Time Ordering

Time ordering arranges products of time-labeled operators so that later times appear to the left. It is required when Hamiltonians or interaction operators at different times do not commute.

Definition

For two operators,

$$\mathcal T[A(t_1)B(t_2)] = \begin{cases} A(t_1)B(t_2), & t_1\gt t_2,\\ B(t_2)A(t_1), & t_2\gt t_1. \end{cases}$$

For more operators, $\mathcal T$ orders the product with the latest time on the left and the earliest time on the right.

Why It Appears

Time evolution is a product of short-time steps. For a time-dependent Hamiltonian,

$$U(t+\Delta t,t) \approx I-\frac{i}{\hbar}H(t)\Delta t.$$

Multiplying many such factors gives later time steps on the left because they act after earlier time steps. If the Hamiltonians do not commute, this order cannot be rearranged.

Time-Ordered Exponential

The compact notation is

$$U(t,t_0) = \mathcal T \exp\left[ -\frac{i}{\hbar} \int_{t_0}^{t}H(t')\,dt' \right].$$

This is shorthand for a series with ordered operator products. It is not an ordinary exponential unless the ordering becomes irrelevant.

Dyson Series

The time-ordered exponential expands as

$$U(t,t_0) =I +\left(-\frac{i}{\hbar}\right) \int_{t_0}^{t}dt_1\,H(t_1) +\cdots.$$

The second-order term is

$$\left(-\frac{i}{\hbar}\right)^2 \int_{t_0}^{t}dt_1 \int_{t_0}^{t_1}dt_2\, H(t_1)H(t_2).$$

The ordered integration region $t_1\gt t_2$ is what enforces the operator order.

Correlation Functions Preview

Time ordering also appears in correlation functions, such as

$$\langle\psi\rvert \mathcal T[A(t_1)B(t_2)] \lvert\psi\rangle.$$

This notation becomes central in perturbation theory, Green functions, and field-theory propagators.

From Evolution Operators to Time-Ordered Products explains how this ordering rule governs field insertions, Wick expansions, and scattering amplitudes.

Common Mistakes

• Treating $\mathcal T$ as decorative notation.
• Pulling noncommuting operators through a time-ordered product.
• Forgetting that time ordering is tied to the order in which operators act.
• Assuming the ordinary exponential of an integral is valid for noncommuting $H(t)$.
• Confusing time ordering with normal ordering, which is a different operation.

Cross-Links

• Time-Dependent Hamiltonians
• Time-Evolution Operator
• Dyson Expansion as Formal Evolution
• Interaction Picture
• From Evolution Operators to Time-Ordered Products
• Path Integrals
• Commutators
• Time-Dependent Hamiltonian Notebook

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.

Exercises

1. Suppose $t_1\gt t_2$. What is $\mathcal T[B(t_2)A(t_1)]$?

Solution

Time ordering places the later-time operator on the left. Since $t_1\gt t_2$,

$$\mathcal T[B(t_2)A(t_1)] =A(t_1)B(t_2).$$

This may differ from the original product if the two operators do not commute.