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.

Helpful background. The Time-Evolution Operator supplies the operator being represented; Wavefunctions as Representations supplies the generalized position basis.

For a closed system with a well-defined unitary propagator, the usual time-ordering notation is

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

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.

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 self-adjoint 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 a time-independent Hamiltonian has a complete 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	Explanation	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

Begin with amplitudes, kernels, and composition, then calculate the free-particle kernel. The later examples explain how spectra, boundaries, and nonrelativistic support affect the answer.

From kernels to further calculations

The free-particle notebook and oscillator notebook compare explicit propagation with analytic results. Use Green Functions and Resolvents to move from time evolution to spectral inverses, or From Propagators to Path Integrals to build repeated composition into a time-sliced integral.

Imaginary-time evolution is different: $e^{-H\tau/\hbar}$ is not unitary. It is contractive for $H\ge0$. For $E_0=\inf\sigma(H)>-\infty$ and $\tau\ge0$, its norm is $e^{-E_0\tau/\hbar}$; shifting the energy to make $H-E_0\ge0$ gives a contraction. This qualification matters when comparing real-time kernels with heat kernels or ground-state projection.

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.

Exercises

Let a qubit undergo two identical Hadamard transformations, $U=(\sigma_x+\sigma_z)/\sqrt2$, starting from $|0\rangle$. Compare the final probability of $0$ when the intermediate alternatives are unmeasured and when a computational-basis measurement is made between the two transformations and its outcome is ignored.

Solution

Without an intermediate measurement, $U^2=I$, so the final probability is one. After the first transformation, the state is $|+\rangle$. Measuring and ignoring the outcome changes it to $I/2$, which remains $I/2$ after the second transformation; the final probability is $1/2$. Summing amplitudes retains interference, whereas summing the probabilities of distinguished alternatives describes a different experiment.

References

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