Path Integral Formulation

The path-integral formulation represents quantum amplitudes as regulated sums over histories. For a particle with fixed endpoint coordinates, the central object is the propagator kernel

$$K(q_f,t_f;q_i,t_i) = \langle q_f\rvert U(t_f,t_i) \lvert q_i\rangle,$$

written formally as

$$K(q_f,t_f;q_i,t_i) = \int_{q(t_i)=q_i}^{q(t_f)=q_f} \mathcal Dq\, e^{iS[q]/\hbar}.$$

This is not a new quantum theory. It is a formulation of the same dynamics described by states, operators, and evolution kernels. Its special strength is that the action, interference between histories, semiclassical saddles, sources, and the extension from particle coordinates to fields become structurally visible.

Helpful background. Propagator kernels explain composition, the Trotter product formula justifies splitting kinetic and potential evolution, and action principles explain stationary action.

The word regulated is essential. The compact continuum notation suppresses time slicing, normalization, operator ordering, boundary conditions, and convergence prescriptions. This chapter develops those ingredients before using the shorthand.

From short-time kernels to paths

Consider one Cartesian coordinate on the real line and a time-independent, self-adjoint Hamiltonian with a potential for which the product formula applies. For

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

the real-time action is

$$S[q] = \int_{t_i}^{t_f}dt\, \left[ \frac{m}{2}\dot q^2 - V(q) \right].$$

At finite time slicing, the kernel is an ordinary multiple integral over intermediate coordinates. If $t_j=t_i+j\epsilon$, then

$$K(q_f,t_f;q_i,t_i) = \lim_{N\to\infty} \int dq_1\cdots dq_{N-1} \times \prod_{j=0}^{N-1} K_\epsilon(q_{j+1},q_j),$$

with $q_0=q_i$ and $q_N=q_f$. Here $\epsilon=(t_f-t_i)/N$. If $K_\epsilon$ denotes the exact short-time kernel, composition is already exact at any $N$; in a useful calculation one instead substitutes a short-time approximation and takes its limit. For example, a left-endpoint potential prescription gives formally

$$K_\epsilon(q',q)\approx \left(\frac{m}{2\pi i\hbar\epsilon}\right)^{1/2} \exp\left[\frac{i m(q'-q)^2}{2\hbar\epsilon} -\frac{i\epsilon V(q)}{\hbar}\right].$$

The Gaussian prefactor is essential: it fixes the delta-function limit and the units of the kernel. Oscillatory integrals require a consistent convergence prescription. See Time Slicing for the limiting construction, rather than interpreting this approximation as an ordinary probability density.

For $H$ bounded below and $au_f> au_i$, the Euclidean evolution operator is well defined. Its coordinate kernel is

$$K_E(q_f,\tau_f;q_i,\tau_i) = \langle q_f\rvert e^{-H(\tau_f-\tau_i)/\hbar} \lvert q_i\rangle,$$

with formal representation

$$K_E = \int\mathcal Dq\, e^{-S_E[q]/\hbar}.$$

Sources extend a kernel or vacuum functional into a generator of insertions:

$$Z[J] = \int\mathcal Dq\, \exp\left[ \frac{i}{\hbar} \left( S[q] + \int dt\,J(t)q(t) \right) \right].$$

Functional derivatives with respect to $J$ produce coordinate factors. Their operator interpretation depends on the state, endpoints, contour, time ordering, and normalization chosen for $Z[J]$.

These three objects organize the chapter:

Object	Weight or operation	Primary use
Real-time kernel	$e^{iS/\hbar}$	amplitudes, interference, propagators
Euclidean kernel	$e^{-S_E/\hbar}$	spectral projection, thermal traces, tunneling saddles
Generating functional	source-dependent action	ordered insertions and correlation functions

