Chaos\ Solitons + Fractals Vol[ 8\ No[ 7\ pp[ 0334Ð0360\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Pergamon Printed in Great Britain \ 9859Ð9668:87 ,08[99¦9[99
PII] S9859!9668"87#99019!8
COBE Satellite Measurement\ Hyperspheres\ Superstrings and the Dimension of Spacetime
M[ S[ EL NASCHIE
DAMTP\ Cambridge\ U[K[
"Accepted 10 April 0887#
Abstract*The _rst part of the paper attempts to establish connections between hypersphere backing in in_nite dimensions\ the expectation value of dim E"# spacetime and the COBE measurement of the microwave background radiation[ One of the main results reported here is that the mean sphere in S"# spans a four dimensional manifold and is thus equal to the expectation value of the topological dimension of E"#[ In the second part we introduce within a general theory\ a probabolistic justi_cation for a com! pacti_cation which reduces an in_nite dimensional spacetime E"# "n# to a four dimensional one "# 2 "DTn3#[ The e}ective Hausdor} dimension of this space is given by ðdimH E ŁdH3¦f where f20:ð3¦f2Ł is a PV number and f"z4−0#:1 is the Golden Mean[ The derivation makes use of various results from knot theory\ four manifolds\ noncommutative geometry\ quasi periodic tiling and Fredholm operators[ In addition some relevant analogies between E"#\ statistical mechanics and Jones polynomials are drawn[ This allows a better insight into the nature of the proposed compacti_cation\ the associated E"# space and the Pisot!Vijayvaraghavan number 0:f23[125956866 representing it|s dimension[ This dimension is in turn shown to be capable of a natural interpretation in terms of Jones| knot invariant and the signature of four manifolds[ This brings the work near to the context of Witten and Donaldson topological quantum _eld theory[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
INTRODUCTION
With supergravity\ superstrings and supermembranes ð0Ł\ it is clear that the issue of dimensionality has become of crucial importance\ yet none of these theories have come anywhere near to giving a credible explanation let alone a mathematical derivation of why the world appears to us to be 2¦0 dimensional or why the alleged extra dimensions should spontaneously compactify ð0Ð4Ł[ There have of course been numerous attempts in the not so distant past to make our familiar 2¦0 space plausible using various heuristic arguments[ These could be divided into two main groups\ physical and mathematical[ An example for a heuristic physical explanation would be H[ Weyl|s argument that Maxwell|s theory exists only in 2¦0 space time dimensions[ The argument used by P[ Ehrenfest to reason\ based on stability considerations\ that only in 2¦0 could we have stable atoms and consequently chemistry and life belong to the same category[ An example for mathematical arguments would be those related to the property of Riemannian spaces and the number of independent components of the Weyl tensor where it can be shown that only the non!trivial conformal structure of three dimensions leads to the properties of Einstein|s gravity in four dimensional space time ð2\ 3Ł[ A relatively more recent mathematical argument which is somewhat more pertinent to the spirit of our present derivation is that of Fake R 3[ This important result in topology implies that
0334 0335 M[ S[ EL NASCHIE while for nÄ3 there is always a corresponding unique di}erentiable structure\ in n3 there are uncountably many such structures ð5\ 6Ł[ In several previous publications we have shown how we could derive the four dimensionality of spacetime based on pure geometrical and set theoretical considerations ð6Ð01Ł[ To do that we had to make one main assumption namely that micro space time is a hierarchical in_nite dimensional trans_nite set resembling the Ord!Nottale fractal spacetime or the so called Bethe lattice model of branched polymers ð6Ð01Ł[ Proceeding that way it was found that this space which is called E"# although strictly speaking in_nite dimensional\ has a _nite topological and
Hausdor} dimension equal to n3 and ðdcŁ3[125 respectively[ In addition the expectation 2 value of n is ½ðnŁ3¦f ðdcŁ where f"z4−0#:1 is the Golden Mean[ Consequently the e}ective core of E"# is basically a four dimensional {fractal| manifold[ In fact it can easily be shown that E"# is a random set and represents a four dimensional version of the one dimensional random Cantor set of Mauldin!Williams theorem ð09Ð01Ł[ 2 2 Noting the remarkable fractional representation of ½ðnŁðdcŁ3¦f 0:f 3¦"3¹#we can see that E"# seems to suggest an imaginative picture of a four dimensional space containing another smaller four dimensional space and so on ad in_nitum in selfsimilar fractal fashion[ In addition the entire fractal space is imbedded in a space with a topological dimension n3[ The _rst part of the present analysis attempts to develop an analogous picture to E"# however this time based on the classical result of the volume of n dimensional hypersphere[ The space which we will attempt to construct will be referred to as S"# space[ We will be showing that S"# has many features in common with E"# and few even more unexpected properties[ However it will be shown here that S"# may have a more direct connection to physics as will be argued using the results of the COBE satellite measurement of the microwave background radiation ð02Ł[ It has been noted frequently for instance\ by M[ Du} in ð0Ł that superstrings have as yet no answer to the question of why our universe appears to be four dimensional\ let alone why it appears to have a signature ð0^ 2Ł[ In the main part of the present work we intend to give an answer to this question[ In the course of doing that\ we will be utilizing and also discovering various analogies and some nontrivial relations between our probabolistic approach to {compacti_cation| and several other branches of current research in pure and applied mathematics\ in particular knot theory ð08Ł and non! commutative geometry ð19\ 10Ł[ Our main thesis is that the dimensionality of spacetime\ as occasionally speculated in the past by some notable scientists\ is a derivable property akin to temperature ð20Ł[ In fact we will be showing that the apparent dimensionality of our every day space time DT3 is derivable from more primitive assumptions and follow from the same law of statistical distribution used by M[ Planck to derive his well known formula for black body radiation[ This formula we may recall was at least historically speaking the beginning of quantum physics[ Naturally\ to undertake such analysis we will have to introduce some radical deviation from our classical notion of spacetime[ The most decisive point in that respects is the introduction of randomness and scale invariance to the very concept of spacetime geometry which inturn shows that spacetime loses it|s smoothness when we sharpen the resolution of observation as is essential for the micro spacetime of quantum physics[ This space time which we refer to as Cantorian space time E"# for obvious reasons have some remarkable properties ð10Ð14Ł[ First\ it is an in_nite dimensional hierarchical and random geometrical manifold with in_nite numbers of equivalent pathes "connections# between any two points[ Second\ any so called point in this space will always reveal a structure on a close examination\ so that strictly speaking the concept point does not exist in E"# which is a resolution dependent zoom space[ It then turned out that E"# is basically a form of noncommutative geometry[ The simplest and best studied example for such a geometry would be the famous Penrose tiling[ It is therefore not surprising that Penrose tiling was presented from the very beginning as a low dimensional example for E"# as well as NCG which obeys a noncommutative C !Algebra ð19Ł[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0336
