Propagators and Kernels

A propagator is the time-evolution operator viewed through its matrix elements. It answers a precise question: with an initial alternative prepared at one time, what complex amplitude is assigned to a final alternative at another time? In the position basis, those matrix elements form a kernel that evolves wavefunctions by integration.

For a closed system,

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(s)\,ds \right],$$

and a general transition amplitude is

$$\mathcal A_{b\leftarrow a}(t,t_0) = \langle b|U(t,t_0)|a\rangle.$$

The position-space propagator kernel is the special case

$$K(x,t;x',t_0) = \langle x|U(t,t_0)|x'\rangle.$$

The chapter develops this operator, amplitude, and kernel language without conflating it with a probability density, a Green function, or a relativistic causal propagator.

What This Chapter Owns

This chapter is the canonical home for

• transition amplitudes as matrix elements of time evolution;
• coordinate-space propagator kernels and integral evolution of wavefunctions;
• composition through complete sets of intermediate alternatives;
• spectral decomposition into discrete, continuous, and mixed spectra;
• exact free-particle and harmonic-oscillator kernels;
• products, separability, and measures in multiple dimensions;
• the dependence of a propagator on domains and boundary conditions;
• support and instantaneous tails in nonrelativistic Schrödinger evolution;
• distinctions among amplitudes, probabilities, kernels, Green functions, and QFT propagators.

The chapter does not own the abstract construction of $U(t,t_0)$, which belongs to Foundations of Time Evolution. It does not own the resolvent or boundary-condition choices for energy-domain Green functions, which belong to Green Functions and Resolvents. It also does not own the regulated sum over histories, developed in the Path Integral Formulation.

From Operators to Kernels

Insert position resolutions of the identity into

$$\lvert\psi(t)\rangle = U(t,t_0)\lvert\psi(t_0)\rangle.$$

The coordinate wavefunction evolves as

$$\psi(x,t) = \int dx'\, K(x,t;x',t_0) \psi(x',t_0).$$

This is an integral representation of the same unitary operator equation. The kernel depends on the basis, configuration-space measure, operator domain, and endpoint times. It is not itself a state and need not be square-integrable as a function of either endpoint.

For a time-independent Hamiltonian,

$$U(t,t_0) = e^{-iH(t-t_0)/\hbar},$$

so the kernel depends only on the time difference when the domain and background are time independent. For a genuinely time-dependent Hamiltonian, both times matter separately.

Four Defining Tests

A proposed kernel should satisfy a connected set of operator identities. No single check is sufficient in every problem.

Initial condition

At equal times, the evolution operator is the identity:

$$K(x,t_0;x',t_0) = \delta(x-x').$$

The limit is distributional. Pointwise divergence or rapid oscillation near $t=t_0$ is compatible with convergence to a delta distribution.

Evolution equation

Acting on the final endpoint,

$$i\hbar\frac{\partial}{\partial t} K(x,t;x',t_0) = H_x(t) K(x,t;x',t_0).$$

The Hamiltonian acts on $x$, with the boundary conditions and domain appropriate to the final coordinate. There is a corresponding endpoint equation involving the initial time and the adjoint action.

Composition

For $t_2\ge t_1\ge t_0$,

$$K(x_2,t_2;x_0,t_0) = \int dx_1\, K(x_2,t_2;x_1,t_1) K(x_1,t_1;x_0,t_0).$$

The intermediate alternatives are summed as amplitudes when no measurement distinguishes them. Replacing the integrand by a product of probabilities changes the physical question.

Unitarity

For unitary closed-system evolution,

$$\int dx\, K(x,t;y,t_0)^* K(x,t;y',t_0) = \delta(y-y').$$

This is the coordinate representation of $U^\dagger U=I$. On a discrete grid or truncated basis, the delta distribution and measure must be replaced consistently by their finite representations.

Together these tests link normalization, dynamics, composition, and reversibility. A formula that solves the differential equation but implements the wrong boundary condition is not the propagator for the intended Hamiltonian.

Spectral and Geometric Views

If the Hamiltonian has a discrete orthonormal eigenbasis,

$$H\lvert n\rangle = E_n\lvert n\rangle,$$

then

$$K(x,t;x',t_0) = \sum_n e^{-iE_n(t-t_0)/\hbar} \psi_n(x)\psi_n(x')^*.$$

Continuous spectra replace sums by integrals with the normalization measure appropriate to the generalized eigenstates. Mixed spectra require both bound-state sums and continuum integrals. Omitting one sector breaks completeness and generally spoils the equal-time delta function.

For quadratic systems, exact kernels can also be written in terms of the classical action. Schematically,

$$K \sim \mathcal N(t,t_0) \exp\left( \frac{i}{\hbar}S_{\rm cl} \right).$$

The phase makes the classical path visible, while the prefactor contains fluctuation, normalization, and caustic information. This structure is exact for the free particle and harmonic oscillator, but it becomes a semiclassical approximation for a generic nonquadratic potential.

