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Re: Toa'n tu+? De Rham Re: Gelfand-Kolmogorov, Najmark & Stone Re: Gelfand-Naimark
Hello Hai et all,
--- Phung Ho Hai <phung@msri.org> wrote:
> Ba'c Ai' Vie^.t a., theo to^i ba'c ne^n nghie^n
cu+'c ky~ ho+n ve^`
> Non-commutative geometry truo+'c khi ba`n lua^.n.
> De^'n ba^y gio+` thi`to^i hie^?u ra(`ng kie^'n
thu+'c cu?a ba'c ve^`
> Non-comm geom ra^'t la` "cuo+~i ngu+a. xem hoa".(
Va^.y ma` ba'c vie^'tma^'y ba`i ve^` AlainConnes hay
dao' de^?!)
>
Co^' nhie^n. Nhu+ng kho^ng pha?i rie^ng to^i, ma`
ai cu~ng va^.y tho^i.
Co`n pha?i la`m va^.y tru+o+'c khi ba`n lua^.n thi`
to^i e i't ngu+o+`i da'm no'i gi` ve^` NCG :-)) Well,
bi`nh lua^.n ve^` tri`nh ddo^. ngu+o+`i kha'c la` du+
thu+`a tho+`i gian va` thie^'u tho^ng tin . Ne^'u
thi'ch ta no'i chuye^.n Toa'n ba(`ng argument tho^i.
Tui ti`nh co+` la.i u+'ng du.ng NCG va`o Physics
kha' nhie^`u. Ne^'u ddo' cu~ng la` "cu+o+~i ngu+.a xem
hoa" thi` ...:-)) Anyway, ne^'y gia? thie^'t nhu+
va^.y mang la.i good feeling khi no'i chuye^.n thi`
OK ddo^'i vo+'i mo^.t so^' ngu+o+`i ye^'u ve^` ba?n
la~nh. Vi` la`m nhu+ va^.y ta nghie^~m nhie^n assume
mo^.t authority dde^? thay the^' no^.i dung.
Tui kho^ng thi'ch no'i ve^` dde^` ta`i nghie^n
cu+'u he.p cu?a mi`nh, vi` tre^n VNSA la^'y authority
la`m argument thi` ho+i funny. Dda^y la` bi. ba('t
buo^.c mo+'i pha?i no'i. Co' le~ Ha?i tu+o+?ng
Physicist thi` bie^'t toa'n ta^`m pho+. Cu~ng co'
the^? to^i du`ng metaphore ( theo mo^.t ca'ch kha'c)
ho+i nhie^`u. Ddie^`u ddo' encourage Ha?i "khuye^n
ba?o" :-)). Thank anyway.
Tho^i bo?. Ta no'i chuye^.n tie^'p. Hie^?u sai
cu~ng la` chuye^.n nho? tho^i :-)) Dda^y la` thread
ve^` Toa'n chu+' kho^ng pha?i lu'c bi`nh ba^`u phong
GS.
> Theo Gelfand thi` mo^.t kgian topo (compact,
> Hausdorff) duo.c xa'c di.nh
> bo+?i mo^.t (commutative) C^*-da.i so^' ca'c ha`m
> tre^n no'. Ca'i na`y
> duo.c go.i la` do^'i nga^~u Gelfand
>
> Va^.y ne^'u consider mo^.t da.i so^' kho^ng giao
> hoa'n thi` ta thu duo.c
> -qua do^'i nga^~u gelfand (gia? di.nh)-mo^.t
> non-comm. geometry theory.
Tu+'c la` ba^y gio+` dda~ nha?y le^n geometry
level ngay? Kho^ng ca^`n toa'n tu+? De Rham? Tu+'c la`
khi dda.i so^' giao hoa'n thi` ta o+? topo level, khi
chuye^?n sang dda.i so^' kho^ng giao hoa'n thi` ta
le^n geometry level.
I don't think so, boy. Connes ddi.nh nghi~a NCG
ba(`ng Spectral Triple
(A, D, H). Kho^ng pha?i chi? ba(`ng dda.i so^' A. Ddai
so^' A du` la` kho^ng giao hoa'n cu~ng kho^ng co'
noncommutative geometry theory na`o.
