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Imaginary Physics
Philip J. Carter
First published: 22 October 2012
Version 5: 8 December 2014
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$op%ri#ht & 'hilip $arter( 2012)2014
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Abstract
The ubi,uit% of comple! numbers throu#hout fundamental ph%sics has never been satisfactoril%
e!plained" -oreover( the mathematical primac% of comple! and ima#inar% numbers su##ests the
primac% of comple! and ima#inar% structures in .ature( /hile further impl%in# the e!istence of
ima#inar% spatial dimensions precedin# real dimensions" On this basis a consistent cosmolo#ical
frame/or is erected( #uided b% a direct readin# of the empirical and theoretical evidence( embracin#
essential principles of ,uantum theor%( relativit% theor%( and strin#-)theor%"
A Note on the Manuscript
This manuscript is e!cerpted from a more #eneral /or approachin# these principles from both ph%sical
and philosophical perspectives" The ph%sical ar#ument is presented here on its o/n terms for the timel%
benefit of those /orin# in the foundations of ph%sics"
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Contents
Introduction' The Problem o% Com#le0ity /
Part ) Real and Imaginary Dimensions
)) +ymmetry and com#le0ity *)1 2lements o% s#ace
)3 The structure o% real 3-s#ace ))
Part 1 Quantum Mechanics and Minkoski !"#pace
1) What is the uantum wave%unction4 )3
11 The enigma o% uantum nonlocality )/
13 +#ecial 5elativity and 6inkowski s#acetime )7
1. 2uclidean s#acetime and imaginary time )*
1/ The domain o% the wave%unction )8
17 9raneworlds )
1* The wave%unction is a wave o% what4 1!
18 :uantum nonlocality unveiled 1)
1 The transactional inter#retation o% uantum mechanics 13
1)! TI and the As#ect e0#eriment 1/
Part 3 $%tra Dimensions& #ymmetry& and 'ime
3) ;alu1? and the .-brane 31
37 SU>1? 0 U>)? and the /-brane 31
3* SU>3? and the 7-brane 33
38 +#atial motion and time 33
3 The /-dimensional wave%unction 37
3)! 5elativity and nonlocality in the three worlds 3*
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Part . Physics in the (")rane
.) "nitary evolution and state reduction .!
.1 The success and %ailure o% uantum %ield theory .)
.3 :uantum attributes and measurement .3
.. The wine glass analogy ..
./ @#erator eigenstates and state reduction ./
.7 +tate reduction and wave-#article duality .7
.* :uantum attributive %ields .8
.8 2nergy and time .8
. Charge and %lavor .
.)! The %our attributive %ields /)
Part / Nature*s +ields and the Platonic Polyhedra
/) :uantum s#acetime /1
/1 Continuous and discrete s#ace /3
/3 +#in and the double icosahedron /3
/. ,ield dualities /.
// The metric %ields //
/7 Charge and the tetrahedron /7
/* Dual symmetries /7
Part 7 A Conte%t ,or #tring 'heory
7) +-dualities and the Platonic solids /8
71 +tring %ields /
73 Ty#e II9 and the tetrahedron 7!
7. +#in strings 7!
7/ 6etric strings 7)
77 6-theory and the 7-brane 71
7* Plato=s Cave and the hologra#hic #rinci#le 73
Conclusion 7*
Notes and 5e%erences 78
9ibliogra#hy *!
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+igures
) ,undamental algebra o% s#ace )!
1 Cross #roduct o% vectors in imaginary s#ace )!
3 The uaternionic structure o% real 3-s#ace )1
. +im#li%ied schematic %or real 3-s#ace )1
/ The abstract wave%unction >momentum state? )3
7 6inkowski s#acetime )*
* The obective wave%unction )
8 The s#atial domain o% uantum mechanics )
6inkowski .-s#ace 1)
)! The inner structure o% 6inkowski .-s#ace 13
)) The wave%unction as a standing wave in 6inkowski .-s#ace 1.
)1 The As#ect e0#eriment in 6inkowski .-s#ace 1/
)3 +#atial dimensions o% the lower three branes 1
). Inter#enetrating s#aces 3!
)/ The tenth >intrinsic? dimension within real 3-s#ace 3)
)7 +#atial con%igurations o% the .-brane 31
)* +#atial con%igurations o% the /-brane 33
)8 +#atial con%iguration o% the 7-brane 33
) +#atial motion on a higher dimension 3.
1! The genesis o% time and energy 3/
1) The uni%ying /-dimensional wave%unction 3*
11 Null sur%ace in 6inkowski /-s#ace 38
13 The action o% the wave%unction .!
1. 2volution o% a uantum system in time ./
1/ The %ive Platonic solids /.
17 Dualities among the Platonic solids /.
1* The cube re#resenting a s#atial metric %ield //
18 Dualities and symmetries among the %undamental %ields /*
1 A cosmological conte0t %or string(6-theory 7)
3! A conte0t %or 6-theory 71
3) Bologra#hic inter#retation o% Ad+(C,T 7/
31 The hologra#hic #rinci#le reversed 77
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Introduction
'he Problem o, Comple%ity
The m%ster% of ima#inar% numbers can be appreciated b% mathematicians and nonspecialists alie" The
eminent mathematical ph%sicist ir 3o#er 'enrose introduces /hat he calls the ma#ic numberi( the
s,uare root of minus one( as follo/s: 617
Bow is it that ) can have a suare root4 The suare o% a #ositive number is always #ositive& and the suare o%
a negative number is again #ositive >and the suare o% ! is ust ! again& so that is hardly o% use to us here? It
seems im#ossible that we can %ind a number whose suare is actually negative
e can appreciate the earl% mathematicians callin# such numbers impossible and simpl% disre#ardin#
them /hen the% appeared" hile real numbers can represent some notion of quantityin our ph%sical
*91+ spacetime our three spatial dimensions and one time dimension each bein# measured in real
units there can be no such interpretation of ima#inar% numbers" $omple! numbers *combinin# both
real and ima#inar% numbers+ abound in fundamental ph%sical theor% and are reno/ned for their
po/erful and ma#ical properties( %et no satisfactor% ontolo#% of comple! and ima#inar% numbers has
come forth" .evertheless( so lon# as a calculation %ields real numbers( one doesn;t need to ,uestion ho/ima#inar% ,uantities can so po/erfull% model ph%sical phenomena one mi#ht " $ramer( the ori#inator of the Transactional ?nterpretation of
,uantum mechanics *T?+( sums up the problem of complexityin ,uantum mechanics as follo/s: 627
@ne o% the serious obections to +chrdinger=s early semiclassical inter#retation o% the +E Fstate vectorGH is that
the +E is a com#le0 uantity Com#le0 %unctions are also %ound in classical #hysics& but are invariably
inter#reted either >)? as an indication that the solution is un#hysical& as in the case o% the $orent1? as a shorthand way o% dealing with two inde#endent and eually valid
solutions o% the euations& one real and one imaginary& as in the case o% com#le0 electrical im#edance In the
latter case the com#le0 algebra is essentially a mathematical device %or avoiding trigonometry& and the #hysicalvariables o% interest are ultimately e0tracted as the real >or imaginary? #art o% the com#le0 variables Never in
classical #hysics is the %ull com#le0 %unction Jswallowed wholeK as it is in uantum mechanics This is the
#roblem o% com#le0ity
This dichotom% cannot be overstated" ?n a nutshell( modern ph%sical theor% is built upon nonph%sical
numbers( ,uantities havin# no representation in our 91 spacetime" hile /orin# ph%sicists appl%
comple! numbers on a dail% basis /ithout a second thou#ht( the more thou#htful stop to ponder" To set
the sta#e ? /ish to ,uote 3o#er 'enrose at some len#th( /ritin# here in conclusion to his masterful tome
The Road to Reality( the passa#e appearin#
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magic in the mathematics o% these numbers& but that Nature hersel% a##ears to harness this magic in weaving
her universe at its dee#est levels Let we may well uestion whether this is really a true %eature o% our world& or
whether it is merely the mathematical utility o% these numbers that has led to their e0tensive use in #hysical
theory 6any #hysicists would& I believe& lean towards this second view 9ut& to them& there is still something o%
a mystery needing some kind o% e0#lanation as to why the role o% these numbers should a##ear to be so
universal in the %ramework o% uantum theoryH To such #hysicists& the real numbers seem Mnatural= and the
com#le0 numbers Mmysterious= 9ut %rom a #urely mathematical stand#oint& there is nothing es#ecially more
Mnatural= about the real numbers than the com#le0 numbers Indeed& in view o% the somewhat magicalmathematical status o% the com#le0 numbers& one might well take the o##osite view and regard them as being
distinctly more Mnatural= or Mod-given= than the reals
,rom my own #eculiar stand#oint& the im#ortance o% com#le0 numbersH in the basis o% #hysics is indeed to be
viewed as a Mnatural= thing& and the #u
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Part -
Real and Imaginary Dimensions
ith the discover% that universal numerical patterns are embodied in #eometric forms( the >ree
#eometers understood numbers as emer#in# from the properties of space( so called Cuclidean )space(
havin# three real dimensions" C!tendin# this concept to the ima#inar% domain(
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To make visualisation even trickier& there is an additional subtle #iece o% geometrical structure that does not
e0ist when one is ust dealing with real numbers In the com#le0 #lane& multi#lication by the imaginary unit
corres#onded geometrically to a !-degree counterclockwise rotation In %our dimensions& there is not ust one
#ossible a0is o% rotation as in two dimensions& but an in%inity o% a0es one could imagine rotating about
counterclockwise by ! degrees The identi%ication o% the %our dimensions with two com#le0 numbers #icks out
these a0es' it is the a0is one rotates about when one multi#lies the two com#le0 numbers by the imaginary
unit +o& two com#le0 dimensions have both one more real dimension than one can visuali1? has a very s#ecial role that brought it into #lay %rom the earliest days o% uantum
mechanics It turns out& %or not at all obvious reasons >but then& not much about the geometry o% #airs o%
com#le0 numbers is obvious?& that the grou# SU>1? is very closely related to the grou# o% rotations in three real
dimensions 5otations o% three dimensions are e0tremely im#ortant since they corres#ond to symmetry
trans%ormations o% the three s#ace dimensions o% our real world& so #hysical systems can #rovide
re#resentations o% this grou# 6athematicians call the grou# o% rotations o% three-dimensional s#ace the grou#
o% s#ecial orthogonal trans%ormations o% three >real? variables and denote this grou# SO>3? The #recise relation
between SO>3? and SU>1? is that each rotation in three dimensions corres#onds to two distinct elements o%
SU>1?& so SU>1? is in some sense a doubled version o% SO>3?
