Propagator Composition Law

Purpose

The propagator composition law expresses evolution over a long interval as evolution through any intermediate time. At the operator level,

$$U(t_b,t_a) = U(t_b,t_c)U(t_c,t_a), \qquad t_a\lt t_c\lt t_b.$$

In a position basis on the line,

$$\begin{aligned} K(x_b,t_b;x_a,t_a) &= \int_{-\infty}^{\infty}dx_c\, K(x_b,t_b;x_c,t_c)\\ &\qquad\times K(x_c,t_c;x_a,t_a). \end{aligned}$$

The integral is continuous-basis matrix multiplication. Each value of $x_c$ labels an unobserved intermediate alternative, so quantum mechanics sums the complex amplitudes before applying the Born rule.

The derivation and measurement interpretation live at Composition Law. This card collects the forms, normalization checks, and extensions used in calculations.

At a glance

Task	Formula
Define the kernel	$K(b,t_b;a,t_a)=\langle b\rvert U(t_b,t_a)\lvert a\rangle$
Propagate a wavefunction	$\psi(b,t_b)=\int da\,K(b,t_b;a,t_a)\psi(a,t_a)$
Compose discrete amplitudes	$K_{ba}=\sum_cK_{bc}K_{ca}$
Compose continuous kernels	$K(b,t_b;a,t_a)=\int d\mu(c)\,K(b,t_b;c,t_c)K(c,t_c;a,t_a)$
Equal-time kernel	$K(b,t;a,t)=\delta_\mu(b,a)$
Reverse a unitary kernel	$K(b,t_b;a,t_a)^*=K(a,t_a;b,t_b)$
Check unitarity	$\int d\mu(b)\,K(b;a')^*K(b;a)=\delta_\mu(a',a)$
Slice repeatedly	$K_{N0}=\int\prod_{j=1}^{N-1}d\mu(q_j)\prod_{j=0}^{N-1}K_{j+1,j}$

The compact notation in the unitarity row suppresses common endpoint times. The order of the kernel factors follows the order of operator action: the rightmost factor propagates first.

Kernel definition and wavefunction evolution

Let $\lvert q\rangle$ be a generalized configuration basis with identity resolution

$$I = \int_Q d\mu(q)\, \lvert q\rangle\langle q\rvert.$$

The evolution kernel is

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

If

$$\psi(q,t)=\langle q\vert\psi(t)\rangle,$$

then

$$\psi(q_f,t_f) = \int_Qd\mu(q_i)\, K(q_f,t_f;q_i,t_i)\psi(q_i,t_i).$$

The kernel is therefore the integral-operator representation of $U$. On $Q=\mathbb R^d$ with $d\mu(q)=d^dq$, a position-normalized kernel has units of length$^{-d}$ so that the propagated wavefunction retains units of length$^{-d/2}$.

Different basis normalizations move weight factors between $d\mu$ and $K$. The invariant statement is the identity resolution, not a memorized bare $dq$.

Derivation by inserting the identity

Start with operator composition:

$$U(t_b,t_a) = U(t_b,t_c)U(t_c,t_a).$$

Taking a matrix element and inserting the intermediate identity gives

$$\begin{aligned} K(q_b,t_b;q_a,t_a) &= \langle q_b\rvert U(t_b,t_c)I U(t_c,t_a) \lvert q_a\rangle\\ &= \int_Qd\mu(q_c)\, \langle q_b\rvert U(t_b,t_c)\lvert q_c\rangle\\ &\qquad\times \langle q_c\rvert U(t_c,t_a)\lvert q_a\rangle. \end{aligned}$$

Thus

$$\begin{aligned} K(q_b,t_b;q_a,t_a) &= \int_Qd\mu(q_c)\, K(q_b,t_b;q_c,t_c)\\ &\qquad\times K(q_c,t_c;q_a,t_a). \end{aligned}$$

This is exact for time-independent and time-dependent Hamiltonians. For a driven Hamiltonian, time ordering is already contained in each $U(t_2,t_1)$. No commutation assumption between Hamiltonians at different times is needed for composition itself.

