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E-LE-VEN
THEPOSSIBILITYOFSTRINGTHEORY
E X T R A DI M E N S IO N S ?
T H E O R Y O F E V E R Y T H I N G ?
V I A
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EN
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by SWIN HUANG
MOHAWK FINE PAPERS INC.
465 Saratoga StreetCohoes, NY 12047465
TEL: 1-800-THE MILL TEL: 1-800-843-6455FAX: 1-518-237-7394
THEPOSSIBILITYOFSTRINGTHEORY
01
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E-LE
-VEN
: The
pos
sibi
lity
of s
trin
g th
eory
. Cop
yrig
ht ©
2
010
by M
ohaw
k Fi
ne P
aper
Inc.
Man
ifac
ture
d in
Am
eric
a.
All
righ
ts r
eser
ved.
No
oth
er p
art
of
thi
s bo
ok m
ay b
e re
prod
uced
in a
ny o
ther
for
m o
r by
any
ele
ctro
nic
or
mec
hani
cal m
ean
s in
clud
ing
info
rmat
ion
stor
age
and
retr
ieva
l sys
tem
s w
itho
ut p
erm
issi
on o
f co
py r
ight
hol
der.
I dedicate this book to my family, who will always be the love of my life.
FOR MOST OF US, OR PERHAPS ALL OF US, IT ’S IMPOSSIBLE TO IMAGINE A WORLD CONSISTING OF MORE THAN THREE SPATIAL DIMENSIONS. WITH THE BEST OF MOHAWK V ELLUM, SATIN 2.0 AND TOMOHAWK FELT PLUS EIGHT NEW SHADES FOR A NEW PERSPECTIV E ON VALUE PAPERS. THE NEW VIA PORTFOLIO IS MORE COMPREHENSIV E—Y ET STILL SIMPLE. WITH A V ERSATILE PALET TE OF CONTEMPOR ARY COLORS, VIA COMBINES ALL THE RIGHT FINISHES, EXCEPTIONAL PRINT QUALIT Y, 30% AND 100% POSTCONSUMER (PCW) RECYCLED ITEMS, DIGITAL I-TONE ITEMS, AND 25% COT TON WRITING—ALL MANUFACTURED GREEN-E CERTIFIED WINDPOWER, “VIA” CA N FULLFILL YOUR IMAGINATION OF GOING TO OTHER DIMENSIONS.
INTRO-DUCTION
Di-MEN-SIONS
IMAGINEOTHER
FOR MOST OF US, OR PERHAPS ALL OF US, IT’S IMPOSSIBLE TO IMAGINE A WORLD CONSISTING OF MORE THAN THREE SPATIAL DIMENSIONS. ARE WE CORRECT WHEN WE INTUIT THAT SUCH A WORLD COULDN’T EXIST? OR IS IT THAT OUR BRAINS ARE SIMPLY INCAPABLE OF IMAGINING ADDITIONAL DIMENSIONS—DIMENSIONS THAT MAY TURN OUT TO BE AS REAL AS OTHER THINGS WE CAN’T DETECT?
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An early attempt to explain the concept of extra dimensions came in 1884 with the publication of Edwin A. Abbott’s Flatland: A Romance of Many Dimensions. This novel is a “f irst-person” account of a two-dimensional square who comes to appreciate a three-dimensional world. + The square describes his world as a plane populated by lines, circles, squares, triangles, and pentagons. Being two-dimensional, the inhabitants of Flatland appear as lines to one another. They discern one another’s shape both by touching and by seeing how the l ines appear to change in length as the inhabitants move around one another. + One day, a sphere appears before the square. To the square, which can see only a slice of the sphere, the shape before him is that of a two-dimensional circle. The sphere has visited the square intent on making the square understand the three-dimensional world that he, the sphere, belongs to. He explains the notions of “above” and “below,” which the square confuses with “forward” and “back.” When the sphere passes through the
plane of Flatland to show how he can move in three dimensions, the square sees only that the l ine he’d been observing gets shorter and shorter and then disappears. No matter what the sphere says or does, the square cannot comprehend a space other than the two-dimensional world that he knows. + Only after the sphere pulls the square out of his two-dimensional world and into the world of Spaceland does he f inally understand the concept of three dimensions. From this new perspective, the square has a bird’s-eye v iew of Flat land and is able to see the shapes of his fellow inhabitants (including, for the f irst time, their insides). + Armed with his new understanding, the square conceives the possibility of a fourth dimension. He even goes so far as to suggest that there may be no limit to the number of spatial dimensions. In trying to convince the sphere of this possibility, the square uses the same logic that the sphere used to argue the existence of three dimensions. The sphere, now the shortsighted one of the two, cannot comprehend this and does not accept the square’s arguments—just as most of us “spheres” today do not accept the idea of extra dimensions.