ere /e mae contact /ith our real /orld( our three)dimensional space in /hich rotations come under
the la/ ofSO*+( /hich is in turn intimatel% connected toSU*2+" hile previousl% /e ma% have been
considerin# abstract mathematical spaces( here /e are talin# about ourspace( the space of our universe"
inceSO*+ rules over rotations in our three)dimensional space( /e infer that each of the three real
variables corresponds to a real spatial dimension in .ature" H% analo#%( the most direct inference is that
SU*2+( bein# in some sense a doubled version ofSO*+( also corresponds to spatial dimensions in.ature"
$orroboration of this con1? o% trans%ormations on two com#le0
variables Three-dimensional geometry thus has a subtle and non-obvious as#ect& since to really understand it
one must study not ust the obvious three-dimensional vectors& but also #airs o% com#le0 numbers These #airs
o% com#le0 numbers& or s#inors& are in some sense more %undamental than vectors @ne can construct vectors
out o% them& but can=t construct s#inors ust using vectors
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Once a#ain /e are reminded that the comple! numbers are more fundamental than the real numbers"
The e% point( ho/ever( is that threedimensional geometry has a su!tle and nono!vious aspect" ince
this conclusion is forced upon us b% both the theoretical and empirical evidence( /e ma% assume that
/hat oit calls three)dimensional #eometr% corresponds directl% to our ob
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+igure -/ +undamental algebra o, space
The essential paradi#m shift implicit in this model is that real space is not fundamentalL rather( /e
arrive at the some/hat radical conclusion that real spatial dimensions emer#e from the interaction
*product+ of ima#inar% dimensions" Technical readers mi#ht understand real dimensions as emer#in#
from the cross productof ortho#onal ima#inar% bases( /hile the spatial polarities mi#ht in turn be
understood in terms of handedness"
?n three real dimensions( the cross product %ields a vector perpendicular to the vectors bein# multiplied(
/ith ma#nitude determined b% the positive area of a parallelo#ram havin# sides defined b% the t/o
vectors" Fi#ure 2 illustrates this principle /hen applied to ortho#onal vectors in ima#inar% space" ince
the area of the rectan#le is a product of t/o ima#inar% numbers( the ma#nitude of the cross product is
real" C!tendin# this principle to space itself( each real dimension is a pro
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-.( 'he #tructure o, Real ("#pace
?n 184 the ?rish mathematician ir illiam 3o/an amilton /as focused on the problem of e!tendin#
the comple! plane to three)dimensional space /hen famousl% the solution came to him durin# a /al in
Dublin" ?n an event celebrated annuall% to this da%( he carved the follo/in# into the stone/or of
Hrou#ham Hrid#e:
i2
Mj2
M k2
M ijkM 1The formula represents the multiplicative rules for /hat amilton called quaternions( /hich he studied
and tau#ht for the rest of his da%s 657" Nuaternions ,uicl% found man% applications in ph%sics /ith #reat
success( before bein# lar#el% replaced b% alternate( more intuitive mathematical methods" ubse,uentl%
,uaternions remained out of vo#ue until bein# revived in the late t/entieth centur% for their utilit% in
calculatin# orientation and rotations in three)dimensional space" Due to their computational efficienc%
and immunit% to #imbal loc( toda% ,uaternions pla% an essential role in applications such as computer
#raphics( computer #ames and spacecraft control soft/are" hile there is no ,uestion that ,uaternions
provide an ele#ant and precise model of orientation and rotation in ph%sical space( mathematicians have
%et to full% come to terms /ith them" 3o#er 'enrose offers the follo/in# perspective: 6B7
F:uaternions haveG a very beauti%ul algebraic structure and& a##arently& the #otential %or a wonder%ul calculus%inely tuned to the treatment o% the #hysics and geometry o% our 3-dimensional #hysical s#ace Indeed&
Bamilton himsel% devoted the remaining 11 years o% his li%e attem#ting to develo# such a calculus Bowever&
%rom our #resent #ers#ective& as we look back over the )th and 1!th centuries& we must still regard these
heroic e%%orts as having resulted in relative %ailure
hat /as it about ,uaternions that so captivated amilton And /h% have the% not lived up to their
mathematical promise amilton described his discover% in a letter to a friend: 6G7
And here there dawned on me the notion that we must admit& in some sense& a %ourth dimension o% s#ace %or
the #ur#ose o% calculating with tri#lesH An electric circuit seemed to close& and a s#ark %lashed %orth
amilton;s e% insi#ht /as that /hen the comple! plane is #eneraliIed to three)dimensional space( the
resultin# mathematical ob
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dimensions into a positive real isotropic manifold( denoted b% the #reen trian#le" The arro/s represent
interactions bet/een the positive intrinsic dimension and the three ne#ative dimensions( as described
mathematicall% b% the cross product( pro
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Part
Quantum Mechanics and Minkoski !"#pace
Armed /ith our ne/found appreciation for comple! and ima#inar% numbers( /e can no/ loo s,uarel%
at some essential facts of ,uantum mechanics and relativit% theor% /hile demonstratin# that nonph%sical
theoretical models can provide consistent e!planations for previousl% une!plained empirical phenomena"
.- 2hat is the Quantum 2a0e,unction3
As pointed out b% =ohn $ramer *pa#e 5+( comple! numbers appear at the ver% heart of ,uantum
mechanics( in the mathematical description of the primar% ,uantum entit% no/n as the wavefunction"
state vector" or quantum state" Cver since the arrival of ,uantum mechanics the ontolo#% of the
/avefunction has been a sub
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From this perspective /e can understand =ohn $ramer;s statement that comple! functions are
s/allo/ed /hole b% ,uantum theor% *the problem of comple!it%+( since the primar% entit% of a
,uantum s%stem is itself a comple! function" Accordin#l%( the /avefunction is sho/n here as a dashed
line to emphasiIe that it is #enerall% considered an abstraction( a purel% mathematical construct( often
called apro!a!ility waverepresentin# no/led#e of the s%stem" 'enrose e!plains the illustration as
follo/s: 627
Thexdirection in my #icture corres#onds to some actual direction in ordinary s#ace& but the uand vdirections
are not ordinary s#atial directionsO they are #ut in to re#resent the com#le0 #lane o% #ossible values o% the
wave%unctionH To get thefull#icture o% these waves& we should have to try to imagine that this is going on in
all the three dimensions o% s#ace at once& which is hard to do& because we would need two e0tra dimensions
>%ive in all? in order to %it in the com#le0 #lane as well as the s#atial dimensions
Pie 'eter oit in the conte!t of s%mmetr% #roups( 'enrose clearl% does not consider the /avefunction;s
comple! plane to represent obiven that /e accept the assumption of preparation independence( /e have arrived
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at a cathartic moment for canonical ph%sics" o/ can /e reconcile a comple! /ave /ith an ob
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#rivate arrangement between our two #hotons When one is measured its twin is a%%ected& but no other
#article in the universe need beH The uantum connection de#ends on history @nly #articles which have
interacted with each other in the #ast seem to retain this #ower o% #rivate communication No classical %orce
e0hibits this kind o% e0clusivity
3 The uantum connection is %aster than light >instantaneous?H
The +#ecial Theory Fo% 5elativityG con%ers u#on light& or rather u#on the s#eed o% light in a vacuum& a uniue
role in the s#ace-time structure It is o%ten said that this s#eed constitutes an absolute #hysical limit whichcannot be broached I% so& then no relativistic theory can #ermit instantaneous e%%ects or causal #rocesses We
must there%ore regard with grave sus#icion anything thought to out#ace light The uantum connection
a##ears to violate this %undamental lawH
It is sur#rising that the communication between #articles is unattenuated and discriminating& but o%ten our best
counsel is sim#ly to acce#t the sur#rising things our theories tell us The s#eed o% the communication is another
matter We cannot sim#ly acce#t the #ronouncements o% our best theories& no matter how strange& i% those
#ronouncements contradict each other The two %oundation stones o% modern #hysics& 5elativity and uantum
theory& a##ear to be telling us uite di%%erent things about the world
The reader /ill appreciate /h% man% consider the foundations of ,uantum theor% to be the most
important and challen#in# problem in science" 'h%sics and philosoph% both stand be/ildered before the
facts" To find our /a% for/ard( let us tae a closer loo at pecial 3elativit%( /hich appears to contradict
the theoretical and e!perimental facts presented b% ,uantum mechanics"
.( #pecial Relati0ity and Minkoski #pacetime
Cinstein;s special theor% of relativit% provides a classic e!ample of the po/er of inference( tain# the most
direct readin# of the facts despite the philosophical conse,uences" The theor% is derived from
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+igure 7/ Minkoski spacetime
-ino/si spacetime connects our three spatial dimensions and one time dimension b% /a% of the
*in+ows+i metric( bein# the rule definin# displacementsin -ino/si spacetime" The metric is defined
in alternate /a%s( the most ph%sical formulation bein# as follo/s *measurin# from the ori#in+:
s2M t2x2 y2%2
/here trepresents time( andx( y( and%are the three spatial dimensions" The displacement s is
interpreted as the time e!perienced *or measured b% an ideal cloc+ /hile traversin# that particular
/orldline( or path throu#h spacetime" Displacements are real *s,is positive+ onl% /ithin the past and
future li#ht cones( these re#ions bein# no/n as timeli+e" An alternative formulation of the -ino/si
metric %ields positive *real+ displacements for spaceli+ere#ions( bein# those outside the li#ht cones(
denotedland e!pressed as follo/s:
l2M t29x29 y29%2
Displacements onthe li#ht cones are said to be lightli+e( /here both metrics become Iero hence the
term nullcone" On the null cone( the time contribution to the metric is e,ual and opposite to the
resultant space contribution( %ieldin# Iero net displacement in -ino/si spacetime" ence( time
e!perienced becomes Iero at the speed of li#ht( impl%in# that photons do not e!perience time" Further(
the spacelie displacement lalso becomes Iero at the speed of li#ht( impl%in# that a photon does not
e!perience space in its direction of motion or rather( there is no distance bet/een an% t/o points on its
/orldline" ence -audlin;s statement that pecial 3elativit% confers upon the speed of li#ht a uni,ue
role in the structure of spacetime"
.! $uclidean #pacetime and Imaginary 'ime
3eaders /ill note the resemblance bet/een both -ino/si metrics and the theorem of '%tha#oras in
four dimensions( onl% the si#ns bein# different" hen e!tended to four dimensions the '%tha#orean
theorem is no/n as the distance metric in Cuclidean 4)space" This resemblance inspired earl% relativit%
theorists to complete the analo#% b% tain# the time coordinate tto be ima#inar%( accordin# to tM iw.