Reading Path

Question	Start here	What to retain
Why use histories at all?	Why Path Integrals?	Histories contribute amplitudes, not classical probabilities.
How does the formulation follow from ordinary quantum evolution?	From Propagators to Path Integrals	Repeated composition and completeness create the intermediate integrations.
What does $\mathcal Dq$ mean?	Time Slicing	The regulator, prefactors, endpoints, and ordering convention define the continuum shorthand.
Why does the action appear as a phase?	Action and Phase	The phase controls interference; stationary action organizes a limit, not a path probability.
What is the simplest exact calculation?	Free-Particle Path Integral	Gaussian fluctuations and normalization reproduce the known kernel.
How do quadratic potentials generalize the method?	Harmonic-Oscillator Path Integral	Classical action plus a fluctuation determinant gives an exact quadratic result.
How are insertions generated systematically?	Sources and Generating Functionals in QM	Source derivatives depend on sign, ordering, and normalization conventions.
Which correlator does a path integral compute?	Correlation Functions in Path Integrals	Time-ordered, connected, Euclidean, and retarded objects are not interchangeable.
What changes in imaginary time?	Euclidean and Imaginary-Time Path Integrals	Unitary phases become damping, with new boundary and continuation questions.
How does classical mechanics emerge from the integral?	Stationary Phase and the Classical Limit	Saddles, fluctuations, and competing paths form the semiclassical expansion.
How can tunneling appear as a saddle?	Instantons in Quantum Mechanics	Instantons are Euclidean saddles, not literal real-time trajectories.
What should be checked before trusting a formula?	Common Pitfalls in Path Integrals	Check the measure, normalization, operator ordering, and convergence prescription.
What survives when coordinates become fields?	Path Integrals from QM to QFT	The saddle and source logic survives; gauge, ultraviolet, and renormalization issues are new.

Exact, Formal, and Semiclassical Statements

The phrase “the path integral equals the propagator” can hide several claims with different status.

At finite time slicing, one has an ordinary multiple integral built from short-time operator kernels. Under suitable assumptions, its limit reproduces the exact operator evolution. The continuum symbol is then a compact representation of that regulated limit.

For quadratic actions, expanding about the classical path leaves a Gaussian fluctuation integral, so the saddle calculation can be exact after normalization and caustic phases are handled correctly. For a generic nonquadratic action, stationary phase is an asymptotic approximation:

$$K \sim \sum_\gamma \mathcal A_\gamma e^{iS_\gamma/\hbar},$$

where $\gamma$ labels contributing saddles and $\mathcal A_\gamma$ contains fluctuation information.

An instanton contribution has the Euclidean structure

$$\mathcal A_{\mathrm{inst}} e^{-S_{\mathrm{inst}}/\hbar}.$$

The exponent, determinant prefactor, zero modes, and sum over saddle sectors are separate parts of the calculation. A saddle approximation therefore requires more than finding a classical solution: its fluctuations and competing saddles determine the amplitude.

Choosing This Formulation

Path integrals are especially effective when the action has useful symmetries, several classical or Euclidean saddles compete, source derivatives organize many insertions, or the problem is preparing for many-body and field-theory methods.

They are not automatically the shortest route to an energy spectrum, a finite-dimensional matrix evolution, or an elementary commutator identity. Operator methods, the Schrödinger equation, spectral decompositions, and numerical diagonalization remain equal partners. A good calculation uses the formulation that makes the requested observable and approximation regime most transparent.

Exercises

For a free particle, write the positive imaginary-time kernel as

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

Use Gaussian convolution to show that two steps of duration $\tau$ give one of duration $2\tau$. What fails if the square-root prefactor is omitted?

Solution

Completing the square gives

$$(x-z)^2+(z-y)^2 =2\left(z-\frac{x+y}{2}\right)^2+\frac{(x-y)^2}{2}.$$

Thus integrating $K_E(x,\tau;z,0)K_E(z,\tau;y,0)$ over $z$ supplies $\sqrt{\pi\hbar\tau/m}$ and leaves

$$\sqrt{\frac{m}{4\pi\hbar\tau}} \exp\left[-\frac{m(x-y)^2}{4\hbar\tau}\right],$$

which is the $2\tau$ kernel. Without the original prefactors, an unwanted factor with units of length appears at every composition; normalization and the delta-function limit are lost.

Explain why the trace of a Euclidean kernel uses closed paths while a transition kernel uses fixed, generally distinct endpoints.

Solution

A transition kernel is a matrix element,

$$K_E(q_f,\tau;q_i,0) = \langle q_f\rvert e^{-H\tau/\hbar} \lvert q_i\rangle,$$

so its paths obey $q(0)=q_i$ and $q(\tau)=q_f$.

A trace sums diagonal matrix elements:

$$\operatorname{Tr}e^{-\beta H} = \int dq\, \langle q\rvert e^{-\beta H} \lvert q\rangle.$$

Thus the endpoint coordinates are identified and then integrated. In imaginary time, the interval has length $\beta\hbar$, and the paths are closed:

$$q(0) = q(\beta\hbar).$$

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, Dover, 2005.
• B. Simon, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea, 2005.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.