Imaginary Physics v5

8/21/2019 Imaginary Physics v5

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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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Imaginary Physics

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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Imaginary Physics

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*

2


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Imaginary Physics

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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Imaginary Physics

+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

4


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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

+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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Imaginary Physics

-.( '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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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

14


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Imaginary Physics

at a cathartic moment for canonical ph%sics" o/ can /e reconcile a comple! /ave /ith an ob


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Imaginary Physics

#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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Imaginary Physics

+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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Imaginary Physics

*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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Imaginary Physics

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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Imaginary Physics

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


22/72

Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

possible ans/er is su##ested b% the inner structure of real )space" 3ecall that the three real dimensions

are pro


25/72

Imaginary Physics

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


26/72

Imaginary Physics

*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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Imaginary Physics

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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Imaginary Physics

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


29/72

Imaginary Physics

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


30/72

Imaginary Physics

+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


31/72

Imaginary Physics

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


32/72

Imaginary Physics

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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Imaginary Physics

(.4 SU56 and the !")rane

This implies( for instance( that theSU*2+ s%mmetr% #roup codifies transformations in an ob


34/72

Imaginary Physics

+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

+

=+

=

+

+

= = ++

+


35/72

Imaginary Physics

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(


36/72

Imaginary Physics

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


37/72

Imaginary Physics

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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Imaginary Physics

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


39/72

Imaginary Physics

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


40/72

Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

!. '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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

!.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


47/72

Imaginary Physics

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


48/72

Imaginary Physics

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


49/72

Imaginary Physics

!.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


50/72

Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

!.-= '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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

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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Imaginary Physics

.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


57/72

Imaginary Physics

4.7 Charge and the 'etrahedron

e are left /ith


58/72

Imaginary Physics

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()


59/72

Imaginary Physics

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


60/72

Imaginary Physics

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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Imaginary Physics

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

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Imaginary Physics v5