FROM2-DTO 3-D
ON
LY AF
TE
R T
HE
SP
HE
RE P
ULL
S
TH
E SQ
UA
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OU
T OF H
IS T
WO
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ION
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WO
RLD
AN
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INT
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HE W
OR
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OF S
PA
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LA
ND
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OE
S H
E FIN
AL-
LY UN
DE
RS
TAN
D
TH
E CO
NC
EP
T OF
TH
RE
E DIM
EN
-S
ION
S. F
RO
M
TH
IS N
EW
PE
R-
SP
EC
TIV
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SQ
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RE H
AS
A
BIR
D’S
-EY
E VIE
W
OF F
LA
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D
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D IS
AB
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S
EE T
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HA
PE
S
OF H
IS F
ELLO
W
INH
AB
ITAN
TS
.
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I t ’s dif f icult for us to accept the idea because when w e t r y t o im ag ine e ven a s ing le add i t ion a l sp a t i a l d imension—much less six or seven—we hi t a br ick wal l . There’s no going beyond i t , not w ith our brains apparently. + Imagine, for instance, that you’re at the center of a hollow sphere. The distance between you and every point on the sphere’s surface is equal. Now, t r y mov ing in a d ir ec t ion that a l lows you to move away from al l points on the sphere’s sur face w hi le maint a in ing that equid is t ance. You c an’ t do i t . T her e’s no w her e t o go —no w her e t h a t w e k no w any way. + The square in F lat land would have thes a m e t r o u b l e i f h e w e r e i n t h e m i d d l e o f a c i r c l e . He c an’ t be a t t he c en t er o f a c i r c le and mo ve in a d ir ec t ion that a l lows h im to r emain equid is t ant to every point of the circle’s circumference—unless he moves into the third dimension. A las, we don’t have the four-dimensionsal equivalent of Abbott’s three-dimensional sphere to show us the way to 4-D.
FROM3-DTO 4-D
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I n 1 9 1 9 , P o l i s h m a t h e m a t i c i a n T h e o d o r K a l u z a p r o p o s e d t h a t t h e e x i s t e n c e o f a f o u r t h s p a t i a l dimension might allow the linking of general relativity a n d e l e c t r o m a g n e t i c t h e o r y . T h e i d e a , l a t e r w a s ref ined by Oskar K lein, the Swedish mathematician, w a s t h a t s p a c e c o n s i s t e d o f b o t h e x t e n d e d a n d c u r l e d - u p d i m e n s i o n s . T h e e x t e n d e d d i m e n s i o n s are the three spatial dimensions that we’re famil iar w i t h , a n d t h e c u r l e d - u p d i m e n s i o n i s f o u n d d e e p within the extended dimensions and can be thought of as a circle. Experiments later showed that Kaluza and Klein’s curled-up dimension did not unite general relat iv i t y and electromagnet ic theor y as or iginal ly hoped, bu t dec ade s l a t er, s t r ing t he or i s t s found the idea usefu l , even mor e, necessar y. + T he mathemat ics used in superstr ing theor y requires a t le a s t 10 d imen s ion s . T h a t i s, for t he eq u a t ion s that descr ibe superstr ing theor y to begin to work out—for the equations to connect general relativity to quantum mechanics, to explain the nature of particles, to uni f y forces, and so on—they need to make use of
additional dimensions. These dimensions, string theorists believe, are wrapped up in the curled-up space f irst described by Kaluza and Klein. + To extend the curled-up space to include these added dimensions, imagine that spheres replace the Kaluza-Klein circles. Instead of one added dimension we have two if we consider only the spheres’ surfaces and three if we take into account the space within the sphere. That’s a total of six dimensions so far. So where are the others that superstring theory requires? + It turns out that, before superstring theory existed, two mathematicians, Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University, described six-dimensional geometrical shapes that super str ing theor is t s say f i t the b i l l for the k ind of s tr uc tur es their equations call for. If we replace the spheres in curled-up space with these Calabi-Yau shapes, we end up with 10 dimensions: three spatial, plus the six of the Calabi-Yau shapes, plus one of time. + If superstring theory turns out to be correct, the idea of a world consisting of 10 or more dimensions is one that we’ l l need to become comfor table w ith. But w i l l there ever be an explanat ion or a v isual representation of higher dimensions that wil l truly satisfy the human mind? The answer to this quest ion may forever be no. Not unless some four-dimensional l i fe-form pul ls us from our three-dimensional Spaceland and gives us a view of the world from its perspective.
