Jones Invariant, Cantorian Geometry and Quantum Spacetime

\

Chaos\ Solitons + Fractals Vol[ 09\ No[ 6\ pp[ 0130Ð0149\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:, ! see front matter

Pergamon

PII] S9859!9668"87#99191!0

Jones| Invariant\ Cantorian Geometry and Quantum Spacetime M[ S[ EL NASCHIE Department of Applied Mathematics and Theoretical Physics\ Silver Street\ Cambridge CB2 8EW\ UK "Accepted 00 August 0887#

Abstract*The relation between Jones| knot polynominals and statistical mechanics is discussed in the light of Cantorian geometry[ It is further shown that von Neumann|s continuous geometry may be regarded as being a quantum spacetime akin to Cantorian space E "# and noncommutative geometry[ Þ 0888 Elsevier Science Ltd[ All rights reserved[

0[ KNOT THEORY AND STATISTICAL MECHANICS*INTRODUCTORY REMARKS

We start here with a very short review of the connection between knot theory and statistical mechanics[ Our discussion is based largely on an excellent review by V[ Jones ð03Ł[ The path connecting knot theory and statistical mechanics leads to von Neumann algebra which is closely related to the mathematical structure of quantum physics[ The main characteristic of this algebra is the notion of a continuous dimension[ Thus\ while it was taken for granted that the dimension of an abstract space is a natural number\ in von Neumann algebra\ a dimension could be an irrational number for instance ð0\ 1Ł[ It was around 0873 when Jones was working on von Neumann|s algebra and noticed that certain algebraic expression for the topological relation of braids were quite similar[ Soon afterwards\ he realized that he had discovered a new knot invariant which he called V"t# and which he found to have some super_cial similarity to the Alexander polynomial D"t#[ Both V"t# and D"t# can be calculated using the so!called crossing "or skein# relation which was found by the British mathematician J[ H Conway[ Jones| polynomial has considerable advantages over any other knot invariants such as D"t# because it can\ for instance\ distinguish between a knot and its mirror image[ However\ the most interesting thing about it is its surprising relation to statistical mechanics because\ at _rst glance\ knots and statistical mechanics seem to have hardly anything in common[ There are mainly two surprising features of statistical mechanics which deals with a huge number of particles when contrasted with classical Newtonian mechanics[ The _rst is irre! versibility and the second is phasetransition[ To study phasetransition\ there are many theoretical models\ the simplest and most well known of which is probably the {ising model|[ This model can have an enormously large number of spin states which makes the actual mathematical calculation of the model the true problem[ An important result in this direction is the so! called {star!triangular relation|[ This states that two ising models which are de_ned on certain corresponding graphs\ di}er from each other by a proportionality factor only[ Using this result\ 0130

