Temporal Green's Functions - Time Travel, Quantum Entanglement and Quantum Computing

Temporal Green’s Function

The temporal Green’s function is the Green’s function that solves a differential equation involving time — typically the evolution equation of a dynamical system — for a delta-function disturbance in time.
It tells you how the system responds at later (or earlier) times to an instantaneous impulse applied at a specific time.

1. Definition

Consider a linear time-dependent differential operator $L_t$ acting on some function $f(t)$ :

$L_t f(t) = s(t),$

where $s(t)$ is a source term.

The temporal Green’s function $G(t, t')$ is defined as the solution to:

$L_t G(t, t') = \delta(t - t').$

Once $G(t, t')$ is known, the solution for any source $s(t)$ is given by:

$f(t) = \int_{-\infty}^{\infty} G(t, t')\,s(t')\,dt'.$

2. Physical Meaning

$G(t, t')$ represents the response of the system at time $t$ due to a unit impulse applied at time $t'$ .
If the system is **causal**, the Green’s function vanishes for $t < t'$ :

$G(t, t') = 0 \quad \text{for} \quad t < t'.$

This ensures that the response cannot precede the cause.

3. Example 1 – First-Order Decay (Damped System)

Consider the simple equation:

$\frac{df}{dt} + \alpha f = s(t),$

where $\alpha > 0$ .

The Green’s function satisfies:

$\left(\frac{d}{dt} + \alpha\right) G(t, t') = \delta(t - t').$

For a causal solution ( $G=0$ when $t < t'$ ):

$G(t, t') = e^{-\alpha (t - t')} \, \Theta(t - t'),$

where $\Theta$ is the Heaviside step function.

Thus, the general solution is:

$f(t) = \int_{-\infty}^{t} e^{-\alpha (t - t')} s(t')\,dt'.$

This is essentially the *temporal convolution* of the source with an exponentially decaying kernel — widely used in electronic RC circuits, population decay, or thermal relaxation models.

4. Example 2 – The Free Particle Propagator (Quantum Mechanics)

In quantum mechanics, the Green’s function (often called the propagator) is defined as the solution to:

$\left(i\hbar \frac{\partial}{\partial t} - \hat{H}\right) G(x, t; x', t') = i\hbar\,\delta(x - x')\delta(t - t').$

For a free particle with Hamiltonian $\hat{H} = \frac{\hat{p}^2}{2m}$ :

$G(x, t; x', t') = \sqrt{\frac{m}{2\pi i \hbar (t - t')}} \exp\!\left[\frac{i m (x - x')^2}{2\hbar (t - t')}\right] \Theta(t - t').$

This temporal Green’s function tells you how a wave packet at position $x'$ and time $t'$ evolves to position $x$ at a later time $t$ .

5. Example 3 – The Wave Equation (Retarded Green’s Function)

For the 1D wave equation:

$\frac{\partial^2 \psi}{\partial t^2} - c^2 \frac{\partial^2 \psi}{\partial x^2} = s(x,t),$

the temporal Green’s function satisfies:

$\left(\frac{\partial^2}{\partial t^2} - c^2 \frac{\partial^2}{\partial x^2}\right)G(x,t; x',t') = \delta(x-x')\delta(t-t').$

The **retarded Green’s function** is:

$G_R(x,t;x',t') = \frac{1}{2c}\,\Theta\!\big(t-t'-|x-x'|/c\big).$

It represents a pulse traveling at finite speed $c$ : a disturbance at $(x',t')$ affects point $x$ only after sufficient time for the wave to propagate.

6. When the Temporal Green’s Function is Useful

Quantum Mechanics: Time-evolution kernels (propagators) for Schrödinger equations.
Electrical Circuits: RC and RLC circuit response to impulses.
Heat Conduction: Temporal evolution of temperature fields under time-varying sources.
Acoustics and Electromagnetism: Retarded potentials and causal field propagation.
Control Theory: Impulse response functions in linear dynamical systems.

Summary

The temporal Green’s function $G(t, t')$ is the system’s impulse response in time.
It converts differential equations into integral equations through convolution:

$f(t) = \int G(t, t')\,s(t')\,dt'.$

Its key property — causality — ensures that $G(t,t')=0$ for $t<t'$ .
Whether in quantum mechanics, heat diffusion, or classical wave propagation, temporal Green’s functions provide a unifying tool to compute how systems evolve in time under arbitrary driving forces.

Temporal Green’s Functions