Log24

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:

Friday, December 20, 2019

Identity Theory*

Filed under: General — Tags: — m759 @ 1:56 AM

The van Dam cited by Polster should not be confused
with the fictional   Vandamm of "North by Northwest."

See Pursued by a Biplane (Log24, May 23, 2017).

* For the title, see posts tagged March 8, 2018.

Thursday, May 26, 2016

Dorje

Filed under: General,Geometry — m759 @ 6:40 PM

Images suggested by the previous post

Note the name "Dorje" in the first image above.

Remarks related to the name "Dorje," as well as to
 "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
a 2009 paper by Y. Jack Ng  and the late Hendrik van Dam

Dorje C. Brody, 'Consistency of PT-Symmetric Quantum Mechanics'

Remarks for the Church of Synchronology from December 16, 2015,
the date of the above Dorje  arXiv upload —

Expanding the Spielraum

Filed under: General — m759 @ 1:45 AM

(Continued)

The physicist Hendrik van Dam was mentioned in recent posts.

He reportedly died at 78 on February 11, 2013.

A post from that date, and a followup —

Wednesday, May 25, 2016

Framework

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

"Studies of spin-½ theories in the framework of projective geometry
have been undertaken before." — Y. Jack Ng  and H. van Dam
February 20, 2009

For one such framework,* see posts from that same date 
four years earlier — February 20, 2005.

* A 4×4 array. See the 19771978, and 1986 versions by 
Steven H. Cullinane,   the 1987 version by R. T. Curtis, and
the 1988 Conway-Sloane version illustrated below —

Cullinane, 1977

IMAGE- Hypercube and 4x4 matrix from the 1976 'Diamond Theory' preprint, as excerpted in 'Computer Graphics and Art'

Cullinane, 1978

Cullinane, 1986

Curtis, 1987

Update of 10:42 PM ET on Sunday, June 19, 2016 —

The above images are precursors to

Conway and Sloane, 1988

Update of 10 AM ET Sept. 16, 2016 — The excerpt from the
1977 "Diamond Theory" article was added above.

Kummer and Dirac

From "Projective Geometry and PT-Symmetric Dirac Hamiltonian,"
Y. Jack Ng  and H. van Dam
Physics Letters B , Volume 673, Issue 3,
23 March 2009, Pages 237–239

(http://arxiv.org/abs/0901.2579v2, last revised Feb. 20, 2009)

" Studies of spin-½ theories in the framework of projective geometry
have been undertaken before. See, e.g., Ref. [4]. 1 "

1 These papers are rather mathematical and technical.
The authors of the first two papers discuss the Dirac equation
in terms of the Plucker-Klein correspondence between lines of
a three-dimensional projective space and points of a quadric
in a five-dimensional projective space. The last paper shows
that the Dirac equation bears a certain relation to Kummer’s
surface, viz., the structure of the Dirac ring of matrices is 
related to that of Kummer’s 166 configuration . . . ."

[4]

O. Veblen
Proc. Natl. Acad. Sci. USA , 19 (1933), p. 503
Full Text via CrossRef

E.M. Bruins
Proc. Nederl. Akad. Wetensch. , 52 (1949), p. 1135

F.C. Taylor Jr., Master thesis, University of North Carolina
at Chapel Hill (1968), unpublished


A remark of my own on the structure of Kummer’s 166 configuration . . . .

See that structure in this  journal, for instance —

See as well yesterday morning's post.

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