What Inspired Keats' Vision on Darien's Peak?

[marcus]

Today is Saturday and it is an unwritten law that on Saturdays
one is permitted to post sonnets. Here are the last 6 lines
of one by Keats:
...
Then felt I like some watcher of the skies
When a new planet swims into his ken;
Or like stout Cortez when with eagle eyes
He stared at the Pacific—and all his men
Looked at each other with a wild surmise—
Silent, upon a peak in Darien.


frpm John Keats’ sonnet “On First Looking into Chapman’s Homer” (1816)

 

[marcus]

Then felt I like some watcher of the skies
When a new planet swims into his ken;

what reminded me of these two lines was seeing how the quantum state of space is written as a knot in a theory of Quantum Gravity currently in the works, here's a quote to illustrate:

-------
The geometry of the universe is $$|s\rangle = |K,c \rangle$$
where K is a knot and c is a quantum number. Or rather, that is how the pure quantum states are written.

All the things defined in or on space are defined on this knot K
by further colorings of the links and nodes.
like particles that are constituents of matter, like stars etc.

The actual geometry is a quantum cloud of these pure states, a particular
pure state is one of a countable orthogonal basis of the hilbert space of quantum gravity (gravity = geometry, quantum gravity = quantum geometry) and a generic quantum state, a vector in that hilbert space, is a mixture of pure states----a linear combination of the basis elements.

The knot is abstract, not embedded in some prior space. It is space...
------end quote---

this is an exerpt from
https://www.physicsforums.com/showthread.php?p=171132#post171132
where I was trying to paraphrase a basic definition in the Fairbairn/Rovelli
paper:
http://arxiv.org/gr-qc/0403047

 

[marcus]

The knot is abstract, not embedded in some prior space. It is space...

and oddly enough one can calculate areas and volumes from the knot.

as I recall the Greeks were very interested in areas and volumes, Archimedes asked to be remembered with a stone carved to show a sphere inside a cylinder---he had discovered something about the ratio of their volumes and the ratio of their surface areas---was that the beginning of calculus? And now we have a way, given a quantum geometrical state of space, given a knot, to find the surface area and volume of a region described in that space: sum the (colorings of the) nodes contained in the region, sum the (colorings of the) links that penetrate the surface.

The knot is space...or a pure quantum state of the geometry of space...and we are back to basics suddenly: considering area and volume just as someone might have years ago in Samos or Syracuse.

One of the interesting little theorems in the F/R paper is that extending the diffeomorphisms shouldn't make any difference. One should get essentially the same area and volume operators.

In view of how very reassuring math symbols are, especially self-adjoint operators on a hilbert space, let's write S (following the notation used in the paper) for a surface and R for a region. The surface can have a finite number of conical singularities. Let's write $$A_S$$ for the operator which is the area of the surface S and $$V_R$$ for the operator which is the volume of the region R.

The two operators are defined on the abstract knot states forming the basis for the kinematical hilbert space. Where is the manifold? It left a long time ago.

The theorem says that if you bring the manifold back just to run a check, and embed the knot in the manifold, just to see if everything is all right, then you get the earlier area and volume. And the fact that extended diffeomorphisms are being used, instead of plain kind, doesn't make any difference.

That would be proposition 3 on page 12.

"An extended diffeomorphism may generate singular points in the surface or in the spin network, but does not affect the topological relation between a surface and the spin network, and the area depends only on this relation..."
and something similar for volume.

well that's a partial paraphrase or discussion of some stuff on pages 11 and 12 of the paper.

 

[marcus]

the paper reconnects with the book at draft pages 104 and 199.
On page 104 there is a form of the classical hamiltonian of General Relativity (equation 4.16) which is carried over to the development of Quantum Gravity in Chapter 7 "Dynamics and Matter" section 7.1 "Hamitonian operator" and appears as equation 7.1.

This form of the Hamiltonian is convenient because it depends only on holonomies and on the volume operator. In the quantum version of the theory, where the state of space is |K, c>, holonomy information is explicit in the coloring of the knot's links and a region's volume depends on which of the knot's nodes it contains. None of these things are changed by extending the diffeomorphisms.