0131

M[ S[ EL NASCHIE

J[ Baxter was able to solve this ising model and to generalize it to spin values other than plus one and minus one ð05Ł[ Unfortunately\ not every spin!model satis_es the star!triangular relation and\ in order to apply Baxter|s method\ one must de_ne expressions for the energy interaction of adjacent spin states[ The simplest such situation is the so!called {Potts model| with q states\ as contrasted with the two states of the ising model[ The connection between Potts model and knot theory stems from the connection between the graph of the spin model and the diagram of a Link entanglement[ This diagram is transformed by colouring it black and white in a chessboard manner[ On such a graph\ every Reidemeister movement of type III corresponds exactly to a star!triangular relation[ That means it leaves the knot invariant and\ consequently\ also the ising model which we gained from this diagram[ That way\ it is possible to show that when a model satis_es the star!triangular\ then the energy function of the model gives the invariant of its entanglement[ In particular\ the ising model gives a knot invariant which is known as the {Arf| or {Kevaire| invariant[ Even more important is the fact that the q states of the Potts model also gives an invariant[ This invariant is nothing else but the Jones polynomial\ V"t#\ which is related to q by a relation which will be discussed later on and which is given by q1¦t¦0:t[ Thus\ the chessboard colouring does indeed lead to a connection between knot theory and statistical mechanics and it may be useful to have a new look at the Feynman chessboard model in this context ð2\ 6\ 8Ł[ There are\ of course\ other models than the ising and Potts models which give further knot invariants[ The vertex model is an example of this ð05Ł[ It was E[ Witten who showed that there are even more interesting invariants in more general spaces than our 2!D Euclidean space which are referred to as {three manifold|[ Such conceptions have led to a fusion of quantum _eld theory\ topology and knot theory[ In fact\ we will be showing towards the end of this work how three manifold can give a geometrical model for our complex time as well as Nagasawa|s dual di}usion process in quantum physics ð09Ð04Ł[ In order to be able to outline the connection between knot theory\ statistical mechanics and Cantorian spacetime\ we will have to give\ next\ a short account of von Neumann geometry followed by a somewhat longer exposition of Cantorian geometry[ After that\ we will return to the main objective of the article which is the connection between Jones| invariant V"t# and the dimensionality of Cantorian space time ðdimT E "# Ł3¦f2 \ where f is the Golden Mean ð00Ł[

1[ VON NEUMANN GEOMETRY

In this section\ we give a short summary of von Neumann geometry and its connection to Cantorian spacetime[ It may have been due to the work of K[ Menger ð0Ł and to von Neumann|s interest in quantum mechanics that he decided around 0824 to generalize his work on rings of operators in Hilbert space ð1Ł[ In the course of this\ von Neumann discovered a new mathematical structure which resembles a lattice Ln of all linear subsets of an n−0 dimensional projective geometry[ However\ von Neumann found that this new structure possesses a dimension function taking all real numbers of the unit interval 9¾dimLn ¾0[

"0#

The best analogy to this may be the lattice!like pattern which one obtains by averaging over a large numbers of instantaneous chaotic images of waves or\ say\ the Faraday instability\ as discussed in previous publications ð2Ł[ In other words\ the continuous dimensional function of von Neumann leads ultimately to fractal geometry with a noninteger Hausdor} dimension ð3Ł[ This already indicates a very strong kinship between von Neumann geometry and Cantorian geometry E "# ð2Ł[

Jones| invariant\ Cantorian geometry and quantum spacetime

0132

It is easy\ of course\ to see that in von Neumann|s continuous geometry\ there can be no monadic elements or points[ In fact\ von Neumann nicknamed his geometry {pointless geometry|\ a property which it shares with the Cantorian proposal E "# in which every point\ no matter how small\ possess a structure[ On the other hand\ the Gel|fand theorem states that noncommutative C!algebra cannot be the function algebras on spaces with points[ This would then mean that von Neumann geometry\ as well as the Cantorian space E "# geometry\ is ideally suited for modelling the geometry of the quantum world where the noncommutative algebra of Q[M[ can be translated to a noncommutative geometry ð4Ł[ Von Neumann did not seem to have followed his own theory to its logical end\ namely] to what is called today {topological quantum theory| and {noncommutative geometry|[ However\ he must have been well aware of these possibilities and their crucial importance to the controversial quantum mechanics of his day\ as witnessed by the title of a manuscript which is\ as far as we are aware\ still unpublished\ namely] {{Continuous Geometry with Transition Probability||[ Nonetheless\ it is evident that the work on noncommutative geometry and E "# space may be rightly regarded as a continuation of von Neumann|s work ð5Ł applied to quantum physics ð6Ł[ Another very important condition in von Neumann|s geometry is that dimensionality\ n\ should be equal or larger than four "1#

n−3[

Now this condition is satis_ed generally in E "# by the fact that the expectation value of the Hausdor} and the topological dimension of E "#\ as given by the gauge and the gamma dis! tribution formulae\ respectively\ are ½ðnŁðdim E "# Ł T