> Trong co+ luong tu+?, ke^'t qua? ca'c phe'p thu+?
duo.c mie^u ta? ba(`ng ca'c toa'n tu+?-chu'ng kho^ng
giao hoa'n.
>
Aha! Ba^y gio+` to^i mo+'i hie^?u vi` sao la.i co'
ca'i metaphore sai la^`m nhu+ the^'. Phe'p thu+?
kho^ng bao gio+` mo^ ta? bo+?i toa'n tu+? trong co+
lu+o+.ng tu+?. Cu~ng nhu+ ha`m so^' kho^ng mo^ ta?
phe'p so+` na`o ca?. Too bad a metaphore originated
from a wrong understanding !
Trong co+ lu+o+.ng tu+? operators la` ca'c physical
quantities ( Observables --kho^ng co' nghi~a la` phe'p
thu+? ma` co' nghi~a la` mo^.t dda.i lu+o+.ng se~ quan
sa't ddu+o+.c). Thi' du. ddo^. mi.n cu?a da, hay
ddo^. da`i cu?a to'c....Ddo' la` ca'c toa'n tu+? ma`
ca'c phe'p "so+`" se~ su+? du.ng.
Phe'p thu+? ca^`n hai thu+': toa'n tu+? no'i le^n
mu.c tie^u cu?a phe'p thu+? muo^'n ddo ca'i gi` va`
tra.ng tha'i cu?a phe'p ddo. Vi` va^.y mo^~i phe'p
thu+? ( so+`) cho ta mo^.t gia' tri. ( co' the^? la`
la` gia' tri. rie^ng
cu?a toa'n tu+? ne^'u phe'p ddo thu+.c hie^.n trong
tra.ng tha'i rie^ng, hoa(.c gia' tri. trung bi`nh
trong ca'c tra.ng tha'i kho^ng pha?i tra.ng tha'i
rie^ng)
Dda^'y la` no'i kha'i nie^.m co+ ba?n cu?a co+
lu+o+.ng tu+? cho mathematicians hie^?u chu+' kho^ng
pha?i tranh lua^.n. Co`n ba`n lua^.n hay kho^ng khi
chu+a hie^?u la` quye^`n cu?a a certain mathematician
:-))
> Logic cu?a o^ng Connes trong quye^?n NCG deep ho+n
va` kho^ng the^? gia?i thi'ch mo^.t ca'ch do+n gia?n
duo.c. Hie^?N nhie^n
> no' lie^n quan de^n Geometry nhung kho^ng pha?i in
the way ma` ba'c Ai' vie^.t hie^?u.
>>
Kho^ng bie^'t Ha?i hie^?u la` to^i hie^?u the^'
na`o :-)) Ne^'u ddu'ng la` hie^?u thi` bao gio+` cu~ng
gia?i thi'ch gia?n ddo+n tho^i. Co^' nhie^n ddo^'i
vo+'i nhu+~ng ngu+o+`i kho^ng pha?i geometrist thi`
bao gio+` cu~ng phu+'c ta.p.
Connes ddi.nh go beyond the usual differential
geometry ba(`ng ca'c bo? ca'c kha'i nie^.m local
coordinates ddi.nh nghi~a qua ca'c kho^ng gian tie^'p
xu'c ( tangent). Ca'c dda.i lu+o+.ng na`y lie^n quan
to+'i ca'c ddie^?m. Nhu+~ng hi`nh ho.c ma` va^.t ly'
ddo`i ho?i o+? Planck scale
nhu+ string, M-theory...kha'i nie^.m ddie^?m ma^'t
he^'t y' nghi~a.
NCG la` mo^.t attempt ( no guarantee of success)
dde^? co' the^? mo^ ta? ddu+o+.c nhu+~ng kho^ng gian
nhu+ the^'. Ca'ch attack tho^ng qua dda.i so^' toa'n
tu+? tre^n mo^.t kho^ng gian Hilbert to? ra thoa't
ddu+o+.c notion of a point dde^? co' nhu+~ng ca^'u
tru'c to^?ng qua't ho+n.
