Natural Science Forum / Physics / Research / February 2008

-I just have to add some information to my former posting, I feel:

Ok. here an explanation would be fine: I'm sorry, I thought that my
posting was to long already, so I didn't add the text about the
definitions with the octonions. -But another explanation will probably
be welcome, I hope..? -f you accept this posting, then here is the
long version:

Q: Why octonions, aren't they just 'out' after Einstein, even though
both Hamilton and Maxwell really liked them?

A: First of all, about modern physics: Both Hamilton and Maxwell very
remarkable persons in physics. -There shouldn't be any doubt, because
some of the most important concepts in physics still bear their names:
The Hamilton Equations and Maxwell Equations, If it wasn't for PAM
Dirac, and his remarkable accomplishments, then the Hamilton Equations
would still be the most fundamental known by man. (AFAIK). But
personally, I still believe in Hamilton and Maxwell, just let this be
said. Does their theroies look somehow strange today? - Honestly I
don't agree about this argument, and I will do my best to explain why
not...

Should everything important in physics (in this "Golden Age of
Physics"), have been done already; honestly, I don't think so! It's
like when somebody in the beginning of the 20th century said that
"everything important in theoretical physics already has been thought
out" or that "semiconductors are just a filthy mess, that no scientist
would like to have anything to do with". -We know a bit better
today...

Q: "Time or Proper Time, that's really the same stuff today, isn't
it?"

A: Well, a lot of people would really think so, but my answer is: No!.
-And honestly, i feel, that's it's hard to make the exclamation point
big enough on this one. One of the most important issues in PSR8,
which I shortly mentioned before, is the difference between "time" and
"proper time". Should the author of this article give somebody who are
studying (relativistic) physics a friendly advise, then this would
probably be the one: "Learn to see the difference between 'time' and
'proper time'".

It's a hard thing, but here are basically the difference in 'time' and
'proper time' in PSR8: First of all the concept of 'proper time' is
usually used. The usual concept of 'time' is used as a property
describing the LENGTH of a line in space-time. -Here is just a little
example of the property of 'time' which should really be seen as a
length in space-time: Should we really move a very long distance in a
few microseconds? - The distance could be so large, that we could be
talking about maybe a whole second! The quantity 'time' as usually
described, should (here in PSR8) be seen as an actual length in space-
time.

Q: But what is 'proper time'?

A: First of all, let's assume that you are living a normal human life
here on Earth: For example 1905 could have been the year, where you
were born, and maybe you would have made it until 2005. Shouldn't we
just say here, that we approximte a human lifetime with 100 years?
(sounds optimistic, but still somehow fair enough.)

In PSR8, here is a little diagram, that should show the conceptual
difference between 'proper time' (on the axis) and the usual time (a
length of a line in space-time): http://psr8-ha.googlegroups.com/web/PSR8-Diagrams.jpg,
the upper diagram with t=50y as an example.
[Sorry about the quality, I really don't know what's wrong with my
scanner this time: :-<) ! ]

Please notice, that in the general version, PSR8, then acceleration IS
actually allowed.

Q: Does your 'PSR8 Model' fit with the general theories of relativity?

A: Yes, there is no conflict at all. The transformations used in PSR8,
have actually been derived from normal relativity.  The theory is
consistent with both SR and GR. However the conception is slightly
different:

* In the so-called R-Space: The unit vectors are expressing "proper
time" as well as spatial position.
* In the so-called P-Space: The unit vectors are expressing
"Mass" (and NOT energy) and momentum.

In PSR8, then it's specially important to notice:
*Time would be considered as the length of a "worldline".
*The same is somehow the case with "Energy".

Q: So aren't 'time' and 'energy' the main parameters on the axes?

A: No, actually not!

-The main parameters are 'proper time' and 'mass'. But both 'time' and
'energy' are still well-defined quantities.

Q: Paradoxes, -how many?

A: Basically none, but of course some really tricky problems can be
constructed; specially there can be a severe confusion about what is
'time' (using 't') and what is 'proper time' (using 'tau').

* 'Proper time' (tau) reminds very much about the usual Newtonian time
concept. This appears to be in best agreement with the usual human
intuition.
* 'Time' (t) in PSR8: That's a bit harder to describe! -But let's try:
It's actually a distance in space-time. It's really well-defined, but
we need a diagram to illustrate it: again http://psr8-ha.googlegroups.com/web/PSR8-Diagrams.jpg
(The upper drawing)

Please have a look at my new board on Google Groups: PSR8-HA. I will
do my best to illustrate the difference between time (t) and proper
time (tau), because this is really the clue to understanding my
socalled PSR8 Theory.

Q: Why Octonions; what's different in PSR8?