1G
Time, t
Spacex
!imeli"e#isplacements
$nergy real%ass real
&orldline
'or(ig)
t
&orldline'o
r(ig)t Spaceli"e#isplacements
$nergy real%ass imaginary
(!acyons)
l=s=*
l=s=* %etric
ds= d tdxdydz
REAL
REAL%etric
d l=dt+dx+dy+dz
t
x
y
,-t-re
.# #
Past
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*As else/here( ima#inar% ,uantities are set in bold"+ The alternate formulation of the -ino/si metric
then becomes
s2M w29x29 y29%2
correspondin# to the distance metric for Cuclidean 4)space" ince wis ima#inar%( the first term becomes
ne#ative /hen s,uared( %ieldin# the spacelie metric for -ino/si spacetime" This procedure later led
to the so calledEuclideani%ationof spacetime" Accordin# to this scheme( the time coordinate is rotatedon the comple! plane into M it( /here is no/n as ima#inar% time 687" This scheme has been
particularl% successful in providin# consistent solutions /ithin the conte!t of 3ichard Fe%nman;s sum
over histories or path inte#ral formulation of ,uantum mechanics( as tephen a/in# e!plains: 6K7
To avoid the technical di%%iculties with ,eynman=s sum over histories& one must use imaginary time That is to
say& %or the #ur#oses o% the calculation one must measure time using imaginary numbers& rather than real ones
This has an interesting e%%ect on s#ace-time' the distinction between time and s#ace disa##ears com#letely A
s#ace-time in which events have imaginary values o% the time coordinate is said to be 2uclidean& a%ter the
ancient reek 2uclid& who %ounded the study o% the geometry o% two-dimensional sur%aces What we now call
2uclidean s#ace-time is very similar e0ce#t that it has %our dimensions instead o% two In 2uclidean s#ace-time
there is no di%%erence between the time direction and directions in s#aceH As %ar as everyday uantum
mechanics is concerned& we may regard our use o% imaginary time and 2uclidean s#ace-time as merely a
mathematical device >or trick? to calculate answers about real s#ace-time
?ma#inar% time has found man% po/erful applications in modern ph%sical theor%" ?n 1K8 a/in# and
=ames artle invoed the concept in a cosmolo#ical model no/n as the no boundar% proposal 6107(
/hile relativit% theorists and ,uantum field theorists routinel% pass into ima#inar% time to simplif% their
calculations" Tellin#l%( not all of ph%sics is captured in ima#inar% time( /hile no satisfactor% ontolo#% of
ima#inar% time has come forth" hat is ima#inar% time( and /h% does it appear in fundamental
ph%sics
.4 'he Domain o, the 2a0e,unction
Pet us tae stoc" On a strictl% lo#ical basis *the 'H3 theorem+ /e have surmised that the /avefunction isan ob
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manifestation( is re,uired to occup% a hi#her)dimensional space" hile a three)dimensional
representationorpro)ectionof the /avefunction ma% e!ist in our 91 spacetime( the complete entit% *as
formulated b% ,uantum mechanics toda%+ is e!tended in four spatial dimensions( one of /hich is
ima#inar%" ?n Fi#ure G the abstract comple! plane of Fi#ure 5 is replaced b% the real direction yand the
ima#inar% direction w( each correspondin# to spatial dimensions present in .ature" =ust t/o real
dimensions are depicted( of course( there bein# no /a% to sho/ the third *%+ real dimension"
+igure 8/ 'he ob9ecti0e a0e,unction
ince the /avefunction is spatiall% correlated /ith ph%sical phenomena in our 91 spacetime( the hi#her)
dimensional space is re,uired to interpenetrate our )space( as depicted in Fi#ure 8" hile the spaces are
delineated verticall% for clarit%( eep in mind that the t/o real manifolds are in fact superimposed" That
is( the three real dimensions of each space coincide the% are the same dimensions manifestin# in
distinct spatial manifolds"
+igure :/ 'he spatial domain o, 1uantum mechanics
.7 )raneorlds
The concept of !ranes*more precisel%(-!ranes+ provides an apt metaphor and a suitable mechanism
for these interpenetratin# spaces 6117" Cmer#in# unambi#uousl% from the mathematics of strin# theor%(
D)branes are essentiall% subspaces of a hi#her)dimensional space called the !ul+" The bul includes atotal of ten spatial dimensions *nine plus a tenth more subtle dimension+( /hile a )brane includes three
spatial dimensions( a 4)brane four dimensions( and so on( up to a ma!imum of nine" Accordin# to strin#
theor%( branes confine all fields e!cept #ravit%" That is( matter fields *therefore matter+ on branes cannot
lea into hi#her dimensions or into other branes( /hile #ravit% can travel freel% throu#h the bul"
The mathematical prediction of branes has led theorists to speculate that our universe could in fact be a
)brane" o called !raneworldscenarios t%picall% picture our )brane as one of man% *more or less
similar+ )branes floatin# in a hi#her)dimensional space( as strin# theorist Hrian >reene e!plains in his
1K
x
y
w/i/=y+
w
01Space IMAG
INARY
REAL
REAL
01space
.1space
x
x
y z w
y z
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boo The #idden Reality: 6127
FA three-brane that is enormous& #erha#s in%initely bigG& wouldfillthe s#ace we occu#y& like water %illing a huge
%ish tank +uch ubiuity suggests that rather than think o% the three-brane as an obect that ha##ens to be
situated within our three s#atial dimensions& we should envision it as the very substrate o% s#ace itsel% Qust as
%ish inhabit the water& we would inhabit a s#ace-%illing three-brane +#ace& at least the s#ace we directly
inhabit& would be %ar more cor#oreal that generally imagined +#ace would be a thing& an obect& an entity a
three-brane As we run and walk& as we live and breathe& we move in and through a three-brane +tringtheorists call this the braneworld scenarioH
In string theory there are more than ust three s#atial dimensions And a higher-dimensional e0#anse o%%ers
am#le room %or accommodating more than one three-brane +tarting conservatively& imagine that there are
two enormous three-branes Lou may %ind it di%%icult to #icture this I certainly do 2volution has #re#ared us to
identi%y obects& those #resenting o##ortunity as well as danger& that sit suarely withinthree-dimensional
s#ace Conseuently& although we can easily #icture two ordinary three-dimensional obects inhabiting a region
o% s#ace& %ew o% us can #icture two coe0isting but se#arate three-dimensional entities& each o% which could %ully
%ill three-dimensional s#ace
>reene maes the crucial point that t/o )branes could theoreticall% occup% the same )space /hilst
remainin# separate on a hi#her dimension" $onse,uentl%( since branes are transparent to #ravit%( /e
have the fascinatin# specter of t/o materiall% isolated /orlds seein# the same #ravitational field" Thispicture closel% reflects our model of interpenetratin# spaces( as demanded b% the comple! /avefunction
the difference bein#( of course( that the interpenetratin# brane is re,uired to be of hi#her dimension"
ithout further ado( this not bein# the moment to discuss the merits of strin# theor% or the ob
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Cinstein;s #eneral theor% of relativit%( a #ravitational /ave can be of an% fre,uenc% and travels at the
speed of li#ht" On this basis( the follo/in# con
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Fi#ure K depicts /hat ? /ill call*in+ows+i .space" *hile the metric for Cuclidean spacetime ma% be
considered Cuclidean( a space havin# an ima#inar% dimension certainl% could not"+ C!perts /ill
reco#niIe it essentiall% as Cuclidean spacetime( but /ith all four dimensions interpreted as spatial" There
is no time dimension in -ino/si 4)space"
Follo/in# the CuclideaniIation procedure( time is rotated on the comple! plane accordin# to:
wM it/here wis the fourth spatial dimension( /hich is ima#inar%" *As in the depiction of the /avefunction in
Fi#ure G( in Fi#ure K the ima#inar% dimension wis divided b% ito render it real( to remind us that
ima#inar% values cannot be depicted directl% on the #raphic+ -ino/si 4)space is home for the
comple! ,uantum /avefunction( /hich the reader can lucidl% illustrate b% ima#inin# Fi#ure G overlaid on
Fi#ure K /ith the respective dimensions ali#ned"
Technical readers /ho are familiar /ith -ino/si dia#rams depictin# -ino/si spacetime and
Cuclidean spacetime ma% have to retrain themselves to properl% interpret Fi#ure K" ?n particular( note
the follo/in#:
E First and foremost( eep in mind that all four dimensions are spatial" ince the /avefunction evolves
in time( clearl% an additional time dimension is re,uired( %ieldin# a 491 spacetime" This shortcomin#/ill be addressed in due courseL first /e must investi#ate the properties of -ino/si 4)space itself"
E hile the metric is unchan#ed from that of Cuclidean spacetime( all four terms are no/ spatial( /ith
the important conse,uence that the displacement scan onl% be interpreted spatiall%"
E ince all four dimensions are spatial( the orientation of a vector in -ino/si 4)space relates not to
velocit% as it does in spacetime( but to a particular #eometrical orientationor directionrelative to the
real and ima#inar% dimensions"
E hile the null cone is defined b% sM 0( eometricall%( it means that the len#th of an% 4)vector in -ino/si 4)space is dependent on its
orientation relative to the real and ima#inar% dimensions" hen the real and ima#inar% components
correspond( the vector has no len#th at all: hence it occupies
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possible ans/er is su##ested b% the inner structure of real )space" 3ecall that the three real dimensions
are pro
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absorbers( each of /hich in turn sends a confirmation /ave !ac+ in timeto the emitter" This implies
that the emitter receives the confirmation /aves at the same instant that it emits the offer /ave $ramer
describes the interaction as a handshae bet/een the emitter and absorber( occurrin# in /hat he calls
pseudo)time" The offer /ave is analo#ous to the /avefunction( /hile the confirmation /ave is an
attenuated con
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*thou#h clearl% not ph%sical time+" C!pressed another /a%( accordin# to the standin# /ave
