The three tenors: SR, GR and quantum

Subject: The three tenors: SR, GR and quantum
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/04/19
Message-Id: <E8w8rB.Aqq@world.std.com>
Newsgroups: sci.physics.research

Hello:

In the initial assessment of any new approach to physics, one
would like a sense of how powerful it might be before lifting
an intellectual pencil. String theory promises a vast space
of 10 (or perhaps 11 or 26) dimensions to explore. If the
right needles can be found - and some of the best minds on
the planet are looking - perhaps the fundamental laws may be
sewn together.

I prefer the opposite approach: start with something
exceedingly simple and see if connections to fundamental
methods in physics can be built. As readers of this group
know, my current approach is to treat events as quaternion
which can be added or multiplied together. No other starting
assumptions are needed, but every conceivable tool for the
analysis of mathematical fields applies.

In this post, three events will be analyzed, trying to get a
measure of the separation between each of the three pairs. I
will not be trying to make connections to actual laws in
physics as I did for the post concerning Maxwell equations.
Rather, I will be playing with quaternions to see if there
are parallels to three methods currently in use. Special
relativity involves the use of the interval

     tau^2 = t^2 -  x^2 - y^2 -  z^2     (c=1).

General relativity says the Minkowski metric must not be
treated as an absolute, but rather as a dynamic variable like
any other in physics. Quantum mechanics makes use of Hilbert
spaces, a type of Banach space. The parallels I will try to
build to quaternions cannot be exact because objects involved
are different. Quaternions are not the scalars, 4-vectors
and metrics of SR, the second rank tensors for GR, or Hilbert
spaces for quantum. With these caveats in mind, we can
proceed.

Consider the three events O, A, and B:


             B
      A     /
       \   /
        \ /
         O

A few simplifications are made for discussion. Lump the 3
spatial directions into one, the I direction. Define O as
the origin at 0 + 0 I, A at 2 - 2 I and B at 3 + 3 I. The
way to calculate the interval squared with quaternions
involves squaring the difference between two events:

      O-A interval:  ((2 - 0) +  (-2 - 0) I)^2 = 0  - 8 I

      O-B interval:  ((3 -  0) + (3 -  0) I)^2 = 0 - 18 I

The real value is the square of the interval. In both cases,
the interval is zero, or light like. This information is
identical to using the Minkowski metric of special
relativity. Yet the approach is distinct because some type
of vector information is retained. Quaternions thus have a
connection to the interval of special relativity, but can
anything else be done with the extra vector information?

What would be a valid measure of the separation of events A
and B? The easiest approach involves repeating the
calculation and the answer is -24 - 10 I, a spacelike
interval. It might be more interesting to develop a measure
of the separation between A-B from the intervals O-A and O-B
just calculated. This is the way we will proceed, hoping to
build intriguing connections.

The square of the difference of two events can be written
symbolically as:

     (delta t)^2 -  (delta x)^2 + 2 (delta t)(delta  x) I

For the intervals O-A and O-B, the real parts are zero. This
means

     (delta t) = (delta x)

Starting from the intervals, that means event A is at 2 -2I
or -2 + 2 I and B is at 3 + 3 I or -3 -3 I. ... Calculate the
four possible intervals:

     ((2 -  3) + (-2 -  3) I)^2 = -24 - 10 I

     ((2 + 3) +  (-2 + 3) I)^2 =  24 -  10 I

     ((-2 -  3) + (2 - 3) I)^2 = - 24 + 10 I

     ((- 2 + 3) + (2 + 3) I)^2  =  24 + 10 I

The magnitude of the interval squared is 24 (correct!), but
it cannot be determined if the interval is spacelike or
timelike.

There is a way to generate one number to characterize the
separation of A and B no matter which pair is chosen. Take
the difference between any pair of A and B, then multiply
that by its conjugate like so (where "*" means complex
conjugate):

     ((2 -  3) + (-2 -  3) I)* ((2 - 3) + (- 2 - 3) I)

   = (-1 + 5 I) (- 1 + - 5 I) = 26 + 0 I

Symbolically, the calculation is

     (delta t)^2 +  (delta x)^2 + 0 I

This is the Euclidean norm of a quaternion, and the norms for
all pairs of A-B are the same. Note that q* q' behaves just
like an inner product. If q and q' are quaternion-valued
functions, it might be possible to construct a Hilbert space
and do quantum mechanics.

Why care about this ad hoc exercise in mathematics? A
quaternion that contains the information identical to the
Minkowski metric was used for a calculation. This "metric-
like" quaternion was used just like any other quaternion. It
was not treated as a dynamic variable, but there is nothing
in theory to stop such a practice. There is no way to a
priori distinguish the "metric-like" quaternions from any
other in this discussion. I find that property very exciting
because it could be a direct bridge to general relativity :-)

Hopefully I've sketched that the following moves are legal
with quaternions: to calculate an interval (SR), to use that
interval in another (possibly dynamic) calculation (GR) and
to use those in a Banach space (quantum). Using quaternions,
the moves required to work on relativistic quantum general
relativity are legal. This looks like a valid chessboard to
play with the fundamental methods of physics. Now I must see
if I can get any of the grand masters to put down their silly
strings and come play a game. Until then, this semi-pro
player is having a wonderful time gazing at the board by
himself.


Doug Sweetser
http://world.std.com/~sweetser
(for a less ad hoc analysis of quaternions)


Theoretical Physics AP exam question.
The standard model in physics is:
A. U(1) x SU(2) x SU(3)
B. Something that works, so don't fix it
C. A big theory that bullies other, smaller theories
D. The ONE theory of elementary particles:
scalar x unitary complex number x unitary quaternion
number x unitary Cayley number
E. All of the above
F. None of the above, because no such AP exam exists