In Wikipedia's surfaces of transitivity (of Lorentz group $G$, it says
"Standard vectors on the one-sheeted hyperbolas would correspond to tachyons. Particles on the light cone are photons, and more hypothetically, gravitons. The "particle" corresponding to the origin is the vacuum."
By definition of the Lorentz group, it preserves the quadratic form
$$Q(x) = x_0^2 – x_1^2 – x_2^2 – x_3^2$$
of the Minkowski space.
According to surfaces of transitivity (of Lorentz group $G$):
If a group $G$ acts on a space $V$, then a surface $S ⊂ V$ is a surface of transitivity if $S$ is invariant under $G$, i.e., $∀ g ∈ G$, $∀s ∈ S : gs ∈ S$, and for any two points $ s_1 , s_2 ∈ S$ there is a $g ∈ G$ such that $gs_1 = s_2$.
I thought
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tachyons are time-like, instead of space-like.
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$Q(x) < 0$ is a hyperboloid of one sheet. Points on this sheet are space-like separated from the origin.
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$Q(x) > 0, x_0 > 0$ is the upper branch of a hyperboloid of two sheets. Points on this sheet are separated from the origin by a future time-like vector. $Q(x) > 0, x_0 < 0$ is the lower branch of this hyperboloid. Points on this sheet are the past time-like vectors.
If my understanding is correct, then Wiki's "Standard vectors on the one-sheeted hyperbolas would correspond to tachyons" is wrong, but "Standard vectors on the two-sheeted hyperbolas would correspond to tachyons."
=> Is the statement in Wikipedia incorrect? Am I correct or incorrect?
Best Answer
Tachyons are "faster than light particles",
and, thus, have space-like 4-momentums (one is shown as the red segment below),
which point outside of the light-cone.
Constant-magnitude tachyon 4-momenta have their tips lying on the hyperboloid of one sheet (for example, on the red hyperbola below) when the diagram is expanded to include the suppressed spatial dimensions.
(Timelike vectors have tips in the blue-shaded region.)
In the diagram below, time $t$ runs along the vertical axis (Desmos's $y$-axis).
https://www.desmos.com/calculator/ceo1eymwqx