representation of T?( pseudo)time is reall% a dimension of space( w" Follo/in# this line of reasonin#( it
could be ar#ued that T? requiresthe e!istence of -ino/si 4)space"
.-= 'I and the Aspect $%periment
Follo/in# our previous ar#ument su##estin# the presence of a 4)brane interpenetratin# our )brane($ramer;s independent line of reasonin# has brou#ht us to the same conclusion from the perspective of
,uantum phenomena rather than the ,uantum formalism" A 491 spacetime is re,uired to e!ist within
our 91 spacetime( coincident/ith our spacetime" To test out these ideas( let us see /hat /e can mae of
the Aspect e!periment *and similar e!periments+ in the conte!t of T? and -ino/si 4)space" For the
sae of e!pedienc%
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the ima#inar% spatial dimension has moved do/n/ards *thexa!is( correspondin# to the present
moment( has moved up+" Cmpiricall%( it is no/n that the /avefunction either ad
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Part (
$%tra Dimensions& #ymmetry& and 'ime
avin# come to terms /ith deep m%steries concernin# space and the ,uantum /avefunction( the reader
ma% be feelin# that /e have found the hol% #rail( that potentiall% all of ph%sics can be e!plained on the
basis of a comple! 491 spacetime interpenetratin# our ph%sical universe" ince pecial 3elativit% and,uantum mechanics meld so naturall% /ithin this frame/or( surel% this interpenetratin# space can
e!plain our ph%sical /orld Alas( a direct readin# of theoretical and mathematical evidence reveals that a
4)brane permeatin# our )brane is still not enou#h to contain all of ph%sics( nor to e!plain the nature and
ori#in of time"
(.- >alu?a*s 4"dimensional $instein"Ma%ell theory
?n 1K1K the >erman)'olish mathematician Theodor @aluIa made a remarable discover%" ?n those da%s
#ravit% and electroma#netism /ere the onl% forces no/n to ph%sics( and Cinstein;s success in describin#
#ravit% in purel% #eometric terms inspired efforts to inte#rate electroma#netism into a similar
frame/or" H% formulatin# Cinstein;s >eneral 3elativit% theor% in five dimensions *four real spatial
dimensions plus one time dimension+ @aluIa derived t/o sets of field e,uations( one bein# Cinstein;s
#ravitational field e,uations( the other bein# -a!/ell;s e,uations of electroma#netism" ?n a nutshell(
both #ravit% and electroma#netism /ere seen to emer#e from the #eometr% of empt% spacetime of a
hi#her dimension" Pe#end has it that the normall% reserved @aluIa danced about lie an ebullient
schoolbo%( convinced he had unified ph%sics" Pater he /rote that his mathematical result revealed
virtuall% unsurpassed formal unit%J /hich could not amount to the mere allurin# pla% of a capricious
accident" 617 @aluIa sent his /or to Cinstein( /ho /as impressed but for one #larin# detail: our ph%sical
universe appears to have
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chemical atoms /ould be unstable in a four)dimensional space( for instance" o /e are faced /ith a
m%ster%" hile the ele#ance and econom% of @aluIa;s theor% are undeniable( nobod% has been able to
mae it /or in the real /orld" .ine decades after its discover%( this beautiful mathematical result still
has not found its place in ph%sics"
@aluIa;s 5)dimensional Cinstein)-a!/ell theor% is essentiall% >eneral 3elativit% formulated in a 491
spacetime *havin# four real spatial dimensions and one real time dimension+( /hich %ields both #ravit%
and electroma#netism in 234 spacetime" Accordin#l%( the fourth spatial dimension( /hile real( is treated
differentl% from the first three dimensions" ?n a comprehensive revie/ of @aluIa)@lein >ravit%( ph%sicists
=" -" Overduin and '" " esson e!plain this distinction as follo/s: 647
;aluthe cylinder condition? on the coordinates& essentially barring the %i%th one a #riori %rom
making a direct a##earance in the laws o% #hysics
As a result of this mathematical slei#ht of hand( all fields *includin# #ravit%+ are confined to the first three
dimensions" 3o#er 'enrose( follo/in# a technical discussion about constraints upon the fourth spatial
dimension re,uired b% @aluIa;s scheme( and the need for a5illing vectorto impose U*1+ s%mmetr% on
the fourth dimension( adds: 657
All that one needs& in addition& is that the ;illing vector have a constant non-in %act negative? norm This
eliminates an unwanted scalar %ield& and the e0act .-dimensional 2instein-6a0well theory is thereby e0#ressed
Pet us pause to consider the profundit% of this result" e are not talin# about the prediction of
phenomena( but the derivation of fundamental ph%sical la/" ?ndeed( as @aluIa observed( the result
displa%s unsurpassed formal unit%( and /e share @aluIa;s vie/ that it could not amount to a capricious
accident" Accordin#l%( rather than tr%in# to shoehorn @aluIa;s theor% into our ph%sical /orld( /here
clearl% it does not belon#( /e acno/led#e that it must appl% to some other space( havin# properties
su##ested b% the theor% itself" e no/ it cannot appl% to our 91 spacetime( nor can it appl% to the 4)
brane( since in @aluIa;s theor% the fourth spatial dimension is real( in contrast to the ima#inar% fourth
dimension of -ino/si 4)space"
The c%linder condition imposed on the fourth spatial dimension /hich sin#les it out as different from
the other three has led to criticism that @aluIa;s theor% is arbitrar% and contrived( there bein# no
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+igure -(/ #patial dimensions o, the loer three branes
.ote that the first three dimensions of each brane constitute coincident manifolds all three manifolds
see the same #ravitational field and /aves" hile not appearin# directl% in the ph%sics of 91 spacetime(
the fourth *ne#ative real+ dimension lis everywhere presentin the 5)brane( as are the t/o ima#inar%
dimensions wand von a more fundamental level" ?n the conte!t of this model( @aluIa;s mathematical
treatment reflects a mechanism b% /hich positive real fields *includin# #ravitational fields+ are confined
to the three positive real dimensions" -oreover( as previousl% discussed( a real field cannot lea into anima#inar% dimension the% are of a different order" ?n each of these three branes( therefore( the need for
compactification disappears"
The alert reader /ill be asin# a crucial ,uestion" H% derivin# Cinstein;s #ravitational field e,uations in
the 5)brane( >eneral 3elativit% manifests spontaneousl% in our )brane( since each real manifold shares
the same space and therefore the same #ravit%" Clectroma#netism is another stor%( ho/ever( since fields
other than #ravit% are confined to a particular brane" o /e have derived an electroma#netic field ruled
over b% -a!/ell;s e,uations in the 5)brane /hile apparentl% havin# no contact /ith our )brane" hat
#ood is a derivation of electroma#netism that is confined to another /orld .ature indeed has an ele#ant
ans/er to this ,uestion( to be addressed as /e tae another spiral into the depths of natural la/"
(. $%tra Dimensions
uperstrin# theor%( incorporatin# a s%mmetr% frame/or no/n as supersymmetry*UR+( fi!es the
number of spatial dimensions at nine( /hile -)theor% reveals a tenth spatial dimension hidden in the
mathematics" .ote that these numbers are fixedb% the mathematicsL /ithin the current strin# formalism
the% cannot be more or less 6B7"
e have determined that at least three superimposed spaces are re,uired to account for no/n ph%sics:
1" Our ph%sical universe of three real spatial dimensions( understood as a )brane" This constitutes
ever%thin# /e can empiricall% no/"
2" A 4)brane( consistin# of three real dimensions coincidin# /ith our ph%sical )space( plus one
ima#inar% spatial dimension" This is home to the /avefunction *as currentl% formulated+"
" A 5)brane( consistin# of three real dimensions coincidin# /ith our ph%sical space( plus t/o ne#ative
ima#inar% dimensions" The t/o ima#inar% dimensions combine to pro
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same pattern continues into the hi#her dimensions" To account for the spatial dimensions as fi!ed b%
strin# theor% re,uires seven interpenetratin# spaces( the seventh bein# a K)brane"
+igure -!/ Interpenetrating spaces
Accordin# to this model( the seven branes occup% the same hi#her)dimensional space *the bul+( /ith
each brane e!cludin# spatial dimensions be%ond its o/n particular dimensionalit%" -ost importantl%(
rather than bein# staced as depicted in Fi#ure 14( the seven branes are superimposed" ?t follo/s that
correspondin# manifolds in different branes see the same #ravitational field( /hile all other fields are
confined to a particular brane" $onse,uentl%( the seven branes are materially isolated there can be no
interaction bet/een the matter of the various branes( meanin# in principle that /e cannot empiricall%
observe an% brane other than the )brane *our ph%sical universe+" hile ever%/here present( the hi#her
branes are forever inaccessi!leto our ph%sical senses and instruments" Onl% #ravit% *the #eometr% of
space itself+ is shared b% the various branes"
(.( 'he 'enth Dimension
hile superstrin# theor% re,uires nine spatial dimensions( the undisputed intellectual leader of strin#
theor%( Cd/ard itten( found a tenth spatial dimension loced up in his advanced mathematics /hich
had previousl% #one undetected b% *appro!imate+ perturbative methods" Hut this tenth dimension is not
lie the other nine" hile strin#s are re,uired to vibrate in nine dimensions( itten found that under
certain conditions a particular t%pe of strin# called the#eteroticEcould itself become e!tended in a
tenthdimension" Hrian >reene maes this distinction as follo/s: 6G7
FThe constraint o% nine s#atial dimensionsG arises %rom counting the number o% inde#endent directions in which
a string can vibrate& and reuiring that this number be ust right to ensure that uantum-mechanical
#robabilities have sensible values The new dimension we have ust uncovered is notone in which theBeterotic-2 string can vibrate& since it is a dimension that is locked u# within the structure o% the JstringsK
themselves
ithin our model of seven interpenetratin# spaces( the hi#hest space is a K)brane" here is the tenth