Another point of importance is the connection between the basic formulas de_ning E"# and
Jones knot polynomial VL ð10Ł"e#[ In fact it turned out that a fundamental upperbound for VL is given by a formula which is a special form of the basic bijection formula of E"# and that the well known Hausdor} dimension of a quantum path dH1 may be derived from the properties of knots in R 3[ Finally we will show that the set of Penrose tiling and quasi crystallography provides a unique space where the classical and quantum description of spacetime meet ð10Ł"c#[ Adding the results of our S"# space analysis to that found in analysing the E"# space in connection with knot theory\ Noncommutative geometry and Penrose tiling\ we feel that our main thesis\ namely that whatever is generally perceived as strangely nonclassical quantum mechanical is in fact reducable to the nonclassical Cantorian!fractal geometry of spacetime itself is reinforced and justi_ed[
0[ PART I] THE SPACES S "# AND E"# 0[0[ Remarks to the mathematic of hyperspaces
To gain easy access to the present analysis it is essential to become _rst familiar with some basic facts related to the mathematics of hyperspaces[ Our approach to the subject is informal and di}ers from the customary one usually encountered in the relevant literature on geometric combinatorics ð06Ł[
0[0[0[ Metric and hypersurface of 2!`eometry[ We start from the well known metric of hyper! surface with positive curvature ð07Ł
ds1r1 ðdx¦sin x"du1¦sin u df1#Ł "0[0# To visualize the 2!geometry involved\ one imagines embedding the metric in four dimensional Euclidian space[ This is done by setting
vr cos x^ Zr sin x cos u "0[1# xr sin x sin u cos f^ yr sin x sin u sin f9 with 9¾x¾p\9¾u¾p\9¾f¾1p the volume of the hypersurface becomes
V "r dx#"rsin x du#"rsin x sin u df# gp g 3p1r1 sin x"r dx#1p1r21p1 "0[2# 9 where we have set r0 to obtain the unit sphere[ We recall for later use that the expression for the classical sphere in 2D Euclidian space is V3pr13p
0[0[1[ Generalization of the notion of a sphere[ It is useful to introduce the following notation and terminology] We denote the classical sphere as two spheres\ thus referring to the dimension of the surface so that we may write AS"2−0#A"S"1## to mean the surface area of the classical sphere[ On the 0337 M[ S[ EL NASCHIE other hand when we are referring to the volume V of the sphere\ then we write V"S"2## where n2 is the dimension of the space in which the sphere is embedded[ Let us generalize now the notion of sphere to higher and lower Euclidean space than the classical 2 dimensional one[ The _rst generalization we have already encountered by determining the surface area of a three dimensional hypersphere in four dimensions[ However we could regard the circle as a one dimensional sphere in two dimensional space[ The area of the circle pr1 would correspond then to the volume of this one sphere while the length of the circumference 1pr will correspond to the surface area[ In other words we have for n1 a unit sphere with
A"S "n−0##A"S "0##1pr1p and
V"S "n##"S "1##pr1p
For n2 we have a similar situation namely
A"S "n−0##"S "1##3pr13p and
3 3 V"S "n##V"S "2## pr2 p 2 2 we see there is a pattern here because
A"S "0##1V"S "1## and
A"S "1##2V"S "2## so one may guess that in four dimensions
A"S "2##3V"S "3##
Now we have found earlier on from our analysis of the hypersurface with positive curvature that V1p1[ That means A"S"2##1p1 and therefore\ if the above relation is correct then we must be able to deduce that
A"S "2## 1p1 p1 V"S "3## 3 3 1
More generally we should have
A"S "n−0##nV"S "n## "0[3#
Now in order to verify the correctness of these formulas we need to derive a general formula for determining the volume of n dimensional sphere[ This we do next[ Let us start by the following improper integral
A more conventional terminology is to speak of the unit ball rather than sphere when considering the volume[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0338
¦ e−x1 dx "0[4# g− Generalizing to two directions "dimensions# it is evident that
¦ 1 ¦ ¦ e−x1 dx e−x1 dx e−y1 dy 0g− 1 0g− 10g− 1
¦ 1 1 e−"x ¦y # dxdy g− Setting rzx1¦y1 one _nds
¦ 1 1p e−x1 dx e−r1 r dr d8 0g− 1 g g 9 9
0 1 1p − e−r $ 1 % 9 p That means
¦ e−x1 dxzp "0[5# g− Generalizing to n dimensions we obtain
¦ n "p#"n:1# e−xdx 0g− 1
¦ 1 1 1 −"x0¦x1¦===¦xn # e dx0dx1===dxn g−
− ¦ e−r1 rn−0dr dA "0[6# gA Sn−0# g " # 9 and since A"S"n−0##nV"S"n## one _nds that
¦ "p#"n:1#nV"S "n##g "e−r1 #"rn−0#dr 9 Setting yr1 one obtains
¦ n "n:1# n "n# −x −0 "p# V"S # e y01 1dx "0[7# 1 g 9 However we know that the gamma function is de_ned as
¦ G"n#g e−xxn−0dx "0[8# 9 Therefore we can write that 0349 M[ S[ EL NASCHIE
Table 0[ Numerical values for the volume V"S"n## of n!dimensional spheres and
the density of sphere packing D"n# using n!dimensional laminated lattices "see also Fig[ 0#[ The values for n13 correspond to the famous leech lattices[ The D"n# values are calculated using eqn[ "0[08# while V"S"n## are calculated using eqn[ "0[09#[
"n# "n# "n# n V"S # D"n# "V"S #D"n# h"n#n"V"S #D"n#
−0 9 * 9 9 90 0 0 9 01 0 1 1 1 2[0304 9[8958 1[738915 4[587941 2 3[0776 9[6393 2[09020 8[29282 3 3[8237 9[5057 2[93267 01[064027 4 4[1526 9[3541 1[33756 01[132255 5 4[0566 9[26184 0[816924 00[45107 6 3[61365 9[1842 0[28419280 8[655310 7 3[9476 9[14256 0[918175 7[12310 8 2[1874 9[03466 9[379480 3[214208 09 1[4490 9[98191 9[1235981 1[23598 00 0[7730 9[9477 9[009674 0[107524 01 0[2241 9[9306 9[9445667 9[5570425 02 9[80951 9[9173 9[9147505 9[22507 03 9[48815 9[9105 9[901833 9[070105 04 9[27033 9[9057 9[9953 9[985 05 9[12422 9[9036 9[99234 9[9441 06 9[039870 9[9977 9[990139 9[91097 07 9[971034 9[9948 9[9993735 9[9976117 08 9[935510 9[9930 9[999080 9[992518 19 9[914795 9[9921 9[9999714 9[99054 10 9[902838 9[9913 9[9999223 9[9996903 11 9[99626 9[9910 9[9999043 9[9992277 12 9[99270 9[9908 9[9999961 9[9999054 13 9[990818 9[99082 9[9999926 9[99977 S 36[229110 5[01640 08[62165 79[1351730