0¦d c"9# 0−d c"9#

b

3¦f2 ¹3

"2#

b

"3#

f

and E "# Ł ðdc Łðdim H

0 "0−d

"9# c

#d

"9# c

3¦f2 ¹3\

f

where d c"9# is the null set and f"z4−0#:1 is the Golden Mean[ It may also be important to point out that the condition D"0#0

"4#

of the dimension function\ D"a#\ for which von Neumann gives a proof\ is automatically ful_lled in E "# because the bijection for E "# ð6Ł d c"n# "0:d c"9# # n−0 \

"5#

which relates n with its subspace projection\ n−0\ gives n0 as the value of d c"0# \ d c"0# "0:d c"9# # 9 0

"6#

and that is regardless of the magnitude of d c"9# [ We conclude our very short revision of von Neumann continuous geometry by stating the following central theorem\ for which von Neumann has given the proof[

0133

M[ S[ EL NASCHIE

2[ THE MAIN THEOREM

Every function D?"a#g0 D"a#¦g1 \

"7#

where g0 and g1 are real numbers\ is a dimension function and every dimension function is of this form[ Again\ eqn "7# is reminiscent of our equation for the mean dimension of Cantorian spacetime E "#\ E "# Łg0c D c ¦g1c \ ðdim H

"8#

which gives for g0c 1\ g1c 0\

0 z4¦0 Dc  \ f 1

"09#

our\ by now\ well!known PV number ð6Ł

0

ðdim E "# Ł1 H

z4¦0 ¦0 1

1

3¦f2 3[125

"00#

as the e}ective dimension for E "#[ It is important to note that eqn "00# is identical to that of A[ Connes| semigroup used in his noncommutative geometry ð7Ł[ It should be mentioned at this point that the Penrose tiling is one of the best explicit illustrations of a low dimensional\ noncommutative geometry and that it is a direct realization of a two dimensional fractalÐCantorian space[

3[ CANTORIAN SPACETIME E "#] SOME FUNDAMENTAL CONCEPTS

In order to understand the connection between E "# and von Neumann|s continuous geometry\ we look closely at some of the fundamental concepts involved in the E "# theory[ This theory depends fundamentally on the notion of the null set[ This set\ which we denote as S c"9# \ is a trans_nite Cantor set with Hausdor} dimension ð6Ł 9¾d c"9# ¾0

"01#

and corresponds to E "#[ Equation "01# should be compared to eqn "0#[ It should also be noted that S c"9# does not need to be embedded in a topological space\ DT0[ However\ whenever it is convenient or useful to do so\ we may think of S c"9# as being embedded in one!dimensional space[ The normality set of E "#\ on the other hand\ which will be denoted S c"0# \ must have a dimension d c"0# 0 and is thus necessarily identical to DT0 space for the reasons explained earlier on in connection with eqn "6#[ The set d c"0# 0 stands thus on the other side\ so to speak\ of the completely empty set S c"−# 0S c9 \ which\ by virtue of eqn "6#\ has a Hausdor} dimension zero ð6Ł

"02#

Jones| invariant\ Cantorian geometry and quantum spacetime

0134

dim S c"−# d c"−# "0:d c"9# # − d 9c 9[ H

"03#

In other words\ all the elementary Cantor sets or monades of E "# lie somewhere between the completely empty set S 9c and the normality set S c"0# ] dim S c9 ¾ dim S c"9# ¾dim S c"0# [ H H H

"04#

This means that we deviate here from the classical MengerÐUrysohn de_nition of the empty set where all Cantor sets are considered to be of dimension zero\ while the empty set is of dimension −0[ In our case\ only the set of dimension − is the truly empty set[ In this connection\ we have termed the distance between d c"9# and d 9c the degree of emptiness of an empty set ð6Ł[