So we have the observation on page 12 of the paper, at the end of section 4.2 "Volume and hamiltonian":

"...the hamiltonian can be defined entirely in terms of the volume operator and holonomy operators, and is not affected by the modifications of the theory considered here."

the upshot is that chunkymorphisms make the Hilbert space separable (a big reduction in dimensionality) without changing the hamiltonian or other basics like the (discrete spectrum) area and volume operators.


For the latest on the melodrama surrounding the Loop hamiltonian see the elusive survey paper by Astekar and Lewandowski "Background Independent Quantum Gravity: a Progress Report" (if you can find a copy*) or, lacking this,
see Rovelli pages 201-205, especially the section 7.1.3 called "Variants". There are a lot of possible variations of the quantum hamiltonian operator to be considered and uncertainty as to which, if any, of those being considered will yield the right classical limit. There is a further section on variants of the theory on pages 211 and 212.

*[edit: It is noted with sincere gratitude that the A&L paper posted at arxiv on April 5. The ink of impatience was hardly dry.]

 

[marcus]

$$H = \int N Tr(F \wedge \{V,A\})$$

this is Rovelli's equation (7.1) on page 199
"The form of the hamiltonian...most convenient for the quantum theory...
The reason is that we have already defined the quantum operator V, and the operators F and A can be defined as limits of holonomy operators of small path, while the classical Poisson bracket can readily be realized in the quantum theory as a quantum commutator..."
What follows is a discussion of regularizing the hamiltonian which ends around the bottom of page 202 with a double exclamation mark ! (I believe this is the only instance of a double-shriek in the whole book, and Rovelli's editors at Cambridge may look upon it with distaste, given the British penchant for understatement, although it serves efficiently to clarify the meaning.)

 

[marcus]

To recapitulate:

The geometry of the universe is $$|s\rangle = |K,c \rangle$$
where K is a knot and c is a quantum number. Or rather, that is how the pure quantum states are written.

All the things defined in or on space are defined on this knot K
by further colorings of the links and nodes.
like particles that are constituents of matter, like stars etc.

The actual geometry is a quantum cloud of these pure states, a particular
pure state is one of a countable orthogonal basis of the hilbert space of quantum gravity (gravity = geometry, quantum gravity = quantum geometry) and a generic quantum state, a vector in that hilbert space, is a mixture of pure states----a linear combination of the basis elements.

The knot is abstract, not embedded in some prior space. It is space...

https://www.physicsforums.com/showthread.php?p=171132#post171132

At this point the pure quantum states of space become countable and combinatorial.

What does combinatorial mean? Well a graph, consisting of a finite set of nodes and a list of which nodes are linked, is a combinatorial object. And a good example---a lot of other types of structure can be represented by graphs. Another way to think of a graph is a finite set of points with a matrix of zeros and ones called the "adjacency matrix" which tells which points are joined to which others. And you can get more elaborate graphs by labeling the nodes and links. Spin networks are examples of colored or labeled graphs. As long as the colorings are from a discrete set of labels the resulting objects are combinatorial.

A combinatorial object, consisting of a finite set or sets, and some relations, some lists of n-tuples, or an integer-valued matrix or two, doesn't need physical room to live in, it can live in some bytes of computer-memory.

If you just use diffeomorphisms, in quantum gravity, then the spatial states don't boil down to a countable list. Two networks which are combinatorially the same can be embedded in a test manifold so that at the same node their links meet at different angles, and no diffeo can change one network into the other. Where 4 or more links meet, the node is stiff in that sense.

Figure 1 in the Fairbairn/Rovelli paper shows a case of this---two embedded graphs that look as if they should be the same but are not diffeo-equivalent
because their links meet at different angles.

It's a revealing picture and shows pretty well what is unsatisfactory about diffeos and why they upgraded to chunkies ("extended diffeos")

http://arxiv.org/gr-qc/0403047

there is more to the story for sure, but a big part of it is getting this state of the geometry of the universe which is combinatorial, because it is a knot together with a discrete quantum number describing the coloring of the knot.
a knot can be described in a purely combinatorial way

so when I go downstairs and out into the garden to look at the goldfish in the pond I will be changing the adjacency matrix that says which points in me are adjacent to which other points----points where particles live or on which fields are defined, particles of air, water or goldfish.