Discrete and mixed bases

For a complete orthonormal discrete basis,

$$I=\sum_c\lvert c\rangle\langle c\rvert,$$

and the rule becomes ordinary matrix multiplication:

$$K_{ba}(t_b,t_a) = \sum_c K_{bc}(t_b,t_c)K_{ca}(t_c,t_a).$$

If the basis has a continuous coordinate and a discrete internal label,

$$I = \sum_\gamma \int_Qd\mu(q)\, \lvert q,\gamma\rangle \langle q,\gamma\rvert.$$

The matrix-valued kernel composes as

$$\begin{aligned} K_{\alpha\beta}(q_b,t_b;q_a,t_a) &= \sum_\gamma\int_Qd\mu(q_c)\, K_{\alpha\gamma}(q_b,t_b;q_c,t_c)\\ &\qquad\times K_{\gamma\beta}(q_c,t_c;q_a,t_a). \end{aligned}$$

The sum over $\gamma$ is as essential as the position integral. It accounts for unobserved spin, band, channel, or other internal alternatives.

Equal-time kernel

Because $U(t,t)=I$,

$$K(q_f,t;q_i,t) = \langle q_f\vert q_i\rangle = \delta_\mu(q_f,q_i).$$

The measure-adapted delta distribution is defined by

$$\int_Qd\mu(q_i)\, \delta_\mu(q_f,q_i)f(q_i) = f(q_f).$$

For Cartesian position on $\mathbb R^d$, $\delta_\mu(q_f,q_i)=\delta^{(d)}(q_f-q_i)$. In curvilinear coordinates, the coordinate expression includes the density required to invert the measure. For example, a radial delta paired with $r^2dr$ is not the same coordinate distribution as a delta paired with $dr$.

The equal-time limit is a strong normalization check. A proposed kernel that does not reproduce the identity distribution has an incorrect prefactor, measure, boundary condition, or limiting prescription.

Adjoint and unitarity identities

For closed-system unitary evolution,

$$U(t_f,t_i)^\dagger=U(t_i,t_f).$$

Taking matrix elements yields

$$K(q_f,t_f;q_i,t_i)^* = K(q_i,t_i;q_f,t_f).$$

This is an adjoint identity. It is not a claim that the Hamiltonian or the kernel is invariant under physical time reversal.

From $U^\dagger U=I$,

$$\begin{aligned} &\int_Qd\mu(q_f)\, K(q_f,t_f;q_i',t_i)^*\\ &\qquad\times K(q_f,t_f;q_i,t_i) = \delta_\mu(q_i',q_i). \end{aligned}$$

From $UU^\dagger=I$,

$$\begin{aligned} &\int_Qd\mu(q_i)\, K(q_f,t_f;q_i,t_i)\\ &\qquad\times K(q_f',t_f;q_i,t_i)^* = \delta_\mu(q_f,q_f'). \end{aligned}$$

These relations preserve inner products and total probability under kernel evolution. They also test numerical kernels more sharply than checking a single propagated state.

One-step and two-step wavefunction evolution

Using the composition law,

$$\begin{aligned} \psi(q_b,t_b) &= \int d\mu(q_a)\, K(q_b,t_b;q_a,t_a)\psi(q_a,t_a)\\ &= \int d\mu(q_c)\, K(q_b,t_b;q_c,t_c)\psi(q_c,t_c). \end{aligned}$$

The result is independent of where the interval is split. This associativity is a useful computational check: direct propagation from $t_a$ to $t_b$ and two-step propagation through $t_c$ must agree within the approximation and quadrature errors of a numerical scheme.

For an approximate short-time kernel, composition need hold only to the order claimed. Repeated composition can accumulate local errors, so the one-step error estimate and the continuum-limit scaling both matter.

Repeated composition and time slicing

Choose

$$t_a=t_0\lt t_1\lt\cdots\lt t_N=t_b$$

and denote $q_0=q_a$, $q_N=q_b$. Repeated identity insertion gives

$$\begin{aligned} K(q_N,t_N;q_0,t_0) &= \int\prod_{j=1}^{N-1}d\mu(q_j)\\ &\quad\times \prod_{j=0}^{N-1} K(q_{j+1},t_{j+1};q_j,t_j). \end{aligned}$$

The product is ordered so that later-time factors appear to the left. This finite identity is the starting point for a time-sliced path integral. The continuum path integral requires additional work: a controlled short-time kernel, normalization, an operator-ordering prescription when needed, and a definition of the limiting measure. See From Propagators to Path Integrals.