4[ THE EXPECTATION VALUE OF dim E "# AND OUR 2¦0 DIMENSIONAL REALITY

The visible\ tangible universe is manifestly a 2¦0 dimensional space and\ while time may be something of an enigma\ there is little doubt whatsoever about the three dimensionality of our real spatial world[ Given that our assumptions about E "# and noncommutative geometry hold true for our real world and\ considering the dual role of d c"9# as being a Hausdor} dimension as well as a geometrical probability\ then we could say that the {spatial| probability of _nding a Cantorian {point| in E "# must be given by the intersection "multiplication# rule as s

P"d c"9# # "D T # "d c"9# # 2 \

"05#

s T

where D is the topological spatial dimension of the classical world[ Now\ P could be interpreted as a Hurst exponent and\ consequently\ the Hausdor} dimension of the fractal path of a Cantorian point "Cantorion# would be Dpath 

2

0 1

0 0 0   "9# H P dc

[

"06#

Next\ and in keeping with the fundamental nature of black body radiation\ we will assume that the elementary "monadic# sets\ S c"9# \ are all random Cantor sets[ Consequently\ and following the MauldinÐWilliams theorem of random {one dimensional| Cantor sets\ the Hausdor} dimen! sion of S c"9# must be ð6Ł z4−0 [ 1

dim S c"9# d c"9# f

"07#

Setting eqn "07# in eqn "06#\ one _nds Dpath 

0

z4¦0 1

2

1

 3¦f2 ¹ 3\

"08#

which is identical to the results of eqn "07#[ In other words\ a Cantorian in 2D gives the impression of being 3D[ In some strange sense\ it sweeps a 3D world sheet[ CantorianÐFractal space creates the time dimension and\ in this sense\ time is a product of fractal space and\ consequently\ gravity is caused by the fractality of time as conjectured in various earlier contributions ð8Ł and discussed and developed further by several authors ð02\ 03Ł[

0135

M[ S[ EL NASCHIE

It should be mentioned that the result of eqn "08# coincides with that obtained from the fundamental equation of E "#[ This is so because the expectation value of the dimension of the formally in_nite dimensional Cantorian space is given by ½ðnŁ

0¦d c"9# 0−d c"9#

3¦f2 ¹3

"19#

and this equation was\ in turn\ derived using the statistical distribution\ namely] a gamma distribution which M[ Planck used to derive his famous formula for black body radiation ð6Ł[ The fact that the equality ½ðnŁðdc Łd c"n#

"10#

leads to the condition that z4−0 and n3 1

d c"9# f

"11#

is a con_rmation of the MauldinÐWilliams theorem as well as the von Neumann condition[ In particular\ eqn "19# lends considerable strength to the thesis of von Weizsacker\ Finkelstein and others that space and dimensions are statistical constructs similar to temperature[ Seen that way\ K[ Menger|s statistical geometry\ as well as von Neumann|s continuous geometry can help us see quantum physics in a di}erent light and deepen our understanding of some of the present day|s physical theorems[ We may also mention that the same formalism of E "# leads to the conclusion that any connection in E "# must have ðd c"1# Łð0:d c"9# Ł1

"12#

as a Hausdorf dimension[ The intimate relation between Cantorian spacetime\ noncommutative geometry\ the theory of four manifolds and quasiperiodic tiling\ as well as knot theory\ was discussed in detail in various recent publications ð09\ 00Ł[ Next\ it is important to look closely at the continuous fraction representation of ½ðnŁ and ðdcŁ[ 0

½ðnŁðdc Ł3¦"3¹#3¦

0

3¦ 3¦

[

"13#

0 3¦[ [ [

We could interpret this interesting result as follows] This _rst term in eqn "13# is clearly our ordinary 2¦0 spacetime dimension fused together\ as in relativity\ to 3D with the only distinction that the signature in our case is identical to that of Euclidean quantum mechanics[ The second set of 3 {compacti_ed| dimensions are not really dimensions in the ordinary sense\ but something between dimension and internal symmetries[ In that sense\ {dimensions| may be regarded as the superspace of supersymmetry[ The next set of 3 more compacti_ed dimensions take us obviously to a space which is formally equivalent to the space of F superstrings[ Further higher dimensional spaces take us completely outside the present observational possibilities and exclude any possible experimental veri_cation in the foreseeable future[