spatial dimension of -)theor% 3ecall that /e are countin# our three real dimensions as three( /hen in
fact the% consist of four ima#inar% dimensions" The tenth dimension is none other than the intrinsic
dimension hidden /ithin the three real dimensions ri#ht before us( %et so hard to see
0
!e :-l"
;1brane
1brane
91brane
01brane
.1brane
?*
;
9
0
Imaginary#imensions
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Accordin#l%( the Universe includes ten ima#inar% spatial dimensions( but /hen /e count our real space
as three dimensions( the total is nine" ?t follo/s that each brane includes one more ima#inar% dimension
than its desi#nation /ould su##est" .evertheless( to avoid confusion /e /ill continue to count our real )
space as three dimensions( /ith the implicit understandin# that the )manifold is constructed from four
ima#inar% dimensions"
+igure -4/ 'he tenth 5intrinsic6 dimension ithin real ("space
(.! +oundations o, #ymmetry3ecall the insi#htful /ords of 3o#er 'enrose *,uoted in the introduction+ su##estin# that t/o #reat
m%steries rise up above all others in the theoretical foundations of ph%sics( these bein# complexityand
symmetry" avin# reduced the problem of comple!it% to the presence of ima#inar% spatial dimensions(
on the basis of our interpenetratin# brane model /e are no/ read% to tacle the problem of s%mmetr%"
%mmetr% itself is not the problem( of course the problem is that /e don;t understand /h% certain
s%mmetries appear in fundamental ph%sics and not others" -ost of us relate the idea of s%mmetr% to
transformations in ph%sical space( such as under reflection in a mirror or b% spinnin# around full circle
and returnin# to our ori#inal state" H% comparison( the idea of s%mmetr% in modern ph%sics is abstract
and obscure" The #au#e s%mmetr% #roups appearin# in the standard model of particle ph%sics each
involve transformations of comple! variables( /hich are not so eas% to visualiIe nobod% no/s ho/ to
visualiIe an ima#inar% dimension( let alone a comple! one" Ret /e no/ that these s%mmetries e!ist in
.ature" The follo/in# three s%mmetr% #roups are of central importance to the standard model:
E SU*2+( the special unitar% #roup of t/o comple! variables( /hich rules over ,uantum spinphenomena"
E SU*2+ ! U*1+( combinin#SU*2+ /ith the unitar% #roup of one comple! variable( under /hich the
electroma#netic and /ea nuclear forces are unified as the electrowea+interaction"
E SU*+( the special unitar% #roup of three comple! variables( rulin# over quantum chromodynamics(
the theor% of stron# interactions involvin# ,uars and #luons"
?n 'art 1 /e ar#ued for the e!istence of ima#inar% spatial dimensions on the basis of theSU*2+ s%mmetr%
#roup and its role in ob
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(.4 SU56 and the !")rane
This implies( for instance( that theSU*2+ s%mmetr% #roup codifies transformations in an ob
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+igure -8/ #patial con,igurations o, the 4"brane
(.8 SU5(6 and the 7")rane
The B)brane introduces a third ima#inar% dimension( as sho/n in Fi#ure 18( allo/in# each of the
additional ima#inar% dimensions to ali#n itself /ith a real dimension" The resultin# spatial confi#uration
of three bound comple! dimensions is almost perfectl% s%mmetrical( but not ,uite" On the basis that
space includes a total of ten ima#inar% dimensions five positive and five ne#ative all five ne#ativedimensions are alread% accounted for in the 5)brane" ?n the B)brane( therefore( the additional ima#inar%
dimension is re,uired to be positive" ?t is clear that theSU*+ s%mmetr% #roup( so crucial to the standard
model( can in principle be supported b% this beautiful spatial confi#uration of the B)brane"
$learl%( the spatial structure of the B)brane represents a radical departure from that of the lo/er three
branes" hile the lo/er branes each include a real )manifold( the B)brane constitutes a complex)
manifold( havin# three comple! dimensions *represented b% the blue trian#le+" This distinction /ill prove
of utmost importance( since it imposes a natural dividin# line bet/een the lo/er three branes *each of
/hich includes a real )manifold+ and the hi#her branes */hich do not+"
+igure -:/ #patial con,iguration o, the 7"brane
(.: #patial Motion and 'ime
The attentive reader /ill have noted a #larin# omission in the current model" Thus far /e have been
discussin# the spatialdimensions of the lo/er four branes /ithout re#ard to time" Ret /e no/ that time
e!ists in our )brane( /hile the evolvin# /avefunction re,uires time in the 4)brane( and @aluIa;s
Cinstein)-a!/ell theor% is formulated in 491 spacetime( impl%in# time in the 5)brane" Finall% /e can
address this omission: /e no/ have all the pieces /e need to approach the puIIle of time"
=
+
+
= = ++
+++
= = ++
=
+
=
+
!ree real pl-s t2o
imaginary dimensions !2o bo-nd complexdimensions
x
+
One complexdimension
=
+
+
= = ++
+++
= +
!ree real pl-s treeimaginary dimensions
!ree bo-nd complexdimensions
+
=+
=
+
+
= = ++
+
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ithin this frame/or( time emer#es as a natural and unambi#uous conse,uence of t/o factors: the
spatial structures of the interpenetratin# branes( and the m%sterious phenomenon of spatial motion" ?n
an% particular space *brane+( time corresponds to motion of the entire space over a hi#her dimension"
This principle can be established on lo#ical principles alone" Fi#ure 1K depicts the reduced case of a t/o)
dimensional space *surface+ movin# alon# a hi#her *third+ dimension" At /hatever coordinate on the
hi#her dimension wthe t/o)dimensional space is located( the entire t/o)dimensional space is present"
On the other hand( at an% location in the t/o)dimensional space(
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renderin# the B)brane theprecursorto spacetime in the lo/er branes" The rotatin# circle represents the
ver% heart of the process( the en#ine room of ob
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ortho#onal ima#inar% motion evolvin# in ima#inar% time *t4+( %ieldin# real displacements" hat could
this mean e interpret these displacements as real ener#% manifestin# spontaneousl% throu#hout real
)space in the 4)brane( constitutin# a universal ener#etic field underpinnin# obubser describes as the gauge9string duality*no/n also as the Ad$FT correspondence( to bediscussed later+( as applied to heav% ion collisions: 6107
+ubseuent develo#ments seem to indicate that many as#ects o% heavy ion collisions have close analogies in
gravitational systems The gravitational systems in uestion always involve an e0tra dimension It=s not like the
e0tra dimensions o% string theory in its theory-o%-everything guise This e0tra dimensionH is not rolled u# It=s at
right angles to our usual ones& and we can=t move into it in the usual way What it describes is energy scale
meaning the characteristic energy o% a #hysical #rocess 9y combining the %i%th dimension with the ones we
know and love& you get a curved %ive-dimensional s#acetime
>ubser;s description closel% reflects @aluIa;s model in the conte!t of the current frame/or" A full
understandin# of Fi#ure 20 promises to e!plain the appearance of real ener#% in the 5)brane /hile
illuminatin# the relationship bet/een ener#% and time( so fundamental to ph%sics"
e are left /ith a foundational ,uestion: hat is the ori#in of these spatial motions hat moves the
ima#inar% dimensions vand w And /hat e!actl% is ima#inar% motion an%/a% hile a satisfactor%
ans/er to these ,uestions /ould tae us into cosmolo#ical and philosophical territor% be%ond the scope
of this paper( the reader mi#ht contemplate the follo/in# /ords from 'lato;s dialo#ue Timaeus: 6117
Now the nature o% the ideal being was everlasting& but to bestow this attribute in its %ullness u#on a creature
was im#ossible Where%ore he resolved to have a moving image o% eternity& and when he set in order the
heaven& he made this image eternal but moving according to number& while eternity itsel% rests in unityO and
this image we call time
(.< 'he 4"Dimensional 2a0e,unctionavin# established that each of the lo/er three branes includes a real )manifold and a time dimension(
alon# /ith energy( /e are brou#ht to a remarable conclusion" First( #iven these properties( each of these
branes can be considered an o!)ective world( containin# three)dimensional forms and processes evolvin#
in time" Further( since branes are transparent to #ravit%( the /avefunction appears identicall% in each of
the three *coincident+ real )manifolds /hile e!tendin# also into the ima#inar% dimension*s+ of the 4)
brane and 5)brane" hile real fields *includin# #ravitational fields+ are confined to the real )manifolds(
the /avefunction itself e!tends over all available dimensions( su##estin# that the canonical formulation
B
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of the /avefunction is incomplete"
The /avefunction;s fifth spatial dimension v/e mi#ht assume relates specificall% to the interface
bet/een the 5)brane and 4)brane( perhaps e!plainin# /h% its presence has not been missed in the )
brane" hile ,uantum mechanics provides consistent *stochastic+ results /ithout considerin# the v
dimension of the /avefunction( it seems reasonable to speculate that this fifth dimension ma% provide
the hidden variables re,uired to determine the outcome of a sin#le ,uantum event( be%ond stochastic
predictions over an ensemble"
The presence of the /avefunction on each dimension of the three lo/er branes #rants it profound
unif%in# po/er( as illustrated in Fi#ure 21" @eep in mind that each depicted /avefunction is the same
wavefunction appearin# in the three branes" From its humble be#innin#s as an insubstantial probabilit%
/ave( the /avefunction has become the cornerstone of unification in and of the three /orlds *branes+"
+igure -/ 'he uni,ying 4"dimensional a0e,unction
'erhaps the most startlin# conse,uence of this model is as follo/s:
E Accordin# to ,uantum theor%( each ob
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location in -ino/si 4)space( adherin# to a null cone /here the real and ima#inar% contributions to the
metric correspond( %ieldin# Iero displacement *spatial distance+ in the 4)brane" ince all four dimensions
