Space Reversal Uses Mirrors

summary:

Reversal in space is only about mirrors.

description:

There are three directions in space, and three complex basis vectors. Flipping a space variable, or equivalently taking the conjugate of a complex basis vector, means using a mirror around the point (t, 0, 0, 0). Whatever time it is, a space reflection makes another point appear on the "other side". What was left handed now looks right handed. If you see pairs of points swing together around the origin, suspect the work of space reflection or conjugation.

command:
q_graph -dir vp -out space_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 5 5 5 0.005 -0.005 -0.005 -0.005 1000' -color blue
Outside video:
math
equation:

$q\to q'=q^*$

tags
Physics Tag:
space reversal
Math Tag:
conjugates
conjugates
Programming Tag:
command line quaternions
q_conj

Time Reversal Need Memory

summary:

Time reversal requires memories of where something was, so it can be undone.

description:

The look of time reversal is familiar. It requires a linear sequence of ordered events to be played back in exactly the reverse order. Time represents the real numbers, a totally ordered set. This set property grants the ability to run things backwards.

command:
q_graph -dir vp -out time_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green
Outside video:
math
equation:

$q\to q'=-q^*$

tags
Physics Tag:
time reversal
Math Tag:
conjugates
conjugates
Programming Tag:
command line quaternions
q_conj
q-x_scalar

Space and Time Reversal Are Different

summary:

Time and space reversal do not look the same which makes the difference between real and imaginary numbers concrete.

description:

Time requires memory, space needs mirrors. These do not look the same animated. If animation starts off like it ends up, then time reversal is in play. While time reversal can involve a minimum of one event on a screen, space reflection requires matching pairs of events. In math, imaginary basis vectors are represented as a 90 degree rotation in the complex plane. There is no difference except in label between real and complex numbers. The complex plane misleads. Real numbers have a graph that is undirectional - one times one is one - so real numbers can live without the imaginaries. Imaginary graphs have a directional graph - 1 times i is i, while i times -i is 1 - and there is no way to do multiplication with only an imaginary basis.

command:
q_graph -dir vp -out space_and_time_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj' -color blue1
Outside video:
math
equation:

\begin{align*} q\to q' &= q^* \quad \textup{space reversal}\\ q\to q' &= -q^* \quad \textup{time reversal} \end{align*}

tags
Physics Tag:
space reversal
time reversal
Math Tag:
conjugates
conjugates
Programming Tag:
command line quaternions
q_conj
q_x_scalar

Spacetime Reversal Goes Forward

summary:

While space reflections require a mirror, and time reflection need memory, a reflection in both space and time can look the same as no reflection at all!

description:

The input is in yellow, going from (-5 -5 -5 -5) to zero. The space reflection in blue, (-5, 5, 5, 5) to zero, is a mirror operation around the origin. The time reflection, (5, -5, -5 -5) to zero, in green requires you recall how the yellow input collection of events came onto the stage, so the green back it out. The reflection of both time and space, (5, 5, 5, 5) to zero, in red looks like a continuation of the yellow. Deep in quantum field theory they tell the odd tale of antiparticles going backward in time looking like particles going forward in time. Such an animated story now looks more reasonable.

command:
q_graph -dir vp -out spacetime_reversal -loop 0 -box 5 -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj' -color blue -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_conj | q_x_scalar -1' -color green -command 'q_add_n -5 -5 -5 -5 0.005 0.005 0.005 0.005 1000 | q_x_scalar -1' -color red1
Outside video:
math
equation:

\begin{align*} q\to q' &= q* \quad\textup{space reversal}\\ q\to q' &= -q* \quad\textup{time reversal}\\ q\to q' &= -q \quad \textup{spacetime reversal} \end{align*}

tags
Physics Tag:
space reversal
time reversal
spacetime reversal
Math Tag:
conjugates
conjugates
Programming Tag:
command line quaternions
q_conj
q_x_scalar