Jones| invariant\ Cantorian geometry and quantum spacetime

0136

5[ JONES| POLYNOMIAL[ POTTS MODEL AND THE EXPECTATION VALUE OF THE DIMENSION OF E "#

We indicated in our introductory remarks Section 0 that the Potts model with q admissible spin states gives an invariant which is identical to Jones polynomial V"¦#\ where q is given by the relationship "14#

q1¦t¦0:t[ Now\ setting tf\ one _nds the surprising result that q1¦f¦0:f3¦f2 \

"15#

E "# Łðdim E "# Ł[ q= f ½ðdim T H

"16#

which means

It is also clear that within this interpretation\ q cannot simply be the number of spin states[ Nevertheless\ the meaning as a number of spin states is restored within the mean_eld approxi! mation for which we may ignore f2 compared to 3 and set qððdim E "# ŁŁ3[

"17#

Similarly\ we see that in the theory of type III factors\ one can easily calculate from JLit 9 HH HH l HH HH 0¦l jl

K F 0 H H  H& H0¦l Hi0H H H 9 k f that F 0 H 0¦l det−0 H H H 9 f

JH 9 HH l HH  l HH "0¦l# 1 HH 0¦l jH f

$

−0

%

3¦f2 [

f

Finally\ for the 3×3 matrix\ F H9 H 0 H9 0¦l H9 H H9 f

9

9

0

zt

zt

t

9

9

J 9H H 9H \ 9H H 9H j

it follows that for 0¾i\ j¾n−0\ e1i ei \ ei ei 20ei  where

t "0¦t# 1

ei \ ei ej ej ej if =i−j=−1\

0137

M[ S[ EL NASCHIE

t "0¦t#

1

b

 "3¦f2 # −0 [

f

6[ THREE MANIFOLDS AND COMPLEX TIME

We have mentioned on previous occasions the crucial role which higher dimensional geometry may play in modelling the quantum universe ð08Ł[ In the present introduction\ we have also hinted at the connection between knot theory\ statistical mechanics and Witten|s work on quantum _eld theory and geometry[ In this section\ we would like to reconsider the possibilities outlined for instance by P[ Thurston and others regarding the use of the geometry of three manifolds to model the universe ð06\ 07Ł[ An example of such a three manifold is the three dimensional torus[ This is basically a knotted hypercube which has all its sides glued together\ so to speak[ The net result of this convoluted and intricate geometry and topology is that a particle which may be seen as going out of the cube from\ say\ a left hand side direction will be seen at precisely the same moment entering the structure from the right hand side direction\ which is\ of course\ classically impossible[ Never! theless\ it is such classically impossible topology and geometry which lies at the heart of the three interrelated important interpretations of quantum mechanics\ namely] Nagasawa|s dual di}usion\ Cramer|s transactional interpretation and the concept of conjugate imaginary time\ 92it\ referred to\ somewhat inaccurately\ as complex time ð6\ 8\ 02Ł[ In all of these three interpretations\ we have always and simultaneously\ a process going forward and another going backwards in time[ It would be most interesting to try to _nd out exactly which higher dimensional structure possesses such 92it duality as a natural consequence of its very geometry and topology[

7[ DIXMIER TRACE AND THE GAMMA EXPECTATION VALUE

In noncommutative geometry ð7Ł\ which is closely related to von Neumann|s continuous geometry and Cantorian spacetime\ the integral of an in_nitesimal of order one is replaced by the so!called Dixmier trace\ TVw "T# ð7Ł[ In what follows\ we show\ using the same statistical distribution of E "#\ that an expectation value can be determined for TVw "T# in the very important case of the powers of the Laplacian operator D on the n!dimensional sphere S "n#[ Subsequently\ we will show that the result coincides with that of the Wodzicki Residue of the inverse of the one dimensional harmonic oscillator[ Starting from D on S "n# which has eigenvalues l"l¦n−0#\ it is not di.cult to see that TVw "D−n:1 #1:n;[