are spatial( the null cone represents a particular orientationor directionrelative to the real and
ima#inar% dimensions" hile appearin# as an e!tended ob
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current model provides the basis for nonlocalit% on a more #eneral level" 3ecall from section 2"8 that
the propa#ation speed *phase velocit%+ of a particle /avefunction is #iven as c2/( /here /is the
velocit% of the particle itself" ?t follo/s that a li#htlie /avefunction adheres to a null cone in
-ino/si 4)space( /hile the /avefunction of a particle /ith rest mass *no/n as a de Hro#lie /ave+ is
confined to re#ions outside the null cone /hile travelin# faster than li#ht in the )brane *the
/avefunction of a particle at rest propa#ates at infinite speed+" Accordin#l%( an appropriate orientation
in the vdirection places the /avefunction of a massive particle on a null surface in the 5)brane" Onthis basis( #iven the spatial metric for -ino/si 5)space and the /avefunction propa#ation formula(
the correct PorentI transformation for mass ma% be derived( in accord /ith pecial 3elativit%"
E .ote that /avefunctions of massive particles are oriented in the vdirection *in the 5)brane+ /hile
li#htlie /avefunctions are not" >iven our previous conclusion that real ener#% ori#inates in the 5)
brane( /e find here a further clue re#ardin# the ori#in of mass( alon# /ith a possible mechanism
couplin# mass /ith curvature of 91 spacetime *or( of the 4)brane+( hence providin# a #eometrical
connection bet/een ,uantum mechanics and #ravit%"
To sum up( there are t/o levels of ,uantum entan#lement in our ob
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Part !
Physics in the (")rane
Pet us tae stoc of /hat has been accomplished thus far" First and foremost( the t/o problems sin#led
out b% 3o#er 'enrose as important but lar#el% unaddressed ,uestions of principle in our ph%sical
theor%( complexityand symmetry( have each been elucidated on the basis of ima#inar% spatialdimensions and interpenetratin# hi#her)dimensional spaces *branes+" Follo/in# from the 'H3 theorem
/e have surmised that the comple! /avefunction is an ob
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!. 'he #uccess and +ailure o, Quantum +ield 'heory
hen 'aul Dirac formulated his relativistic theor% of the electron in the 1K20s he /as led to a field
theor%( /hich subse,uentl% evolved into the first ,uantum field theor%( ,uantum electrod%namics" ince
then( ,uantum field theor% *NFT+ has become a cornerstone of fundamental ph%sics( as 3o#er 'enrose
e!plains: 617
:uantum %ield theory constitutes the essential background underlying the standard model& as well as #racticallyall other #hysical theories that attem#t to #robe the %oundations o% #hysical realityH
In %act& :,T a##ears to underlie virtually all the #hysical theories that attem#t& in a serious way& to #rovide a
#icture o% the workings o% the universe at its dee#est levels 6any >and #erha#s even most? #hysicists would
take the view that the %ramework o% :,T is Mhere to stay=& and that the blame %or any inconsistenciesH lies in the
#articular scheme to which :,T is being a##lied& rather than in the %ramework o% :,T itsel%
hat are these inconsistencies faced b% NFT ere /e sin#le out three issues /hich tend to be lar#el%
s/ept under the carpet in the /orin# lives of ph%sicists:
1" 3enormaliIation" NFT is mathematicall% inconsistent"
2" The #rossl% /ron# vacuum ener#% *space densit%+ calculation"
" Field proliferation"
NFT has the curious distinction of producin# the most accurate calculation ever in science *the ma#netic
moment of the electron( correct to ten si#nificant di#its+ and the /orst result in the histor% of science *the
vacuum ener#%( off b% some 120 orders of ma#nitude+ 627" -ean/hile( the dubious mathematical
procedure no/n as renormali%ation( re,uired to e!tract finite ans/ers from NFT( remains controversial
amon# ph%sicists and mathematicians alie" tephen a/in# and Peonard -lodino/ describe the
procedure as follo/s: 67
The #rocess o% renormali
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The picture of ph%sical realit% presented b% the standard model of particle ph%sics includes some si!t%
fields e!tended throu#hout 91 spacetime( /ith each particle species *alon# /ith its antiparticle+ bein#
understood as e!citations of a uni,ue ,uantum field( in addition to fields of a more substantial nature( as
.obel laureate Fran ilcIe e!plains in his boo The (ightness of &eing: 657
9esides the %luctuating activity o% uantum %ields& s#ace is %illed with several layers o% more #ermanent&
substantial stu%% These are ethers in something closer to the original s#irit o% Aristotle and Descartes they are
materials that %ill s#ace In some cases& we can even identi%y what they=re made o% and even #roduce little
sam#les o% it Physicists usually call these material ethers condensates @ne could say that Fthe ethersG
condense s#ontaneously out o% em#ty s#ace as the morning dew or an all-envelo#ing mist might condense out
o% moist invisible air
The most /idel% no/n of these material ethers is the#iggs condensate( /hile the best understood is
no/n as chiral symmetry!rea+ing condensate"The -ichelson)-orle% e!periment and Cinstein;s
special theor% of relativit% are /idel% re#arded as havin# done a/a% /ith the ether( but ilcIe points out
that Cinstein later chan#ed his mind on this issue" ?n fact( Cinstein himself claimed that >eneral
3elativit% is ver% much an ethereal *ether)based+ theor% of #ravitation: 6B7
According to the general theory o% relativity s#ace without ether is unthinkableO %or in such a s#ace there not
only would be no #ro#agation o% light& but also no #ossibility o% e0istence %or standards o% s#ace and time>measuring-rods and clocks?& nor there%ore any s#ace-time intervals in the #hysical sense
The ether measurin# space and time is no/n as a metric field( permittin# the notion of intervalsin
space and time" 'h%sicists spea of the metric field #ivin# ri#idit% to space and time( permittin#
consistent measurements of both" He%ond these conventional fields( variations of the standard model
*such as those incorporatin# supers%mmetr%+ add man% further fields" -ean/hile( strin# theor% */hich
incorporates the principles of ,uantum field theor%+ does no better( as strin# theorist Peonard ussind
e!plains: 6G7
The $aws o% Physics are like the Jweather o% the vacuumK& e0ce#t instead o% the tem#erature& #ressure& and
humidity& the weather is determined by the values o% %ieldsH +tring theory has an une0#ected answer to the
uestion o% how many %ields control the local vacuum weather ,rom the current state o% knowledge& it seemsthat it is in the hundreds or even thousands
?f ph%sics is indeed the pursuit of order and ele#ance in .ature( somethin# is clearl% /ron#" The problem
of field proliferation #oes be%ond the sheer number of fields permeatin# space and time:
E -an% of the fields of the standard model are near)duplicates" For instance( the ei#ht #luons are all
similar e!cept for their color char#e( %et each re,uires a uni,ue field( leadin# to field properties bein#
duplicated man% times over" This paints a ver% uneconomical picture of .ature"
E Discrete attributes match for the various fields" For instance( multiple particle species mi#ht share the
same precise electric char#e of 1 or spin one)half" The fact that attributes correspond or are
consistentl% related amon# the various fields implies a deeper order /hich informs each field *be it
ob
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o let us sum up" Despite its man% successes( NFT is no/n to be mathematicall% inconsistent" The
#rossl% /ron# vacuum ener#% calculations provide further notice that NFT does not paint a true picture
of .ature" 'erhaps most tellin# is the proliferation of fields populatin# the standard model( man% bein#
lar#el% duplicates( surel% settlin# the issue on the basis of econom% and aesthetics alone" Hut there is a
deeper reason /h% NFT does not and cannot correctl% reflect .ature: ,uantum field theor% is the lo#ical
outcome of brin#in# to#ether pecial 3elativit% and ,uantum mechanics in 234 spacetime"
e have alread% discovered that the holistic /avefunction occupies a 4)brane and a 5)brane( outsideour
91 spacetime( /here it coe!ists ver% happil% /ith pecial 3elativit%" ?t follo/s that NFT is a
mathematical abstraction representin# a limitin# case of a hi#her)dimensional process bein# shoehorned
into our ph%sical 91 spacetime" e /ill come across corroboration of this fact from an une!pected
source later on" Accordin#l%( the multitudinous fields of the standard model are no more than
mathematical artifacts the% do not e!ist ob
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The calculated amplitude relates to the probabilit% of an observed particle havin# that particular
momentum value" Attributes other than momentum are measured in an analo#ous fashion( but
appl%in# different harmonics( different sets of pure tones" Cach d%namic attribute *observable+
corresponds to a particular#ermitian operator( bein# a mathematical operation applied to the
/avefunction" Cach operator represents a uni,ue famil% of pure tones *no/n as eigenstates( from the
>erman /ord for selfor innate+ /hich form a complete !asis set( meanin# that an% reasonable
/avefunction can be represented as a linear superposition */ei#hted sum+ of these pure tones*ei#enstates+" Cach ei#enstate is associated /ith an eigenvalue( representin# the ph%sical value of the
correspondin# attribute" -easurement operators in ,uantum mechanics are #enerall% re,uired to be
ermitian *technicall%( selfad)oint+ due to their rather ma#ical propert% that the ei#enstates( /hile
themselves representin# comple! /aves( al/a%s have real ei#envalues"
To measure a particular attribute *observable+( one applies the correspondin# operator to the
/avefunction" ?n practical terms one is doin# harmonic anal%sis( /ritin# out the /avefunction as a
/ei#hted sum of operator ei#enstates" ?t is instructive to depict this mathematicall% as:
M c1W
19 c
2W
29 c
W
9 " " " 9 c
.W
.
