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My understanding so far:

  1. A wave is a vector field defined on the space-time. i.e. mathematically wave is just a mapping which for every point in the space-time maps it to a vector.

  2. A world-line is function which maps an event (or a particle) on the space-time. In case the event (or the particle) "exists" only for an instant then the world-line will just be a point in the space-time diagram.

A few questions now (basically I want to check if I understand the concepts correct as I self-study these topics):

Q1 - Are the above definitions correct (and generic enough)?

Q2 - Based on above then (and if they are correct) there is nothing like a world-line of a wave. I'm getting quite confused here (maybe I'm unable to visualize) but it appears to me that only "particles" can have world-lines defined

Thanks

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  • $\begingroup$ Just as in classical physics, you can define the geometric approximation for waves, where the worldlines are the flow lines of the momentum vector $\endgroup$ – Slereah Jun 1 at 23:23
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Calling something a wave usually carries the connotation that there is some kind of periodic variation. However, this concept is not rigorously defined, as far as I'm aware. A wave can vary over space, over time, or both. For example, the wave $f(t,x)=\sin(x)$ varies in space but not in time, $f(t,x)=\sin(t)$ varies in time but not in space, and $f(t,x)=\sin(t)\sin(x)$ varies in both time and space.

You are correct in your definition of world-lines. In the case of a vector field, there is a generalization of the concept of world-lines. If you get a snapshot of the vector field at a point in time, then the "world-line" of the vector field is a function that gives the value of the field at every point in space and time. (You might call it a "world-sheet," except that term is already defined to mean something else in string theory.) If you already know how your vector field varies in time, then you already have your "world-line."

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Normally an EM plane wave is taken as sinusoidal vector field in space time. But it is not required to have this form to solve the wave equation.

An electric field $E_y = e^{-u^2}$ where $u = k(x+/-ct+a)$ also solves the wave equation:

$$\frac {\partial^2 E_y }{\partial t^2} = c^2k^2(4u^2 - 2)e^{-u^2}$$

$$\frac {\partial^2 E_y }{\partial x^2} = k^2(4u^2 - 2)e^{-u^2}$$

It is a world line, except to have some thickness because it fades quickly to zero when $u \neq 0$.

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