Time reversal with quaternions

Subject: Re: principles of statistical physics
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/04/03
Message-Id: <E81tFq.3HA@world.std.com>
Newsgroups: sci.physics.research
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Let's see if we can build what Greg Weeks wants: an operator that
almost exactly reverses time for any interval between a pair of events.
This operator should look very, very time symmetric, but not quite. The
tiny amount that it misses should only amount to something significant
if enough are added together.

I will treat the interval I(AB) between two events as a quaternion (a
4-dimensional mathematical skew field, antecedent to vector calculus).
The interval is

I(AB) = DT + dx I + dy J + dz K (d is for delta, DT >>> dx, dy, dz)

Can I find a quaternion R that reverses the time order of these two
events without changing their spatial locations? Since quaternions are
a mathematical field, the answer is necessarily yes. The quaternion R
that reverses the time to first order in dn/DT is:

R = -1 + 2 dx/DT I + 2 dy/DT J + 2 dz/DT K

Verify this works:

R I(AB) = -DT + O(dn^2) + (2 dx - dx + O(dy dz)) I +
(2 dy - dy+ O(dx dz)) J + (2 dz - dz + O(dx dy)) K

~= -DT + dx I + dy J + dz K

If we think about reversing the intervals between an ensemble of events,
they will all require almost the same quaternion R: -1. Laws of
physics should respect this global symmetry (and they do : ) However, the
dx/DT, dy/DT and dz/DT will be slightly different. There is no nice way
to rearrange these R's because they don't commute. This is true on the
micro scale as well. The quaternion R to reverse events A to B is not
the same quaternion to reverse B to A (it's R's transpose). The
dn/DT will often be in the range of 10^-8 or even orders of magnitude
lower, so R really is almost -1 for all. However, it ain't exactly minus
one, they don't all commute, and we need some kind of law to account for
this statistical detail. If the interval between a pair of events is
viewed as a quaternion, the math required to reverse time dictates a
statistically based arrow of time.

Reversing an interval is usually done with the following member of the
Lorentz group:

diag(-1,1,1,1)

No way I can see of answering Greg's question using this tool.


Doug
http://world.std.com/~sweetser/quaternions

4 D fields of numbers are fun
4 D fields of operators are dangerous