Part IX, Chapter 1 | Page 1 of 3

Feynman Path Integral Formulation

The third pillar of quantum mechanics: summing over all histories

The Third Formulation of Quantum Mechanics

Quantum mechanics can be formulated in three equivalent but conceptually distinct ways. The Schrodinger picture evolves state vectors via the Schrodinger equation. The Heisenberg picture evolves operators while states remain fixed. The path integral formulation, developed by Richard Feynman in his 1942 PhD thesis (inspired by a remark from Dirac), takes a radically different approach: rather than solving differential equations, we sum over all possible trajectories a particle could take between two spacetime points. Each path contributes a complex phase proportional to the classical action, and the resulting interference produces all of quantum mechanics.

Three equivalent formulations at a glance:

Schrodinger (1926): States evolve, operators fixed. Central equation: $i\hbar\partial_t|\psi\rangle = \hat{H}|\psi\rangle$
Heisenberg (1925): Operators evolve, states fixed. Central equation: $i\hbar\dot{\hat{A}} = [\hat{A}, \hat{H}]$
Feynman (1948): Sum over all paths. Central formula: $K = \int\mathcal{D}x\,e^{iS[x]/\hbar}$

The path integral is not merely an alternative computational tool. It reveals deep connections between quantum mechanics and classical mechanics, provides the most natural language for quantum field theory and gauge theories, and offers intuitive access to non-perturbative phenomena such as tunneling and instantons.

The Quantum Propagator

The central object in quantum dynamics is the propagator (or transition amplitude), which gives the probability amplitude for a particle at position $x_a$ at time $t_a$ to be found at position $x_b$ at time $t_b$:

$$K(x_b, t_b;\, x_a, t_a) = \langle x_b | \, e^{-i\hat{H}(t_b - t_a)/\hbar} \, | x_a \rangle$$

This matrix element of the time-evolution operator contains the complete dynamical information of the system.

Given the propagator, the wave function at any later time is obtained by integration:

$$\psi(x_b, t_b) = \int_{-\infty}^{\infty} K(x_b, t_b;\, x_a, t_a)\, \psi(x_a, t_a)\, dx_a$$

The propagator satisfies the composition (semigroup) property: if we insert a complete set of position states at an intermediate time $t$, then

$$K(x_b, t_b;\, x_a, t_a) = \int_{-\infty}^{\infty} K(x_b, t_b;\, x, t)\, K(x, t;\, x_a, t_a)\, dx$$

This is the quantum analog of the Chapman-Kolmogorov equation in probability theory.

Time-Slicing Derivation

The path integral emerges naturally from repeated application of the composition property. Divide the time interval $[t_a, t_b]$ into $N$ equal slices of width $\varepsilon = (t_b - t_a)/N$, with intermediate times $t_j = t_a + j\varepsilon$. Inserting a complete set of position states $\int dx_j \, |x_j\rangle\langle x_j| = \hat{1}$ at each time $t_j$:

$$K(x_b, t_b;\, x_a, t_a) = \int \prod_{j=1}^{N-1} dx_j \;\prod_{k=0}^{N-1} \langle x_{k+1} | \, e^{-i\hat{H}\varepsilon/\hbar} \, | x_k \rangle$$

For each infinitesimal propagator, with $\hat{H} = \hat{p}^2/2m + V(\hat{x})$, we insert a momentum completeness relation and evaluate:

$$\langle x_{k+1}| e^{-i\hat{H}\varepsilon/\hbar} |x_k\rangle \approx \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{1/2} \exp\!\left[\frac{i\varepsilon}{\hbar}\left(\frac{m}{2}\!\left(\frac{x_{k+1}-x_k}{\varepsilon}\right)^{\!2} - V(x_k)\right)\right]$$

Valid to first order in $\varepsilon$. The exponent is $\frac{i}{\hbar}\varepsilon\, L(x_k, \dot{x}_k)$ where $L = T - V$ is the Lagrangian.

The Path Integral Formula

Assembling all $N$ infinitesimal propagators and taking $N \to \infty$, we arrive at Feynman's path integral:

$$\boxed{K(x_b, t_b;\, x_a, t_a) = \int \mathcal{D}x(t)\; e^{iS[x]/\hbar}}$$

where $S[x] = \int_{t_a}^{t_b} L(x, \dot{x}, t)\, dt$ is the classical action functional evaluated along the path $x(t)$.

Every possible path from $(x_a, t_a)$ to $(x_b, t_b)$ contributes equally in magnitude but with a phase $e^{iS[x]/\hbar}$. The paths interfere: nearby paths with slowly varying action add constructively, while paths with rapidly oscillating action cancel out.

