Log24

Thursday, June 21, 2018

Dirac and Geometry (continued)

"Just fancy a scale model of Being 
made out of string and cardboard."

Nanavira Thera, 1 October 1957,
on a model of Kummer's Quartic Surface
mentioned by Eddington

"… a treatise on Kummer's quartic surface."

The "super-mathematician" Eddington did not see fit to mention
the title or the author of the treatise he discussed.

See Hudson + Kummer in this  journal.

See also posts tagged Dirac and Geometry.

Monday, December 14, 2015

Dirac and Geometry

Filed under: General,Geometry — Tags: , — m759 @ 10:30 am

(Continued)

See a post by Peter Woit from Sept. 24, 2005 — Dirac's Hidden Geometry.

The connection, if any, with recent Log24 posts on Dirac and Geometry
is not immediately apparent.  Some related remarks from a novel —

From Broken Symmetries by Paul Preuss
(first published by Simon and Schuster in 1983) —

"He pondered the source of her fascination with the occult, which sooner or later seemed to entangle a lot of thoughtful people who were not already mired in establishmentarian science or religion. It was  the religious impulse, at base. Even reason itself could function as a religion, he supposed— but only for those of severely limited imagination. 

He’d toyed with 'psi' himself, written a couple of papers now much quoted by crackpots, to his chagrin. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand— for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox. Quantum theory was inextricable from the uncertainty relations. 

The classical fox knows many things, but the quantum-mechanical hedgehog knows only one big thing— at a time. 'Complementarity,' Bohr had called it, a rubbery notion the great professor had stretched to include numerous pairs of opposites. Peter Slater was willing to call it absurdity, and unlike some of his older colleagues who, following in Einstein’s footsteps, demanded causal explanations for everything (at least in principle), Peter had never thirsted after 'hidden variables' to explain what could not be pictured. Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once. It was a thin nectar, but he was convinced it was the nectar of the gods. 

The psychic investigators, on the other hand, demanded to know how  the mind and the psychical world were related. Through ectoplasm, perhaps? Some fifth force of nature? Extra dimensions of spacetime? All these naive explanations were on a par with the assumption that psi is propagated by a species of nonlocal hidden variables, the favored explanation of sophisticates; ignotum per ignotius

'In this connection one should particularly remember that the human language permits the construction of sentences which do not involve any consequences and which therefore have no content at all…' The words were Heisenberg’s, lecturing in 1929 on the irreducible ambiguity of the uncertainty relations. They reminded Peter of Evan Harris Walker’s ingenious theory of the psi force, a theory that assigned psi both positive and negative values in such a way that the mere presence of a skeptic in the near vicinity of a sensitive psychic investigation could force null results. Neat, Dr. Walker, thought Peter Slater— neat, and totally without content. 

One had to be willing to tolerate ambiguity; one had to be willing to be crazy. Heisenberg himself was only human— he’d persuasively woven ambiguity into the fabric of the universe itself, but in that same set of 1929 lectures he’d rejected Dirac’s then-new wave equations with the remark, 'Here spontaneous transitions may occur to the states of negative energy; as these have never been observed, the theory is certainly wrong.' It was a reasonable conclusion, and that was its fault, for Dirac’s equations suggested the existence of antimatter: the first antiparticles, whose existence might never have been suspected without Dirac’s crazy results, were found less than three years later. 

Those so-called crazy psychics were too sane, that was their problem— they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry."

Particularly relevant

"Mathematical relationships were enough to satisfy him,
mere formal relationships which existed at all times,
everywhere, at once."

Some related pure  mathematics

Anticommuting Dirac matrices as spreads of projective lines

Saturday, November 12, 2022

“Dirac’s Hidden Geometry”

Filed under: General — Tags: — m759 @ 1:00 pm

See also other Log24 posts tagged Dirac and Geometry.

Tuesday, March 24, 2020

The Amsterdam Connection

Filed under: General — Tags: , , — m759 @ 11:22 am

Some mathematics from Ghent —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

Earlier, in Amsterdam . . .

First page of 'Configurations in Quantum Mechanics,' by E.M. Bruins, 1959

See as well Dirac and Geometry.

Monday, October 21, 2019

Algebra and Space… Illustrated.

Filed under: General — Tags: , — m759 @ 4:26 pm

Related entertainment —

Detail:

   George Steiner

"Perhaps an insane conceit."

 

Perhaps.

 

See Quantum Tesseract Theorem .

 

Perhaps Not.

 

 See Dirac and Geometry .

Saturday, August 10, 2019

Schoolgirl Space* Revisited:

Filed under: General — Tags: , , — m759 @ 10:51 pm

The Square "Inscape" Model of
the Generalized Quadrangle W(2)

Click image to enlarge.