n n p"n:1# V"S "n##G 1 01 1 or n p"n:1#V"S "n##G ¦0 01 1 Thus we have the volume of n sphere in terms of the gamma function and the dimensions of the space n V"S "n##p"n:1#:G ¦0 "0[09# 01 1 The numerical evaluation of this formula is displayed in the second column of Table 0[ Using this result we can verify immediately that A"S"2## is given correctly by A"S"2##1p1r21p1 and that V"S"3## is indeed V"S "3##A"S "2##:n"1p1:3#"p1:1#3[8237[ Naturally the above equation holds for all n[
1[0[2[ The empty set and V"S"−0##[ In E"# space the truly empty set was found to be associated "−# "n# with n−and is given by d c 9 where d c is a Hausdor} dimension[ This is of course COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0340 di}erent from the classical Menger!Urysohn de_nition of the empty set which has a dimension n−0[ Following Table 0 it is interesting to see that for n9 the volume is non zero\ in fact it is unity V"S"9##0[ Consequently we would like to determine V"S"−0##[ To do that we start by the elementary fact that the volume of the unit sphere may be determined by integrating over "n−0# dimensional sections\ so that we may write
0 n−0 "n# "n−0# 1 0 1 V"S #1g V"S #"0−x # 1 dx "0[00# 9 Setting xsin u and dxcos u du leads to
p 1 V"S "n##1V"S "n−0##g cosnu du "0[01# 9 For n9 we have thus V"S "9## V"S "−0## "0[02# p 1 1g "cos u#−0du 9 Noting that V"S"o##0 and integrating one _nds
"−0# p:1 V"S #0:1ðln sec x¦tan xŁ9 0:1ðln"tan p:1#−ln"tan p:3#Ł 0 1 9 "0[03# Consequently the {empty| sphere S"−0# has a zero volume[ So in case of n³9 we have the identity V"S "−0##V"S "−1##[[[V"S"−##9 "0[04#
0[1[ The relation between A and V of hyperspheres
We have established the relationship
A"S "n−0##nV"S "n## It works as we have shown for the circle and the classical sphere[ For the circles we have n1[ Thus A"S "0##1V"S "1## and since V"S"1##1p which is the area of a circle\ then we _nd A"S"0##1p which is the length of the circumference of the circle[ The con~icting terminology is of course a little confusing at the very beginning[ For the classical sphere we have n2[ Thus
A"S "1##A of the sphere2V"S "2## 3 and since V"S"2##3p:2 is the volume of the classical sphere then one _nds A"S "1##2[ p3p 2 which is the surface area[ 0341 M[ S[ EL NASCHIE
For the hypersurface with positive curvature we have n3 and therefore
A"S "2##3V"S "3## Now following our Table 0 we have
V"S "3##p1:1 Thus
p1 A"S "2##3 1p1 1 which is the _rst result obtained earlier[ Next we go further and take n9[ Consequently
A"S "−0##"9#"V"S "9### which is always zero[ Thus we may write
A"S "−0##9 However we have already established that V"S"−0##9[ Consequently we must have
A"S "−0##V"S "−0##9 "0[05# Let us take n0[ Thus
A"S "9##"0#"V"S "0### Since V"S"0##1\ then
A"S "9##1 "0[06# That means A"S"9##1 but V"S"9#0 This con_rms the basic fact that in S"# we have for n³9
A"S "−0##===A"S "−##V"S "−0##===V"S "−##9 "0[07#
"n# 0[2[ The volume V of S and packin` density D"n#
Next we evaluate V"S"n## using eqn "0[09# as shown in Table 0[ We complement this Table with the values for maximum density of lattice sphere backing D"n# given in ð04Ł by
n "n# r V"S # n:1 n n D"n# p r G ¦0 zdet L "0[08# zdet L >0 01 11
Here rn is taken to be half the minimal distance between lattice points\ zdet LM and M is the generator matrix ð04Ł[ It may be instructive to compare eqn "0[09# and eqn "0[08# with the gamma distribution function
f"x#"0:G"r##lrxr−0e−lx "0[19# which was used to _nd ½ðnŁ3¦f2 ð4Ł[ Notice that in all the three expression we are dividing COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0342
5.263 ) ) n ( S ( V
1
0 5 16 Fig[ 0[ The shrinking volumes V of the unit n dimensional hypersphere S"n# as the dimension n increases[ Notice that for n9 we have V"S"9##0\ while for n: we have V"S"n##:9[ For n4 on the other hand V"S"n## attains it|s maximal value V"S"4##4[1526[ Notice also the strong resemblance between this _gure and the curve of the COBE measurement shown in _g 1[ All values of V"S"n## are taken from Table 0 and are calculated using eqn "0[09#[
by gamma[ In fact in the V"S"n##−n space the curve of the volume versus dimension shown in Fig[ 0 resembles the curve of a gamma distribution ð02Ł[ Inspecting Table 0 we note the following important facts]
0[ The largest unit sphere is V"S"4##4[1526 and for n: we have V"S"n##:9[
1[ The density for n3 is D"3#9[5057 which is very close to the Golden Mean[ This may indicate a connection to the so called space _lling condition ð4Ł[ 2[ For n4 the value of V"S"4## is only slightly larger than 4 so that this space is reminiscent of the condition of space _lling ð4Ł[ "n# 3[ The largest product "n#& "D"n## is 1[356 and corresponds to n3[ The largest product V &D"n# corresponds also to n3[ 4[ For n2 we have V"S"2##¹n¦03 while for n9 we _nd V"S"9##0[ 5[ For n7 we V"S "7##¹"n:1#3 "13# 6[ For n13 we have V"S #D"13#9[99082[ This is the famous leech lattice spheres packing[ 7[ The curve n−V"S"n## of Fig[ 0 resembles a gamma distribution[ It also has a striking similarity to the COBE curve ð02Ł of microwave background radiation "shown in Fig[ 1# to the extent that using appropriate units "x in cycles:centimetres# both curves attain a maximum at exactly 4[ Both curves start at a non zero value for x9 and y tends to zero as x tends to in_nity[ We speculate that there must be a deep relation between V of n dimensional spheres\ our space E"# and the thermodynamics of a blackbody[ The COBE curve would amount in this case to a possible experimental veri_cation of the in_nite dimensionality and hierarchical structure of real micro spacetime[
The preceding observations should also be compared with previous observations reported in ð4Ł[ "13# Another important observation apart of V"S #D"13# is that n4 is appropriate for fermions because having the largest V"S"n##\ it may be regarded in a sense to be what is needed for the {extra dimension| of the {Cantorian| Fermions superspace of E"#[ 0343 M[ S[ EL NASCHIE
1.2
1.0
0.8
0.6
Brightness units 0.4
0.2
0.0 2 4 6 8 10 12 14 16 18 20 Frequency (cycles / cm) Fig[ 1[ The variation of the intensity of the microwave background radiation with it|s frequency as observed by COBE satellite[ The observations display a perfect _t with the curve expected from pure heat radiation with temperature of 1[62>K[ Notice _rst the general likeness to the curve of the shrinking spheres of Fig[ 0 and second that the maximum of both curves in Figs 0 and 1 lies at the numerical value 4[ The correspondence holds\ of course\ only for the unit used in _g 1[ The resemblance is not a coincidence and goes back to the involvement of the gamma function in both curves "see eqns "0[08# and "0[09##[
It may also be possible using Table 0 to show that n3 is a mean value of all dimensions[ This would reinforce our earlier results for E"# namely that n3 and ½ðnŁ3¦f23¦"3¹#[ This and the agreement with the COBE curve of Fig[ 1 would give a rather strong indication of why it appears as if we were living in 3 dimensional spacetime only\ although all modern physical theories seem to require a far larger dimensionality ð0Ð2Ł[ Such analysis is the subject of the next section[
0[3[ The Cantorian space E"#\ the expectation value of the dimension of E"# and S "#
