a galathaean theory of space

On Feb 27, 7:21 am, "galathaea" wrote:
[ the following was rejected from sci.physics.foundations
for attempting to overstimulate interdisciplinary debate
through excessive cross-posting ]

i have learned that to be an effective usenet crank
one must have a crackpot theory of space to defend

whether mike gordge or henri wilson
or any of those fighting for or against various continua
or jack sarfatti evangelising the interesting models
connecting the zero-point to inertia and gravity
or robert israel on euclidean dynamics
( though of course not with the same existential commitments )
every name is throwing their hat in the ring

so i have decided to destroy all remnants of credibility
( meagre as they may be )
and describe some ideas on space i have toyed with over the years

when i first studied relativity
i became very impressed by the writings of ernst mach

i enjoyed the way he related properties of space
to properties of interaction

but both berkeleyan immaterialism
and einsteinian relativity
seemed to fall short of the machian interactionism

in particular in their notions of space

i began to think that models of space should know only
distance between two existents
that there was no good way to define a space of possible points
without introducing counterfactual existents
and other nonoperational existents
space could only be defined
in terms of properties on collections of existents
and i started simply with two-point distance relations

the observation that at the time "cinched" the model for me
was that none of the known forces had angular dependence
except in the presence of 3 or more particles
(and their higher multipole terms)

that all known forces decompose
to interactions between existents

models of space should be in the language of networks
with the edges of the interaction graph
colored by distances and other "interaction intensities"

to start
i used simple distance pairs

d , d , d , ...
12 13 23

for existents

e , e , e , ...
1 2 3

so that a succession of cases could present themselves for analysis

now i was drawn to first find a formulation
that would recapitulate the euclidean / galilean dynamics
so that i could understand better the translation
to this language of space

1 existent models
have no interactions and no notion of distance of pairs
they are very lonely and boring

2 existent models
can define d
12

there are several natural dynamics that i have studied here
but in general any H(d ) = C provides a dynamic
12

if
d = constant
12

this describes a completely correlated and combined state

there is no change with time
and we can see that if we map this to euclidean
we see the natural appearance of circles in this model

in particular
this model only has derived notions of angular arrangement
but not one innate to the two-point description

now if both existents are "inertial"
and i translate in the euclidean model
whose dynamics i want to emulate
i see a point and a line of locations for the other existent

d will decrease to a minumum and then increase
12

the constant of integration
can be taken as that minimum distance d
mu
and focus on the perpendicular d as the initial condition
|_
then

2 2 2
d = d + ( d - v t )
12 mu |_ 2

2
2 d /\ (d ) = - 2 d v /\ t + O( (/\t) )
12 -- 12 |_ 2 -- --

or as the first "fundamental" equation
in this formulation:

.
d d = - d v
12 12 |_ 2

i felt at the time that this seemed to mesh
with the views of poincare on relativity
but i look at the equation now
and cannot quite see why

it is a very elegant conservation equation
in units of area per time

as also found in newtonian mechanics
or revilla's goldbach work
or in quantum black hole thermodynamics
these are beautiful units

there is more structure at 3-existents
here euclidean restriction actually inhibit possible space state

there are the triangle inequalities

d + d d
12 13 = 23

d + d d
12 23 = 13

d + d d
13 23 = 12

constraining possibilities and correlating the d_ij

now following out similar calculations to the two existent case
derived in a euclidean frame where e is stationary
1

http://i16.tinypic.com/3ywvbjk.jpght...om/2e4k86h.jpg
.
d d = - d v
12 12 2 mu alpha 2

.
d d = - d v
13 13 3 mu alpha 3

.
d d = [ ( x - x ) ( v - v ) +
23 23 2 alpha 3 alpha 2 x 3 x
( y - y ) ( v - v ) ]
2 alpha 3 alpha 2 y 3 y

an important point about these formulations
is that the constants on the right side are just constants
call them C
simple conservation symmetries
and the values from the euclidean ontology
simply one way to name the relationships between constants

from here on the generalisation is just repetition
but the structure is lain

for complete graphs
distances form a traceless symmetric matrix D(t)

0 d d d d d d d ...
12 13 14 15 16 17 18
d 0 d d d d d d ...
12 23 24 25 26 27 28
d d 0 d d d d d ...
13 23 34 35 36 37 38
d d d 0 d d d d ...
14 24 34 45 46 47 48
d d d d 0 d d d ...
15 25 35 45 56 57 58
d d d d d 0 d d ...
16 26 36 46 56 67 68
d d d d d d 0 d ...
17 27 37 47 57 67 78
d d d d d d d 0 ...
18 28 38 48 58 68 78
. . . . . . . .
. . . . . . . .
. . . . . . . .

