time reversal | visualphysics.org

Time Reversal in Quantum Mechanics

12 May 2009

Posted by doug

Hello:

I received a question about time reversal for a complex quaternion matrix as appears in scattering problems that have spin-orbit interactions. Not an easy subject!

Tags:

Login or register to post comments
Read more

Spacetime Reversal Goes Forward

Posted by doug

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

youtube link:

Spacetime Reversal

youtube embed:

iPod downloads:

spacetime_reversal.povray.100.1000.animation.scan_.m4v

math

equation:

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

sage notebook:

quaternion conjugates - spacetime reversal

numbers:

spacetime_reversal.povray.100.1000.numbers

Space and Time Reversal Are Different

Posted by doug

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

youtube link:

Space and Time Reversal

youtube embed:

iPod downloads:

space_and_time_reversal.povray.100.1000.animation.scan_.m4v

math

equation:

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

numbers:

space_and_time_reversal.povray.100.1000.numbers

Time Reversal Need Memory

Posted by doug

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

youtube link:

Time Reversal

youtube embed:

math

equation:

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

sage notebook:

quaternion conjugates - time reversal

numbers:

time_reversal.povray.100.1000.numbers