FROM4-DTO OTHERDIMENSIONS
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0 8
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0 4
0 5
0 6
0 7
0 8
0 9
1 0
1 1
I n 1 9 1 9 , P o l i s h m a t h e m a t i c i a n T h e o d o r K a l u z a p r o p o s e d t h a t t h e e x i s t e n c e o f a f o u r t h s p a t i a l dimension might allow the linking of general relativity a n d e l e c t r o m a g n e t i c t h e o r y . T h e i d e a , l a t e r w a s ref ined by Oskar K lein, the Swedish mathematician, w a s t h a t s p a c e c o n s i s t e d o f b o t h e x t e n d e d a n d c u r l e d - u p d i m e n s i o n s . T h e e x t e n d e d d i m e n s i o n s are the three spatial dimensions that we’re famil iar w i t h , a n d t h e c u r l e d - u p d i m e n s i o n i s f o u n d d e e p within the extended dimensions and can be thought of as a circle. Experiments later showed that Kaluza and Klein’s curled-up dimension did not unite general relat iv i t y and electromagnet ic theor y as or iginal ly hoped, bu t dec ade s l a t er, s t r ing t he or i s t s found the idea usefu l , even mor e, necessar y. + T he mathemat ics used in superstr ing theor y requires a t le a s t 10 d imen s ion s . T h a t i s, for t he eq u a t ion s that descr ibe superstr ing theor y to begin to work out—for the equations to connect general relativity to quantum mechanics, to explain the nature of particles, to uni f y forces, and so on—they need to make use of
additional dimensions. These dimensions, string theorists believe, are wrapped up in the curled-up space f irst described by Kaluza and Klein. + To extend the curled-up space to include these added dimensions, imagine that spheres replace the Kaluza-Klein circles. Instead of one added dimension we have two if we consider only the spheres’ surfaces and three if we take into account the space within the sphere. That’s a total of six dimensions so far. So where are the others that superstring theory requires? + It turns out that, before superstring theory existed, two mathematicians, Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University, described six-dimensional geometrical shapes that super str ing theor is t s say f i t the b i l l for the k ind of s tr uc tur es their equations call for. If we replace the spheres in curled-up space with these Calabi-Yau shapes, we end up with 10 dimensions: three spatial, plus the six of the Calabi-Yau shapes, plus one of time. + If superstring theory turns out to be correct, the idea of a world consisting of 10 or more dimensions is one that we’ l l need to become comfor table w ith. But w i l l there ever be an explanat ion or a v isual representation of higher dimensions that wil l truly satisfy the human mind? The answer to this quest ion may forever be no. Not unless some four-dimensional l i fe-form pul ls us from our three-dimensional Spaceland and gives us a view of the world from its perspective.
FROM4-DTO OTHERDIMENSIONS
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TH
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D I M E N S I O N S
5 T H
11 T H
STR-INGTHEO-RY?
WHAT IS
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STRING THEORISTS ARE BETTING THAT EXTRA DIMENSIONS DO EXIST; IN FACT, THE EQUATIONS THAT DESCRIBE SUPERSTRING THEORY REQUIRE A UNIVERSE WITH NO FEWER THAN 10 DIMENSIONS. BUT EVEN PHYSICISTS WHO SPEND ALL DAY THINKING ABOUT EXTRA SPATIAL DIMENSIONS HAVE A HARD TIME DESCRIBING WHAT THEY MIGHT LOOK LIKE OR HOW WE APPARENTLY FEEBLE-MINDED HUMANS MIGHT APPROACH AN UNDERSTANDING OF THEM. THAT IS ALWAYS BEEN THE CASE, AND MAYBE ALWAYS WILL BE.