the dynamics of this matrix
conserve constants element-wise
and thus completely decomposes in the inertial case

but of course
to me
the point was always secretly
to relax and generalise these conditions

noneuclidean spaces
nonmanifold spaces
strange topologies with exotic (un-"real") metrics
distance distributions and stochastic interpretations

discretisations are obvious
and seem to naturally describe petri nets
and other models for a linear logic or pi calculus

i used to think about possibilities
like interpreting quantum mechanics
in "inconsistent" distance sets
but later when i read fotini markopoulou
i would think about how this might be a model structure
on which to map "consistent geometries"
as a dual interpretation to consistent histories

it was a different approach
from connes' deformations into noncommutative spaces
and didn't quite seem to mesh with other physics foundations

yet it had all these nice theoretical properties

- machian
- interactionist
- does not platonise space
existents are not assigned point positions in a space
pairs of existents have a property distance
- dimensions, metrics, connectivity, etc.
all easily generalisable in this model
- provides the ontology necessary for the known interactions

so i would nurse these particular delusions
secretly feeding them by developing relativistic versions
(simple)
or trying to generalise to forms that were in some way quantum
( never could come up with anything intuitive
that i could derive anything with )

i would read loop quantum gravity papers
straining my eyes in this way i do
trying to find an interpretation in their algebra
because their language seemed so suggestively close

but

i haven't touched this in years
and most of it is in one of four boxes of papers
that i still need to go through and organise
because i have moved often over that time

and i have always felt slightly ashamed
and i never wanted to show anyone
because i was afraid of the type of insanity exposed

but i have seen the courage of the cranks on usenet
and in their spirit and in the fullest earnestness
i present the above outline

of the galathaean theory of space

it is a revolutionary theory
despite its lack of concrete testable prediction
like einstien's negation of the ether or preferred frame
for once my theory is appreciated
it will show present-day science for the sham it is

any hidebound reactionary or self-appointed defender of the orthodoxy
can challenge my theory with all their nazi zeal
but i will not be harmed

because i don't really care if it is "right" or not

its just a pastime i have had
that i wanted to finally share

according to
http://math.ucr.edu/home/baez/crackpot.html
you probably should not give any of this much time

just to demonstrate one of the ways
i have taken this foundation in the past
i wanted to describe one of my clumsy approaches to quantisation

if anything
i suspect it will show my awkwardness and make for light comedy

my idea was to start with a path integral formulation

here
however
as in the rest of the formalism
the action S is a function of
configurations of distances d_12, ... through time
ie. S(D) where D(t) is a "path" of d's

so in the standard form the amplitude is

/-/
| | i S(D) / h
| | dD e
/-/

at the time i was very interested in looking at
quantum leaps and tunneling
in terms of alternate topological paths

( or more accurately in terms of alternate paths
in idealisations of the physics that platonise space )

and so I was looking for ways to encode this in the action

the idea i came up with
was that the action could be seen as having contributions
from all manifolds that support a given path

---
\
S(D) = / S (D)
--- M
manifolds M

and each manifold contribution could also be decomposed furthur
by interpretations on the given manifold

---
\
S (D) = / S'(i(D))
M --- M
interpretation i

that maps the existents to points
in a way that allows the distance path to be possible

now these summations are actually measure unions
as manifolds and interpretations are not even countable

so these are "actually" integrals
but i keep this notation because of the discrete differences
that separate case types

for instance
2-manifolds have the classic classification theorem
in terms of tori and projective planes
but there are a number of metrics that may be applied
to any of these topologies

similarly
interpretations might be related by discrete symmetries
or by continuous translations and rotations

i was hoping to separate these at some point
( and although i investigated these issues individually
i don't believe i ever came back with a more explicit
decomposition )

of course at some point i'd have to come to some action
that i could in principle calculate
and the interpretation was my stopping point

here i could borrow from work on the euclidean (ahem cartesian!) case

standard relativistic quantum mechanics
gives the action of a path proportional to its proper time
and the translation to the language of "distance paths D"
gives a [n(n-1)/2]-volume form with some interesting properties

-+-+-+-

anyway
the point here is that this formalism
allows one to exploit the possibilities in topology
in a way very reminiscent of both string theory and lqg

however
as with most of my approaches in this formalism
there is always some point when the equations are derived
in the context of some interpretation on a platonised space

i had always hoped that
i could arrive at a sufficiently beautiful equation
and then magically say:

" i no longer take this as derived
i take this as fundamental
and all else is subsequently derived from it "

but even taking this down to the simplest case of 2 particles
and going through some very long and tedious calculations
i never arrived at anything that was even close to elegant

( though i did derive basic wave mechanics
when i threw much of this out and limited to planes! )

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar

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