* The title refers to the role of PG (3,2) in Kirkman's schoolgirl problem.
For some backstory, see my post Anticommuting Dirac Matrices as Skew Lines
and, more generally, posts tagged Dirac and Geometry.

Sunday, July 14, 2019

Old Pathways in Science:

Filed under: General — Tags: , — m759 @ 12:37 pm

The Quantum Tesseract Theorem Revisited

From page 274 — 

"The secret  is that the super-mathematician expresses by the anticommutation
of  his operators the property which the geometer conceives as  perpendicularity
of displacements.  That is why on p. 269 we singled out a pentad of anticommuting
operators, foreseeing that they would have an immediate application in describing
the property of perpendicular directions without using the traditional picture of space.
They express the property of perpendicularity without the picture of perpendicularity.

Thus far we have touched only the fringe of the structure of our set of sixteen E-operators.
Only by entering deeply into the theory of electrons could I show the whole structure
coming into evidence."

A related illustration, from posts tagged Dirac and Geometry —

Anticommuting Dirac matrices as spreads of projective lines

Compare and contrast Eddington's use of the word "perpendicular"
with a later use of the word by Saniga and Planat.

Monday, February 18, 2019

Quantum Choreography

Filed under: General — Tags: — m759 @ 1:06 pm

In the beginning of the 1954 film "Seven Brides for Seven Brothers,"
Adam, oldest of the brothers, is paired with Milly. The problem then is
to find brides for Adam's six brothers.

Above is a video illustration, published on Aug. 12, 2016, of the six couples. 
Matching them involves some choreography.

See also this  journal on Aug. 12, 2016 — "Dustbucket Physics" —

" through the proposition machine of quantum mechanics
comes pregeometry; pregeometry makes geometry;
geometry gives rise to matter and the physical laws
and constants of the universe." 

— Harvard professor Peter Galison, 
     defender of the faith of Scientism

For a different sort of quantum "proposition machine," see posts tagged
Dirac and Geometry.

Tuesday, November 13, 2018

Blackboard Jungle Continues.

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 6:19 pm

From the 1955 film "Blackboard Jungle" —

From a trailer for the recent film version of A Wrinkle in Time

Detail of the phrase "quantum tesseract theorem":

From the 1962 book —

"There's something phoney
in the whole setup, Meg thought.
There is definitely something rotten
in the state of Camazotz."

Related mathematics from Koen Thas that some might call a
"quantum tesseract theorem" —

Some background —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also posts tagged Dirac and Geometry. For more
background on finite  geometry, see a web page
at Thas's institution, Ghent University.

Friday, July 6, 2018

Something

Filed under: General,Geometry — Tags: , , , , — m759 @ 9:48 am

"… Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

— T. S. Eliot, "Burnt Norton," 1936

"Read something that means something."

Advertising slogan for The New Yorker

The previous post quoted some mystic meditations of Octavio Paz
from 1974. I prefer some less mystic remarks of Eddington from
1938 (the Tanner Lectures) published by Cambridge U. Press in 1939 —

"… we have sixteen elements with which to form a group-structure" —

See as well posts tagged Dirac and Geometry.

Tuesday, July 3, 2018

Lost in Quantum Space

Filed under: General,Geometry — Tags: , — m759 @ 10:45 am

Combining concepts from earlier posts today, we have the above title.

A more concise alternative title

Lost in the Matrix

For some related non -fiction, see posts tagged Dirac and Geometry.

Monday, March 12, 2018

“Quantum Tesseract Theorem?”

Filed under: General,Geometry — Tags: , , — m759 @ 11:00 am

Remarks related to a recent film and a not-so-recent film.

For some historical background, see Dirac and Geometry in this journal.

Also (as Thas mentions) after Saniga and Planat —

The Saniga-Planat paper was submitted on December 21, 2006.

Excerpts from this  journal on that date —

A Halmos tombstone and the tale of HAL and the pod bay doors

     "Open the pod bay doors, HAL."

Saturday, December 23, 2017

The Right Stuff

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:12 pm

A figure related to the general connecting theorem  of Koen Thas —

Anticommuting Dirac matrices as spreads of projective lines

Ron Shaw on the 15 lines of the classical generalized quadrangle W(2), a general linear complex in PG(3,2)

See also posts tagged Dirac and Geometry in this  journal.

Those who prefer narrative to mathematics may, if they so fancy, call
the above Thas connecting theorem a "quantum tesseract theorem ."

Thursday, November 30, 2017

The Matrix for Quantum Mystics

Filed under: G-Notes,General,Geometry — Tags: , , , — m759 @ 10:29 pm

Scholia on the title — See Quantum + Mystic in this journal.