The Functional Measure

The integration measure $\mathcal{D}x(t)$ is defined as the continuum limit of the discretized multiple integral:

$$\mathcal{D}x(t) = \lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{N/2} \prod_{j=1}^{N-1} dx_j$$

The normalization prefactor $(m/2\pi i\hbar\varepsilon)^{N/2}$ ensures the correct dimensions and limiting behavior. With this definition, the full discretized propagator reads:

$$K = \lim_{N\to\infty}\left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{\!N/2}\!\int\!\prod_{j=1}^{N-1}dx_j\;\exp\!\left[\frac{i}{\hbar}\sum_{k=0}^{N-1}\!\left(\frac{m(x_{k+1}-x_k)^2}{2\varepsilon} - V(x_k)\varepsilon\right)\right]$$

where $x_0 = x_a$ and $x_N = x_b$ are the fixed endpoints.

The Classical Limit

The path integral beautifully illuminates the emergence of classical mechanics. In the limit$\hbar \to 0$, the phase $S[x]/\hbar$ oscillates wildly for most paths, causing them to cancel via destructive interference. The dominant contribution comes from the path (or paths) of stationary phase where the action is extremal:

$$\frac{\delta S[x]}{\delta x(t)} = 0 \quad \Longrightarrow \quad \frac{d}{dt}\frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x} = 0$$

This is precisely the Euler-Lagrange equation - Newton's second law in Lagrangian form! Classical mechanics emerges as the stationary-phase approximation to the path integral.

For macroscopic objects where $S \gg \hbar$, the classical path dominates overwhelmingly. For microscopic particles where $S \sim \hbar$, many paths contribute and quantum interference effects become significant.

Recovering the Schrodinger Equation

The path integral is fully equivalent to the Schrodinger equation. To see this, consider the wave function at time $t + \varepsilon$:

$$\psi(x, t+\varepsilon) = \left(\frac{m}{2\pi i\hbar\varepsilon}\right)^{1/2}\!\int\! dy\;\exp\!\left[\frac{im(x-y)^2}{2\hbar\varepsilon} - \frac{i\varepsilon}{\hbar}V(y)\right]\psi(y, t)$$

Substituting $y = x + \eta$, expanding $\psi(x+\eta, t)$ in a Taylor series, performing the Gaussian integral over $\eta$, and keeping terms to order $\varepsilon$ yields:

$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi$$

The time-dependent Schrodinger equation emerges exactly. The path integral is not an approximation; it is a complete reformulation of quantum mechanics.

Path Integral View of the Double Slit

The double-slit experiment is beautifully illuminated by the path integral. A particle traveling from source to detector has paths passing through slit 1, paths passing through slit 2, and (in principle) paths that do far more exotic things. The propagator is:

$$K_{\text{total}} = K_{\text{slit 1}} + K_{\text{slit 2}} + K_{\text{other}}$$

The "other" paths (going around the barrier, through the wall, etc.) contribute negligibly because their actions are far from stationary and they destructively interfere. The surviving interference between $K_{\text{slit 1}}$ and $K_{\text{slit 2}}$ produces the famous diffraction pattern.

If we detect which slit the particle passes through, we restrict the sum to paths through one slit only, and interference disappears — wave-particle duality emerges naturally from the structure of the path sum.

Boundary Conditions and the Action Principle

The path integral connects different boundary conditions to different physical quantities:

Fixed endpoints: $K(x_b,t_b;\,x_a,t_a) = \int_{x(t_a)=x_a}^{x(t_b)=x_b}\mathcal{D}x\;e^{iS/\hbar}$ gives the propagator
Trace (periodic): $Z = \oint_{x(0)=x(\beta\hbar)}\mathcal{D}x\;e^{-S_E/\hbar}$ gives the partition function
Source terms: Adding $\int J(t)x(t)\,dt$ to the action generates correlation functions via functional derivatives

The generating functional $Z[J] = \int\mathcal{D}x\;\exp\!\left[\frac{i}{\hbar}(S[x] + \int Jx\,dt)\right]$ encodes all $n$-point functions:

$$\langle x(t_1)x(t_2)\cdots x(t_n)\rangle = \frac{1}{i^n}\frac{\delta^n \ln Z[J]}{\delta J(t_1)\cdots\delta J(t_n)}\bigg|_{J=0}$$

This machinery generalizes directly to quantum field theory, where it becomes the foundation for computing Feynman diagrams and scattering amplitudes.

Historical Note: Feynman developed the path integral formulation in his 1942 PhD thesis at Princeton, supervised by John Archibald Wheeler. He was inspired by a paper of Dirac's from 1933, which noted that the short-time propagator is "analogous to" $e^{iL\,\delta t/\hbar}$. Feynman realized the analogy is exact, and that the propagator is obtained by multiplying such factors and integrating over intermediate positions — summing over all paths.

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