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.

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.

What This Chapter Owns

This chapter is the canonical home for the path-integral formulation of ordinary quantum mechanics:

the derivation of sums over histories from propagator composition;
time slicing and the meaning of the formal measure;
the action phase and stationary-phase interpretation;
exact Gaussian examples for the free particle and harmonic oscillator;
sources, generating functionals, and coordinate correlation functions;
Euclidean kernels, ground-state projection, and thermal traces;
semiclassical saddles and an introductory instanton example;
a diagnostic guide to normalization, ordering, rigor, and continuation;
the conceptual bridge from coordinate paths to field configurations.

Detailed WKB matching, determinant technology, and quantitative instanton calculus belong to Approximation and Semiclassical Methods. Full field-theory functional integrals, gauge fixing, renormalization, and field-theory instantons belong beyond this quantum-mechanical chapter. Rigorous existence results also require hypotheses more specific than the formal symbol $\int\mathcal Dq$ can display.

Core Objects

For the standard Cartesian Hamiltonian

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

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

with $q_0=q_i$ and $q_N=q_f$ . The short-time kernels contain the normalization and discretization rule that the continuum measure symbol hides.

After imaginary-time continuation, the corresponding operator 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 Preview	Instantons are Euclidean saddles, not literal real-time trajectories.
What should be checked before trusting a formula?	Common Pitfalls in Path Integrals	Audit measure, normalization, ordering, status, contour, and method choice.
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.

Suggested Routes

First systematic pass

Read Why Path Integrals?, From Propagators to Path Integrals, and Time Slicing before relying on continuum notation. Then use the free particle and harmonic oscillator to see how normalization, classical paths, and Gaussian fluctuations work in complete examples. Finish with Common Pitfalls in Path Integrals.

Correlators and the QFT bridge

After time slicing, read Sources and Generating Functionals in QM and Correlation Functions in Path Integrals. Add the Euclidean page to distinguish real-time, imaginary-time, vacuum, and thermal objects. The final bridge page then translates paths, sources, and insertions into field language.

Semiclassical and tunneling route

Begin with Action and Phase, then study Stationary Phase and the Classical Limit. The instanton preview shows how Euclidean saddles encode exponentially small tunneling effects. Continue to the dedicated semiclassical volume for WKB matching, determinant prefactors, and multi-instanton calculations.

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. Labeling which steps are exact, regulated, asymptotic, or formal is part of a trustworthy derivation.

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.

Common Mistakes

Writing $\int\mathcal Dq$ without a regulator, boundary conditions, or normalization.
Treating $e^{iS/\hbar}$ as a probability weight on real-time paths.
Dropping short-time prefactors because they are independent of a particular path.
Assuming a continuum classical action uniquely fixes quantum operator ordering.
Calling a stationary path the “most probable path.”
Confusing a fixed-endpoint Euclidean kernel with a thermal trace.
Assuming Wick rotation is always a reversible symbol substitution.
Treating exact Gaussian results as representative of generic nonquadratic actions.
Using path integrals by habit when an operator or differential-equation method answers the question more directly.

Cross-Links

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, Dover, 2005.
J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.
H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed., World Scientific, 2009.
B. Simon, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea, 2005.

Exercises

A formula contains $\int\mathcal Dq\,e^{iS[q]/\hbar}$ and nothing else. List the minimum information needed to interpret it as a fixed-endpoint propagator.

Solution

One must specify:

the initial and final times and coordinates;
the Hamiltonian or action and its domain;
the time-slicing or other regulator;
which intermediate variables are integrated;
the short-time normalization factors;
the discretization and operator-ordering convention;
the real-time contour or convergence prescription;
the limiting procedure.

The propagator interpretation also requires the equal-time delta-function limit and the kernel composition law.

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).

Classify each statement as exact, regulated, or asymptotic: the composition law for $U$ ; an $N$ -slice multiple integral; and a generic stationary-phase sum over classical paths.

Solution

The evolution-operator composition law

U(t_f,t_i) = U(t_f,t)U(t,t_i)

is an exact operator identity.

The $N$ -slice multiple integral is a regulated finite-dimensional expression. Depending on the short-time approximation, it may equal a finite product of approximate kernels or represent a controlled product-formula approximation whose continuum limit is taken afterward.

For a generic nonquadratic action, the saddle sum

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

is asymptotic in a semiclassical regime. It becomes exact for special quadratic systems only after the fluctuation normalization and caustic structure are handled correctly.