Free-particle check

For a free particle on the line and positive elapsed time $T$,

$$K_0(x_f,T;x_i,0) = \left( \frac{m}{2\pi i\hbar T} \right)^{1/2} \exp\!\left[ \frac{im(x_f-x_i)^2}{2\hbar T} \right].$$

Let

$$T_1=t_c-t_a, \qquad T_2=t_b-t_c, \qquad T=T_1+T_2.$$

The exponent in the product of the two short-interval kernels satisfies

$$\begin{aligned} &\frac{(x_b-x_c)^2}{T_2} +\frac{(x_c-x_a)^2}{T_1}\\ &\quad= \frac{T}{T_1T_2}(x_c-\bar x_c)^2 +\frac{(x_b-x_a)^2}{T}, \end{aligned}$$

where

$$\bar x_c = \frac{T_2x_a+T_1x_b}{T}.$$

Using the oscillatory Gaussian integral with the branch fixed continuously for positive time,

$$\int_{-\infty}^{\infty}dx_c\, \exp\!\left[ \frac{imT}{2\hbar T_1T_2}(x_c-\bar x_c)^2 \right] = \left( \frac{2\pi i\hbar T_1T_2}{mT} \right)^{1/2},$$

the normalization factors combine to give

$$\int dx_c\, K_0(x_b,T_2;x_c,0) K_0(x_c,T_1;x_a,0) = K_0(x_b,T;x_a,0).$$

The labels in this last line use time-translation invariance to display only elapsed times. The Gaussian phase and prefactor are both required; checking the exponent alone misses normalization and branch errors.

Amplitudes are not probabilities

Suppose a discrete intermediate basis contains alternatives $c$. Coherent composition gives

$$\mathcal A_{b\leftarrow a} = \sum_c \mathcal A_{b\leftarrow c} \mathcal A_{c\leftarrow a},$$

followed by

$$P(b\mid a) = \left\lvert\mathcal A_{b\leftarrow a}\right\rvert^2.$$

For two alternatives with path amplitudes $A_1$ and $A_2$,

$$\begin{aligned} P_{\rm coherent} &= \lvert A_1+A_2\rvert^2\\ &= \lvert A_1\rvert^2+\lvert A_2\rvert^2 +2\operatorname{Re}(A_1A_2^*). \end{aligned}$$

If an actual intermediate measurement records which alternative occurred and destroys the coherence between records, the probability becomes

$$P_{\rm recorded} = \lvert A_1\rvert^2+\lvert A_2\rvert^2.$$

The two expressions describe different experimental procedures. Inserting a mathematical identity does not perform a measurement.

Boundaries, geometry, and spectra

The composition law is basis independent, but its coordinate expression is not:

• On a finite interval, use the kernel satisfying the physical endpoint boundary conditions.
• On a curved configuration space, use the Hilbert-space measure and its corresponding delta distribution.
• In angular coordinates, include factors such as $\sin\theta$.
• In radial problems, specify whether radial measure factors are carried by the wavefunction, the kernel, or $d\mu$.
• For mixed discrete and continuous spectra, include both sums and integrals.
• For identical particles, compose within the properly symmetrized or antisymmetrized Hilbert space.

A convenient spectral representation for a time-independent Hamiltonian is

$$K(q_f,t_f;q_i,t_i) = \sum_n e^{-iE_n(t_f-t_i)/\hbar} \phi_n(q_f)\phi_n(q_i)^*,$$

with continuum contributions added when present. Composition follows from orthonormality, while the equal-time limit follows from completeness. The full construction is at Spectral Decomposition of the Propagator.

Numerical checks

For a discretized kernel and quadrature rule:

1. Propagate the same test state directly and through an intermediate time.
2. Include quadrature weights at every intermediate integration.
3. Check preservation of weighted inner products, not only pointwise values.
4. Refine the grid and time step to distinguish discretization error from a structural normalization error.
5. Test the equal-time limit on several smooth functions rather than comparing a sampled delta distribution pointwise.

If the Hamiltonian is time dependent, construct each interval with its own time ordering. If the evolution is approximated by matrices $U_{j+1,j}$, their product must use chronological operator order:

$$U_{N0} = U_{N,N-1}\cdots U_{2,1}U_{1,0}.$$

Common mistakes

• Reversing the two kernel factors.
• Omitting the intermediate integration measure or discrete-index sum.
• Treating $\lvert K\rvert^2$ as a normalized transition density for exact position eigenstates.
• Summing probabilities over coherent, unobserved alternatives.
• Interpreting identity insertion as a physical measurement.
• Using a full-line free-particle kernel for a system with boundaries or a potential.
• Forgetting that a time-dependent Hamiltonian still has exact propagator composition.
• Dropping short-time prefactors during repeated time slicing.
• Confusing the adjoint relation with physical time-reversal symmetry.
• Testing a delta-function limit pointwise instead of distributionally.

