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From: Archimedes Plutonium on 27 Mar 2010 16:10


Archimedes Plutonium wrote:
> --- quoting Wikipedia on symmetry breaking ---
> Symmetry breaking in physics describes a phenomenon where
> (infinitesimally) small fluctuations acting on a system crossing a
> critical point decide a system's fate, by determining which branch of
> a bifurcation is taken. For an outside observer unaware of the
> fluctuations (the "noise"), the choice will appear arbitrary. This
> process is called symmetry "breaking", because such transitions
> usually bring the system from a disorderly state into one of two more
> ordered, less probable states. Since disorder is more symmetric in the
> sense that small variations to it don't change its overall appearance,
> the symmetry gets "broken".
>
> Symmetry breaking is supposed to play a major role in pattern
> formation.
>
> In particular, we can distinguish between:
>
> An explicit symmetry breaking happens when the laws describing a
> system are themselves not invariant under the symmetry in question.
>
> Spontaneous symmetry breaking describes the case where the laws are
> invariant but it appears the system isn't because the background of
> the system, its vacuum, is noninvariant. Such a symmetry breaking is
> parametrized by an order parameter. A special case of this type of
> symmetry breaking is dynamical symmetry breaking.
>
> In 1972, Nobel laureate P.W.Anderson used the idea of Symmetry
> breaking to show some of the drawbacks of Reductionism in his paper
> titled More is different in Science[1].
>
> --- end quoting Wikipedia ---
>
> The above is a liberal quote, and I did it because so few places in
> books, cares to define
> symmetry breaking. Atkins talks about "symmetry" in some length in his
> book QUANTA
> but no mention of symmetry-breaking. And Wikipedia may have hit the
> date of first analysis
> of symmetry breaking starting with Anderson.
>
> I am going to try to make a start into a precision definition of
> symmetry-breaking, better than
> that of Wikipedia. But I am going to depart before 15 April.
>
> What I am thinking is that the finest definition of symmetry-breaking
> is to spot the finest
> example in mathematics and then call that example the definition of
> symmetry-breaking.
> Unlike Wikipedia's definition which is a culling together of piecemeal
> traits or aspects
> rolled up into some ball for a concept. This is like giving some
> traits of a animal species,
> such as (a) have a backbone (b) 4 chambered heart (c) has a tail
> without ever being specific
> as to what species it is.
>
> So here, what if I were to define Symmetry Breaking as to the best
> example I can find in
> mathematics. It would be the Euclidean geometry model = Elliptic geom
> model unioned
> with the Hyperbolic geometry model. So that if I were to break the
> symmetry of Eucl model
> such as a rectangular solid leaving a sphere model of elliptic geom
> and what is not the sphere
> model is the remaining model as hyperbolic geometry.
>
> Now the question is, if my definition of Symmetry-Breaking is
> equivalent to the Wikipedia
> definition? Well in some features we can see some compatibility.
> Because we have a bifurcation of Eucl into two separate elliptic and
> hyperbolic models.
>
> So in simple terms, symmetry-breaking would simply mean that we split
> apart one
> independent system into two separate independent systems.
>
> And this would make alot more sense for physics in something like
> duality. In the Double
> Slit Experiment, we are always worried that the electron is either a
> wave or particle, but
> we seem to be disturbed to think it is both simultaneously. And so
> when we check the
> screen we see particles coming, but when we check the screen we see
> wave interference.
> Likewise, the electron is Euclidean and when it reaches the screen it
> is particle as elliptic
> and is a wave as hyperbolic. So our confusion is not understanding
> that existence must be
> two things simultaneously. The electron must be both particle and
> wave. The electron
> must be both elliptic and hyperbolic geometry wrapped up into one.
>
> I think my definition is equivalent to the Wikipedia. But that mine
> would be more precise,
> since it gives a mathematical truth, not a cobbled together ball of
> piecemeal traits one happens to see in experiments.
>
> So Symmetry Breaking is simply the recognition of pair dualities in
> quantum mechanics.

Do I have any confirmation of Symmetry-Breaking as defined as this:

Eucl geom = Elliptic geom unioned Hyperbolic geom

Is there any sort of confirmation in Physics that such a formula is
Symmetry
Breaking in Quantum Mechanics?

I suspect so. Notice that the formula is truly unlike any other math
formula because
it has a "union" rather than it has addition, for which no other math
formula has
a union rather than the operator addition.

But if you look in Physics there is a formula that sticks out like a
sore thumb which has
a set-theory-operator union rather than that of math operators of add,
subtract, multiply
and divide. I am speaking of the Uncertainty Principle in its
"uncertainty operator"

del_x * del_P > h

where x is position, P is momentum, and h is Planck's constant, and
where del stands
for Uncertainty operator.


So, here, all I have to do is show that

Eucl = Ellipt unioned Hyperb

is the same as

del_x * del_P > h

The literal translation is easier than the mathematical manipulations.
Literally, it
means that the Uncertainty Principle gets to a "smallness in size"
that where
the physics can no longer tell whether the geometry is Elliptic,
Hyperbolic or
Euclidean. The Uncertainty Principle for Mathematics, means that at
some small
size we cannot convert Euclidean geometry into its component
NonEuclidean
geometries.

In this book, I covered the idea of the limits of concavity of
triangles in that a
equilateral triangle of 60 degree arc longitude is the upper limit of
reversing the
concavity in forming the hyperbolic triangle.

Well, the Uncertainty Principle in Math means that the conversion of
Ellipt and
Hyperbolic geometries into forming Euclidean is also limited or
restricted by
Planck's constant, and where reversing concavity is one limiting
factor amoungst
many limiting factors.

So I would have to show, mathematically that these two equations are
the same

A = B union C
B' * C' = A'

First I would have to show that set theory "union" is turned into a
set-theory
multiplication. Then I would have to show that position/momentum are
geometries.
But if I simply showed that the above two equations are "in general,
the same"
will be sufficient proof.

P.S. the fact that Planck's constant comes in so many different
"flavors and stripes"
is a sign or indicator that the above is true, because the many
flavors of Planck's
constant is a sign that it is Euclidean geometry with Elliptic and
Hyperbolic bursting
out the seams of trying to come alive.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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