The basic concept and equations of the space E"# were introduced in some details on numerous previous occasions so that we may be con_ned here to a short introduction coupled with a summary of the main equations ð10Ð14Ł[ For simplicity we may start with Mauldin!Williams theorem which states that with a probability of one\ a randomly constructed triadic Cantor set "9# Sc will have a Hausdor} dimension equal to the Golden Mean[
"9# "9# dim S c d c f"z4−0#:1 "0[10# H
The question is now\ how could we lift this formula to n dimensions\ i[e[ what is the Hausdor} "n# dimension of Sc [ It turned out that\ a very simple formula is all that we need[ This formula is a generalization of the relationship between the triadic Cantor set dcln1:ln2 and the Sierpinski gasket ds0:dc0:"ln1:ln2#ln2:ln1 and is termed the bijection formula ð10Ð14Ł[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0344
"n# "n# "9# n−0 "9# n−0 dim S c d c "0:dim S c # "0:d c # "0[11# H H
Consequently\ in four dimensions\ Mauldin!Williams theorem would be
"3# "3−0# 0 2 dim S c "0:f# 3¦f 3[125956866 "0[12# H f2 where 0:f2 is known as a Pisot!Vijayvaraghavan number ð29Ł Next let us construct the E"# space ab initio[ "9# We do that using an in_nite number of Cantor sets Sc with all conceivable Hausdor} dimensions in the unit interval[ Suppose these sets are all {mixed| together in all possible forms of union and intersections to form one large space made of in_nitely many weighted dimensions[ Now we ask the following question[ What is the expectation value for the dimensionality of this formally in_nite dimensional space< To answer this question we need to know the distribution function according to which it was constructed[ Assuming a gamma distribution G\ the expec! tation value of this Gaussian distribution is ð10Ð14Ł[
EG "n#ðnŁr¹:l "0[13#
Setting the shape factor r¹1 and substituting for the mean value of a Poission distribution of "9# the elementary Cantor sets i[e[ lln"0:dc #inEG"n# one _nds
"# "9# "1# ðdim E ŁðnŁ1:ln "0:d c #dimH:ln "dimS c # "0[14# T H where H is the Hilbert space of Witten theory for two spheres with four points[ Expanding and retaining the linear terms only\ we obtain
"9# "# "9# "9# 0¦d c ½ðdim E Ł½ðnŁ"0¦dimS c #:"0−dimS c # "0[15# T H H "9# 0−d c
It was shown in previous work that the last expression is exact within a genuinely discrete and in_nite dimensional space ð13Ł[ Now to have a space _lling it is clear that we must satisfy the following condition ð13Ł
"# "n# "9# n−0 ½ðdimE Ł½ðnŁd c "0:d c # "0[16# T
It is an elementary matter to solve the above equation and show that it can be satis_ed if and "9# only if d c f"z4−0#:1 and n3[ In other words the expectation value for the dimensionality of E"# is identical to Mauldin! Williams theorem in four dimensions[ Consequently our E"# space although of in_nite dimen! sions it has an e}ective _nite expectation value for the topological dimension "n3# and appears therefore as if it were four dimensional[ This is a very similar situation to that of a Bethe lattice ð10Ł" f#[ It is not particularly di.cult to show that the same space has also an expectation value for the Hausdor} dimension given by
Equation "0[14# is identical to the equation derived by V[ Jones and A[ Connes q exp "21Z# when solving for 0:Z 1:q[ Consequently ðnŁ 1:q where q 0:d"z4¦0#:Z[ Again using the notation of the theory of subfactors it is easily shown that Jones| t−0 3 cos h1Z is nothing but our eqn "0[15# where one can show that −0 "9# "9# 2 −0 2 t dc ¦0:dc ¦13¦f [ Thus for q 0:fwe have t ½ðnŁ 3¦f [ 0345 M[ S[ EL NASCHIE
"# "# 0 0 ðdimE ŁðdcŁ½ðdimE Ł "0[17# H T "9# "9# 2 "0−d c #dc f The above equation will be shown in paragraph 3 to correspond to eqn "0[10# of the theory of subfactors ð19Ł[ We have also given elsewhere some arguments for the fact that the signature of spacetime must be perceived as "2\0# which we will not repeat here ð13Ł[ To sum up\ Cantorian spacetime E"# has in_nite dimensions but has an e}ective Hausdor} dimension dHðdcŁ¹3 while the topological dimension of the core is exactly 3[ We have thus reduced the in_nite dimension to a hierarchical four dimensional space[ The idea is very di}erent from the original three main methods of compacti_cation used in superstrings namely Toroidal compacti_cation\ orbifold compacti_cation and Calabi!Yau compacti_cation[ There a _nite internal manifold is left over after compacti_cation where the vibration of the strings take place[ Such a manifold could be a six dimensional orbifold for instance[ A detailed discussion of our approach to this point will be given elsewhere[ It seems therefore that we in the macro world are aware of the 3 dimensions only while the in_nite rest are very unlikely to be observed in a sense similar to the extremely tiny compacti_ed dimensions of all types of Kaluza!Klein theories[ It is important to note the remarkable con! tinuous fractial representation of ½ðnŁ namely
½ðnŁ3¦f23¦"3¹#0:"3¹# "0[18# It suggests an imaginative picture of a 3D fractal universe containing a much smaller 3D fractal universe and so on ad in_nitum[ We may mention that the preceding analysis was inspired by an old proposal due to A[ Wheeler[ For more details of the analysis and a discussion of it|s implications we refer the reader to ð10Ð14Ł[ 2 Finally applying the so called mean _eld approximation\ it is easily shown that ðdcŁ3¦f reduces to ð4\ 6Ł[
ðdcŁdH1$11&13 "0[29# Now in the case of the present space of hyperspheres S"#\ it is clear that the corresponding equations are far more complicated than in the case of E"# and a closed form exact expression for the expectations will not be easily obtainable[ For this reason we are forced to calculate our "# mean value of ³dimT S Ł numerically\ that is to say if it exists at all[ Luckily this value\ as will be shown shortly\ does indeed exist and is easily estimated numerically once the values of V"S"n## and D"n# are determined with su.cient accuracy[ In fact a glance at Table 0 will convince us that we can con_ne our numerical calculations to say n13 without a}ecting the accuracy of the results appreciatively[ The numerical procedure to be used here is elementary and amounts to nothing more than _nding a centre of gravity[ For instance following Table 1 we _nd in the case of the E"# space that the expectation value of n is given by
n 13 s n"nfn# sn1fn 9 9 ½ðdim E"#Ł½ðnŁ ¹ ¹3[151 "0[20# T n 13 s nfn snfn 9 9 which is a rather good approximation to the exact result ½ðnŁ"0¦f#:"0−f#3[1259568 "0[21#
"# In a similar way we can determine ðdimT S Ł using Table[ 0 and _nd that COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0346
Table 1[ Weighted moments of the dimensions in a Can! "# 1 n torian space E [ The expression n f corresponds to h"n# of the hypersphere space S"# given in Table 0[ The expectation value for n is given by 06[83316080 ½ðnŁSn1fn:Snfn¹ ¹3[125956866[ 9 9 3[125956866
nnfn n1fn