 

[marcus]

This exerpt from page 5 clarifies the idea of a knot state. F&R say that in going from from the spin network state $$|S\rangle$$
to the knot state
"...we preserve the entire information in $$|S\rangle$$
except for it’s location [in the test manifold] Σ. This is the quantum analog to the fact that physically distinguishable solutions of the classical Einstein equations are not fields, but equivalence classes of fields under diffeomorphisms.

It reflects the core of the conceptual revolution of general relativity: spatial localization concerns only the relative location of the dynamical fields, and not their location in a background space..."

the combinatorial object, graph, network, knot, is a device for describing
the relative location of dynamical fields...without resort to a background space or precommitment to some standard type of geometry.

the process for getting from spin network states to knot states is to equate any two spin networks where one can be morphed into the other by a diffeo (or even more effectively by a chunky)
because you want to get rid of all traces or memory of the location in the test manifold, which is arbitrary. all that should be left is abstract spatial relationships---no trace should remain of the test manifold in which the spin networks were once embedded.

Fairbairn and Rovelli's point about chunkies is that mere unextended diffeos fail to erase all imprint of the geometry in which the network was originally defined.

 

[marcus]

Today a new poster, Stevo, posted some evocative thoughts about background independence, that kind of re-energized my interest in that.
It's a major issue.

We live in a time in science-history when BI is coming into its own, in a certain sense.

A lot of what interests people nowadays and what you hear about----accelerated expansion, gravitational lensing, black holes, inflation scenario, cosmological constant, dark energy, bumps in the CMB, certain aspects of gammarayburst astronomy----arise from a BI theory or in conjunction with it.

The Friedmann model of expanding U, and black hole models, and models of stellar collapse---these arise from the BI theory with its dynamic un-fixed geometry and oftentimes extreme and changing curvature.

Friedmann and Schwarzschild were right around 1920 on the heels of Einstein's 1915 breakthrough. Whatever has to do with expanding universe or black holes arose from that.

It seems significant to me that Einstein was hung up for three years about BI. He had general relativity within reach but spent three years (1912-1915) wrestling with "the meaning of the coordinates" because he couldn't accept the fundamental BI message. Then in 1915 he decided to go with it, and published.

For me I guess, Background Independence is the real name of the theory, not Gen. Rel., or maybe Background Independence is the bigger theory of which GR is just a part, the only part we currently see and can check out.

--------------
take any specific result and you can sometimes ad hoc modify some other non-BI theory to copy that result or produce something suggestively like it.
so there are alternative ways of modeling an expanding universe or a black hole which don't use GR, but the ideas are natural to GR---at home in it.

Any model which at some stage or location involves extremes of curvature
(and its evolution) is likely to be asking for BI treatment because the alternative,
perturbation theory, works through approximation assuming small perturbations from a fixed, typically flat, background.

In particle theory there is an habitual focus on approximating the universe in small flat patches of it, instead of on the whole picture. So engrained and so pervasive is the habitual mindset that I rarely hear it even mentioned. As to how all the local flat approximations fit together? wave your hands.
-------------------

Nowadays people seem to be interested in the shape of the universe, or its beginnings or its longrange future. It is just an aspect of this historical period that there is so much curiosity and new information about these things.
Partly it must be new ground and satellite observatory technogy, Keck, COBE, Hubble, WMAP. Maybe there are other reasons. Zeitgeist.
These are global issues----shape is global.
Global models are BI.

or anyway the ones we have that work are BI
-------------------

heres an example of Zeitgeist. the team in the Pyrennees led by that frenchwoman just found a z=10 galaxy
redshift 10, you must be kidding, you heard right redshift 10.
how could they see it?
gravitational lensing, a massive cluster of galaxies is between us and it
and it acts as a magnifying glass

in one sense an accidental aspect of some recent news item
in another sense part of an emerging set of ideas and awareness

that bumpy funhouse optics of the gravitational field is another part of the shape of the universe

part of a gradual accumulation of detail affecting how people think and
what physical models appeal to them

 

[wolram]

heres an example of Zeitgeist. the team in the Pyrennees led by that frenchwoman just found a z=10 galaxy
--------------------------------------------------------------------------
i thought a "galaxy" couldn't form that early, excuse interruption.