"18#

Following the same procedure used previously in the E "# theory ð00Ł\ we can write the following mean and expectation values for n[

$0 1>0 1% 

ðnŁ0 lim n: and

s 9

1n n;



s 9

1 n;

0

"29#

Jones| invariant\ Cantorian geometry and quantum spacetime

0138

Table 0 n

1:n;

1n:n;

1n1:n;

9 0 1 2 3 4 5 6 7 8 09 00

1 1 0 0:2 0:01 9[905555 9[9916666 9[99285714 9[999938592 9[999994400 9[999999440 9[99999994

9 1 1 0 0:2 9[97222 9[905555 9[991666 9[999285714 9[999938592 9[999994400 9[999999440

9 1 3 2 0[2222 9[305555 9[0 9[908333 9[992063592 9[999335317 9[999944003 9[999995951

S

4[325452502

4[325452502

09[762015530

4[325452502 09[762015530 0 and ðnŁ03 30[888887[ 4[325452502 4[325452502

Note that ðnŁ03

ðnŁ1 lim n:

$0



s 9

1n1 n;

1>0 1% 

s 9

1n n;

1[

"20#

Following Table 0\ the above result means that ðTVw "D−n:1 #Ł= ðnŁ0 1

"21#

EG "TVw "D−n:1 ##= ðnŁ1 0[

"22#

and

It is interesting to note that the W!residue of an inverse one dimensional harmonic oscillator with a Hamiltonian\ H0:1"j1¦x1#\ is given by WR"H −0 #

0 p

g

R

1 0¦x1

1[

"23#

That means ðTVw "D−n:1 #Ł= ðnŁ0 WR"H −0 #1[

"24#

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0149

M[ S[ EL NASCHIE

8[ El Naschie\ M[ S[\ Fractal gravity and symmetry breaking in hierarchical Cantorian space[ Chaos\ Solitons and Fractals\ 0886\ 7"00#\ 0754Ð0761[ 09[ El Naschie\ M[ S[\ Knot theory in R3 and the Hausdor} dimension of a quantum path in E "#[ Chaos\ Solitons and Fractals\ 0887\ 8"5#\ 0994Ð0996[ 00[ El Naschie\ M[ S[\ COBE satellite measurement\ hyperspheres\ superstrings and the dimension of spacetime[ Chaos\ Solitons and Fractals\ 0887\ 8\ 0334Ð0360[ 01[ Agob\ M[\ Nica\ P[\ On the Cantorian structure of time in relativity[ Chaos\ Solitons and Fractals "in press#[ 02[ Argyris\ J[\ Ciubotariu\ C[ and Andreadis\ I[\ Complexity in spacetime and gravitation[ Part I[ Chaos\ Solitons and Fractals\ 0887\ 8\ 0540Ð0690[ 03[ Jones\ V[ F[ R[\ Knot theory and statistical mechanics[ Scienti_c American\ 0889\ 152"4#\ 87Ð092 04[ Witten\ E[\ Quantum _eld theory and Jones polynomial[ Communications in Mathematical Physics\ 0878\ 010\ 240Ð 288[ 05[ Baxter\ R[ J[\ Exactly Solved Models in Statistical Mechanics[ Academic Press\ 0871[ 06[ Scott\ P[\ The geometrics of 2 manifolds[ In Bulletin of The London Mathematical Society\ 0872\ 04"4#\ No[ 45\ 390Ð 376[ 07[ Thurston\ W[ P[ and Weeks\ J[ R[\ The mathematics of 2!dimensional manifolds[ Scienti_c American\ 0873\ 092Ð019[ 08[ El Naschie\ M[ S[\ On octahedrons in R and the mean dimension of space[ Chaos\ Solitons and Fractals\ 0887\ 8\ 0672Ð0674[