x_f, t_2; x_i, t_0)\) appear.
\(\psi(x_f, t+\epsilon) = \int K(x_f, \epsilon; x_i, 0) \psi(x_i, t) dx_i\). Expanding \(K\) up to second order in \(x_f - x_i\) and first order in \(\epsilon\), and also expanding \(\psi\) in Taylor series, comparing both sides yields the Schrödinger equation.
\(K \approx \int \mathcal{D}\eta \, e^{i(S_{cl} + \frac{1}{2}\delta^2 S \eta^2)/\hbar} = e^{iS_{cl}/\hbar} \int \mathcal{D}\eta \, e^{i(\delta^2 S)\eta^2 / (2\hbar)}\). The subsequent Gaussian integral becomes (Factor).
Solve the solution (satisfying the boundary conditions \(x(0)=x_i, x(T)=x_f\)) of \(m\ddot{x}_{cl} + m\omega^2 x_{cl} = 0\), and calculate \(S_{cl} = \int_0^T L(x_{cl}, \dot{x}_{cl}) dt\).
\(K_E(x_f, \tau; x_i, 0) = \sqrt{\frac{m}{2\pi \hbar \tau}} \exp\left(-\frac{m(x_f - x_i)^2}{2\hbar \tau}\right)\). This is the fundamental solution (Green’s function) of the heat equation. \(\tau\) plays the role of the diffusion time.
\(Z[J] = \int \mathcal{D}\phi (1 + i \int J\phi + \frac{(i)^2}{2!} (\int J\phi)^2 + \dots) e^{iS[\phi]}\). The second derivative with respect to \(J\) extracts \(i^2 \phi(x)\phi(y)\) from the \((\int J\phi)^2\) term, and the path integral \(\int \mathcal{D}\phi \dots e^{iS[\phi]}\) computes the vacuum expectation value \(\langle \dots \rangle_0\), so \(\langle \phi(x)\phi(y) \rangle\) is obtained.
The classical path is \(\ddot{x}_{cl} = F/m\) (uniformly accelerated motion). Find \(x_{cl}(t)\) and calculate \(S_{cl} = \int L(x_{cl}) dt\). \(K \approx (\text{Factor}) \times e^{iS_{cl}/\hbar}\).
\(K \approx \int \mathcal{D}x \, e^{iS_0/\hbar} (1 + i\lambda S_I/\hbar) = K_0 - \frac{i\lambda}{\hbar} \int \mathcal{D}x \, e^{iS_0/\hbar} \int V(x(t)) dt\).
\(\Delta S = \int (q\vec{v}\cdot\vec{A}) dt = q \int \vec{A} \cdot d\vec{l}\). The change in this action causes a phase difference (\(e^{i\Delta S/\hbar}\)) even if the particle passes through a region with no magnetic field (\(B=\nabla \times A = 0\)) but \(\oint \vec{A} \cdot d\vec{l} \neq 0\), shifting the interference pattern between two paths (Aharonov-Bohm effect).