# Time-Ordered Definition & Summary: What Is It?

• Greg Bernhardt
In summary, time-order refers to arranging items in order of time, with earlier items to the right of later ones. This concept is used in quantum field theory, specifically in perturbation theory, to calculate the S-matrix. The time-ordered product is a rearrangement of items into time-order, indicated by the symbol T. This allows for easier calculation and Lorentz covariance.
Definition/Summary

Time-order means in order of time. A sequence or product is in time-order if "earlier" items are placed to the right of "later" ones.

For example, if $t_1\ t_2\ \cdots t_n\text{ are times}$$\text{, and if }t_1>t_2>\cdots t_n$, they are in time-order. And so is the product $V(t_1)\ V(t_2)\ \cdots V(t_n)$, where $V$ is an operator depending on time.

And if $(x_1,y_1,z_1,t_1)\ (x_2,y_2,z_2,t_2)\ \cdots$$(x_n,y_n,z_n,t_n)\text{ are position-time 4-vectors}$$\text{, and if }t_1>t_2>\cdots t_n$, they are in time-order. And so is the product $\mathcal{H}(x_1,y_1,z_1,t_1)\ \mathcal{H}(x_2,y_2,z_2,t_2)\ \cdots \mathcal{H}(x_n,y_n,z_n,t_n)$, where $\mathcal{H}$ is an operator depending on position and time.

Time-ordered integrals, and time-ordered products, are used in perturbation theory in quantum field theory: a time-ordered integral is either the integral of an ordinary product with time-ordered limits, or the integral of a time-ordered product with ordinary limits (and one can be converted to the other by using the time-ordering symbol $T$).

Equations

EXAMPLE OF INTEGRAL OF PRODUCT WITH TIME-ORDERED LIMITS:

$$S\ =\ \sum_{N\ =\ 1}^{\infty} (-i)^N\int_{-\infty}^{\infty}\int_{-\infty}^{t_1}\int_{-\infty}^{t_2}\cdots\int_{-\infty}^{t_{N-1}}V(t_1)\cdots V(t_N)\,dt_1\cdots dt_N$$

THE SAME INTEGRAL, WRITTEN AS AN INTEGRAL OF TIME-ORDERED PRODUCT WITH ORDINARY LIMITS:

$$S\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}T\{V(t_1)\cdots V(t_N)\}\,dt_1\cdots dt_N$$

Extended explanation

Time-ordered product:

The time-ordered product of any items is the ordinary product of the same items, but with the items first rearranged into time-order.

If the items depend on a 4-vector variable, $(x,y,z,t)$, then the rearrangement is in order of the time-components, $t$, only.

The $T$ symbol:

The symbol $T$ placed before a product indicates that the items in the product are to be re-arranged into time-order before multiplying them:

For example:

$T\,\{\mathcal{H}(\boldsymbol{a},3)\mathcal{H}( \boldsymbol{b},5.5)\mathcal{H}(\boldsymbol{c},7)\}\ =$$\ \mathcal{H}(\boldsymbol{a},3)\mathcal{H}( \boldsymbol{b},5.5)\mathcal{H}(\boldsymbol{c},7)$

$T\,\{\mathcal{H}(\boldsymbol{a},3)\mathcal{H}( \boldsymbol{b},7)\mathcal{H}(\boldsymbol{c},5.5)\}\ =$$\ \mathcal{H}(\boldsymbol{a},3)\mathcal{H}( \boldsymbol{c},5.5)\mathcal{H}(\boldsymbol{b},7)$

$T\,\{\mathcal{H}(\boldsymbol{a},7)\mathcal{H}( \boldsymbol{b},5.5)\mathcal{H}(\boldsymbol{c},3)\}\ =$$\ \mathcal{H}(\boldsymbol{c},3)\mathcal{H}( \boldsymbol{b},5.5)\mathcal{H}(\boldsymbol{a},7)$

etc

Perturbation theory:

The S-matrix (in quantum field theory) is the limit as $\tau_0\rightarrow -\infty\text{ and }\tau\rightarrow \infty$ of an operator $U(\tau,\tau_0)$ satisfying:

$$U(\tau,\tau_0)\ =\ 1 - i\int_{\tau_0}^{\tau}\,V(t)\,U(t,\tau_0)\,dt$$

and by repeated integration we obtain the Dyson series:

$$S\ =\ \lim_{\tau_0\rightarrow -\infty,\,\tau\rightarrow \infty}\, U(\tau,\tau_0)\ =\ \sum_{N\ =\ 1}^{\infty} (-i)^N\int_{-\infty}^{\infty}\int_{-\infty}^{t_1}\int_{-\infty}^{t_2}\cdots\int_{-\infty}^{t_{N-1}}V(t_1)\cdots V(t_N)\,dt_1\cdots dt_N$$

The limits of integration are time-ordered, which is awkward to calculate , so we change to the following integral, which has the same value, but has easy limits of integration:

$$S\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}T\{V(t_1)\cdots V(t_N)\}\,dt_1\cdots dt_N$$

For the advantage of Lorentz covariance, we further change from integrals over the whole of time to integrals over the whole of space-time, and use a (scalar) Hamiltonian density $\mathcal{H}(x)\ =\ \mathcal{H}(\boldsymbol{x},t)\text{ with }V(t)\ =\ \int\int\int\,d^3\boldsymbol{x}\,\mathcal{H}( \boldsymbol{x},t)$, to obtain:

$$S\ =\ \sum_{N\ =\ 1}^{\infty}\frac{(-i)^N}{N!}\ \int\,T\{\mathcal{H}(x_1)\cdots \mathcal{H}(x_N)\}\,d^4x_1\cdots d^4x_N$$

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

Thanks for the overview of time-ordered in physics

## 1. What is a time-ordered definition?

A time-ordered definition is a type of definition that presents information in a specific chronological order, starting from the earliest event or concept and moving forward in a sequential manner.

## 2. How is a time-ordered definition different from a regular definition?

A regular definition simply provides a general explanation of a term or concept, while a time-ordered definition presents the information in a specific order to help readers better understand the progression or development of the topic.

## 3. When is a time-ordered definition most commonly used?

A time-ordered definition is often used when discussing historical events, scientific processes, or any other topic that involves a progression or sequence of events.

## 4. What is the purpose of a time-ordered summary?

A time-ordered summary is a condensed version of a time-ordered definition, highlighting the key points and events in a specific chronological order. Its purpose is to provide a quick and clear overview of the topic being discussed.

## 5. How can a time-ordered definition and summary be helpful in understanding complex concepts?

A time-ordered definition and summary can break down complex concepts into smaller, more manageable pieces of information. By presenting the information in a chronological order, it can help readers better understand the progression and connections between different events or concepts. It also allows for a more organized and structured approach to learning and comprehending complex topics.

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