Time-Ordered Definition & Summary: What Is It?

[Greg Bernhardt]

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:

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

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[APhysicist]

Thanks for the overview of time-ordered in physics