Composition Law

The composition law says that an amplitude from an initial alternative to a final alternative can be decomposed through a complete set of intermediate alternatives. For times

$$t_a\lt t_c\lt t_b,$$

the central identity is

$$K(x_b,t_b;x_a,t_a) = \int dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a).$$

This looks similar to a classical law for composing transition probabilities, but its meaning is different. Quantum mechanics composes amplitudes over unobserved alternatives, then applies the Born rule.

Operator Composition

The source of the composition law is the time-evolution operator identity

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

The order matters. The factor $U(t_c,t_a)$ acts first, carrying the state from $t_a$ to $t_c$; then $U(t_b,t_c)$ carries it from $t_c$ to $t_b$.

For a time-dependent Hamiltonian this composition law remains true. The time ordering is already built into each propagator $U(t_2,t_1)$. One should not replace the product by a single ordinary exponential unless the relevant Hamiltonians commute or an approximation has been justified.

Taking a matrix element between normalized states gives

$$\langle b\rvert U(t_b,t_a)\lvert a\rangle = \langle b\rvert U(t_b,t_c)U(t_c,t_a)\lvert a\rangle.$$

The right-hand side is not yet a useful sum. To expose intermediate alternatives, insert a resolution of identity.

Insert a Resolution of Identity

Let $\{\lvert c\rangle\}$ be a complete orthonormal basis. Then

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

Inserting this identity at the intermediate time gives

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

In transition-amplitude notation,

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

This formula is basis independent in the following sense: any complete identity resolution may be inserted. Different choices give different intermediate decompositions of the same total amplitude.

For continuous labels, the sum becomes an integral. If the identity is

$$I=\int dc\,\lvert c\rangle\langle c\rvert,$$

then

$$\mathcal A_{b\leftarrow a} = \int dc\, \mathcal A_{b\leftarrow c} \mathcal A_{c\leftarrow a}.$$

For more general configuration spaces or spectra, $dc$ must be replaced by the correct measure.

Position-Space Kernel Composition

The position identity on the line is

$$I=\int_{-\infty}^{\infty}dx_c\, \lvert x_c\rangle\langle x_c\rvert.$$

Insert it into the coordinate-space matrix element

$$K(x_b,t_b;x_a,t_a) = \langle x_b\rvert U(t_b,t_a)\lvert x_a\rangle.$$

Using operator composition,

$$\begin{aligned} K(x_b,t_b;x_a,t_a) &= \langle x_b\rvert U(t_b,t_c)U(t_c,t_a) \lvert x_a\rangle\\ &= \int_{-\infty}^{\infty}dx_c\, \langle x_b\rvert U(t_b,t_c)\lvert x_c\rangle \langle x_c\rvert U(t_c,t_a)\lvert x_a\rangle\\ &= \int_{-\infty}^{\infty}dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a). \end{aligned}$$

This is the kernel composition law. It is the coordinate-space version of matrix multiplication. The intermediate coordinate $x_c$ is summed over because it is not observed.

On a configuration space $Q$ with measure $d\mu(q)$, the same formula becomes

$$K(q_b,t_b;q_a,t_a) = \int_Q d\mu(q_c)\, K(q_b,t_b;q_c,t_c) K(q_c,t_c;q_a,t_a).$$

The measure is not optional. On a sphere, for example, $d\mu=\sin\theta\,d\theta\,d\phi$; for radial problems, the measure depends on the radial convention.

Wavefunction Evolution and Associativity

The same identity ensures that wavefunction evolution is consistent whether one evolves in one step or two. Starting from

$$\psi(x_b,t_b) = \int dx_a\, K(x_b,t_b;x_a,t_a)\psi(x_a,t_a),$$

insert the kernel composition law:

$$\begin{aligned} \psi(x_b,t_b) &= \int dx_a \int dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a) \psi(x_a,t_a)\\ &= \int dx_c\, K(x_b,t_b;x_c,t_c) \psi(x_c,t_c). \end{aligned}$$

Thus the kernel composition law is not an extra rule; it is the associativity of unitary time evolution written in a continuous basis.

Relation to Path-Integral Time Slicing

Path integrals arise by applying the composition law many times. Split the interval into

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

Repeatedly inserting position identities gives

$$\begin{aligned} K(x_b,t_b;x_a,t_a) &= \int dx_{N-1}\cdots dx_1 \prod_{j=0}^{N-1} K(x_{j+1},t_{j+1};x_j,t_j), \end{aligned}$$

with fixed endpoints

$$x_0=x_a, \qquad x_N=x_b.$$

The path-integral derivation then approximates the short-time kernel and studies the continuum limit. This page owns the composition identity; From Propagators to Path Integrals owns the time-sliced path-integral construction.

Comparison with Classical Probability Composition

For a classical Markov process, transition probabilities compose as

$$p(b,t_b\mid a,t_a) = \int dc\, p(b,t_b\mid c,t_c) p(c,t_c\mid a,t_a).$$

The quantities being integrated are nonnegative probabilities. There are no relative phases and no interference terms.