99 9 0 9[507922877 9[507922877 1 9[652821911 0[416753934 2 9[697192818 1[013500686 3 9[444958901 1[22325743 4 9[349738605 1[14313747 5 9[223257426 1[995100114 6 9[130981863 0[576549708 7 9[06917878 0[25120801 8 9[007399446 0[954594906 09 9[970295075 9[702950754 00 9[944163874 9[5979913728 01 9[926156328 9[336198164 02 9[913840812 9[213263888 03 9[905596266 9[121492177 04 9[909886950 9[05384481 05 9[996138550 9[004883473 06 9[99365946 9[979818692 07 9[992004153 9[945963655 08 9[991921291 9[927502643 19 9[990211028 9[915331672 10 9[999746872 9[907906538 11 9[999444402 9[901110185 12 9[999247820 9[997144323 13 9[999120366 9[994444337 S 3[196018325 06[82203762
n n13 "n# s h"n# s n"V"S #D"n# 9 9 79[134 ðdimS "#Ł ¹ ¹ ¹3[9567 "0[22# T n13 "n# 08[615 sk"n# s "V"S #D"n# 9 9 If it is not a surprising result to retrieve our familiar four dimensionality\ then eqn "0[22# is at least a welcome con_rmation of our intuition about the complex fractal!like nature of micro spacetime[ It seems that our _nite dimensional reality arises somehow through averaging and the complex brain process of visual and tangible perception out of an underlying in_nite dimensional surreality[ In fact Tables 0 and 2 could be also used to calculate the mean value ðV"S"n##Ł[ This turned out to be given by
n13 "n# s nV"S #D"n# 9 79[134 ðV"S "##Ł¹ ¹2[7¹3 "0[23# n13 10[1648 s nD"n# 9 This result is also quite interesting because of what may seem at _rst to be a slight contradiction[ 0347 M[ S[ EL NASCHIE
Table 2[
"n# 1 nnD"n# "V"S ## D"n#
99 0 00 3 1 0[70207 7[849104 2 1[1101 01[880764 3 1[3561 04[910573 4 1[215 01[789632 5 1[1266 8[848561 6 1[9560 5[481976 7 1[9125 3[0676062 8 0[2002 0[4748802 09 9[81 9[48739586 00 9[5357 9[19890303 01 9[4993 9[963239887 02 9[2581 9[91244998 03 9[2913 9[996645710 04 9[141 9[9913301050 05 9[1241 9[999700777 06 9[0385 9[999063705 07 9[0951 9[999928796 08 9[9668 9[9999978 19 9[953 9[999991017 10 9[9493 9[999999354 11 9[9351 9[999999002 12 9[9326 9[999999916 13 9[9352 9[999999996 S 10[1648 67[97642
Since ðdimS"#Ł3 means that the average sphere is four dimensional\ then one may have expected following Table 0 that the average volume should be around 4[ However we are getting a value around 3[ Disregarding the real possibility of rounding and numerical errors the result may be interpreted as follows[ Since a volume of about 3 is according to Table 0 the volume of a S"2# sphere\ then we could say that our average 3 dimensional sphere has the {volume| appearance of a three dimensional sphere[ In other words we could use this to justify the division of 3D into a "2¦0#D[ Finally we may mention another approximation to eqn "0[23# which is more accurate
n13 n13 "# "n# "n# "n# ðV"S #Ł s V "S #"V"S #D"n## s V"S #D"n# n9 >n9
67[97642:08[621¹2[846¹3 "0[24#
1[ PART II] THE E"# SPACE\ KNOT THEORY\ NONCOMMUTATIVE GEOMETRY AND RELATED SUBJECTS 1[0[ Knot theory and E"#
A knot is de_ned as a smooth!embedding of a circle in R 2 ð08Ł[ An important result in this subject is the discovery of the following index by Jones ð19Ł[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0348
1 Jind"1 cos p:r# "1[0#
It is easily veri_ed that for r4 we _nd Jind to be numerically identical to the Hausdor} dimension "2# "9# "# of a three dimensional Cantorian space Sc \ when the null set\ i[e[ the Kernel dc of E is taken "9# to be dc f[ This is because
1 Jind =rs"1 cos p:4# 1¦f "1[1# while the bijection formula ð10Ð14Ł[
"n# "n# "9# n−0 dim S c d c "0:d c # "1[2# H
"9# gives the same result for n2 and d c f"z4−0#:1 Another very important result also found by Jones is the following ð15Ł[ If an oriented link L is a closed n braid then
1pi:r n−0 =VLe =¾"1 cos p:r# ^ r2\3\[[[ "1[3# It is equally easily veri_ed that the right hand side of eqn "1[3# is nothing else but our familiar bijection formula of Cantorian Space E"# and that for n3 and r4 one _nds ð15\ 10Ł"e#
1pi:4 2 "9# 2 "# 2 =VLe =¾"1 cos p:4# "0:d c # ðdimE Ł3¦f "1[4# H
"9# "# when setting dc f[ Here we continue to think of n as the topological dimension of E while rK¦1 is a parameter where K can be thought of as the inverse of the Planck constant :[ Equation "1[4# gives thus the expectation value of the dimension of the in_nite dimensional hierarchical Cantorian space in this particular case which is ð10Ð14Ł[
2 ½ðnŁðdcŁ3¦"3¹#3¦f 1¦z4[ "1[5#
1pi:4 Note that in case of r4 and n2 we have =V L =9Jind where 9 means evaluated at the 1pi:r corresponding parameters[ Note also that for n3 we must have 0¾=VLe =¾7 with 0 "9# 1¾d c ¾0[
1[1[ Noncommutative `eometry "NCG# and Cantorian space time
Noncommutative algebra has for a long time been a basic tool in mathematical physics with the most important application in quantum mechanics[ The aim of noncommutative geometry is to extend the idea to algebraic geometry[ It then turned out that the spacetime concept of NCG and Cantorian spacetime "CST# have many common features ð10Ł"e#[ For instance one of the topological invariants of a certain very interesting noncommutative or {quantum| space X which is the dimension group\ is a subgroup of R generated by Z and the inverse of the Golden Mean 0:f0¦f"z4¦0#:1[ It followed then that a certain dimension ð19Ł[ dim"e#c"e# "1[6# can take only values in the subgroup ð19Ł 0 Z¦ Z "1[7# 0f 1
Following Ref[ ð19Ł it can be shown that there is a semigroup K9 such that 0359 M[ S[ EL NASCHIE
0 K ¦ "A# "n\m#$Z 1\ n ¦m−9 "1[8# 9 6 0f 1 7
It is clear from the last equation that for n1 and m0 we have 0 0 n ¦m1 ¦03¦f2 "1[09# 0f 1 0f 1
"3# "# which is identical to what we have obtained for d c ðdimH E Ł in eqns "0[12# and "0[18#[ Furthermore and following the theory of subfactors and the notation of ref[ ð19Ł\ it can be shown that the index ðM]NŁ of N in M ð19Ł
ðM]NŁdim"L1 "M## "1[00# N is also given by 0 dim"L1 "M## "1[01# N "0−l9#l9 where l9TrM"e#[ It is clear from the right hand side of "1[01# that this expression is nothing but the formula for the expectation value of the Hausdor} dimension of E"# namely ð10Ð14Ł
"# "9# "9# ðdim E ŁðdcŁ0:ð"0−dc #dc Ł "1[02# H and we just need to set
"9# "9# l9dim S c d c f"z4−0#:1 "1[03# H in order to _nd again our by now familiar number
0 0 2 ðM]NŁðdcŁ 3¦f "1[04# "0−f#f f2
"9# It should be noted that minimizing ðdcŁ leads to dc 0:1 and
"# ðdcŁ min 3ðdim E Ł min[ H
Another interesting correspondence between the formalism of E"# and that of NCG is evident from looking at the expression for ¹t as given on page 48 in ð19Ł[ There we see that setting qf or 0:f in
¹t"1¦q¦q−0#−0 "1[05# leads to
¹tf2 "1[06# and consequently
2 0:t¹ðM]NŁðdcŁ1¦z4 3¦f "1[07# A similar result is also obtained for Jones polynomial for tf[ That means
This form of Jones polynomial is given on p[ 064 in The Knot Book by C[ Adams\ Freeman\ New York "0883#[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0350
−3 −2 −0 2 V"K#"t −t ¦t #=f3¦"3¹#ðM]NŁ0:t¹1¦z4 3¦f
1[2[ Connections to four manifolds
To illustrate the connections between the geometry of four manifolds and E"# consider _rst the signature c ð5Ł cb¦−b− "1[08# where b¦dim H ¦ and b−dim H − are the dimensions of the maximal positive and negative subspaces of the form H1 respectively ð5Ł[ Setting
b¦dim KerE"#f "1[19# H and