/here represents the /avefunction( W1 W
.represent the operator ei#enstates *pure tones+( and c
1 c
.
are e!pansion coefficients *amplitudes+( /hich are comple! numbers" The squared modulusof a
particular coefficient *bein# the sum of the s,uared real and ima#inar% parts+ is proportional to the
probabilit% of that outcome occurrin#( the result bein# represented b% the associated ei#envalue" This
completes the measurement process 6K7"
!.! 'he 2ine lass analogy
For the sae of nonspecialists /ho ma% have found the previous section tou#h #oin#( the follo/in#
analo#% is offered" ince nothin# in .ature is perfectl% ri#id( ever% ob
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!.4 Bperator $igenstates and #tate Reduction
hile investi#ations into the ontolo#% of the /avefunction appear fre,uentl% in the literature( the
ontolo#% of the measurement operators is rarel% ,uestioned" Accordin# to the conventional vie/ the% are
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hile state reduction is #enerall% considered to be no more than a mathematical procedure( and hence a
purel% abstract process( /e have surmised from the 'H3 theorem and protective measurements that the
/avefunction is an ob
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the emitted /avefunction reflects the ei#enstate and null cone of the measured attribute"
tate reduction( then( is a real physical processreflectin# the e!citation of ob
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!.8 Quantum Attributi0e +ields
The concept of attributive fields fields manifestin# ,uantum attributes rather than particle species
brin#s /ith it a profound econom%" 3ather than havin# multiple fields identical but for their electric
char#e( for instance( the harmonics of one field can( in principle( account for the char#e of ever%
elementar% particle" Accordin#l%( both static and d%namic attributes are considered e!citations of
ener#etic attributive fields *ethers+"
o/ man% fields are re,uired to account for the observed attributes To ans/er this ,uestion /e must
briefl% address the notions of con)ugate wavesand con)ugate attri!utes" -athematicall%( a /ave is
transformed into its con
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a deep relationship bet/een these fundamental principles of .ature"
The notion of an energy operatorenters ,uantum mechanics in t/o different conte!ts" First( there is an
operator called the#amiltonian( representin# the total classical ener#% of a s%stem( meanin# both
inetic ener#% and potential ener#%" Of more relevance to us here is the ener#% operator /hich acts on the
/avefunction( measurin# thefullener#% of the s%stem( includin# both rest ener#% *mass+ and inetic
ener#%" The ener#% operator essentiall% measures the temporal fre,uenc% of a /avefunction b% appl%in#
harmonic anal%sis on the basis of temporal sine/aveforms 617"
Time( mean/hile( remains problematic" ?n an article on the time)ener#% uncertaint% relation( 'aul Husch
points out that time enters ,uantum mechanics in at least three different conte!ts 6147" First there is
externaltime or la!oratorytime( formin# part of the spatio)temporal environment in /hich e!periments
are conducted( /hile servin# as a parameter for input into theoretical models" Then there is intrinsic
time( /herein time is scaled to suit the temporal scale of the phenomenon such as a /ave)pacet
havin# a time unit e,uivalent to ho/ lon# it taes to traverse its o/n len#th" The third notion of time is of
most interest to us here( /hat Husch calls o!serva!letime( /hich essentiall% means treatin# time lie an%
other attribute b% appl%in# a time operatorto the /avefunction( a tas that has eluded the efforts of the
best researchers" ?n fact( in 1K1 olf#an# 'auli proved a famous theorem sho/in# that there is no self)
ad
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s#ace& each o% the color charges o% the Core >red& white& blue& green& and #ur#le? is re#resented by a se#arate
two-dimensional #lane >so there are /V1 R)! dimensions altogether? 9ecause there are rotations that move
any #lane into any other& the Core charges and symmetries get uni%ied and e0#anded in SO>)!?H
In the Charge Account& all the uarks and le#tons a##ear on an eual %ooting Any o% them can be trans%ormed
into any other They %all into a very s#eci%ic #attern& the so-called s#inor re#resentation o% SO>)!? When we
make se#arate rotations in the two-dimensional #lanes& corres#onding to the red& white& blue& green& and
#ur#le charges& we %ind in each case that hal% the #articles have a #ositive unit o% charge& hal% a negative unitH
2ach #ossibility %or combinations o% S and occurs e0actly once& subect to the restriction that the total number
o% S charges is even
The electric charges& which within the Core a##ear to be random decorations& become essential elements in the
harmony o% uni%ication They are no longer inde#endent o% the other charges The %ormula
L R )(3 >5 S W S 9? S )(1 > S P?
e0#resses electric charge more #recisely hy#ercharge in terms o% the others Thus the trans%ormations
associated with electric charge rotation turn each o% the %irst three #lanes through some common angle& and
turn the last two through 3(1 as big an angle& in the o##osite sense
Accordin# to ilcIe;s char#e account( the various char#es of the elementar% particles can each be
e!plained b% some combination of
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!.-= 'he +our Quantum Attributi0e +ields
On the fore#oin# basis it is proposed that the properties *attributes or observables+ manifested b%
elementar% particles can in principle be ascribed to the operation of
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Part 4
Nature*s +ields and the Platonic Polyhedra
$learl%( the four attributive fields are of a ver% special nature" .ot onl% must the% displa% harmonics
reflectin# those of the ,uantum measurement operators( but the entire s%stem of four fields must be
(orent% invariant*the la/s of ph%sics bein# the same in an% inertial reference frame+ as /ell as isotropic*the la/s of ph%sics are the same in ever% direction+" Further( the four fields are energetic*since the%
displa% ener#etic effects+ and ob
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Furthermore( since ima#inar% ,uantities cannot be represented in a real space( there can be no real
points in ima#inar% space *at least as /e understand the term in real space+" The su##estion is that
noncommutative #eometr%( /hen placed in the correct conte!t( ma% provide the mathematical machiner%
to describe the ima#inar% underpinnin#s of spacetime"
4. Continuous and Discrete #pace
e are faced /ith a dichotom%" hile lo#ic dictates that space( in its primar% nature( must be
continuous( technical considerations insist that it must be discrete" This dichotom% lies at the heart of the
problem of ,uantum #ravit%( but is resolved ver% simpl% under the current frame/or"
The unitar% evolution of the /avefunction( under the rule of chrSdin#er;s e,uation( is continuous in
space and time" There is nothin# at all ,uantiIed about the /avefunction( impl%in# that the
/avefunction is an oscillation of a continuum" avin# found that the /avefunction is a #ravitational
/ave( it follo/s that space itself is a continuum" Discrete phenomena enter into ,uantum mechanics onl%
upon the collapseof the /avefunction( /hereupon particles manifest attributes defined b% discrete field
ei#enstates( impl%in# that the attributive fields are themselves discrete" On these #rounds the follo/in#
principles are proposed:
E pace is a continuum"
E The ,uantum attributive fields are discrete"
The discrete nature of the attributive fields is most obvious for the char#e and spin attributes( bein#
confined /aveforms( each reflectin# orientationin space" The remainin# attributes constitute the metric
fields( each reflectin#positionin space *real and ima#inar%+ the spatial metric field *measurin#
position and momentum+ and the temporal metric field *measurin# time and ener#%+ /hich /e also
tae to be discrete" To#ether the four fields manifest all empiricall% observable phenomena in our )
brane( e!haustin# /hat /e can in principle measure" That is( from our perspective in the )brane( /e
do not and cannot observe space and time in their more fundamental *continuous+ natureL rather( /e
observe the effectsof space and time *the #ravitational /avefunction and spatial motions in the 4)brane
and 5)brane+ e!citin# the ,uantum attributive fields in our )brane" $onse,uentl%( since the metric fieldsare discrete( /hen /e loo to the limits of /hat can in principle be observed in our )brane( /e /ill
observe both space and time as discrete"
4.( #pin and the Double Icosahedron
avin# surmised that the four attributive fields are ob
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C!tendin# this idea( it is natural to as if each of the attributive fields mi#ht be characteriIed b% a
particular 'latonic solid" The demands of isotrop% and s%mmetr% in .ature /ould indeed su##est that the
#eometr% of the attributive fields must be re#ular( the five 'latonic solids bein# the onl% re#ular *conve!+
pol%hedra in three real dimensions"
+igure 4/ 'he ,i0e Platonic solids
4.! +ield Dualities
>eometers have lon# reco#niIed various dualitiesamon# the 'latonic pol%hedra" ?n simple terms( t/o
pol%hedra can nest to#ether to form a *concave+ re#ular pol%hedron onl% if one pol%hedron has the same
number of vertices as the other has faces" Accordin#l%( the tetrahedron is dual to itself( the cube is dual to
the octahedron( and the icosahedron is dual to the dodecahedron"
e be#in b% notin# that there are five re#ular pol%hedra and 'aces > vertices
+ =* 'aces * vertices
!etraedron
0 'aces
-be
> 'aces
Octaedron
< 'aces
#odecaedron
? 'aces
Icosaedron
* 'aces
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.aturall%( since the 'latonic solids are e!tended in three real dimensions the% are constrained to a space
of three real dimensions" ince real )manifolds e!ist onl% on the lo/er three branes( /e count a total of
si! real fields underpinnin# .ature the four attributive fields in the )brane( plus one field each in the
4)brane and 5)brane" Fortuitousl%( since the tetrahedron is dual to itself( /e thus have a total of si!