Quantum mechanically, for unobserved intermediate alternatives,

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

and only then

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

For two alternatives, define

$$A_1= \mathcal A_{b\leftarrow c_1} \mathcal A_{c_1\leftarrow a}, \qquad A_2= \mathcal A_{b\leftarrow c_2} \mathcal A_{c_2\leftarrow a}.$$

Then

$$\begin{aligned} P(b\mid a) &= \lvert A_1+A_2\rvert^2\\ &= \lvert A_1\rvert^2+\lvert A_2\rvert^2 {}+ 2\operatorname{Re}(A_1A_2^*). \end{aligned}$$

The last term is the interference term. If an intermediate measurement records which alternative occurred and the record is retained, the interference term is removed. The probability is then built from the recorded alternatives rather than from the coherent amplitude sum.

Measured Intermediate Alternatives

If an intermediate projective measurement in the $\{\lvert c\rangle\}$ basis is actually performed and the result is not ignored coherently, the probability for reaching $b$ through a recorded $c$ is

$$P(b,c\mid a) = \left| \langle b\rvert U(t_b,t_c)\lvert c\rangle \right|^2 \; \left| \langle c\rvert U(t_c,t_a)\lvert a\rangle \right|^2.$$

Summing over recorded intermediate outcomes gives

$$P_{\rm measured}(b\mid a) = \sum_c \left| \langle b\rvert U(t_b,t_c)\lvert c\rangle \right|^2 \; \left| \langle c\rvert U(t_c,t_a)\lvert a\rangle \right|^2.$$

This is generally not equal to

$$\left| \sum_c \langle b\rvert U(t_b,t_c)\lvert c\rangle \langle c\rvert U(t_c,t_a)\lvert a\rangle \right|^2.$$

The difference is not a mathematical contradiction. It is a difference between two physical experiments: one with a coherent unobserved intermediate basis, and one with a recorded intermediate measurement.

Common Mistakes

• Reversing the operator order in $U(t_b,t_a)=U(t_b,t_c)U(t_c,t_a)$.
• Summing probabilities over unobserved alternatives instead of summing amplitudes.
• Forgetting the integration measure in a continuous or curved configuration space.
• Treating the kernel composition law as a classical Markov law. It is amplitude composition.
• Applying coherent composition after an intermediate measurement has recorded which alternative occurred.
• Using the full-line position measure for systems with boundaries, constraints, or angular coordinates.
• Dropping short-time normalization factors when iterating the composition law toward a path integral.

Cross-Links

• Transition Amplitudes
• Propagator Kernel
• Time-Evolution Operator
• Time Ordering
• Born Rule
• Transition Probabilities
• Classical Versus Quantum Probability
• From Propagators to Path Integrals
• Propagator Table

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, Wiley, 1981.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Exercises

1. Derive the discrete-basis amplitude composition law from operator composition.

Solution

Start with

$$\langle b\rvert U(t_b,t_a)\lvert a\rangle = \langle b\rvert U(t_b,t_c)U(t_c,t_a)\lvert a\rangle.$$

Insert the identity

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

between the two evolution operators:

$$\langle b\rvert U(t_b,t_a)\lvert a\rangle = \sum_c \langle b\rvert U(t_b,t_c)\lvert c\rangle \langle c\rvert U(t_c,t_a)\lvert a\rangle.$$

2. Derive the position-space kernel composition law.

Solution

Use

$$I=\int dx_c\,\lvert x_c\rangle\langle x_c\rvert.$$

Then

$$\begin{aligned} K(x_b,t_b;x_a,t_a) &= \langle x_b\rvert U(t_b,t_c)U(t_c,t_a)\lvert x_a\rangle\\ &= \int dx_c\, \langle x_b\rvert U(t_b,t_c)\lvert x_c\rangle \langle x_c\rvert U(t_c,t_a)\lvert x_a\rangle\\ &= \int dx_c\, K(x_b,t_b;x_c,t_c) K(x_c,t_c;x_a,t_a). \end{aligned}$$

3. For two unobserved alternatives with amplitudes $A_1$ and $A_2$, show explicitly where interference enters.

Solution

The total amplitude is

$$A=A_1+A_2.$$

The probability is

$$\begin{aligned} \lvert A\rvert^2 &= (A_1+A_2)(A_1^*+A_2^*)\\ &= \lvert A_1\rvert^2+\lvert A_2\rvert^2 {}+ A_1A_2^* {}+ A_2A_1^*\\ &= \lvert A_1\rvert^2+\lvert A_2\rvert^2 {}+ 2\operatorname{Re}(A_1A_2^*). \end{aligned}$$

The final term depends on relative phase and is the interference contribution.

4. Compare coherent and measured intermediate alternatives for two paths.

Solution

If the two alternatives are unobserved, the probability is

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

If an intermediate measurement records which alternative occurred and the record removes coherence between the alternatives, the probability is

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

The difference is

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

Thus the physical act of recording the intermediate alternative changes the probability rule by removing interference.

5. Explain why the composition law requires the correct measure on a sphere.

Solution

The identity operator must match the Hilbert-space inner product. For wavefunctions on the unit sphere,

$$\langle\phi\vert\psi\rangle = \int_0^\pi d\theta \int_0^{2\pi}d\phi\, \sin\theta\, \phi^*(\theta,\phi)\psi(\theta,\phi).$$

Therefore the position identity on the sphere has the measure

$$d\mu=\sin\theta\,d\theta\,d\phi.$$

Using $d\theta\,d\phi$ instead would fail to reproduce the identity operator and would give the wrong kernel composition law.