A: First what's different?. Here are the two main points:

First of all: We use different units than in SR/GR. Time and energy
does NOT make up the primary physical dimensions; however 'proper
time' and mass actually does.

Second: The usual idea of energy and momentum as a property of a
particle is exchanged with another slightly different idea: We are
talking about "intervals on dimensions" the same way as we are
thinking about space intervals and time intervals in the usual
physical space (Which I prefer to call the "R-Space").

I use 'quats' and octonions simply because I can. They just appear to
be very interesting, and I'm still fascinated by the ideas and
inventions by physicists like Hamilton and Maxwell. Personally, I
believe that the 'quats' (4D complex numbers) does actually reflect
the properties of the physical space-time. That's my believe (Maybe
you won't agree, but that's just my opinion.)

Hope, that this addition might add a bit of info, and maybe also clear
up a few things...

Otherwise, please visit my new forum on PSR8, which is still under
construction, however: http://Groups.Google.com/group/PSR8-HA (please
give me a month or two to add some relevant articles)

That will very soon include the definitions with the EQ's for the
octonian wavefunctions...

If you have a good question, then please just leave a note on my
board; I will do my best to answer within a few days...

Q: -And last but not least: Special versions of this "PSR8 Theory"?

A: Yes indeed. First there are PSR4, which will only handle a problem
that has 1 dimension in space. (Like 'flat spacetime' in SR). -And
both PSR4 and PSR8 exist in socalled 'delta-versions'.

PSR4-DELTA is basically intended for illustration and education. It
probably won't hold for many practical problems, but it's really good
to derive and illustrate the properties of the more general theory.
The delta-versions are like SR, which means, that accelerations can't
be included. 'Delta' means, that only intervals are handled, these
versions does NOT support partial derivatives in general. However, the
general model does...

Hope that I succeded to clear up a few of the things, which I just
felt was missing in the original article.

Rgds,
Peter Christensen
(CPH, DK)

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

 You say this produces the same results as relavity, if so its a nice
new formulism of the old physics, but i wonder if you can reproduces
either quanutum mechanic or doubly special relavitivity from
octonians. You'd need one or two constants added. Can you get

xp - px = ihbar ( qm)

or

x_0.x_i - x_ix_o = x_i /k
p_0 p_i -- p_i p_0 = 0
x_i p_j = - delta_ij
x_0 p_i - p_i x_0 = -p_i /k
x_0 p_0 - p_0 x_0 = 1 - p_0 / k

which is typical of dsr models?
http://lanl.arxiv.org/abs/0711.4053

On 26 Jan., 07:47, Doug Sweetser <dougsweet...@gmail.com> wrote:

Personally, I'm not, but of course I understand your concerns.

It's not about just 'misunderstanding' the 'standard stuff', it's
about doing an attempt to write the same thing in a different way. -A
different way, that might be both more
intuitive and more usefull in practical calculations. That many have
tried, I will honestly see as a support, rather than something
negative. I looks like that it is possible, IMHO. .Many people think
so, that's known, at least.

And honestly, I have the impression that the moderators are just open-
minded conserning this 'trend', as long as we do not get in conflict
with any scientific results, of course. Many important things can be
said in many different ways, IMHO.

Not a problem, :-)

Again, I understand what you mean. -But I do not agree completely. -I
could write a really long text on this one, and if the moderators ask
me to, then of course I will, but here are just a few lines:

I simply have been using another representation, which I do know is
non-standard. It is possible to use proper time and position in a
coordinate system. Things will look different: For example time will
be something like the length of a line in space-time; of course this
IS really non-standard. I know it. -But non-standard does not always
mean wrong, or does it?

I mean quaternions, and sure I've just heard the word 'quats'
somewhere on internet; actually I think that it was on something from
computer-science. So ok, maybe it's not normally used in physics.

If you consider the numbers known as quaternions, then I will claim
that they are actually 4-Dimensional, the same way that I will say
that ordinary complex numbers are "2-Dimensional".

(Somewhere in one of my postings, I somehow mixed up the complex
units, sorry about it, of course it gives a bad impression. -And
usually 1,i,j,k is used, but I've been using 1, j,k,l everywhere in my
postings.)

At last: Could you please tell me about my 'trick', since I do not
really understand what you mean (?).

PC

On 28 Jan., 08:16, "B.Adams" <barry.david.ad...@googlemail.com> wrote:

Hello Mr. Adams,

I must admit, that I have problems understanding the info on your
site.

To write a short answer at the moment: Basically, with the
quaternions, I never had any problems because I never exchanged a*b
with b*a and so on. The quaternions are not commutative, but I just
didn't get any problems with this in practice.

But, I must read more about the DSR-models before I can write a good
reply on this one , IMHO. - I will ASAP,

B Rgds,
PC