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Is Causality Defineable?
Re: Causation, by Professor Anonymous
---------------------------
The Stanford Encyclopedia of Philosophy online is a gem.
I was a bit overwhelmed however. I knew in advance that when you mix
philosophy and physics, esp on the subject of causation, one ends up
with a real impenetrable mess....
[can one define causality?]
I suggest that no one else has either or can even really do so in so
many words or less. All we are left with are lengthy theories and
mathematical techniques.
I think my approach using
the correlation of world-line bundles to loosely define interaction and
causation neatly dispenses with the moving shadow-spot problem as
pseudo-causation. Causation is not the Russel approach of A-to-B along
a world-line (Russel);
it has more to do with how separate world lines tango.
This tango is where the 'Process' and 'Conservation' camps enter the
philosohpical fray.
To complicate matters, the observer effect due to relativity says
the sequencing of A and B are dependent on the position of the observer
in the Space-Time plenum. As macroscopic observers, we percieve only the
woven bundles of world-lines or more precisely we interact as a causal
assemblage itself. The observation process itself might be where the
true essence of the debate lies. Anyway, here's what I wrote two years
ago:
--------------------------CUT--------------------------------------
Thoughts on Correlation, Causality, Non-Locality
and 'Collapsing the Wave'
by Professor Anonymous, Dec. 1995
I) Section One
note: For simplicity we will use one-dimension of space.
All results will apply to higher dimensions.
Check out your set-theory of metric spaces if not
convinced about this.
Consider a one-dimensional space and time. We can construct a
2-dimensional space-time grid for plotting positions as a function of
time. If we normalize everything to the speed of light, then all motion
in this system will have to lie within the so-called 'light-cone'
which is formed by two lines at 45 degrees with respect to the time axis.
Force is defined as the time-rate-of-change-of momentum. A
particle experiencing no force will thus appear as a straight line on
the space-time graph. Force relates directly as the 2nd order derivative
of x. The first order derivative is simply dx/dt = velocity = the slope
of the line at time t. Graphically we can say that force manifests as
a curviture of some particle's time-line. A time-line for a classical
particle is always a curvi-linear affair. Quantum particles in contrast
will have FUZZY time-tubes!
Now for correlation. Lets define correlation as simply the
product of two terms! Consider first a particle experiencing no force.
Its time-line is straight (zero curvature), therefore its auto-correlation
product is zero for all pts along that line. When there is force involved,
however, there is curvature (2nd derivative non-zero). The auto-correlation
product is now non-zero and ALWAYS POSITIVE, as the product of a negative
value with itself is positive. I believe that this allways non-zero
auto-correlation of 2nd order space change for classical particles is
nothing other than inertia - as 'prime' causality. In other words,
a classical particle will trace a straight time-line if left to its
own causes. This will also apply to quantum particles so long as the
space-time interval exceeds the quantum uncertainty range.
But what about that particle whose time-line is curved?
There is some interval or range where the auto-correlation is non-zero.
Why? Well we really can't say, but lets look at two particles which
we know interact with each other in a force-full way i.e say two 'charged'
particles. There will be some space-time region where they are close
enough together for a noticable relationship between their time-lines.
This relationship can be quantified by degree as the correlation
between curvatures of the two time-lines. Either time-line may have a
positive or neative curvature indicating acceleration or deceleration,
with the qualification that which-is-which depends on whether the space
location is on the positive or negative side of the time axis. The net
result is that we end up with a correlation value that is non-zero and
positive or negative for every time at which the two interact. If the
product is positive the particles repel; if negative, they attract;
If one integrates the products over time, he comes up with a
net measure of the interaction between the particles! I think this is an
adequate description of what we mean by CAUSALITY in the classical
sense. Causality reaches across space and weaves time-lines together or
unravels them. Quantum particles confuse the situation because their
time-lines are fuzzy and there is a point at which we resort to
probability wave amplitudes to represent instantaneous possibilities
for correlation. Now wave definitions are an excellent construct to use
because they convey the sense of bounded action which oscillates
periodically over time within some spatial arena. I think a random
generator function would perhaps do just as well or better! The math
might be horrendous though, becuse then we would be dealing with non-
diffentiable functions! Scientists choose the easiest and simplest way
to get good answers, and there was a lot of macroscopic waves to give
credence to such an approach when these theories were developed.
One could define 'locality' very broadly as as simply any
slice of the 'light-cone' perpendicular to the time axis; A more
narrow view would assign locality to the fuzzy boundaries around any
single time-line; Considering systems of interacting particles leads
to viewing locality as a union of fuzzy locations - all of which
must lie inside the light-cone.
When one applies the above egalitarian view of 'locality',
NON-LOCALITY is a relative affair and always operative i.e. causality
implies non-locality. In this light, Bell's Theorem is no more
mysterious than the interaction between the poles of two magnets
separated in space. Also one might note that the concept of force
had its earliest origins in man's ability to sense the exertion in
his own muscles when in direct contact with objects. It is no wonder
that at first glance Bell's Theorem appears counter-intuitive!
II) Collapsing the Wave
If one has read my remarks in section I) and gotten my drift,
I think we can come to an understanding of why probability amplitudes
are squared to arrive at a final real probability for some event.
First off, recall the idea that quantum particles are spread
over a fuzzy region in space-time. The fact that I said space & time
together, implies that within that boundary with its locus at some space-
time pt, we can have the fanciful instantaneous situation where the
particle moves backward in time. This is obvious if one considers that
inside the quantum uncertainty range any particular position is equally
likely. Now in the sense of ordinary mathematical functions, such a
situation cannot be expressed as an ordinary differentiable function.
A way out of the dilema is to create a representation using complex
numbers such that each component (real and imaginary) is expressed
by a wave equation as stated previously.
Considering a complex number graphically as a vector we see
that the instaneous probability amplitude is merely a vector to some
point inside the fuzzy quantum locale. The locus of the locale is
the reference origin and provides the mean position of the particle
on its time-line in space-time. Now when dealing with correlations
between multiple time-lines we perhaps could perform all those
instantaneous correlations between curvatures if we knew the individual
vectors. Unfortunately we don't, and according to quantum theory, we
can't. Remember also, I view the mathematical representations as being
to some degree arbitrary. All is not lost however. The length of an
instantaneous vector is just the deviation from the locus. A statisitcal
analysis gives us just enough information to say something about the
net correlation or interaction between two particles.
To get this length of a complex number we simply multiply it
by its complex conjugate. This result equals the length squared. Valla!
We have squared the probability amplitude and arrived at a concrete
probability. Mathematically speaking, if we had not used complex numbers,
or vectors, we would be faced with taking the square root of a negative
number which is computationally impossible. Thank god for symbolics,
however, which allows us to postpone the inevitable and bypass such
horrors. Then when the process is continued on to calculating the
time-line curvatures, we have real differentiable functions to compute
the correlations.
In my mind I have hereby given a tentative interpretation of what
constitutes a wave-collapse. Its nothing mysterious. I think Feynman
would be proud of me.
DATE: 8/2/97 12:16 AM
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