b−dim Coker E"#0−ff1 "1[10# H one _nds from eqn "0[17# that ð3Ł"e#
¦ − ¦ − 1 1 2 cðb −b Łfðb &b Łff−f f = f f "1[11# Recalling that
cðM]NŁ−0 "1[12# then
dim"L1 "M##ðM]NŁ0:cðdimE"#Ł3¦f2 "1[13# N H
This is exactly the same result obtained earlier on ð10Ł[ It is worth mentioning here that in ð11Ł it was found that the Fibonacci numbers play a role in the topology of four manifolds[ The importance of the PV number\ 1¦z43[12595\ seems to extend far beyond quasicrystallography where it found one of it|s _rst physical application ð10Ł"e#[ The most important conclusion so far however is that Jones| knot invariant has a natural interpretation in terms of dimensionality of E"# and the signature of four manifolds[
1[3[ Quasicrystallo`raphy and E"#
One of the surprises which we have encountered recently is that the expectation value of the e}ective dimensionality of E"# namely ½ðnŁ3¦f21¦z4 as well as it|s inverse f2 occurs in connection with the theory of quasicrystallography[ In ref[ ð17Ł it was found that the Z!module which carries di}raction pattern possesses certain symmetry which is invariant through a group of homotheties ð17Ł[ Noting that Shur|s lemma entails under certain conditions that Lp>¦L=p_ "1[14# for any real numbers L and L?\ one _nds that
Mf−2p>−f2p_ "1[15#
_ where p> and p are certain projective matrices ð17Ł and 0351 M[ S[ EL NASCHIE
−2 2 f 3¦f 1¦z4ðdcŁðM]NŁ "1[16#
1[4[ Penrose universes\ NCG and E"#
One of the most important results in noncommutative geometry is undoubtedly the conclusion by Connes that the Penrose space X represents in e}ect an example of a low dimensional noncommutative space ð19Ł[ To appreciate the importance of this result in nonclassical physics we need just recall that while classical mechanics obeys commutative algebra\ quantum mechanics in the Heisenberg! Born formalism is manifestly a problem in noncommutative analysis[ Thus\ Penrose universe is a unique medium where classical and nonclassical physics meets[ The objective of this section is to show the intimate connection between Penrose universe and Cantorian spacetime E"#\ an undertaking which may lead to a resolution of many paradoxes in quantum physics\ by reducing them to the nonclassical trans_nite nature of the ambient micro spacetime of quantum and subquantum particles[ To see how important such an undertaking is as well as the importance of noncommutative geometry it may be su.cient to mention that it is very likely that A[ Einstein|s opposition to quantum mechanics stem from the fact that all forms of geometries known to him at that time were commutative[ The noncommutative quantum theory of Heisenberg and Bohr may have therefore remained obscure to him because it did not _t into his basically geometric thinking[ In other words it may be reasonable to suppose that had Einstein known about the possibility of noncommutative geometry he would most probably modify his attitude towards quantum mechanics ð19Ł[ With the experimental discovery of quasicrystals which possesses the supposedly forbidden _ve fold symmetry\ the subject of nonperiodic tiling of space acquired a prominent place in mathematical physics\ particularly in the context of the work of R[ Penrose and H[ Conway on Penrose tiling ð19Ł[ The recent work of A[ Connes ð19Ł on noncommutative geometry added a new dimension to the importance of nonperiodic tiling by realizing that X is an example of noncommutative geometry[ The starting point of this realization is to look at the case of measured foliations with continuous dimension ð19Ł[ The general von Neumann projection of the foliations gives a sort of a random Hilbert space[ This is of course reminiscent of Mauldin|s theorem and our Golden Mean theorem ð10Ð15Ł[ It then turned out that the space of Penrose tiling of the plane does indeed give a very clear geometrical intuitive picture of what a leaf of foliation could generally look like[ The reason for that is simply the following ð19Ł[ When analysing X using classical tools\ it appears to be pathological as observed by Connes[ However when we replace the commutative C !Algebra with a noncommutative C !Algebra we _nd that X is readily analysed ð19Ł[ This is a clear indication of the inherently quantum mechanical nature of the space X[ In this sense we understand this space as an example of noncommutative geometry of a low dimensional noncommutative space ð19Ł[ The preceding reasoning yield then two quantitative results which imply quite unexpected connections of the Penrose space to knot theory and Cantorian spacetime[ The _rst result is that X has a natural subfactor which is identical to Johns| index ð19Ł
1 Jind"1 cos p:4# "1[17# This was discussed earlier on as an important quantity in knot theory ð08Ł[ The second point is that Jind is itself numerically identical to the Hausdor} dimension of a three dimensional Cantorian space which represents a generalization of Mauldin|s theorem ð4Ł to three dimensions using the so called bijection formula COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0352
0 1 d "2# "0:d "9##"2−0# 1¦fJ "1[18# c c 0f 1 ind These are by no means the only indications that the Penrose universe can be seen as a realization of projection of a three dimensional Cantorian space with
dim KerE"#dimðf of E"##f "1[29# H
To explain this point we suppose we have a circular region in X of a diameter r[ Suppose further that one is transferred to a randomly chosen parallel Penrose universe[ Then we ask the following question] How far do we need to travel from our initial circular region in order to end in another circular region which can match the initial one< The answer is that we have to travel a distance 3¦f2 l ¾ r"1[007922878===#r "1[20# r 0 1 1 This is the essence of the local isomorphism theorem[ This theorem is of course trivial for a periodic pattern\ but the Penrose universe is nonperiodic and here lays the _rst surprise[ The second surprise is that "3¦f2#:11[007922878 is exactly equal to half of 0:f2 which is the exact expectation value for the dimensionality of the hierarchical Cantorian universe E"#[ As men! tioned earlier on several times this essentially in_nite dimensional universe has an e}ective 2 Hausdor} dimension ðdimH E Ł3¦f 3¦"3¹# and a topological dimension for the e}ective
{core| of exactly DTn3 as can again be seen immediately from the bijection formula introduced earlier on
"n# "9# n−0 dim S c "0:dim S c # "1[21# H H when setting n3 and
"9# "9# dim S c d c f H
"3# "3# "9# 2 2 "# 2 dim S c d c "0:d c # "0:f# ðdim E Ł3¦f "1[22# H H in agreement with our earlier discussion[ It is worth remembering and stressing again that the distribution used to obtain these results is the same distribution used to derive the formula of black body radiation[ In this sense dimensionality seems to share indeed some essential features with temperature as anticipated by D[ Finkelstein ð20Ł[
1[5[ Knot theory and the Hausdorff dimension of a quantum path
