Time Reversal in Quantum Mechanics

12 May 2009

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Tue, 05/12/2009 - 12:03

doug

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Joined: 03/13/2009

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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!

One thing I never work with: complex-valued quaternion matrices. That is too bad for me, since a lot of fine work has been done for a long period of time with such math structures. The Maxwell equations were writen with complex-valued quaternions back in the eighteen hundreds. Six years after Einstein developed special relativity, two different people figured out how to do Lorentz transfomations using complex-valued quaternions. There are fine researchers today working with such tools. I wish them luck.

The reason I banash them from my work was best described by Stephen Adler who was pointing out work by John von Neuman on the logical foundation of quantum mechanics. If you do quantum mechanics over the real number field, then you will not get any of those interference patterns that makes quantum mechanics so challenging to understand. At a minimum, one needs to use complex numbers. Such an approach dominates the work of theoretical physics today. The third choice is to do quantum mechanics over the field of quaternions. Adler has written the book on the topic, "Quaternionic Quantum Mechanics". His efforts so far have been limited to using quaternion numbers in a few select spots. As far as I can tell, few new things come out of his change of math tools.

If you have field equations, one thing one wants to do is invert the field equation to make a propagator useful in quantum field theory calculations. Adler was the one who pointed out this connection to me. If all my tools are part of a division algebra where I know an inverse exisits, then I should be able to invert any field equations I construct, gauranteed. I cannot do so with complex-valued quaternions, so I don't use them ever.

To be honest, the reason I don't use complex-valued quaternions is much simpler. It strikes me as illegal accounting. A quaternion is three complex numbers that share the same real. Each of those separately acts like a good complex number, commuting with numbers just like itself. There are actually more than three quaternions that behave this way - there are an infinite number of them, so long as the 3-vector points in precisely the same direction. With the complex-valued quaternions, there is a fourth imaginary unit beyond the 3 + infinite choices we have which is like the others exect that it never gets into the non-communing trouble. Why? Just because it makes calculations easier? It does accomplish that goal, but it sounds like cheating to me. Non-commuting is an issue that has to be dealt with, not avoided.

People talk about time-reversal operators. This is a bit frustrating because one is reversing time in the context of spacetime. The trick is to flip the sign of time while leaving space untouched. There is a huge mystery about the arrow of time in thermodynamics. That is because the question neglects the spacetime contex. Here is the element of the Lorentz group that will flip the sign of time without altering space:

$L_{tr} = diag(-1, 1, 1, 1)$

$(t, x, y, z) L_{tr} = (-t, x, y, z) \quad eq ~1-2$

This certainly is simple. Now if you want to take (-t, x, y, z) and transform it back, you would use exactly the same element of the Lorentz group. The mystery in time's universal symmetry is right here: the same element of the global Lorentz group takes a 4-vector forward or backward in space.

A way to solve this riddle is to use a local group, one that depends on the local values of (t, x, y, z), instead of the global Lorentz group. There is no "arrow of time" which is a scalar. There is a handedness to spacetime if one wants to flip the sign of time, and that comes from the space part of spacetime. Let's find the quaternion operator 'L' that will do the job locally (using dR as a 3-vector so I can type less, remember more):

(dt, dR) L = (-dt, dR)
$\begin{align*}L &= (dt, dR)^{-1}(-dt, dR)\\ &= (dt, -dR)(-dt, dR)/(dt^2 + dR^2)\\ &=(-dt^2 + dR^2, 2 dt dR)/(dt^2 + dR^2) \quad eq ~3 \end{align*}$

For classical physics where speeds are small, the change in time, dt, may be ten or more orders of magnitude larger than dR/c. Classically, $L \approx (-1, 0, 0, 0)$ . Now lets look for the inverse, a calculation that is darn similar:

$(-dt, dR) L^{-1} = (dt, dR)$
$\begin{align*}L^{-1} &= (-dt, dR)^{-1}(dt, dR)\\ &= (-dt, -dR)(dt, dR)/(dt^2 + dR^2)\\ &=(-dt^2 + dR^2, -2 dt dR)/(dt^2 + dR^2)\end{align*}$
$L \neq L^{-1}$
$L \approx (-1, 0) \approx L^{-1}\quad eq ~4-6$

What changes for the time reversal operator is the sign of the 3-vector. That 3-vector can be ignored in classical physics because changes in time vastly exceed changes in space.

With L, its inverse, and the identity, we do have a local group. All a group needs is an inverse, an identity, and closure - which is straight forward having only 3 players in the group. I don't know the name of this small group.

A large amount of work would be required to show that this observation about quaternion spacetime reversal operators does precisely what is needed to technically resolve the arrow of time problem in thermodynamics. Too bad I don't have those skills.