Ac
co
rd
ing
to
str
ing
theo
ry,
if we c
ou
ld
ex
am
ine th
ese
par
ticles
with
e
ven
gr
ea
ter
pr
ecis
ion
—a
p
rec
isio
n m
an
y o
rd
ers
of m
ag
-n
itud
e be
yon
d
ou
r p
res
ent
tech
no
log
ica
l c
apa
cit
y—w
e w
ou
ld fin
d
tha
t ea
ch
is
no
t po
intlik
e b
ut in
ste
ad
c
on
sis
ts
of
a tin
y, on
e-d
imen
sio
na
l lo
op. Lik
e an
in
finitely th
in
ru
bb
er b
an
d,
ea
ch
par
ticle
co
nta
ins
a
vib
ra
ting
, o
sc
illat-
ing
, da
nc
ing
fila
men
t tha
t p
hys
icis
ts
h
av
e na
med
a
str
ing
.
DIM
EN
SIO
N
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01
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Think of a guitar string that has been tuned by stretching the string under tension across the guitar. Depending on how the str ing is plucked and how much tension i s in t he s t r ing , d i f fer en t mu s ic a l no t e s w i l l be c r e a t ed b y t he s t r ing. T he se music a l notes cou ld be sa id to be exc i t at ion modes of that gu i t ar s tr ing under tension. + In a s imi lar manner, in str ing theor y, the elementary par t ic les we observe in par t ic le accelerators could be thought of as the “musical notes” or excitat ion modes of elementary str ings. + In str ing theory, as in guitar play ing, the string must be stretched under tension in order to become excited. However, the str ings in str ing theor y are f loat ing in spacet ime, they aren’ t t ied down to a gui tar. Nonetheless, they have tension. The str ing tension in str ing theor y is denoted by the quant i t y 1 /(2 p a’), where a’ is pronounced “alpha pr ime” and is equal to the squar e of the str ing length sc ale . + I f s tr ing theor y is to be a theory of quantum gravity, then the average size of a string should be somewhere near the length scale of quantum gravity, called the Planck length, which is about 10-33 cent imeter s, or about a mi l l ionth of a b i l l ionth of a b i l l ionth of a b i l l ionth of a centimeter. Unfortunately, this means that strings are way too small to see by current or expected particle physics technology and so string theorists must dev ise more c lever methods to test the theor y than just look ing for l i t t le str ings in particle experiments. + String theories are classif ied according to whether or not the strings are required to be closed loops, and whether or not the particle spectrum includes fermions. In order to include fermions in string theory, there mu s t be a spec i a l k ind o f s y mme t r y c a l led s uper s y mme t r y, w h ic h me an s for every boson (particle that transmits a force) there is a corresponding fermion (particle that makes up matter). So supersymmetry relates the particles that t r ansmi t for ces to the par t ic les that make up mat ter. + Super sy mmetr ic par tner s to to cur r ent l y k now n par t ic les have not been obser ved in par t ic le experiments, but theorists believe this is because supersymmetric particles are too massive to be detected at current accelerators. Particle accelerators could b e o n t h e v e r g e o f f i n d i n g e v i d e n c e f o r h i g h e n e r g y s u p e r s y m m e t r y i n t h e next decade. Ev idence for supersymmetr y at high energy would be compel l ing ev idence that str ing theor y was a good mathemat ical model for Nature at the smallest distance scales.
BASICPROPERT Y
T here are t wo basic t y pes of str ing
theor ies : those w ith closed str ing
loops that can break into open
str ings , show n above, and those
w ith closed str ing loops that can’t
break into open str ings ,
Ac
co
rd
ing
to
str
ing
theo
ry,
if we c
ou
ld
ex
am
ine th
ese
par
ticles
with
e
ven
gr
ea
ter
pr
ecis
ion
—a
p
rec
isio
n m
an
y o
rd
ers
of m
ag
-n
itud
e be
yon
d
ou
r p
res
ent
tech
no
log
ica
l c
apa
cit
y—w
e w
ou
ld fin
d
tha
t ea
ch
is
no
t po
intlik
e b
ut in
ste
ad
c
on
sis
ts
of
a tin
y, on
e-d
imen
sio
na
l lo
op. Lik
e an
in
finitely th
in
ru
bb
er b
an
d,
ea
ch
par
ticle
co
nta
ins
a
vib
ra
ting
, o
sc
illat-
ing
, da
nc
ing
fila
men
t tha
t p
hys
icis
ts
h
av
e na
med
a
str
ing
.
