From Propagators to Path Integrals

The path integral rewrites a propagator as an integral over intermediate positions. Composition is exact; replacing each short-time evolution by separate kinetic and potential factors introduces an approximation whose limit must be justified. For a Cartesian particle with $H=p^2/(2m)+V(x)$, inserting momentum states then produces both the action phase and its essential Gaussian normalization.

Helpful background. The propagator kernel explains how an amplitude density evolves a wavefunction. The composition law supplies the identity used at each intermediate time.

Propagator composition

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

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

The endpoints are $x_0=x_i$ and $x_N=x_f$. This composition identity is exact for every finite partition. A limit becomes necessary below when exact short-time kernels are replaced by split approximations.

Short-Time Kernel

For

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

take one Cartesian coordinate on the full line, $m>0$, forward time, and a time-independent real potential for which the self-adjoint Hamiltonian and the Trotter product limit are well defined. Write $T_p=p^2/(2m)$ and define the split step

$$U_{\mathrm{split}}(\epsilon) =e^{-iT_p\epsilon/\hbar}e^{-iV\epsilon/\hbar}.$$

This is a unitary operator at every positive $\epsilon$, but generally differs from $e^{-iH\epsilon/\hbar}$. Its coordinate kernel can nevertheless be calculated exactly. Fix the Fourier convention

$$\langle x|p\rangle=\frac{e^{ipx/\hbar}}{\sqrt{2\pi\hbar}}, \qquad I=\int_{-\infty}^{\infty}dp\,|p\rangle\langle p|.$$

The rightmost potential factor acts on $|x\rangle$, while the kinetic factor is diagonal in momentum. With $\Delta x=x'-x$, inserting momentum completeness gives

$$\begin{aligned} K_{\mathrm{split},\epsilon}(x',x) &=e^{-i\epsilon V(x)/\hbar} \int dp\,\langle x'|p\rangle e^{-i\epsilon p^2/(2m\hbar)}\langle p|x\rangle\\ &=e^{-i\epsilon V(x)/\hbar} \int\frac{dp}{2\pi\hbar} \exp\!\left[\frac{i}{\hbar} \left(p\Delta x-\frac{\epsilon p^2}{2m}\right)\right]. \end{aligned}$$

Completing the square exposes the kinetic action:

$$p\Delta x-\frac{\epsilon p^2}{2m} =-\frac{\epsilon}{2m} \left(p-\frac{m\Delta x}{\epsilon}\right)^2 +\frac{m(\Delta x)^2}{2\epsilon}.$$

To fix the oscillatory integral and its phase, replace $\epsilon$ in the kinetic factor by $z=\epsilon-i\eta$ with $\eta>0$, then take $\eta\to0^+$. The momentum integrand now has Gaussian damping. Use

$$\int_{-\infty}^{\infty}dp\,e^{-ap^2+bp} =\sqrt{\frac{\pi}{a}}\,e^{b^2/(4a)}, \qquad \operatorname{Re}a>0,$$

with $a=iz/(2m\hbar)$ and $b=i\Delta x/\hbar$. This gives

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

The continuation fixes $i^{-1/2}=e^{-i\pi/4}$ for positive real time. Restoring the potential phase yields

$$K_{\mathrm{split},\epsilon}(x',x) =\sqrt{\frac{m}{2\pi i\hbar\epsilon}} \exp\!\left[\frac{i}{\hbar} \left(\frac{m(x'-x)^2}{2\epsilon}-\epsilon V(x)\right)\right].$$

The two plane waves supply $1/(2\pi\hbar)$; the momentum integral supplies the square-root prefactor. Neither normalization is optional. This kernel is exact for the split step and is used as an approximation to the kernel of the full Hamiltonian. Time Slicing compares its left-endpoint potential with the endpoint average produced by a symmetric split.

Continuum Limit

With $\epsilon=(t_f-t_i)/N$, the operator product $[U_{\mathrm{split}}(\epsilon)]^N$ converges strongly to the full evolution under the stated product-formula hypotheses. Inserting the calculated kernel in each factor gives the regulated representation

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

The finite-$N$ integrand and prefactor specify the normalization and endpoint data hidden by $\mathcal D x$. Strong operator convergence does not by itself prove pointwise convergence at every pair of coordinates: the kernel limit is understood after integration against suitable wavefunctions, or under additional kernel-convergence assumptions.

For smooth comparison paths the discrete action approaches $S[x]=\int dt\,[m\dot x^2/2-V(x)]$. The integration itself is not restricted to smooth paths. The usual continuum notation summarizes this regulated construction:

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

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.

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965.
• H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
• A. Turner, Quantum Theory I, Lecture 11, notes for Senthil Todadri’s MIT 8.321 course, MIT OpenCourseWare (16 October 2017), pp. 52–53, Eqs. (11.4)–(11.12) — composition, momentum insertion, and the short-time kernel.

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. Reverse the two operator factors, defining $\widetilde U_{\mathrm{split}}(\epsilon)=e^{-iV\epsilon/\hbar}e^{-iT_p\epsilon/\hbar}$. For equal time steps, let $K_N$ and $\widetilde K_N$ be the kernels of the $N$th powers of the original and reversed split steps. Find their relation for fixed endpoints where $V$ is finite. Does their common continuum limit make either finite-$N$ product exact?

Solution

The reversed step contributes the potential at the final endpoint of each slice:

$$\widetilde K_{\mathrm{split},\epsilon}(x',x) =e^{-i\epsilon[V(x')-V(x)]/\hbar} K_{\mathrm{split},\epsilon}(x',x).$$

Multiplication over slices telescopes the potential differences:

$$\sum_{j=0}^{N-1}\bigl[V(x_{j+1})-V(x_j)\bigr] =V(x_f)-V(x_i).$$

The factor is independent of every intermediate coordinate, so it can be pulled through all the integrations:

$$\widetilde K_N(x_f,T;x_i,0) =e^{-i\epsilon[V(x_f)-V(x_i)]/\hbar} K_N(x_f,T;x_i,0).$$

At fixed total time $T=N\epsilon$, the extra phase tends to one as $N\to\infty$. Under the product-formula assumptions, both limits give the same evolution. At finite $N$, noncommuting kinetic and potential terms generally produce splitting error in both approximants. A constant potential is an exact special case because it commutes with $T_p$.