Ca'i notice cu?a to^i la` mo^.t trong Spectral
Triplet (A, D, H) ne^'u ta thay ddo^?i thi` cu~ng dda~
go beyond geometry tho^ng thu+o+`ng.
Cha(?ng ha.n Connes-Lott Model du+.a tre^n C\inf +
C\inf . Dda.i so^' na`y commutative. Ne^'u ta la^'y
toa'n tu+? De Rham tho^ng thu+o+`ng, thi` hi`nh ho.c
perfectly commutative. Cha(?ng co' gi` mo+'i. Ca'i
mo+'i cu?a Connes-Lott model la` ddi.nh nghi~ toa'n
tu+? De Rham (Dirac)
Ne^'u ddi.nh nghi~a toa'n tu+? De Rham mo^.t
ca'ch na`o ddo' thi` ta mo+'i co' NCG ( Ha?i co' the^?
mo+? ma^'y chu+o+ng cuo^'i cu?a NCG ma` ddo.c ve^`
Connes-Lott model ne^'u dda~ ddo.c va` hie^?u ma^'y
chu+o+ng dda^`u)
Tuy nhie^n ra^'t nhie^`u ngu+o+`i vie^'t trong
ca'c introduction
hoa(.c ca'c ba`i review ve^` NCG la` "the notion of
geometry is replaced
by the notion of the noncom algebra of operators" ( Or
as expressed in my word "the algebra of operators
replaces the commutative geometry" )
Ddo' la` mo^.t misunderstanding co' tha^.t trong
Math-Phys, thu+.c te^' vi` ho. kho^ng dde^? y' to+'i
vai tro` cu?a toa'n tu+? De Rham. Tu+o+?ng ra(`ng
phe'p to^?ng qua't ho'a nhu+ va^.y la` xong.
Ngu+o+.c la.i ta co' the^? thay dda.i so^' giao
hoa'n ca'c ha`m phu+'c ba(`ng dda.i so^' ca'c ha`m
quaternion ( hay octonion). DDa.i so^' na`y hie^?n
nhie^n kho^ng giao hoa'n ( tha^.m chi' kho^ng ke^'t
ho+.p). Nhu+ng ne^'u ta du`ng toa'n tu+? De Rham
tho^ng thu+o+`ng ta va^~n chi? co' mo^.t hi`nh ho.c
giao hoa'n ra^'t ta^`m thu+o+`ng. Ca'c loa.i hi`nh
ho.c nhu+ the^' na`y dda~ ddu+o+.c ca'c nha` va^.t ly'
su+? du.ng 20,30 na(m tru+o+'c Connes dde^? xa^y
du+.ng ca'c hi`nh ho.c Riemann va` xa^y du+.ng ca'c
ta'c du.ng Hilbert-Einstein suy ro^.ng va` co^' nhie^n
kho^ng co' gi` tha^.t hay va` mo+'i ve^` ma(.t va^.t
ly'.
To^i dda~ tha^'y ca'c loa.i commutative geometry
du+.a tre^n ca'c dda.i so^' ha`m kho^ng giao hoa'n
nhu+ the^' trong va^.t ly' tu+` ra^'t la^u va` nha('c
nho+? ca'c nha` Math-Phys ddang su+? du.ng NCG ve^`
vai tro` cu?a vie^.c cho.n mo^.t toa'n tu+? De
Rham-Dirac sao cho co' ke^'t qua? kho^ng ta^`m
thu+o+`ng trong nhie^`u seminar va` conference. Thu+.c
te^' notice ddo' giu'p nhie^`u nho'm ra kho?i ca'c
co^' ga('ng lua^?n qua^?n.
Trong khi ho. co^' ga('ng na(.n ra ca'c dda.i so^'
mo+'i dde^? co' ke^'t qua? va^.t ly' na`o ddo', dd'ang
ra ca^`n ta^.p trung nhie^`u ho+n va`o vie^.c ddi.nh
nghi~a mo^.t toa'n tu+? Dirac-DeRham thi'ch ho+.p.
Nhu+ng no' dde^'n level na`y so+. beyond quan ta^m
va` kha? na(ng tie^'p thu cu?a mo.i ngu+o+`i.
Cheers
Aiviet
=====
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