 

[marcus]

heres an example of Zeitgeist. the team in the Pyrennees led by that frenchwoman just found a z=10 galaxy
--------------------------------------------------------------------------
i thought a "galaxy" couldn't form that early, excuse interruption.

I did too.
Let me go and check. Right now I can't remember her name or even
be sure that it was z = 10.

 

[marcus]

Yeah, I got this from the Astronomy "Reference Shelf" stickythread
Her name is Roser Pello
The galaxy would have had to form very quickly, in only 500 or 600 million years. Hard for me to picture, but I choose to defer to expert opinion on this.
Here's the post from Astro Reference thread (to which you contributed the last half dozen links or so IIRC)
--------

Meteor pointed us to the mid-pyrenees observatory
finding a z = 10 galaxy
(and GedankenDonuts gave a link too) then Nereid came up with
the scientific article co-authored by Roser Pello

http://www.edpsciences.org/papers/aa/pdf/press-releases/aaga201.pdf

here is a picture of Roser, she looks pleased to have found the galaxy
http://webast.ast.obs-mip.fr/people/roser/

z=10 means that the universe has expanded 11-fold since
the light issued from that galaxy


so while the light was traveling to get to us, distances between things became eleven times larger.
that means it was a long time that the light was traveling, estimated 13.2 billion years

for a calculator to calculate stuff like that try
Siobahn Morgan's online cosmology calculator

http://www.earth.uni.edu/~morgan/ajjar/Cosmology/cosmos.html

homepage for Siobahn also with photo
http://www.earth.uni.edu/smm.html

putting in the usual 71 for H, 0.73 for Lambda and 0.27 for matter density, we get that the object Roser and the others found is currently 31.5 billion light years from us and receding at a speed of 2.3 times the speed of light

 

[wolram]

i would never contradict an expert, but this is very close to birth of U,
and must be on the precipice of age change.

 

[marcus]

i would never contradict an expert...

never say never
why don't you write her an email asking how it can be
that a galaxy forms so early as 13.2 billion yrs ago
she looks like a nice person in her picture
she might reply referring you to some article about this that we
could read

 

[wolram]

i have sent ROSA an E MAIL as you suggested MARCUS, i hope she
replies, as this topic is important to cosmology.

 

[wolram]

http://www.astro.ucla.edu/~wright/cosmolog.htm

1 Mar 2004 - Pello et al. have found a galaxy much further away from us than any previously known. The evidence comes from a single line observed in the infrared which implies a redshift of z = 10. The source is seen magnified by a cluster of galaxies, Abell 1935, acting as a gravitational lens, and the source location is where sources with 9 < z < 11 should be very highly magnified. The colors of the source are also very consistent with z = 10. The technical paper and the press release both give pictures and spectra of this object. My Cosmology Calculator gives for z = 10 and the WMAP cosmic parameters (Ho=71, OmegaM=0.27 in a flat Universe) an age of the Universe of 0.48 Gyr at the time the light we see was emitted, a light travel time of 13.18 Gyr, and a current distance of 31.5 billion light years. This distance is much greater than the speed of light times the light travel time because the Universe has expanded by factors between 1 and 1+z=11 since the light did its traveling.

 

[marcus]

A recent Baez paper sheds some light on quantum geometry from a different angle:

"...propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime..."
quant-ph/0404040

In GR gravity is geometry
meaning that the gravitational field is not defined on spacetime
rather it is spacetime

the flat space (surnamed Minkowski) is merely one special case of the gravitational field, one where there is no matter in the universe. To the extent that quantum field theory and its wayward children are defineable chielfly on Minkowski space they are approximations anchored to one special case of the gravitational field.

the field is the background
and in a theory of the field, the field had better be independent or free to vary
if the gravitational field is to be a fully dynamic entity then
the background is necessarily dynamic, not fixed
(in algebra you don't start by saying what number X is)
quantum states of the gravitational field
are quantum states of space, i.e. the geometry of the universe
there is no physically meaningful pre-established background

there seem to be strong reasons emerging why one can't expect to have a satisfactory quantum mechanics unless it is background independent, that is to say until it has a theory of spacetime, until it is, in effect, QG.