As mentioned earlier on a knot is by de_nition a smooth!embedding of a circle in R 2 ð08Ł[ A circle C0 is of course a dim c00 geometrical object[ It can be deformed to a true knot only in a space S2 where dim S22[ Consequently the co!dimension must be
Codim C0dim S2−dim C0 2−01 "1[23#
3 There are however no {real| knot in R for which dim S33 because all knots are equivalently trivial and dissolve in this higher dimensional space[ It turned out that the only valid way to 0353 M[ S[ EL NASCHIE generalize the concepts of knot theory to higher dimensions is to keep the Codim1 constant[
Consequently the geometrical object corresponding to the circle must have a dim C11[ Therefore in R 3 we must have
Codim C13−11 "1[24# In other words to have a knot in R 3 we must have
dim C1Codim C11 "1[25# The next point to be reasoned is based on the hypothesis that all particles and all interactions between particles in micro space are manifestation of fractal!like knots in the {fabric| of Cantorian spacetime E"# at an appropriate resolution[ Now since E"# is e}ectively four dimensional by virtue of dim C ½ðdim E "#Ł 1 −04[12595−03¦f2 T dim Coker E "# ¹1&11$13 "1[26# then it follows that we could not have a knot\ a particle and consequently a particle path in E"# unless dim C11 and that means
"1# ðd c Łdim C1Codim C11 "1[27# In this sense a Cantorian spacetime sheet must fall back on itself and thus form an e}ective four dimensional knot[ That is basically why a quasi continuous connection in E"# is essentially a "1# path with a Hausdor} dimension ðd c Ł1 ð10Ð14Ł[ In this context two more side remarks are in order[ First we recall the Frish!Wasserman!DelbruÃck conjecture which states that the prob! ability for a randomly embedded circle to be knotted tends to 0 as the length of the circle tends to in_nity[ For a {fractal| circle\ the length of any part is in_nite provided the circle appears to be continuous at the corresponding resolution of observation and if the FWD conjecture is correct then a fractal circle is everywhere knotted[ Furthermore\ FWD implies also that a self! avoiding polygon must be knotted and since in 3 dimension a polygon is naturally self!avoiding this means our {circles| in E"# must be knotted[ We will regard all form and particle interactions as a manifestation of these trans_nite!fractal knots as indicated earlier on[ Finally we may recall that the universe as a whole may be regarded in some speculative models as a knot complement[ "1# This concludes our topological justi_cation of ðd c Ł1[
1[6[ The so called wave!particle duality\ NCG and E"#
The aim of the present section is to show that the indistinguishability theorem of Cantorian spaces E"# ð10Ł" f#
$ & VVf2\ f"z4−0#:1 "1[28# is conceptually homomorphic to a wave!particle interpretation of the index theorem of Toeplitz operators ð18Ł Ind"T" f##−ððhŁf ðTŁŁ "1[39# In fact\ it can be shown in an elementary fashion that our indistinguishability theorem "equ[ 1[28# is derivable from the index theorem of equ[ "1[35# which we do next[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0354
Let f be a complex continuous and never vanishing periodic function on R with a period 1p\ then we could write
f"x#einx¦c"x# "1[30# where n could be interpreted as a winding number] 0 1p n ð f?"x##:"fx#Łdx "1[31# 1ip g 9 for the closed path gf"T# "1[32# in the complex plan given by the image TR:Z "1[33# The winding number can be also expressed as 0 n dZ h "1[34# gg 1ipZ gg where the closed 0!form h de_nes a de Rham cohomology class in the _rst cohomology group ð18Ł Similarly and since g is a closed path\ and therefore de_nes a homology class we have ðgŁf ðTŁ[ Consequently using de Rham duality\ n can be written as
& ncn−ððhŁf ðTŁ−ððhŁðgŁŁ "1[35#
The right handside of the above equation "1[35# is thus analogus to the Born formula and re~ects therefore the homological structure of the SchroÃdinger wave quantization[ Next let us state the wellknown result that the Toeplitz operator T " f # is Fredholm ð18Ł[ Since a Fredholm operator admits a _nite dimensional Kernel and co!Kernel we can state the well known formula Ind"T#dim"Ker"T##−dim"Coker"T## "1[36# This formula is clearly analogus to the index of four manifolds when we admit continuous dimensions ð10Ð14Ł Now we can state the index theorem Ind"T" f ##−ððhŁðgŁŁn "1[37# In words\ this means\ the index of a Toeplitz operator T"f# is equal to the winding number n of f[ However since the left handside of "1[36# and "1[37# are a winding number in terms of an operator\ it is analogus to the essentially particle picture of the Heisenberg!Born quantization formalism[ Consequently we may write
$ ncnInd"T# "1[38#
Next we like to give a derivation of the index theorem of eqn "1[39# using a purely formal analogy between the formalism of the index theorem and that of Cantorian spaces E"#[ This procedure which holds for knot theory\ noncommutative geometry as well as four manifolds ð5Ł will be referred to rather loosely as {analogical| continuation[ Following ð5Ł\ eqn "1[36# could be rewritten by performing the following replacements 0355 M[ S[ EL NASCHIE
Ind"T" f ##:cb¦−b− "1[49#
dim"Ker"T" f ###:dim"Ker"E "###b¦f "1[40#
dim"Coker"T" f ###:dim"Coker"E"###b−f1 "1[41# where f"z4−0#:1 is the Golden Mean Consequently one _nds
$ nInd"T" f ##cf2 "1[42# where f2 happened to be the inverse of the well known expectation value for the dimension of "# "# 2 E namely ðdimHE Ł3¦f which is a PV number[ On the other hand making the following exchanges in eqn "1[35#
ðhŁ:ð−f1 = "1[43# and
ðgŁ:ðf= "1[44# one _nds
& n−ð−f1=fŁf2 "1[45#
Next we invoke the geometric probability interpretation of the Hausdor} dimension of the Kernel and co!Kernel of E"# as discussed for instance in ð10Ł"a#[ That way we see immediately that n as given by eqn "1[36# and eqn "1[42# are essentially the addition theorem of independent probability events which can be consistent only with a particle picture[ On the other hand eqn "1[35# and eqn "1[45# are a clear statement of the multiplication theorem which make sense only for extended objects such as a SchroÃdinger wave and never for a particle[ In other words
$ & nn "1[46# is entirely consistent with
$ & VV "1[47# as well as the wave!particle duality interpretation of the index theorem ð10Ł"b#ð18Ł[
2[ DISCUSSION AND CONCLUDING REMARKS
The preceding analysis\ interesting as it may be\ would\ in our view\ account to little more than a numerological curiosity if it were not for two things[ First\ it is a true fact that we live in 2¦0 dimensions\ although these dimensions seem occasionally to fail in explaining certain experimental results at least in particle physics[ Second\ the similarity between the curves of COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0356
Fig[ 0\ 2Ð5 and Fig[ 1 of the COBE measurement is truly intriguing to say the least[ We feel that if there were ever serious thoughts to regard spacetime and it|s dimensionality as a process\ then this similarity between the basically thermodynamical curve of Fig[ 1 and the geometry of n dimensional spaces of Fig[ 0 and 2Ð5 must be the best argument to use[ In fact it may well be that we have the following admittedly unconventional alternatives[ Either the COBE curve looks