The Matrix of Lévi-Strauss

"In Vol. I of Structural Anthropology , p. 209, I have shown that
this analysis alone can account for the double aspect of time
representation in all mythical systems: the narrative is both
'in time' (it consists of a succession of events) and 'beyond'
(its value is permanent)." — Claude Lévi-Strauss, 1976

I prefer the earlier, better-known, remarks on time by T. S. Eliot
in Four Quartets , and the following four quartets (from
The Matrix Meets the Grid) —

.

From a Log24 post of June 26-27, 2017:

A work of Eddington cited in 1974 by von Franz

See also Dirac and Geometry and Kummer in this journal.

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Tuesday, October 10, 2017

Another 35-Year Wait

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 9:00 pm

The title refers to today's earlier post "The 35-Year Wait."

A check of my activities 35 years ago this fall, in the autumn
of 1982, yields a formula I prefer to the nonsensical, but famous,
"canonical formula" of Claude Lévi-Strauss.

The Lévi-Strauss formula

My "inscape" formula, from a note of Sept. 22, 1982 —

S = f ( f ( X ) ) .

Some mathematics from last year related to the 1982 formula —

Koen Thas, 'Unextendible Mututally Unbiased Bases' (2016)

See also Inscape in this  journal and posts tagged Dirac and Geometry.

Monday, June 26, 2017

Upgrading to Six

This post was suggested by the previous post — Four Dots —
and by the phrase "smallest perfect" in this journal.

Related material (click to enlarge) —

Detail —

From the work of Eddington cited in 1974 by von Franz —

See also Dirac and Geometry and Kummer in this journal.

Updates from the morning of June 27 —

Ron Shaw on Eddington's triads "associated in conjugate pairs" —

For more about hyperbolic  and isotropic  lines in PG(3,2),
see posts tagged Diamond Theorem Correlation.

For Shaw, in memoriam — See Contrapuntal Interweaving and The Fugue.

Tuesday, June 6, 2017

The Table

Filed under: General,Geometry — m759 @ 12:00 pm

John Horgan and James (Jim) McClellan, according to Horgan
in Scientific American  on June 1, 2017

Me: "Jim, you're a scholar! Professor! Esteemed historian of science! And yet you don’t really believe science is capable of producing truth."

Jim: "Science is stories we tell about nature. And some stories are better than other stories. And you can compare stories to each other on all kinds of grounds, but you have no access to"— he pauses for dramatic effect— "The Truth. Or any mode of knowing outside of your own story-telling capabilities, which include rationality, experiment, explanatory scope and the whole thing. I would love to have some means of making knowledge about the world that would allow us to say, 'This is really it. There really are goddamn electrons.'" He whacks the table.

See also posts tagged Dirac and Geometry and Glitch.

Tuesday, March 21, 2017

Special Topics

Filed under: General,Geometry — Tags: , — m759 @ 12:41 pm

A roundup of posts now tagged "Apollo Psi" led to the name
Evan Harris Walker in the post Dirac and Geometry of
Dec. 14, 2015. That post mentions

" Evan Harris Walker’s ingenious theory of
the psi force, a theory that assigned psi
both positive and negative values in such a way
that the mere presence of a skeptic in the near
vicinity of a sensitive psychic investigation could
force null results. Neat, Dr. Walker, thought
Peter Slater— neat, and totally without content."

— From the 1983 novel Broken Symmetries  
     by Paul Preuss 

It turns out that Walker died "on the evening of August 17, 2006." 

From this journal on that date

Tuesday, November 22, 2016

Jargon

Filed under: General,Geometry — Tags: , , , — m759 @ 4:00 pm

See "sacerdotal jargon" in this journal.

For those who prefer scientific  jargon —

"… open its reading to
combinational possibilities
outside its larger narrative flow.
The particulars of attention,
whether subjective or objective,
are unshackled through form,
and offered as a relational matrix …."

— Kent Johnson in a 1993 essay

For some science that is not just jargon, see

and, also from posts tagged Dirac and Geometry

Anticommuting Dirac matrices as spreads of projective lines

The above line complex also illustrates an outer automorphism
of the symmetric group S6. See last Thursday's post "Rotman and
the Outer Automorphism
."

Thursday, November 3, 2016

Triple Cross

(Continued See the title in this journal, as well as Cube Bricks.)

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168
Related material —

Dirac and Geometry in this journal,
Kummer's Quartic Surface in this journal,
Nanavira Thera in this journal, and
The Razor's Edge  and Nanavira Thera.

See as well Bill Murray's 1984 film "The Razor's Edge"

Movie poster from 1984 —

"A thin line separates
love from hate,
success from failure,
life from death."

Three other dualities, from Nanavira Thera in 1959 —

"I find that there are, in every situation,
three independent dualities…."