D i a m o n d
M o l e c u l e
H y d r o g e n A t o m
P r o t o n
S t r i n g
1 O C E N T I M E T E R S
I M I L L I M E T E R
1 0 P I C O M E T E R
The str ings of str ing theory are unimaginably small. Average string, if it exists, is about 10-33 centimeters long. That's a point fol lowed by 32 zeros and then a 1. It's a millionth of a bill ionth of a bill ionth of a bill ionth of a centimeter. (Physicists st ick to metr ic). Or think of i t th is way: i f an atom wer e magni f ied to the s ize o f the so lar system, a s tr ing would be the s ize of a tree. + Star t ing at an ever yday scale, we travel b y po w er s o f 10 0 do w n in t o t he sh ado w y w or ld o f s t r i n g s . T h a t i s, w e b e g i n 1 0 m e t e r s a w a y f r o m a n diamond, then zoom 100 times closer to 10 centimeters aw ay f r om i t s sk in , t hen 10 0 t ime s c lo ser ag a in t o one mil l imeter from its sk in, and so on, down no fewer than 15 addi t ional powers of 100 unt i l we reach the P lanck length. S ince the P lanck length is roughly 17 or der s of magni tude smal ler than w hat phys ic is t s c a n c u r r e n t l y d e t e c t u s i n g t h e i r l a r g e s t p a r t i c l e accelerators (in fact, to see individual strings we would need an accelerator the size of the Milky Way), we have taken a kind of visual poetic licence in imagining what the world looks like smaller than a quark. We hope you enjoy this journey into the infinitesimally itsy-bitsy.
THESIZE OF THE STRING
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S T R I N G S
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E V E R Y T H I N G
In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and in ter nal quantum number s, and v ice-ver sa. + So far, ther e is on l y indirect evidence for the existence of supersymmetry. Since the superpartners of the Standard Model part icles have not been observed, supersymmetry, i f i t ex ists, must be a broken symmetry, a l lowing the superpar t ic les to be heav ier than the corresponding Standard Model particles. + If supersymmetry exists c lose to the TeV ener g y sc ale, i t a l lows for a so lu t ion of the h ier ar chy pr oblem o f t h e S t a n d a r d M o d e l , i . e . , t h e f a c t t h a t t h e H i g g s b o s o n m a s s i s s u b j e c t t o quantum corrections which—barring extremely f ine-tuned cancellations among independent contr ibutions—would make it so large as to undermine the internal consistency of the theor y. In super sy mmetr ic theor ies, on the other hand, the contributions to the quantum corrections coming from Standard Model particles are naturally canceled by the contributions of the corresponding superpartners. Other attractive features of TeV-scale supersymmetry are the fact that it allows for the high-energy unif ication of the weak interactions, the strong interactions and electromagnetism, and the fact that i t provides a candidate for Dark Matter and a natural mechanism for e lectroweak symmetr y break ing. + Another advantage of supersymmetry is that supersymmetric quantum f ield theory can sometimes be solved. Supersymmetry is also a feature of most versions of string theory, though it can exist in nature even if string theory is incorrect.
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SUPERSYMMETRY
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Be for e t he 19 9 0 s, s t r ing t he or i s t s be l ie ved t her e were f ive dist inct superstr ing theor ies: open t ype I, c losed t y pe I , c losed t y pe I I A , c losed t y pe I IB, and the t wo f lavor s of heter ot ic s tr ing theor y (SO(32) a n d E 8 × E 8) . T h e t h i n k i n g w a s t h a t o u t o f t h e s e f i v e c a n d i d a t e t h e o r i e s , o n l y o n e w a s t h e a c t u a l cor r ec t theor y of ever y th ing, and that theor y was the one whose low energy l imit , w i th ten spacet ime d i m e n s i o n s c o m p a c t i f i e d d o w n t o f o u r, m a t c h e d t he phy s ic s ob ser ved in our w or ld t od ay. I t i s no w be l ie ved t h a t t h i s p ic t ur e w a s inc or r ec t and t h a t the f ive superstr ing theor ies are connected to one ano t her a s i f t he y ar e e ac h a spec i a l c a se o f s ome more fundamental theor y (thought to be M-theor y). These theories are related by transformations that are called duali t ies. I f two theories are related by a duality transformation, it means that the f irst theory c an be t r an s for med in s ome w ay s o t h a t i t end s up look ing just l ike the second theory. The two theories are then said to be dual to one another under that kind of transformation. Put dif ferently, the two theories are mathematically different descriptions of the same phenomena. + These dualit ies link quantities that wer e a lso thought to be separ ate. L ar ge and smal l distance scales, as well as strong and weak coupling strengths, are quant i t ies that have always marked very dist inct l imits of behavior of a physical system in both c lassical f ie ld theor y and quantum par t ic le p h y s i c s . B u t s t r i n g s c a n o b s c u r e t h e d i f f e r e n c e between large and small, strong and weak, and this is how these f ive very different theories end up being related. T-duality relates the large and small distance scales bet ween str ing theor ies, whereas S-dual i t y relates strong and weak coupling strengths between string theories. U-duality links T-duality and S-duality.