pol%hedra to account for si! fields( /ith the si! fields formin# dual pairs as sho/n in Fi#ure 2B"
The #eometr% of the 'latonic solids cannot be applied literall% to the fields( of course( since onl% the cube
uniforml% tiles Cuclidean )space( formin# /hat is no/n as the cu!ic honeycom!;There are also t/o
,uasi)re#ular tilin#s incorporatin# t/o re#ular pol%hedra( no/n as the tetrahedraloctahedral
honeycom!and the gyrated tetrahedraloctahedral honeycom!" ?ntri#uin#l%( ho/ever( an% finite
uniform pol%tope can be pro
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4.7 Charge and the 'etrahedron
e are left /ith
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it is fortuitous that such a connection e!ists /ithin the present #eometrical frame/or" hen t/o
tetrahedrons are superimposed as in Fi#ure 2B their intersection forms an octahedron( su##estin# a
subtle dualit% bet/een the dualtetrahedral fields and the octahedral ener#%)time field" ubtle indeed( it
is difficult to ima#ine that .ature /ould not mae use of this purel% #eometrical device"
Fi#ure 28 brin#s to#ether various elements of our discussion thus far( illustratin# the characteristic
#eometries( dualities( and inherited s%mmetries amon# the seven fundamental fields underpinnin#
.ature" .ote that the four attributive fields are arran#ed in a se,uence reflectin# that of their hi#her
duals" .ote also that the 'latonic solids are e!hausted under this model each re#ular pol%hedron and
dualit% is full% utiliIed"
+igure :/ Dualities and symmetries among the ,undamental ,ields
5G
3-Brane
B-ant-m3
ttrib-tive,ields(eters)
Spatialmetric 'ield
Position Momentum
SU(.)
!emporalmetric 'ield
ner!y Time
SU(.)
?
.
0
arge 'ield"har!e Ma!netic moment
SU() x U(?)
Spin 'ield
S#in $irection S#in ma!n%
SU()
Octaedron
!etraedron
Icosaedron
-be
Time
Space
4-Brane
#odecaedron
5-Brane
6-BraneCiger1dimensional
polytope
!etraedron
D
>1:rane 'ield
SU(.)
91:rane 'ield
SU() x U(?)
01:rane 'ield
SU()
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Part 7
A Conte%t ,or #tring 'heory
trin# theor% has been criticiIed for not bein# a theor% at all( but simpl% a mathematical frame/or /ith
no connection to the real /orld" Ret strin# theorists labor on( entranced b% the storied serendipit% of
the strin# formalism( convinced that such an ele#ant mathematical frame/or must relate in some /a%
to .ature" >iven that mathematics has preceded a scientific conte!t on numerous occasions throu#hout
histor%( /e ma% do /ell to tae heed" istoricall%( mathematicians often #et there first"
As strin# theor% matures( more ph%sicists *not necessaril% strin# theorists+ are /ei#hin# in /ith their
considered insi#hts re#ardin# a possible conte!t for strin# theor%" Follo/in# is the abstract from a recent
paper b% .obel laureate >erard ;t ooft( titled On the 0oundations of Superstring Theory: 617
+u#erstring theory is an e0tension o% conventional uantum %ield theory that allows %or stringlike and branelike
material obects besides #ointlike #articles The basic %oundations on which the theory is built are ama
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The e!istence of five consistent theories /as considered an embarrassment for strin# theorists until the
five theories /ere sho/n to be aspects of a more #eneral theor%( -theor%( encompassin# various
relationships amon# the five theories no/n as dualities" The dualities that concern us here are no/n as
strongwea+dualities( or )dualities( referrin# to coupling strength"/hich sets the stren#th of
interactions" A bi# problem in strin# theor% is that the couplin# constant is an unno/n parameter"
hile calculations assumin# /ea couplin# are relativel% strai#htfor/ard( calculations /ith stron#
couplin# brea do/n in all but a fe/ hi#hl% s%mmetric cases due to the limitations of the perturbativemathematical methods inherited from ,uantum field theor%" The )dualities provide a /oraround( since
calculations in one theor% /ith /ea couplin# can describe the same ph%sics as in a dual theor% /ith
stron# couplin#" trin# theorists are thus able to pass bac and forth bet/een dual theories to perform
calculations that other/ise /ould not be possible 627" The )dualities are as follo/s:
E T%pe ??H is dual to ?tself
E T%pe ? is dual to eterotic)O
E T%pe ??A is dual to eterotic)C *each in ten spatial dimensions+
The first t/o dualities are in nine spatial dimensions *plus one of time+( and each maps a theor% /ith
/ea couplin# to the dual theor% /ith stron# couplin#" For instance( the T%pe ??H theor% /ith couplin# g
/ill %ield ph%sics identical to the same theor% /ith couplin# 1g" imilarl%( T%pe ? /ith couplin# g/ill%ield the same ph%sics as eterotic)O /ith couplin# 1g( and vice versa"
The T%pe ??Aeterotic)C dualit% is more subtle" ?n each theor%( couplin# gis mapped to a tenth spatial
dimension of siIe g" The tenth spatial dimension is interpreted as the spatial e!tension of a strin# to %ield
a membrane( formin# the basis of -)theor% in eleven *1091+ dimensions"
e are in sufficientl% rarefied territor% that the subtlest of clues can prove crucial( and this basic picture
of the strin# theor% )dualities offers important #uidance" First( let us note that there are five strin#
theories and five 'latonic solids" This means little until /e reco#niIe that the pattern of )dualities
correlates /ith the #eometric dualities amon# the five 'latonic solids( as follo/s:
E Tetrahedron is dual to ?tself
E Octahedron is dual to $ubeE ?cosahedron is dual to Dodecahedron
$onsider that /e have noted a correspondin# pattern of dualities bet/een the five strin# theories and
pure geometryin three real dimensions nothin# less" ?s this pure coincidence
7. #tring +ields
Pet us assume that the correspondence holds" The implications are far)reachin#" ?n a nutshell( it implies
that each of the five strin# theories corresponds to a particular ,uantum attributive field or( even more
profoundl%( to a field outsideour /orld( in the 4)brane or 5)brane" The proposal is as follo/s:
E trin# theor% is the mathematical theor% of .ature;s fundamental fields"hat( then( are elementar% particles Are the% e!citations of tin% open or closed strin#s Or( are the%
composite e!citations of up to four ,uantum attributive fields Thus far /e no/
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On the fore#oin# basis it is not difficult to ima#ine a 'lanc)scale mesh or #rid( formed from one)
dimensional filaments of pre)material *strin# stuff+( e!tended throu#hout our real )space" Accordin#
to this model( the fundamental entities of strin# theor% turn out to be directl% related to the fundamental
fields both are strin#s" And( /hether tin% filamentar% loops or snippets( or filamentar% fields someho/
characteriIed b% re#ular pol%hedra( /e could e!pect the harmonics of these t/o varieties of strin#s to be
related" -i#ht the primar% entities of strin# theor% actuall% be filamentar% fields rather than isolated
open or closed strin#s $ould it be that strin# theorists have actuall% been stud%in# filamentar% fields allalon#( onl% approached in the /ron# conte!t ?n support of this con
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7.4 Metric #trings
e are left /ith the T%pe ??A and eterotic)C theories to describe the ener#%)time and position)
momentum fields( bein# the metric fields measurin# time and space in our )brane" Once a#ain /e are
#uided b% the directionalit% of the heterotic theor%" ?n our ph%sical /orld( space loos the same in ever%
direction( /hile time al/a%s flo/s in 1:rane 'ield
$
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E TheSO*2+ s%mmetr% #roup of the eterotic)O and T%pe 1 theories *the spi