also one cannot have a satisfactory theory of spacetime (i.e. a theory of gravity/geometry) until it is quantum

we have been seeing how QG (a la bojowald) removes singularities in classical GR like the bigbang singularity. there is a practical need for quantizing GR just to get rid of obvious glitches. that is one way to see it but it must go deeper.

two main 20th century branches of theory, neither will be right until they get them to fit together

well that is one view.

so these combinatorial things, these knots (as in Fairbairn/Rovelli paper)
will they turn out to serve as quantum states of geometry, of space, of the background on which other stuff is defined?
and so, when quantum field theory becomes a theory of spacetime (a la Baez) and therefore a background independent theory, will it be a theory of knots?

 

[wolram]

i am ever fascinated by the ingenuity of mathematics to extricate
modelers of the universe from obscurity, it seems that maths is
the only savorer to these modelers.

 

[marcus]

i am ever fascinated by the ingenuity of mathematics to extricate
modelers of the universe from obscurity,..

it wouldn't be the first time maths reached the chestnuts out of the fire for some line of inquiry

I agree it is amazing how this happens now and then:
as when Georg Friedrich Bernhard Riemann (1826-1866)
provided the fabric Einstein needed in 1915 to make general relativity
and when David Hilbert (born 1862) thought up Hilbert space,
the basic gadget later used by others to make quantum theories.

the stumbling lurch of progress never quite looks planned
or even dignified
whoever called it a "march" of progress
to me it seems more like the first time I tried to ski
and only reached the bottom of the slope by sheer luck
if humans insisted on dignity they would never get anywhere

well now Baez happens to be asking what is so good about
Hilbertspace that quantum mechanics has been based on it for over 75 years
and plenty of other people have, I guess, wondered about that
and he has a possible answer (that it gives you a star-category
unexpectedly analogous to the basic structure in GR---a suggestive sameness, a hidden similarity, between QM and GR the two theories nobody can put together) he has a possible answer, called star-category, and
I for one will give it a look. You can't let yourself be put off by the
strangeness of a new mathematical idea---and this is not all that bad anyway it even looks sort of normal.

 

[marcus]

wolram I must apologize for insisting on looking at the *-category business
since my perception is that one of your strengths is that you only
accept what you can see, or that astronomers by the farthest stretch of their instruments can see. insisting that things be real is a type of strength.

so you may be offended if I veer off into abstract maths, instead of focusing on the observational astronomy of people like Pello et al.

however two things have come up lately that excite me very much
one is chunkies---the business of allowing spatial mappings to have a finite number of places where they arent smooth (this class of mappings, although very simple and routine to define, has for some reason not been studied, so the consequences of using it in modeling gravity are not yet known)

and the other is star-categories
a group of objects with mappings running (as usual) between them but
with a relation defined on the mappings
so that each map has a buddy, which is a kind of reverse or adjunct, a partner going in the opposite direction.

there is this large branch of mathematics called category theory where
they study the general situation of having a bunch of objects with
mappings to get you from one to the other
and they didnt yet think
to study the situation where for every map there is a reverse map

in Baez definition there are just 3 requirements:

1.the relation is symmetric (f** = f, the buddy of a thing's buddy is the original thing)

2.the relation is reflexive on identities (any identity map is its own buddy)
1* = 1

3.the relation repects the composition of maps (going first by g and then by f, composing the two maps, is a map called fg, and the reverse of that is backing up first by f* and then by g*)
if you do two things in sequence then if you want to revoke or undo them you have to take them back in the opposite order
(fg)* = g*f*

 

[wolram]

wolram I must apologize for insisting on looking at the *-category business
since my perception is that one of your strengths is that you only
accept what you can see, or that astronomers by the farthest stretch of their instruments can see. insisting that things be real is a type of strength.
--------------------------------------------------------------------------
no i just need pushing, nature has plenty of symmetries that can be
seen, but sometimes its hidden under complexity, my skepticism is
more for the untestable, i admire people that can push to the ragged
edge of a theory.

 

[marcus]

... my skepticism is
more for the untestable, i admire people that can push to the ragged
edge of a theory.

true image: the edge is ragged
I admire them too, those who venture to the ragged edge
it is fascinating to watch