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I 10
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BOSONIC 26
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HO 10
Supersymmetry between forces and
matter, with closed strings only, no
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group symmetry is SO.
HE 10
Supersymmetry between forces and
matter, with closed strings only, no
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A n intr igu ing featur e of s tr ing theor y is that i t i n v o l v e s t h e p r e d i c t i o n o f e x t r a d i m e n s i o n s . T h e number of dimensions is not f ixed by any consistency criterion, but f lat spacetime solutions do exist in the so-called “critical dimension”. Cosmological solutions exist in a wider variety of dimensionalities, and these d i f f e r e n t d i m e n s i o n s — m o r e p r e c i s e l y d i f f e r e n t values of the “ef fect ive central charge”, a count of degrees of freedom which reduces to dimensionality in weakly curved regimes—are related by dynamical t r a n s i t i o n s . + O n e s u c h t h e o r y i s t h e 1 1-d imen s ion a l M-t he or y, w h ic h r eq u ir e s sp ac e t ime to have eleven dimensions, as opposed to the usual thr ee spat ia l d imensions and the four th d imension of time. The original string theories from the 1980s d e s c r i b e s p e c i a l c a s e s o f M - t h e o r y w h e r e t h e eleventh dimension is a very small circle or a line, and if these formulations are considered as fundamental, t h e n s t r i n g t h e o r y r e q u i r e s t e n d i m e n s i o n s . B u t the theor y also descr ibes universes l ike ours, w ith f o u r o b s e r v a b l e s p a c e t i m e d i m e n s i o n s, a s w e l l as universes with up to 10 f lat space dimensions, a n d a l s o c a s e s w h e r e t h e p o s i t i o n i n s o m e o f t h e dimensions is not described by a real number, but by a completely different type of mathematical quantity. So the notion of spacetime dimension is not f ixed in string theory: it is best thought of as different in different c ircumstances. + Nothing in Max wel l ’s theor y of electromagnetism or Einstein’s theory of relativity makes this kind of prediction; these theories require physicists to insert the number of dimensions “by hand”, and this number is f ixed and independent of potential energy. String theory allows one to relate the number of dimensions to scalar potential energy. Technically,
th is happens bec ause a gauge anomaly ex is t s for ever y separ ate number of predicted dimensions, and the gauge anomaly can be counteracted by including nontr i v ia l potent ia l ener g y in to equat ions to so l ve mot ion. Fur ther mor e, the absence of potential energy in the “critical dimension” explains why flat spacetime solutions are possible. + This can be better understood by noting that a photon included in a consistent theory (technically, a particle carrying a force related to an unbroken gauge symmetry) must be massless. The mass of the photon which is predicted by string theory depends on the energy of the string mode which represents the photon. This energy inc ludes a contr ibut ion from the Casimir effect, namely from quantum fluctuations in the string. The size of this contribution depends on the number of dimensions since for a larger number of dimensions, there are more possible f luctuations in the string position. Therefore, the photon in f lat spacetime will be massless—and the theory consistent—only for a particular number of dimensions. + When the calculation is done, the critical dimensionality is not four as one may expect (three axes of space and one of time). The subset of X is equal to the relation of photon f luxuations in a linear dimension. Flat space str ing theories are 26-dimensional in the bosonic case, while superstr ing and M-theories turn out to involve 10 or 11 dimensions for f lat solut ions. In bosonic string theories, the 26 dimensions come from the Polyakov equation. Starting from any dimension greater than four, it is necessary to consider how these are reduced to four dimensional spacetime.
