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:

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

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

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