Invariant Thinking
James B. Glattfelder
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"...
James B. Glattfelder
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...
Arkadiusz Jadczyk
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
--
Uncle Al
[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!
James B. Glattfelder
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"...