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EXTRA DIMENSIONS
WHAT’SINSIDETHEBLACK HOLE? T
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In the abstr ac t theor et ic a l model o f b lack ho les, a b lack ho le is s tud ied as i f i t wer e the on l y th ing in the Universe. Using that approximation, the math of g e n e r a l r e l a t i v i t y b e c o m e s d o a b l e , a n d w e c a n make predict ions about black hole behav ior that are usefu l in under st anding the b lack ho les we see. In addi t ion, we lear n a lo t o f th ings about b lack ho les m a t h e m a t i c a l l y t h a t w e m a y n e v e r g e t a c h a n c e to w itness direct ly through obser vat ion. + In general relativity, the paths of light can be calculated f o r m a n y d i f f e r e n t d i s t r i b u t i o n s o f m a t t e r a n d energy using equations call the geodesic equations. The geodesic equations give us the paths that would b e f o l l o w e d b y f r e e l y - f a l l i n g t e s t p a r t i c l e s . F o r example, a basebal l af ter being hi t by Sammy Sosa and before being caught by an eager fan would be a freely fal l ing particle, travell ing on a geodesic path through spacetime. + L ight travels on geodesics p a t h s t h r o u g h s p a c e t i m e . W h e n t h o s e g e o d e s i c paths cross the event hor izon of a b lack hole, they never come back out . In ter est ingl y, in a Univer se w her e t he ener g y den s i t y i s ne ver neg a t i ve, t h i s behav ior of l ight leads mathemat ical ly to t wo very crucial properties of black holes: 1. The surface area of the event horizon of a black hole can only increase, never decrease. This also means that although two black holes can join to make a bigger black hole, one black hole can never spl i t in t wo. 2. The pul l of grav i t y at the event hor izon is constant; it has the same value everywhere on the event horizon. + Note that according to the f irst proper t y, i t is impossible for black holes to decay and go away, because a black hole cannot get smaller or spli t into smaller black holes. This is going to be changed when we add quantum mechanics to the theory. I f a black hole traps al l the l ight that crosses the event horizon, then how can we ever hope to observe one? + In the abstract theoretical model of a black hole, i t sits alone forever in the Universe let t ing us do math on it . In the Nature we observe, the Universe is f i l led with dust and gas in addition to stars, planets and galaxies. When dust and gas fall into a black hole, they can be sucked towards the event horizon so fast that the atoms are ionized and release bright light that esc apes w i thout cr ossing the event hor izon. + So the way astr onomer s and astrophysicists detect black holes in astronomical observations is to look for l ight fr om ion ized dust and gas be ing sucked in to something so fast that it could only be a black hole, not a normal gravitating massive object like a star. + Ho w e ver, t h i s br igh t l igh t c an be h ar d t o see, bec au se mo s t b l ac k ho le s a ls o a t t r ac t g i an t c loud s o f in t er s t e l l ar du s t t h a t h ide m an y o f t he ir fe a t ur e s, as shown on the prev ious page. The suspected black hole shown in the photo abo ve h a s a w ar ped du s t c loud ar ound i t , s o t h a t t he br igh t l igh t f r om t he ionized gas can be seen.
A NO T H E R DI M E N S ION ?
NO T H I NG ?
F U Z Z B A L L S ?
P O I N T
3 T H
S P A C E T I M E
11 T H
It seems fair ly likely that there was a Big Bang. The obvious question that could be asked to challenge or def ine the boundaries between physics and metaphysics is: what came before the Big Bang? + Physicists def ine the boundaries of physics by trying to describe them theoretically and then testing that description against observation. Our observed expanding Universe is very well described by f lat space, with critical density supplied mainly by dark matter and a cosmological constant, that should expand forever. + If we follow this model backwards in t ime to when the Universe was very hot and dense, and dominated by radiat ion, t hen w e h ave t o under s t and t he p ar t ic le phy s ic s t h a t h appen s a t s uc h h igh densities of energy. The experimental understanding of particle physics starts to poop out after the energy scale of electroweak unif icat ion, and theoretical phys ic is t s have to r each for models of par t ic le phys ic s beyond the St andar d Model, to Grand Unif ied Theor ies, supersymmetr y, str ing theor y and quantum cosmology.+ If we follow this model backwards in time to when the Universe was very hot and dense, and dominated by radiation, then we have to understand the particle physics that happens at such high densities of energy. The experimental understanding of part icle physics starts to poop out after the energy scale of e lectroweak uni f icat ion, and theoret ical physic ists have to reach for models o f p ar t ic le phy s ic s be y ond t he St and ar d Mode l , t o Gr and Un i f ied T he or ie s, supersymmetry, str ing theory and quantum cosmology. + A big complicat ing factor in understanding str ing cosmology is understanding str ing theor ies. Str ing theor ies and M theor y appear to be l imit ing cases of some bigger, more fundamental theor y. Unt i l that ’s sor ted out, any thing we think we know today is potentially up for grabs.
