Re: Octonian Wavefunctions -Still Any Research Today?
- From: Doug Sweetser <dougsweetser@xxxxxxxxx>
- Date: Fri, 25 Jan 2008 00:26:27 -0500 (EST)
Hello Peter:
There is an entire cottage industry devoted to finding alternative
ways to write the laws of physics. They will be having a biannual
international meeting devoted to this topic: The 8th International
Conference on Clifford Algebras (ICCA8) and their Applications in
Mathematical Physics, Campinas, Brazil, May 26-30, 2008. Technically,
there is an important difference between octonions and Clifford
algebras: the latter is associative while the former is not. This is
an essential issue to address. I did a quick scan of your group web
site and saw nothing about the non-associative property.
It is standard to write energy and momentum as a 4-vector, since it
transforms under the Lorentz group as a 4-vector:
P^u = (E, Px c, Py c, Pz c)
P_u = (E, -Px c, -Py c, -Pz c)
P^u P_u = (E^2 - Px^2 c^2 - Py^2 c^2 - Pz^2 c^2, 0, 0, 0)
= (m^2 c^4, 0, 0, 0)
This is standard special relativity. This most essential equation may
be poorly formed:
'E' = m c^2 + j p_x c + k p_y c + l p_z c
Energy has units of ML^2/T^2. The momentum term will only have the
right units if the positions are dimensionless. I also consider it
bad form to mix things that are invariant under a Lorentz boost with
things that are covariant (3-momentum), so that the resulting sum
transforms like the first part of a 4-vector. There are people who do
this, but I am not a fan of the practice.
A position 4-vector and the 4-momentum already have connections. By
taking a Fourier transform, one can go from the a position
representation to a momentum representation. I don't understand that
process as well as I would like, but folks who work in quantum field
theory do this sort of operation all the time. Another critical issue
is the uncertainty relations between position and momentum.
As a reader of SPR, I will presume you have spent quality time at
http://math.ucr.edu/home/baez/octonions/
As the owner of quaternions.com, my research is with quaternions. I
may travel to Brazil for the conference. Much of my work has been
rewriting laws, which involves no improvements except in the eye of
the beholder. Take as a concrete example a central Newtonian force in
a plane. If you recall that bit of classical physics, it takes two to
three pages for an author to find a dense way to write Newton's
central force law in a plane. With quaternions, the following
suffices:
F = M (cos(a), 0, 0, -sin(a))(d/dt,0,0,0)^2 (t, r cos(a), r sin(a), 0)
= M (0, L^2/m^2 r^2 + d^2 r/dt^2, 2 (L dr/dt)/m r^2, 0)
where L = m r^2 da/dt is the angular momentum, presuming d^2 a/
dt^2=0. The final result is identical to the standard approach, so I
cannot claim it is better. For me, I can spot what is going on. The
expression for the polar coordinates is kind of cool: (t, r cos(a), r
sin(a), 0). It has a factor of t for time which does not appear in
the final result having been hit by two time derivatives. Yet it is
good to have a place to put events in spacetime, even if the system is
traveling far less than the speed of light. It is just a good
accounting thing :-) Then there is the operator, two time
derivatives. This is classical physics, but we could make the law
relativistic by putting in spatial derivatives too (the result is far
too complicated to write here). The first cos, -sine rotates things
so that the trig functions drop due to trig identities. Nice, at
least in my eyes.
I just read Baez' "easy reading" article on octonions. He made two
points that struck a chord for me. First, it is important to develop
visual tools for all the algebra games you intend to play. This
reminds me to go make more quaternion animations. Second, Hamilton
created his own rule for multiplying quaternions. That is where I am
playing recently. One can make a real representation of quaternions
that is a _commuting_ 4D division algebra so long as (0,0,0,0) is
excluded - like Hamilton's quaternions - and all quaternions where one
element is the sum of the other three. The sum exclusion tosses out
events on the light cone, a cool tie-in to physics.
Doug
.
- Follow-Ups:
- Re: Octonian Wavefunctions -Still Any Research Today?
- From: Doug Sweetser
- Re: Octonian Wavefunctions -Still Any Research Today?
- References:
- Prev by Date: Black holes and observer-dependency
- Next by Date: Re: Octonian Wavefunctions -Still Any Research Today?
- Previous by thread: Re: Octonian Wavefunctions -Still Any Research Today?
- Next by thread: Re: Octonian Wavefunctions -Still Any Research Today?
- Index(es):
Relevant Pages
|
|
