The arrow of time

Subject: The arrow of time
From: sweetser@alum.mit.edu (Doug Sweetser)
Date: 1996/05/17
Message-Id: <4niokn$2p2@noise.ucr.edu>
Newsgroups: sci.physics.research
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Here's a fun way to transform the question: Should an event equivalent to
time reversal be locally unique in spacetime? If I draw a dot on a piece
of paper, then erase the dot, that is equivalent to drawing the dot and
then going backwards in time to undraw the dot. In both cases, the sheet
had a dot for a while, then it disappeared. The event of undrawing the dot
is locally unique in spacetime: it must be at the correct location in
space at a time after the dot is drawn. (for those concerned with macro vs
micro: drawing dots actually involves over a trillion atomic events
involving the atoms of graphite and is therefore not reversible because
there is too long a locally unique string of events to unravel. Please
consider a =B3dot=B2 to be a singular event on an atomic scale.)

Special relativity deals with 4 vectors in spacetime. The member of the
Lorentz group which reverses time is {-1,1,1,1} along the diagonal, zeroes
elsewhere. This works for any and all events in spacetime. It is not
locally unique. Yet the trend from general relativity through modern gauge
theories is for local symmetries.

The Lorentz group has had more success than almost any other mathematical
tool in physics. There is a small cottage industry which denies its
existence, using =B3logic=B2 instead of (or in spite of) mathematics. My
interest is to find a mathematical tool to accomplish the feats of the
Lorentz group, but one that does so _locally_ so that time reversal is
locally unique in spacetime.

My approach involves treating events in spacetime as quaternions. Since
quaternions are a skew field, there exists a time reversal
quaternion =B3R=B2 such that

R . q(t,x,y,z,) =3D q(-t,x,y,z)

R is a unique function of q(t,x,y,z) (it is fun to figure out exactly what
R is). Since quaternions don't commute, R=82R^-1. For events expressed as
quaternions, an event equivalent to time reversal is locally unique in
spacetime. For events expressed as quaternions, time has an arrow on the
smallest scale.

Doug Sweetser
sweetser@alum.mit.edu