WHAT’SBEFORETHEBIG BANG?
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It seems fair ly likely that there was a Big Bang. The obvious question that could be asked to challenge or def ine the boundaries between physics and metaphysics is: what came before the Big Bang? + Physicists def ine the boundaries of physics by trying to describe them theoretically and then testing that description against observation. Our observed expanding Universe is very well described by f lat space, with critical density supplied mainly by dark matter and a cosmological constant, that should expand forever. + If we follow this model backwards in t ime to when the Universe was very hot and dense, and dominated by radiat ion, t hen w e h ave t o under s t and t he p ar t ic le phy s ic s t h a t h appen s a t s uc h h igh densities of energy. The experimental understanding of particle physics starts to poop out after the energy scale of electroweak unif icat ion, and theoretical phys ic is t s have to r each for models of par t ic le phys ic s beyond the St andar d Model, to Grand Unif ied Theor ies, supersymmetr y, str ing theor y and quantum cosmology.+ If we follow this model backwards in time to when the Universe was very hot and dense, and dominated by radiation, then we have to understand the particle physics that happens at such high densities of energy. The experimental understanding of part icle physics starts to poop out after the energy scale of e lectroweak uni f icat ion, and theoret ical physic ists have to reach for models o f p ar t ic le phy s ic s be y ond t he St and ar d Mode l , t o Gr and Un i f ied T he or ie s, supersymmetry, str ing theory and quantum cosmology. + A big complicat ing factor in understanding str ing cosmology is understanding str ing theor ies. Str ing theor ies and M theor y appear to be l imit ing cases of some bigger, more fundamental theor y. Unt i l that ’s sor ted out, any thing we think we know today is potentially up for grabs.
WHAT’SBEFORETHEBIG BANG?
T I M E
X
Y
A ccor d i ng to t he big ba ng model ,
t he u n iv er se de v eloped from a n
e x tremely dense a nd hot state.
Space itsel f has been e x pa nd i ng
e v er si nce, ca r r y i ng ga la x ies (a nd
a l l ot her matter w it h it .
B IGB A NG
1. C
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? 2.
WH
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3. IS
TH
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? 4. W
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AB
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?
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DIM
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DIRECTLY OBSERVING STRINGS IS FAR BEYOND OUR CAPABILITIES NOW AND FOR THE FAR FUTURE. STRING THEORY'S DIVERSIT Y MAKES IT DIFFICULT TO DERIVE ANY CLEAR PREDICTIONS THAT APPLY TO ALL VERSIONS. STRING THEORY HAS ITS SUPPORTERS AND ITS GAINSAYERS AMONG THEORETICAL PHYSICISTS. EVEN ADVOCATES ADMIT THAT THE THEORY COULD BE ENTIRELY WRONG.
IS ITREAL?
What is beyond question, is that even if one accepts the debatable reasoning of the staunch reductionist, principle is one thing and practice quite another. Almost everyone agrees that f inding string theory would in no way mean that psychology, bio logy, geology, chemistr y, or even physics had been solved or in some sense subsumed. The universe is such a wonderfully rich and complex place that the d iscover y of the f ina l theor y, in the sense we ar e descr ib ing her e, would not spe l l the end of science. + Quite the contrary: The discovery of string theory, the ultimate explanation of the universe at i ts most microscopic level, a theor y that does not r e l y on an y deeper ex p l an a t ion— w ou ld pr o v ide t he f irmest foundation on which to build our understanding of the world. Its discovery would mark a beginning, not an end. The ultimate theory would provide an unshak-able pi l lar of coherence forever assuring us that the universe is a comprehensible place.
AFRESH START
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WHETHER OR NOT STRINGS ARE VALID-ATED AS A "THEORY OF EVERYTHING," THEY PROVIDE A UNIQUE SET OF TOOLS TO UND-ERSTAND AND EXPLORE THE DEEP STRUCTURE OF REALITY.
INF
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PRODUCTION NOTES
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