Related formulas

• Propagator Kernel
• Time-Evolution Operator
• Dyson Series
• Free-Particle Propagator
• Harmonic-Oscillator Propagator
• Propagators and Boundary Conditions
• Propagators in Multiple Dimensions

Exercises

1. Discrete amplitude composition

Derive the composition law in an arbitrary complete orthonormal basis $\{\lvert c\rangle\}$.

Solution

Start from

$$U(t_b,t_a)=U(t_b,t_c)U(t_c,t_a)$$

and insert

$$I=\sum_c\lvert c\rangle\langle c\rvert.$$

Then

$$\begin{aligned} \langle b\rvert U(t_b,t_a)\lvert a\rangle &= \sum_c \langle b\rvert U(t_b,t_c)\lvert c\rangle\\ &\qquad\times \langle c\rvert U(t_c,t_a)\lvert a\rangle. \end{aligned}$$

Defining each matrix element as the corresponding $K$ gives $K_{ba}=\sum_cK_{bc}K_{ca}$.

2. Unitarity in kernel form

Derive the first unitarity identity on this page from $U^\dagger U=I$.

Solution

Take the matrix element between $\lvert q_i'\rangle$ and $\lvert q_i\rangle$:

$$\langle q_i'\rvert U^\dagger U\lvert q_i\rangle = \langle q_i'\vert q_i\rangle = \delta_\mu(q_i',q_i).$$

Insert a final-time identity between $U^\dagger$ and $U$:

$$I=\int_Qd\mu(q_f)\, \lvert q_f\rangle\langle q_f\rvert.$$

The left-hand side becomes

$$\begin{aligned} \int_Qd\mu(q_f)\, &\langle q_i'\rvert U^\dagger\lvert q_f\rangle \langle q_f\rvert U\lvert q_i\rangle\\ &= \int_Qd\mu(q_f)\, K(q_f,t_f;q_i',t_i)^*\\ &\qquad\times K(q_f,t_f;q_i,t_i), \end{aligned}$$

which equals the required delta distribution.

3. Free-particle Gaussian

Use the completed-square identity above to verify the free-particle composition law, including its normalization.

Solution

Multiplying the two kernels gives the prefactor

$$\frac{m}{2\pi i\hbar\sqrt{T_1T_2}}$$

times an exponential. After completing the square, the endpoint-dependent factor is

$$\exp\!\left[ \frac{im(x_b-x_a)^2}{2\hbar T} \right],$$

and the remaining integral is

$$\left( \frac{2\pi i\hbar T_1T_2}{mT} \right)^{1/2}.$$

Their product is

$$\left( \frac{m}{2\pi i\hbar T} \right)^{1/2} \exp\!\left[ \frac{im(x_b-x_a)^2}{2\hbar T} \right],$$

which is $K_0(x_b,T;x_a,0)$. The square-root branch is the one obtained by the usual positive-time convergence prescription.

4. Coherent versus recorded alternatives

For two intermediate alternatives with amplitudes $A_1$ and $A_2$, compute the difference between coherent propagation and a recorded intermediate measurement whose outcome is later ignored.

Solution

Without a record,

$$P_{\rm coherent} = \lvert A_1+A_2\rvert^2 = \lvert A_1\rvert^2+\lvert A_2\rvert^2 +2\operatorname{Re}(A_1A_2^*).$$

With a perfectly distinguishing record and no postselection,

$$P_{\rm recorded} = \lvert A_1\rvert^2+\lvert A_2\rvert^2.$$

Therefore

$$P_{\rm coherent}-P_{\rm recorded} = 2\operatorname{Re}(A_1A_2^*).$$

The disappearance of the cross term reflects the changed physical process, not a different way of evaluating the same experiment.

References

• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, 1965, Chs. 2 and 3.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981, Chs. 4 and 6.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, Ch. 8.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, Ch. 2.
• C. Grosche and F. Steiner, Handbook of Feynman Path Integrals, Springer, 1998, for kernels on varied configuration spaces and boundary geometries.