(Click to enlarge.)

Friday, June 3, 2016

Bruins and van Dam

Filed under: General,Geometry — Tags: , — m759 @ 8:00 am

A review of some recent posts on Dirac and geometry,
each of which mentions the late physicist Hendrik van Dam:

The first of these posts mentions the work of E. M. Bruins.
Some earlier posts that cite Bruins:

Monday, February 8, 2016

A Game with Four Letters

Filed under: General,Geometry — Tags: , — m759 @ 2:56 pm

Related material — Posts tagged Dirac and Geometry.

For an example of what Eddington calls "an open mind,"
see the 1958 letters of Nanavira Thera.
(Among the "Early Letters" in Seeking the Path ).

Wednesday, January 20, 2016

Fringe Physics and Beyond

Filed under: General,Geometry — m759 @ 11:00 am

"One day not long ago Oppenheimer stalked
up and down his office and divulged some
startling new discoveries about the 15 fundamental
particles of which the universe is made….

physicists today are wondering if these particles
are themselves actually the final, stark, immutable
and indivisible foundation stones of the universe
that until now they have been thought to be."

—Lincoln Barnett in LIFE magazine,
    Oct. 10, 1949, page 122

Fringe Physics

" astrophysics limits the number of fundamental particles to 15…."

— Franklin Potter at FQXi.org on Sep. 27, 2009

"I agree there can't be more than 15 fundamental particles."

— Lawrence B. Crowell at FQXi.org on Sep. 29, 2009

Beyond

There are, at any rate, 15 "final, stark, immutable* and indivisible*
foundation stones" (namely, 15 points ) of the finite projective
space PG(3,2). See Symplectic  in this journal.

For related physics, see posts tagged Dirac and Geometry.

* Update of Jan. 21, 2016 — I was carried away by Barnett's
   powerful rhetoric. These adjectives are wrong.

Monday, December 14, 2015

The Forking

Filed under: General,Geometry — m759 @ 12:00 pm

From the previous post:

"Neat, Dr. Walker, thought Peter Slater—
neat, and totally without content."

— Paul Preuss's 1983 novel Broken Symmetries

A background check yields

"Dr. Evan Harris Walker died on the evening of
August 17, 2006…."

A synchronicity check of that date in this journal yields a diagram
that, taken by itself, is "neat, and totally without content." —

The image “http://www.log24.com/log/pix06A/060817-Tree.jpg” cannot be displayed, because it contains errors.

The diagram may be viewed as a tribute
to the late Yogi Berra, to the literary
"Garden of Forking Paths," or, more
seriously, to the modular group Γ.

Friday, November 27, 2015

Einstein and Geometry

Filed under: General,Geometry — Tags: , — m759 @ 2:01 pm

(A Prequel to Dirac and Geometry)

"So Einstein went back to the blackboard.
And on Nov. 25, 1915, he set down
the equation that rules the universe.
As compact and mysterious as a Viking rune,
it describes space-time as a kind of sagging mattress…."

— Dennis Overbye in The New York Times  online,
     November 24, 2015

Some pure  mathematics I prefer to the sagging Viking mattress —

Readings closely related to the above passage —

Thomas Hawkins, "From General Relativity to Group Representations:
the Background to Weyl's Papers of 1925-26
," in Matériaux pour
l'histoire des mathématiques au XXe siècle:
Actes du colloque
à la mémoire de Jean Dieudonné
, Nice, 1996  (Soc. Math.
de France, Paris, 1998), pp. 69-100.

The 19th-century algebraic theory of invariants is discussed
as what Weitzenböck called a guide "through the thicket
of formulas of general relativity."

Wallace Givens, "Tensor Coordinates of Linear Spaces," in
Annals of Mathematics  Second Series, Vol. 38, No. 2, April 1937, 
pp. 355-385.

Tensors (also used by Einstein in 1915) are related to 
the theory of line complexes in three-dimensional
projective space and to the matrices used by Dirac
in his 1928 work on quantum mechanics.

For those who prefer metaphors to mathematics —

"We acknowledge a theorem's beauty
when we see how the theorem 'fits' in its place,
how it sheds light around itself, like a Lichtung ,
a clearing in the woods." 
— Gian-Carlo Rota, Indiscrete Thoughts ,
Birkhäuser Boston, 1997, page 132

Rota fails to cite the source of his metaphor.
It is Heidegger's 1964 essay, "The End of Philosophy
and the Task of Thinking" —

"The forest clearing [ Lichtung ] is experienced
in contrast to dense forest, called Dickung  
in our older language." 
— Heidegger's Basic Writings 
edited by David Farrell Krell, 
Harper Collins paperback, 1993, page 441

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