Page Map

Question	Canonical page	Main object
How do complex amplitudes become transition probabilities?	Transition Amplitudes	$\langle b\rvert U\lvert a\rangle$
How does a coordinate kernel evolve a wavefunction?	Propagator Kernel	$K(x,t;x',t_0)$
Why are intermediate alternatives integrated as amplitudes?	Composition Law	$U(t_2,t_0)=U(t_2,t_1)U(t_1,t_0)$
How do bound and continuum eigenstates determine propagation?	Spectral Decomposition of the Propagator	spectral sums and integrals
What is the simplest exact continuous kernel?	Free-Particle Propagator	Gaussian momentum integral
How do caustics and periodic motion appear in a kernel?	Harmonic-Oscillator Propagator	Mehler kernel and classical action
How do products, separability, and coordinate measures enter?	Propagators in Multiple Dimensions	$K(\mathbf x,t;\mathbf x',t_0)$
Why is the operator domain part of the propagator?	Propagators and Boundary Conditions	image, box, periodic, and twisted kernels
Do instantaneous Schrödinger tails imply relativistic signaling?	Causality, Support, and Interpretation	support versus relativistic microcausality

These nine articles form the planned chapter. The first four establish the general language; the next four develop exact examples and geometric qualifications; the last page protects the nonrelativistic interpretation from a common relativistic overreading.

Suggested Routes

First systematic pass

Read Transition Amplitudes, Propagator Kernel, and Composition Law in order. Then derive and check the Free-Particle Propagator. This route establishes the distinction between an amplitude, its modulus squared, and an integral kernel before adding spectral or path-integral machinery.

Spectral and stationary-state route

Begin with Spectral Decomposition and then study the harmonic oscillator. Continue to Resolvent Operator and Energy Green Function to see how time-domain phases become energy-domain poles and cuts.

Path-integral route

Read the kernel definition and composition law before From Propagators to Path Integrals. Time slicing repeatedly applies the same composition identity. The free and oscillator kernels then provide exact normalization standards for their corresponding path integrals.

Geometry and domains route

Use Propagators in Multiple Dimensions before Propagators and Boundary Conditions. This route makes the integration measure, domain, flux condition, topology, and self-adjoint realization explicit. A formal differential expression alone does not specify the propagator.

QFT and causality route

After the general kernel pages, read Causality, Support, and Interpretation and then From Propagators in QM to Propagators in QFT. The same word “propagator” labels related but nonidentical objects. Time ordering, commutators, Green-function boundary conditions, and relativistic locality must be stated rather than inferred from notation.

Computational route

Pair the exact free and oscillator articles with the Free-Particle Propagator Notebook and Harmonic-Oscillator Propagator Notebook. The notebooks test normalization, full-line versus periodic evolution, spectral truncation, caustic phases, and convergence.

Representation and Domain Checklist

Before using a propagator formula, state

1. the Hilbert space and basis used for its matrix elements;
2. the endpoint times and whether the Hamiltonian is time dependent;
3. the coordinate measure and normalization of generalized basis states;
4. the operator domain and boundary conditions;
5. whether the spectrum is discrete, continuous, or mixed;
6. the branch and phase convention for square roots and caustics;
7. whether the formula is exact, regulated, or semiclassical;
8. whether an intermediate alternative is unobserved, measured, or traced over;
9. which Green-function or QFT object, if any, is being compared;
10. how the initial condition, composition law, and unitarity are checked.

These declarations prevent most apparent disagreements among kernels that actually describe different domains, measures, or endpoint prescriptions.

Common Distinctions

Object	Definition	Interpretive warning
transition amplitude	$\langle b\rvert U\lvert a\rangle$	complex and basis dependent
transition probability	$\lvert\langle b\rvert U\lvert a\rangle\rvert^2$ for normalized discrete alternatives	requires the measurement question and normalization
propagator kernel	$\langle x\rvert U\lvert x'\rangle$	distributional and measure dependent
energy Green function	matrix element of a resolvent with a boundary prescription	not obtained by merely renaming $t$ as energy
heat or Euclidean kernel	$\langle x\rvert e^{-H\tau/\hbar}\lvert x'\rangle$	contractive rather than unitary
QFT propagator	field correlator or Green function with specified ordering	locality and particle interpretation differ from one-particle QM

Common Mistakes

• Calling $K$ a probability density. It is a complex amplitude kernel; probabilities arise only after forming the appropriate measured quantity.
• Taking the equal-time limit pointwise. The kernel approaches a delta distribution.
• Dropping the integration measure. Curvilinear coordinates and constrained spaces require the correct measure.
• Summing probabilities over unobserved intermediate alternatives. Coherent alternatives compose as amplitudes.
• Using only the differential equation. Boundary conditions and the initial condition select the intended propagator.
• Keeping only bound states in a mixed spectrum. The continuum is required for completeness.
• Ignoring caustic phase changes. A square-root prefactor needs a continuous branch or equivalent spectral prescription.
• Treating a wall as a large finite barrier without qualification. The domains and limiting procedures need not be interchangeable automatically.
• Equating temporal retardation with light-cone support. Nonrelativistic Schrödinger kernels generally have instantaneous spatial tails.
• Assuming every object called a propagator has the same ordering and causal meaning. State the operator or correlator explicitly.

Cross-Links

• Time-Evolution Operator
• Spectral Theorem
• Green Functions and Resolvents
• Path Integral Formulation
• Phase-Space Formulation
• Notebooks and Worked Examples
• Propagator Table
• From Propagators in QM to Propagators in QFT

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended ed., Dover, 2010.
• L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness, Academic Press, 1975.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013, doi:10.1007/978-1-4614-7116-5.