1. Path Integral Formulation

The path integral formulation of quantum mechanics, developed by Richard Feynman, provides an alternative to the operator formalism based on the principle that a quantum particle explores all possible paths between two points.

Feynman's Path Integral

The fundamental path integral formula for the propagator:

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

Where:

$K(x_f, t_f; x_i, t_i)$ is the propagator (probability amplitude)
$\mathcal{D}[x(t)]$ is the functional integration measure over all paths
$S[x(t)] = \int_{t_i}^{t_f} L(x, \dot{x}, t) \, dt$ is the classical action
$L$ is the Lagrangian

Physical interpretation: Each path contributes a phase $e^{iS/\hbar}$, and all paths interfere quantum mechanically.

Connection to Schrödinger Equation

The wave function evolves according to:

$$\psi(x_f, t_f) = \int K(x_f, t_f; x_i, t_i) \psi(x_i, t_i) \, dx_i$$

For infinitesimal time $\epsilon = t_f - t_i$:

$$K(x_f, t+\epsilon; x_i, t) \approx \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{1/2} \exp\left[\frac{i}{\hbar}\left(\frac{m(x_f - x_i)^2}{2\epsilon} - V(x_i)\epsilon\right)\right]$$

Taking $\epsilon \to 0$ recovers the Schrödinger equation

Free Particle Propagator

For a free particle ($V = 0$), the path integral can be computed exactly:

$$K_0(x_f, t_f; x_i, t_i) = \left(\frac{m}{2\pi i\hbar(t_f - t_i)}\right)^{1/2} \exp\left[\frac{im(x_f - x_i)^2}{2\hbar(t_f - t_i)}\right]$$

Key properties:

Classical limit: As $\hbar \to 0$, rapid oscillations enforce stationary phase → classical path dominates
Composition: $K(x_f, t_f; x_i, t_i) = \int K(x_f, t_f; x, t) K(x, t; x_i, t_i) \, dx$
Normalization: $K(x, t; x_i, t_i) \to \delta(x - x_i)$ as $t \to t_i$

Harmonic Oscillator

For the harmonic oscillator $V = \frac{1}{2}m\omega^2 x^2$:

$$K(x_f, t_f; x_i, t_i) = \left(\frac{m\omega}{2\pi i\hbar\sin(\omega T)}\right)^{1/2} \exp\left[\frac{im\omega}{2\hbar\sin(\omega T)}S_{cl}\right]$$

Where $T = t_f - t_i$ and $S_{cl}$ is the classical action:

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

Remarkable result: Only the classical path contributes (Gaussian fluctuations exactly cancel)

Discretization and Lattice Formulation

To make sense of $\mathcal{D}[x(t)]$, divide time into $N$ intervals:

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

Where $\epsilon = T/N$ and $x_0 = x_i, x_N = x_f$

This discretization:

Makes the path integral mathematically rigorous
Provides numerical evaluation methods (Monte Carlo techniques)
Connects to lattice field theory formulations

Semiclassical Approximation

In the limit $\hbar \to 0$, use stationary phase approximation around classical path $x_{cl}(t)$:

$$K \approx \left(\frac{1}{2\pi i\hbar}\right)^{1/2}\left|\det\left(-\frac{\delta^2 S}{\delta x^2}\right)\right|^{-1/2} e^{iS_{cl}/\hbar}$$

This is the WKB approximation in path integral language:

Amplitude: Determined by determinant of action's second variation
Phase: Given by classical action $S_{cl}$
Caustics: Occur when determinant vanishes (classical trajectories focus)

Path Integral in Phase Space

Alternative formulation integrating over both position and momentum:

$$K = \int \mathcal{D}[x]\mathcal{D}[p] \exp\left[\frac{i}{\hbar}\int_{t_i}^{t_f} (p\dot{x} - H(p,x)) \, dt\right]$$

Advantages:

Treats $x$ and $p$ symmetrically
More natural for gauge theories and constrained systems
Connects directly to canonical quantization

Integrating out $p$ recovers the configuration space path integral

Imaginary Time and Statistical Mechanics

Wick rotation $t \to -i\tau$ ($\tau$ real) gives Euclidean path integral:

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

Where $S_E = \int (\frac{m\dot{x}^2}{2} + V(x)) \, d\tau$ is the Euclidean action

Connection to statistical mechanics:

$$Z = \text{Tr}[e^{-\beta H}] = \int K_E(x, \beta\hbar; x, 0) \, dx$$

Partition function = path integral with periodic boundary conditions

Applications:

Thermal field theory and finite temperature QFT
Quantum tunneling (instantons are classical solutions in imaginary time)
Numerical simulations (Monte Carlo methods work better with real exponential)

Gaussian Path Integrals

For quadratic actions $S = \frac{1}{2}\int x \, A \, x$:

$$\int \mathcal{D}[x] \, e^{iS/\hbar} = (\det A)^{-1/2}$$

This extends the finite-dimensional Gaussian integral:

$$\int \frac{dx}{\sqrt{2\pi}} e^{-\frac{1}{2}ax^2} = a^{-1/2}$$

Applications:

Harmonic oscillator (exact solution)
Quadratic fluctuations around classical path
Free field theories
One-loop quantum corrections

Path Integrals and Quantum Field Theory

Path integrals generalize naturally to field theory:

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

Now integrating over field configurations $\phi(x,t)$ in spacetime

Advantages over canonical quantization:

Manifest Lorentz invariance: No preferred time direction
Gauge theories: Natural framework for gauge fixing and BRST symmetry
Non-perturbative: Instantons, solitons, and topological effects
Generating functionals: Efficient calculation of correlation functions

Advantages and Applications

Conceptual advantages:

Makes quantum-classical correspondence transparent
No operator ordering ambiguities
Natural framework for symmetries and conservation laws
Elegant treatment of constraints and gauge symmetries

Practical applications:

Quantum field theory: Standard tool for perturbative and non-perturbative calculations
Condensed matter: Effective theories, phase transitions, critical phenomena
Quantum chemistry: Path integral molecular dynamics
Quantum computing: Variational quantum algorithms
Cosmology: Quantum fluctuations in early universe

Comparison: Operator vs. Path Integral

Aspect	Operator Formalism	Path Integral
Starting point	Hamiltonian, operators	Lagrangian, classical action
Time evolution	Schrödinger equation	Sum over all paths
States	Vectors in Hilbert space	Boundary conditions on paths
Observables	Hermitian operators	Insertions in path integral
Best for	Bound states, eigenvalues	Scattering, field theory
Classical limit	Commutators $\to 0$	Stationary phase

Historical Note: Richard Feynman developed the path integral formulation in his 1942 PhD thesis, inspired by a remark from P.A.M. Dirac about the Lagrangian in quantum mechanics. It became a cornerstone of modern theoretical physics, particularly in quantum field theory and gauge theories.

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