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Jul 8, 2002, 10:39:25 PM7/8/02

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[The nature of this post is somewhat philosophical, so it probably

will get filtered out by the moderators. Although, one could argue

that science is slowly getting very close to epistemology and ontology

(e.g. what is the nature space and time and such - or see the book:

"Physics Meets Philosophy at the Planck Scale"). And obviously our

philosophical make-up will have an impact on the way we mathematically

try and grapple with these border-line issues.]

will get filtered out by the moderators. Although, one could argue

that science is slowly getting very close to epistemology and ontology

(e.g. what is the nature space and time and such - or see the book:

"Physics Meets Philosophy at the Planck Scale"). And obviously our

philosophical make-up will have an impact on the way we mathematically

try and grapple with these border-line issues.]

[Moderator's note: This post is OK, but respondents are urged to

confine themselves to physics aspects of the question. -MM]

Arguably the most fruitful principle in physics has been the notion of

symmetry. Covariance and gauge invariance - two simply stated symmetry

conditions - are at the heart of GR and the SM. This is not only

aesthetically pleasing it also illustrates a basic fact: in coding

reality into a formal system, we should only allow the most minimal

reference to be made to this formal system. I.e. reality likes to be

translated into a language that doesn't explicitly depend on its own

peculiarities (coordinates, number bases, units, ...). This is a

pretty obvious idea and allows for physical laws to be universal.

But what happens if we take this idea to the logical extreme? Will the

ultimate theory of reality demand: I will only allow myself to be

coded into a formal framework that makes *no* reference to itself

whatsoever. Obviously a mind twister. But the question remains: what

is the ultimate symmetry idea? Or: what is the ultimate invariant?

(Does this imply "invariance" even with respect to our thinking?) How

do we construct a system that supports itself out of itself, without

relying on anything external? Can such a magical feat be performed by

our thinking?

The problems concerning quantum gravity could be seen to arise within

the context of background dependence and in constructing

background-free theories one is trying just such an ultimate

"invariance trick"...

Jul 11, 2002, 10:20:55 PM7/11/02

to

j_...@gmx.net (James B. Glattfelder) wrote in message news:<ed220336.02070...@posting.google.com>...

I mean isn't this what we are trying to achieve, to reduce the ballast

of our mathematical formalism to the minimum, so it can mirror a

maximum of reality?

Take category theory. Reduce mathematics to two abstractions: objects

(stuff) and morphisms (relations of stuff) *without* impairing its

power (e.g. topos theory). I love these words (from John's home page):

> We can also express the principal of general covariance and the principal of

> gauge-invariance most precisely by saying that observables are functorial.

> So physicists should regard functoriality as mathematical for "able to be defined

> without reference to a particular choice of coordinate system."

I.e. another "invariance trick".

*****

And what is the holographic principle telling us about reality?

String/M-theory: "[the holographic principle is] the assertion that

the number of possible states of a region of space is the same as that

of a system of binary degrees of freedom distributed on the boundary

of the region" (Susskind). A bulk theory of gravity is equivalent to a

non-gravitating theory on the boundary, as in AdS/CFT.

However, quantum gravity wants to go deeper: "The conclusion is that

the [weak] holographic principle is not a relationship between two

independent sets of concepts: bulk theories and measures of geometry

vrs boundary theories and measures of information. Instead,it is the

assertion that in a fundamental theory the set of concepts must be

completely reduced to the second" (Smolin). And: "The familiar picture

of bulk space-times with fields and geometry must emerge in the

semiclassical limit, but these concepts can play no role in the

fundamental theory." Again a distinction between reality, our

fundamental description thereof and mathematics.

Finally: "This is what we found about Nature's book keeping system:

the data can be written onto a surface, and the pen with which the

data are written has a finite size" (t'Hooft).

Apart from its implications, this quantum information problem seems

again to spell out the issues of symmetry/invariance: area and volume

are not distinguishable concepts on a fundamental level. One is

reminded of Gauss' theorem in vector analysis: all the information of

the source (infinitesimal point) of a vector field is coded into any

surface enclosing this source. So maybe the holographic principle is

natures way of shielding itself from mathematical idealizations...

*****

And while we are in these outlandish regions of abstract thought and

fundamental reality, what about time? Obviously something very

annoying in physics, because it appears so real to our senses, yet is

so poorly implemented in theories. And just as we are getting used to

mental pictures of higher dimensions, why not go for the ultimate

nightmare: more than one time dimensions.

To start, these ideas have (at least in two dimensions) been

implemented in 12-dimensional F-theory and (implicitly) in the

transactional interpretation of quantum mechanics (Wheeler-Feynman

Absorber theory). While the first explicitly introduces two degrees of

freedom, the second allows for a propagation from the the future to

the past (advanced wave solutions), i.e. invoking the notion of two

time directions one one time axis.