2.334 n φ 2 n 1
0 4 16 n Fig[ 2[ The gamma distribution of n1fn where n is the dimension of the Cantorian space E"# and f is the Golden Mean "see eqns "0[20# and "0[21##[ Notice the similarity to the curve of the shrinking hyperspheres of Fig[ 0 as well as the COBE curve of Fig[ 1[ However\ unlike Figs 0 and 1 the curve starts at xy9 and has a maximum at xn3 rather than at 4[ All values of n1fn are taken from Table 1[
15.02 2 )) ) n ( S ( V ( ) n ( ∆ = ) n ( ξ
1
0 2 4 n Fig[ 3[ The weighted "V"S "n###1 as a function of n used in calculating the mean value of V"S "n##inS"# "see eqn "0[24##[ Note the similarity to the corresponding Fig[ 2[ 0357 M[ S[ EL NASCHIE
Fig[ 4[ The curve h"n#−n also resembles Figs[ 0Ð3[ The values are taken from Table 0 and used to estimate the mean "# dimension ðdimT S "#Ł in S "see eqn "0[22##[
3.101 ) n ( ) S ( V ) n ( ∆ = ) n ( k 1
0 1 3 12 n
"# Fig[ 5[ The curve k"n#−n is used to determine the mean value of n in S using the expression ðnŁS h"n# S k"n# as given 9 9 by eqn "0[22#[ >
the way it looks because the geometry of micro spacetime forces it to look like that^ or micro spacetime looks as it does because it is ultimately the creation of a thermodynamics!like process[ In conclusion we may reiterate the words of A[ Wheeler ð05Ł which encapsulates the quintessence of our analysis of the E"# and S"# spaces] Recall the notion of a Borel set[ Loosely speaking a Borel set is a collection of points {bucket of dust| which has not yet been assembled into a manifold of any particular dimensionality[ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime 0358
Recalling the universals way of the quantum principle one can imagine probability amplitudes for the points in a Borel set to be assembled into points with this\ that and the other dimensionality[ More conditions have to be imposed on a given number of points!as to which has which for a nearest neighbour!when the points are put together in a _ve!dimensional array than when these same points are arranged in a two dimensional pattern[ Thus one can think of each dimensionality as having a much higher statistical weight than the next higher dimensionality[ On the other hand\ for manifolds with one\ two and three dimensions the geometry is too rudimentary\ one can suppose\ to give anything interesting[ Thus Einstein|s _eld equations applied to manifold of dimensionality so low\ demand ~at space\ only when the dimensionality is as high as four do really interesting possibilities arise[ Can four therefore be considered to be the unique dimensionality which is at the same time high enough to give any real physics and yet low enough to have great statistical weight< We hope that our result for the expectation value of the dimensionality of both\ the dust!like E"# space and the sphere!like S"# space
½ðdim E"#Ł¹ðdim S "#Ł¹3 T T has con_rmed Wheeler|s idea to a reasonable extent[ To be able to make stronger statements we must _rst have an exact solution to the expectation values of S"#[ In the second part of this paper and starting with a trans_nite hierarchical spacetime E"# of in_nite dimensions and following some ideas due to A[ Wheeler\ D[ Finkelstein and C[ von WeizsaÃcker\ we derive a _nite expectation value and an e}ective Hausdor} dimension for E"# using a gamma distribution[ This is the same distribution used to derive the Maxwell velocity distribution law as well as Planck black body radiation formula[ This may be seen as an indication for a conceptual link between temperature and dimensions[ The transformation E"#¹E"3# can then be used as a basis for developing a form of trans_nite hierarchical superstrings theory which is closely related to knot theory\ non!commutative geometry and quasi periodic tiling[ The "# hierarchical dimensional formula ðdcŁ3¦"3¹#3[12[[[is obtained for E without the need of suppressing any terms such as the lovelace term ð0!"D!1#:13Ł or invoking supersymmetry and it allows for a wide spectrum of possible quantized {vibration| on a _ner and _ner {dimensional| scales[ In addition the relativistic demand on a minimum string|s world surface can be met in our model in an elementary fashion by minimizing ðdcŁ as given by eqn "1[10# and eqn "1[11#[ The "9# result follows then from Di}er ðdcŁ9 to d c 0:1 and consequently ðdcŁmin3 in a natural way[ Important insight into the meaning of the connection between topological quantum _eld theory\ knots and E"# may be gained by contemplating the meaning of n in both eqn "1[1# for "n# 1pi:r dimS c and eqn "1[02# for =VLe =[ In the _rst n clearly denotes topological dimension[ In the second the meaning of n is slightly more involved[ An n braid is a braid group on n strands Bn[
This Bn can be de_ned formally as a fundamental group in con_guration space Cn of n distinct points[ The braid can then be viewed as the spacetime graph of motion along a closed connection "# in Cn and that establish the analogy to our E spacetime[ The next step is to look at the classical limit of the corresponding TQF theory by letting ::9 in the usual way[ Now we know that rk¦1 and that the level of the theory\ k\ plays the same role of 0::[ Consequently for ::9we have K: and thus r:[ This means cosp:r goes to unity and we are left with 1pi:r n−0 1pi:r "2# =VLe =¾"1# [ This means for a space behaviour "n2# one _nds =VLe =¾d c 3 which is our classical spacetime dimension indictating that one dimension "3−20# will remain invisible in the three dimensional space giving rise to n2¦0 spacetime[ It is also clear that 1cosp:r1 "9# "9# corresponds in the bijection formula to ðd c Ł0:1[ In turn for d c 0:1 we have ½ðnŁ2 and 0369 M[ S[ EL NASCHIE
Table 3[ Page 351 of {{The Curves of Life|| by T[A[ Cook _rst published in 0803 showing the remarkable Table of f scale[ With the help of this Table Cook analysed Master Works of painting by S[ Botticelli "Venus#\ Franz Hals {{The Laughing Cavalier|| and Turner "Ulysses deriving polyphemas#[ The Table is readily repro! "9# 0 1 2 duced using the bijection formula for d c 0:f "z4−0#:1[ Note that f and f "# are identical to Johns index and ðdimHE Ł respectively[
PLATE VII[*THE LAUGHING CAVALIER[ By Franz Hals[ Reproduced by permission of the Trustees\ The Wallace Collection\ London
Table 3"ii#
Separate spaces Successive additions
Symbol Number of units Number of units Symbol[
f0 0[507 0[507 f0 f9 0[999 1[507 f1 f0 0[507 3[125 f2 f1 1[507 5[743 f3 f2 3[125 00[989 f4 f3 5[743 06[833 f5 f4 00[989 18[923 f6 f5 06[833 35[868 f7 f6 18[923 65[902 f8
ðdcŁ3 which reinforces our conclusion of why classical spacetime is 2¦0 rather than simply 3 dimensional[ It may be quite amazing to note that the rather interesting numbers of Johns| index "1¦f# and the exact dimension of E"# "3¦f2# were included in a book published as long ago as 0803 the relevant page of it is displayed in its original form in Table 3[
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