From Propagators to Path Integrals

The path integral follows formally from the propagator composition law by slicing time into many short intervals and inserting intermediate positions.

Start with the Propagator

The kernel is

$$K(x_f,t_f;x_i,t_i) = \langle x_f\rvert U(t_f,t_i)\lvert x_i\rangle.$$

Choose $N$ time steps of size

$$\epsilon=\frac{t_f-t_i}{N}.$$

Then

$$U(t_f,t_i) = U(t_f,t_{N-1})\cdots U(t_1,t_i).$$

Insert Intermediate Positions

At each intermediate time, insert

$$I=\int dx_j\,\lvert x_j\rangle\langle x_j\rvert.$$

This gives

$$K(x_f,t_f;x_i,t_i) = \lim_{N\to\infty} \int dx_1\cdots dx_{N-1} \prod_{j=0}^{N-1} K(x_{j+1},t_{j+1};x_j,t_j).$$

The endpoints are $x_0=x_i$ and $x_N=x_f$.

Short-Time Kernel

For

$$H=\frac{p^2}{2m}+V(x),$$

the short-time kernel is approximated by

$$K(x_{j+1},t_{j+1};x_j,t_j) \approx \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{1/2} e^{i\epsilon L_j/\hbar},$$

where

$$L_j = \frac{m}{2} \left(\frac{x_{j+1}-x_j}{\epsilon}\right)^2 -V(x_j).$$

More careful derivations track operator ordering and the precise point at which $V$ is evaluated.

Continuum Limit

Before taking that limit, the regulated expression is

$$\begin{aligned} K(x_f,t_f;x_i,t_i) &=\lim_{N\to\infty} \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{N/2} \int\prod_{j=1}^{N-1}dx_j\\ &\quad\times \exp\!\left\{ \frac{i\epsilon}{\hbar} \sum_{j=0}^{N-1} \left[ \frac{m}{2} \left(\frac{x_{j+1}-x_j}{\epsilon}\right)^2 -V(x_j) \right] \right\}. \end{aligned}$$

This finite-$N$ formula defines the normalization and endpoint data that the symbol $\mathcal D x$ suppresses. A midpoint prescription or a different operator splitting changes finite-slice details and is required for more general Hamiltonians.

Multiplying the short-time kernels gives a phase

$$\exp\left[ \frac{i}{\hbar} \sum_{j=0}^{N-1}\epsilon L_j \right].$$

In the formal continuum limit,

$$\sum_j\epsilon L_j \longrightarrow S[x],$$

so

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

For a Hamiltonian formulation the corresponding formal expression is

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

Integrating each momentum slice reproduces the configuration-space measure for $H=p^2/(2m)+V(x)$. For nonquadratic momentum dependence or curved configuration spaces, the measure and ordering cannot be inferred by merely copying the Cartesian formula.

Normalization and Exact Checks

The path-integral kernel must satisfy

$$\lim_{t_f\to t_i^+}K(x_f,t_f;x_i,t_i) =\delta(x_f-x_i)$$

and the composition law. For the free particle,

$$K_0(x_f,t_f;x_i,t_i) =\sqrt{\frac{m}{2\pi i\hbar T}} \exp\!\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right], \qquad T=t_f-t_i\gt0.$$

The prefactor has dimensions of inverse length in one dimension, as a coordinate kernel must. Quadratic actions, including the harmonic oscillator away from caustics, can also be evaluated exactly; their fluctuation determinant and phase require more care than the free result.

Under a justified continuation $T=-i\tau$, the free Euclidean kernel becomes

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

It is a heat kernel, not a real-time transition amplitude. Wick rotation is not a purely algebraic replacement in every potential or contour problem.

Interpretation

The intermediate variables $x_1,\ldots,x_{N-1}$ become a discretized path. The path integral sums amplitudes over all such intermediate histories. Interference, not probability addition, is the key.

Limitations and Rigor Notes

The derivation is formal in real time. The continuum measure, convergence, normalization, boundary conditions, and operator ordering require care. Euclidean path integrals are often better behaved, but Wick rotation has its own assumptions.

Common Mistakes

• Treating time slicing as optional decoration rather than the origin of the formal measure.
• Dropping the short-time normalization factors.
• Ignoring the order of noncommuting kinetic and potential operators.
• Forgetting endpoint conditions.
• Treating the continuum expression as an ordinary integral over smooth paths.

Cross-Links

• Propagator Kernel
• Composition Law
• Why Path Integrals?
• Action Principles
• Time Ordering
• Path Integrals
• Fourier Transforms

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.

Exercises

1. In the time-sliced expression, why are there $N-1$ integrations for $N$ short intervals?

Solution

The endpoints $x_0=x_i$ and $x_N=x_f$ are fixed by the propagator. The intermediate positions $x_1,\ldots,x_{N-1}$ are summed over, giving $N-1$ integrations.

2. Check the dimensions of the free-particle kernel in one spatial dimension.

Solution

The kernel acts as $\psi(x_f,t_f)=\int dx_i\,K(x_f,t_f;x_i,t_i)\psi(x_i,t_i)$, so $K$ must have dimensions $\mathsf L^{-1}$. In the prefactor,

$$\left[\frac{m}{\hbar T}\right] =\mathsf L^{-2},$$

and its square root has the required dimension. The exponent is dimensionless.

3. Integrate one short-time momentum slice and recover its normalization.

Solution

For $\Delta x=x_{j+1}-x_j$,

$$\int\frac{dp}{2\pi\hbar} \exp\!\left[ \frac{i}{\hbar} \left(p\Delta x-\frac{\epsilon p^2}{2m}\right) \right] =\sqrt{\frac{m}{2\pi i\hbar\epsilon}} \exp\!\left[ \frac{im(\Delta x)^2}{2\hbar\epsilon} \right].$$

Completing the square gives both the kinetic action and the Gaussian normalization factor.