One can always argue that, of course, we psychologically only perceive

one time dimension directed into the future, but maybe our

mathematical description of reality needs this additional degrees of

freedom to work. Einstein spent the last years of his life trying to

unite electrodynamics with gravity, without real success (although,

admittedly, I don't know what teleparallelism is about). Kaluza, with

not much fuss, pulls this trick by allowing for more degrees of

freedom to his theory: (1,4) space-time dimensions (hence starting the

higher dimensions thing).

So why not promote t to a tensor, or at least a 3-vector;-) You can

always draw simple diagrams with two time axes and one space axis and

think about going from a worldline to a "temporal" world-sheet. And if

you really feel like it you can try all those QM puzzles with this new

idea (double slit: same particle located at two different points in

time, or whatever).

I think I just got a headache...

Jul 13, 2002, 11:16:23 PM7/13/02

to

On Tue, 9 Jul 2002 02:39:25 GMT, j_...@gmx.net (James B. Glattfelder)

wrote:

>But what happens if we take this idea to the logical extreme? Will the

>ultimate theory of reality demand: I will only allow myself to be

>coded into a formal framework that makes *no* reference to itself

>whatsoever.

I think one should avoid all extremes. The devil is always in the

details, and for anything good to function - the details must not be

too simple.

Bohm, for instance, in his early book "Causality and Chance in Modern

Physics" stressed that there are levels of description, and that these

levels are not always compatible and/or reducible to each other.

Every bright idea in ophysics seems to have its limits. To understand

why is it so, we would have to discuss the question: "what are ideas

and where do they come from?" - but this is not the right place for

this kind of inquiry (at least not yet). Therefore let me just point it

out that there is something important missing in the way you stated

your question, and this something is the "quantum factor" or, better,

the principle of "partnership" that J.A. Wheeler described visually

with his U picture - where on one end is the "creation" and on the other

end is the "observation" that makes the "virtual creation" real.

It seems that this kind of duality in the laws of physics is necessary

for the universe to "work".

You mntioned symmetry principles: coordinate invariance and gauge

invariance. They are useful principles but, perhaps, only to some

degree. Perfect symmetry need not be the best solution. Physicists

get much more from reseraching symmetry breaking mechanisms.

Not always they are able to explain the detailed mechanisms - so

they created the concept of "spontaneous symmetry breaking".

Perfect symmetry (diffemorphism invariance and gauge invariance)

hold, perhaps, in an ideal world. In our "sample" these symmetries

may well be broken.

To answer your question, to make a step in this direction, we will

probably need to know more about complexity and how physics

attempts to analyze (perhaps infinitely) complex universe in relatively

simple terms, and what is the price that must be paid for it.

Gell-Mann and Hartle introduced the term IGUS-es - information gathering

and utilizing systems. Perhaps the physics of XXI century will have to

pay more attention to these ideas of Wheeler, Gell-Mann and Hartle,

Wolfram and others.

In our paper (Blanchard and Jadczyk, Ann. der Phys. 4 (1995) 583-599)

http://www.cassiopaea.org/quantum_future/jadpub.htm#blaja95a

we wrote

"So, how can we manipulate states without being able to manipulate

Hamiltonians? We can only guess what could be the answer of other

interpretations of Quantum Theory.

Our answer is: we have some freedom in manipulating

$C$ and $V$. We can not manipulate dynamics, but binamics is open.

It is through $V$ and $C$ that we can feedback the processed information

and knowledge - thus our approach seems to leave comfortable space

for IGUS-es. In other words, although we can exercise little if any

influence on the continuous, deterministic evolution\footnote{ Probably

the influence through the damping operators $\Lambda_\alpha$ is

negligible in normal circumstances}, we may have partial

freedom of intervening, through $C$ and $V$, at bifurcation points, when

die tossing takes place. It may be also remarked that the fact that more

information can be used than is contained in master equation of standard

quantum theory, may have not only engineering but also biological

significance. "

The concept of "information" will be, perhaps, important in what you

call "ultimate theory of reality".

ark

--

Arkadiusz Jadczyk

http://www.cassiopaea.org/quantum_future/homepage.htm

--

Jul 14, 2002, 11:50:14 AM7/14/02

to

"James B. Glattfelder" wrote:

[snip]

> Arguably the most fruitful principle in physics has been the notion of

> symmetry. Covariance and gauge invariance - two simply stated symmetry

> conditions - are at the heart of GR and the SM. This is not only

> aesthetically pleasing it also illustrates a basic fact: in coding

> reality into a formal system, we should only allow the most minimal

> reference to be made to this formal system. I.e. reality likes to be

> translated into a language that doesn't explicitly depend on its own

> peculiarities (coordinates, number bases, units, ...). This is a

> pretty obvious idea and allows for physical laws to be universal.

