Path Integrals: Mathematical Constructions

Path integrals connect quantum evolution with action principles, probability, and asymptotic analysis. Their mathematical meaning depends on the construction being used. This chapter supplies precise quantum-mechanical examples: strong limits of operator products, Brownian representations of heat semigroups, regulated oscillatory kernels, and stationary-phase expansions with controlled remainders. It also explains which conclusions do not follow from those theorems alone.

The physical derivations and standard applications belong to Path Integral Formulation. The present route is useful when reading a proof, extending a calculation to a singular potential, or deciding whether two limiting procedures can be interchanged. It concerns quantum mechanics with finitely many configuration coordinates; field-theoretic continuum constructions require additional results.

Enter this chapter

Begin with Why Path Integrals Are Subtle to distinguish the objects hidden by the same continuum notation. An operator can be well defined before any path representation is available. Convergence on wave packets does not automatically establish pointwise kernel convergence, trace convergence, or the existence of a measure on paths.

The most useful prerequisites are Functional Calculus, Quadratic Forms, Measure Theory Minimum, and Tracking Topologies. Readers interested primarily in semiclassical calculations can instead begin with the oscillatory-integral route below.

Operator products identify the limiting evolution

Trotter Product Formula separates two theorems. The unitary formula uses a self-adjoint closure of an operator sum. The positive-time form-sum theorem can construct a semigroup from a dense intersection of form domains. Their hypotheses and conclusions should not be exchanged by a formal substitution of imaginary time.

The page includes a graph-norm proof for a bounded perturbation of an unbounded generator. This example is particularly useful because convergence remains valid for some discontinuous potentials even when the formal commutator does not define a bounded operator. A splitting theorem and a numerical error rate are separate achievements.

Probability constructs a Euclidean representation

Wiener Measure versus Feynman Measure explains how Gaussian increments define a continuous path law and why typical paths do not have finite classical kinetic action. It also gives a precise obstruction to representing free real-time coordinate marginals by an ordinary finite complex path measure.

Feynman–Kac Formula then proves the Brownian expectation formula for bounded continuous real potentials. Monotone form limits extend the construction to continuous potentials bounded below. Fixed-endpoint kernels use Brownian bridges with an explicit free heat-kernel factor; Dirichlet boundaries use a killed process.

This route gives genuine positive-measure statements under stated hypotheses. It does not treat every Euclidean-looking functional expression as a probability measure or every real-time expression as its analytic substitution.

Oscillation and semiclassical limits need controlled prescriptions

Oscillatory Integrals defines Fresnel phases using Gaussian damping and recovers the free kernel as a distributional and strong operator boundary value. Its moment example shows why differentiation and integration need not remain ordinary convergent operations after a regulator is removed.

Stationary Phase and Semiclassical Rigor gives a quantitative remainder in terms of Fourier moments of an amplitude. It identifies the derivative and Hessian bounds needed for uniform estimates. A separate argument is required when the number of time-slice variables tends to infinity.

Path Integrals for Quadratic Actions verifies Gaussian kernels through Fourier factorization and canonical covariance. At caustics, the coordinate kernel changes to a distributional chart while the quantum operator remains unitary. Normalization, phase, and composition are each part of the construction.

Continuation and regulator limits complete the comparison

Euclidean Continuation Caveats proves the spectral analytic continuation for semibounded Hamiltonians. It distinguishes exact uniqueness from stable reconstruction using imperfect data and gives a simple reflection-positivity test for candidate Euclidean correlations.

What Is Constructed and What Is Formal? works through shrinking regular potentials. Their almost-everywhere pointwise limit vanishes, but their norm-resolvent limit is a nontrivial delta interaction. This example shows why the topology, domain, and limiting operator matter more than the appearance of the formal Hamiltonian expression.

After this chapter, a useful way to read a path-integral calculation is to identify its target observable, regulator, convergence theorem, and allowed exchanges of limits. The resulting statement may be exact, asymptotic, or a justified representation on a restricted class of states. Each can be useful when its scope is explicit.

References

• Reed, Michael, and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980 — operator constructions and convergence.
• Simon, Barry, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, 2005, doi:10.1090/chel/351 — probability and path-integral representations.
• Zworski, Maciej, Semiclassical Analysis, Graduate Studies in Mathematics 138, American Mathematical Society, 2012 — oscillatory analysis and rigorous semiclassical approximations.