[snip]

> Arguably the most fruitful principle in physics has been the notion of

> symmetry. Covariance and gauge invariance - two simply stated symmetry

> conditions - are at the heart of GR and the SM. This is not only

> aesthetically pleasing it also illustrates a basic fact: in coding

> reality into a formal system, we should only allow the most minimal

> reference to be made to this formal system. I.e. reality likes to be

> translated into a language that doesn't explicitly depend on its own

> peculiarities (coordinates, number bases, units, ...). This is a

> pretty obvious idea and allows for physical laws to be universal.

The heart of physics is Noether's theorem: every symmetry is

associated with a conserved quantitiy and the reverse. A physical

system with a Lagrangian invariant with respect to the symmetry

transformations of a Lie group has, in the case of a group with a

finite (or countably infinite) number of independent infinitesimal

generators, a conservation law for each such generator, and certain

"dependencies" in the case of a larger infinite number of generators

(General Relativity and the Bianchi identities). The reverse is true.

A non-geometric fundamental theory of matter is unimaginable given our

perception of symmetries and their consequences.

Relativity removes all background coordinates. It models continuous

spacetime, going beyond conformal symmetry (scale independence) to

symmetry under all smooth coordinate transformations - general

covariance (the stress-energy tensor embodying local energy and

momentum) - resisting quantization. However, if reality is quantized

then there is an intrinsic scale to things (e.g., the Planck length,

1.616x10^(-26) nm). This conflict remains unresolved.

> But what happens if we take this idea to the logical extreme? Will the

> ultimate theory of reality demand: I will only allow myself to be

> coded into a formal framework that makes *no* reference to itself

> whatsoever. Obviously a mind twister. But the question remains: what

> is the ultimate symmetry idea? Or: what is the ultimate invariant?

> (Does this imply "invariance" even with respect to our thinking?) How

> do we construct a system that supports itself out of itself, without

> relying on anything external? Can such a magical feat be performed by

> our thinking?

>

> The problems concerning quantum gravity could be seen to arise within

> the context of background dependence and in constructing

> background-free theories one is trying just such an ultimate

> "invariance trick"...

John Baez is our doyen of fundamental mathematization of physics.

Neither he nor any of his ilk can bell the cat. This is not for lack

of ability, but for lack of constraint. Theory gropes hopelessly and

abundantly without observation to guide it to physical solutions.

Contemporary physics is experiment driven by theory. It is incredibly

hobbled by being forced to look "in the right places." Serendipity is

looking for a needle in a haystack and finding the farmer's daughter.

Relativity and quantization are contradictory and incompatible. One

or both must be incomplete in the manner that Newton fell to both.

Empirical anomalies must exist! The link below explores an

unambiguous test in existing apparatus that addresses both the

validity of a General Relativity founding postulate and the concept of

"point phenomenon" in physics as a whole. Somebody should look.

One way or another we will then move forward.

--

Uncle Al

http://www.mazepath.com/uncleal/eotvos.htm

(Toxic URL! Unsafe for children and most mammals)

"Quis custodiet ipsos custodes?" The Net!

Aug 5, 2002, 10:07:31 PM8/5/02

to

j_...@gmx.net (James B. Glattfelder) wrote in message news:<ed220336.02070...@posting.google.com>...

This thread started with going on about the question of the relation

of our thinking with respect to reality. E.g. the usefulness of

mathematical frameworks that exhibit a minimal amount of referencing

(i.e. maximally invariant) to express universal features.

But is this all? Is the feature of analytical coding of reality the

only formal possibility to grapple with reality? Until recently I

would have suggested a "yes", but something made me rethink.

Although I personally thought of things such as object-oriented

structures as being a very powerful approach to problem-solving, I

never gave it any thought beyond being a pragmatic engineering tool.

(Here one could already emphasize that OO programming is an

implementation of systems theoretic thinking and that computer

languages in general hold linguistic issues.) But I never conceived of

computation per se as a framework to formally deal with the workings

of reality. However, looking at the contents of "A New Kind of

Science" really made me wonder. And not just about an algorithmic

approach to complex phenomenon (although Wolfram also claims to shed

light on issues of fundamental physics as well, next to much more).

It appears as though the whole industry of mathematics can be seen to

spring from this foundation of computation. Assertions like "category

theory can be viewed as a formalization of operations on abstract data

types in computer languages" (p. 1154) are only the tip of the

iceberg: "This emphasis on theorems has also led to a focus on

equations that statically state facts rather than on rules that define

actions, as in most of the systems in this book. But despite all these

issues, many mathematicians implicitly tend to assume that somehow

mathematics as it is practiced is universal, and that any possible

abstract system will be covered by some area of mathematics or

another. The results of this book, however, make it quite clear that

this is not the case, and that in fact traditional mathematics has

reached only a tiny fraction of all the kinds of abstract systems that

can in principle be studied" (p. 860).

If verified this would really shift a paradigm I held, that

mathematical models are the best probe of reality. I liked the image

of a map labeled "mathematics" with an arrow pointing to some region

declaring "you are here", meaning that our reality would be described

by this kind of mathematics. So perhaps the map should be called

"computational possibilities"...

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