真实之路 宇宙法则的完全指南 (彭罗斯)16无限遥远.pdf

16 The ladder of infinity 16 1Finite fields It appears to be a universal feature of the mathematics normally believed to underlie the workings of our physical universe that it has a fundamental dependence on the inWnite In the times of the ancient Greeks even before they found themselves to be forced into considerations of the real number system they had already become accustomed in eVect to the use of rational numbers see 3 1 Not only is the system of rationals inWnite in that it has the potential to allow quantities to be indeWnitely large a property shared with the natural numbers themselves but it also allows for an unending degree of reWnement on an indeWnitely small scale There are some who are troubled with both of these aspects of the inWnite They might prefer a universe that is on the one hand Wnite in extent and on the other only Wnitely divisible so that a fundamental discreteness might begin to emerge at the tiniest levels Although such a standpoint must be regarded as distinctly unconven tional it is not inherently inconsistent Indeed there has been a school of thought that the apparently basic physical role for the real number system R is some kind of approximation to a true physical number sys tem which has only a Wnite number of elements This kind of approach has been pursued particularly by Y Ahmavaara 1965 and some co workers see 33 1 How can we make sense of such a Wnite number system The simplest examples are those constructed from the integers by reducingthemmodulop wherepissomeprimenumber Recall that the prime numbers are the natural numbers 2 3 5 7 11 13 17 which have no factors other than themselves and 1 and where 1 is itself not regarded as a prime To reduce the integers modulo p we regard two integers as equivalent if their diVerence is a multiple of p that is to say a b mod p if and only if 357 a b kp for some integer k The integers fall into exactly p equivalence classes see the Preface for the notion of equivalence class according to this prescription so a and b belong to the same class whenever a b These classes are regarded as the elements of the Wnite Weld Fpand there are exactly p such elements Here I am adopting the algebraists use of the term Weld This should not be confused with the Welds on a manifold such as vector or tensor Welds nor a physical Weld such as electromagnetism An algebra ist s Weld is just a commutative division ring see 11 1 Ordinary rules of addition subtraction commutative multiplication and division hold fortheelementsofFp 16 1 However wehavetheadditional curious property that if we add p identical elements together we always get zero and of course the prime number p itself has to count as zero Note that as Fphas been just described its elements are themselves deWned as inWnite sets of integers since the equivalence classes arethemselvesinWnitesets suchastheparticularequivalence class 7 2 3 8 13 which deWnes the element of F5 p 5 that we would denote by 3 Thus we have appealed to the inWnite in order to deWne the quantities that constitute our Wnite number system This is an example of the way in which mathemati cians often provide a rigorous prescription for a mathematical entity by deWning it in terms of inWnite sets It is the same equivalence class procedure that is involved in the deWnition of fractions as referred to in the Preface in relation to the cancelling that my mother s friend found so confusing I imagine that to someone convinced that the number system Fp for some suitable p is really directly rooted in nature the equivalence class procedure would be merely a mathe matician s convenience aimed at providing some kind of a rigorous prescription in terms of the more historically familiar inWnite procedures In fact we do not need to appeal to inWnite sets of integers here it is just that this is the most systematic procedure In any given case we could alternatively simply list all the operations since these are Wnite in number Let us look at the case p 5 in more detail just as an example We can label the elements of F5by the standard symbols 0 1 2 3 4 and we have the addition and multiplication tables 16 1 Show how these rules work explaining why p has to be prime 358 16 1CHAPTER 16 01234 01234 001234000000 112340101234 223401202413 334012303142 440123404321 and we note that each non zero element has a multiplicative inverse 1 1 1 2 1 3 3 1 2 4 1 4 in the sense that 2 3 1 mod 5 etc From here on I use rather than when working with the elements of a particular Wnite number system There are also other Wnite Welds Fq constructedin asomewhatmore ela borate way where the total number of elements is some power of a prime q pm Letmejustgivethesimplestexample namelythecaseq 4 22 Here we can label the diVerent elements as 0 1 o o2 where o3 1 and where eachelementxissubjecttox x 0 Thisslightlyextendsthemultiplicative groupofcomplexnumbers1 o o2thatarecuberootsofunity describedin 5 4andmentionedin 5 5asdescribingthe quarkiness ofstronglyinteract ing particles To get F4 we just adjoin a zero 0 and supply an addition operation for which x x 0 16 2 In the general case Fpm we would have x x x 0 where the numberofxs inthe sum isp 16 2A finite or infinite geometry for physics It is unclear whether such things really have a signiWcant role to play in physics althoughthe idea hasbeen revivedfrom time totime If Fqwere to take the place of the real number system in any signiWcant sense then p wouldhavetobeverylargeindeed sothat the x x x 0 wouldnot show up as a serious discrepancy in observed behaviour To my mind a physical theory which depends fundamentally upon some absurdly enor mous prime number would be a far more complicated and improbable theory than one that is able to depend upon a simple notion of inWnity Nevertheless itisofsomeinteresttopursuethesematters Muchofgeometry survives in fact when coordinates are given as elements of some Fq The ideasof calculusneed morecare nevertheless manyof thesealso survive 16 2 Make complete addition and multiplication tables for F4and check that the laws of algebra work where we assume that 1 o o2 0 359 The ladder of infinity 16 2 It is instructive and entertaining to see how projective geometry with a Wnite total number of points works and we can accordingly explore the projective n spaces Pn Fq over the Weld Fq We Wnd that Pn Fq has exactly 1 q q2 qn qn 1 1 q 1 diVerent points 16 3 The projectiveplanesP2 Fq areparticularlyfascinatingbecauseaveryelegant constructionforthemcanbegiven Thiscanbedescribedasfollows Takea circulardiscmadefromsomesuitablematerialsuchascardboard andplace a drawing pin through its centre pinning it to a Wxed piece of background card so that it can rotate freely Mark 1 q q2points equally spaced around the circumference on the background card labelling them in an anticlockwisedirection bythenumbers0 1 2 q 1 q Ontherotat ing disc mark 1 q special points in certain carefully chosen positions These positions are to be such that for any selection of two of the marked pointsonthebackground thereisexactlyonepositionofthediscforwhich the two selected points coincide with two of these special points on the disc Another way of saying this is as follows if a0 a1 aqare the successive distances around the circumference between these special points taken cyclically where the distance around the circumference between successive marked points on the background circle is taken as the unit distance then every distance 1 2 3 q can be uniquely represented as a sum of a cyclically successive collection of the as I call such a disc a magic disc In Fig 16 1 I have depicted magic discs for q 2 3 4 and 5 for which a0 aqcan be taken as 1 2 4 1 2 6 4 1 3 10 2 5 1 2 7 4 12 5 respectively 16 4 In the cases q 7 8 9 11 13 and 16 we can make magic discsdeWnedby1 2 10 19 4 7 9 5 1 2 4 8 16 5 18 9 10 1 2 6 18 22 7 5 16 4 10 1 2 13 7 5 14 34 6 4 33 18 17 21 8 1 2 4 8 16 32 27 26 11 9 45 13 10 29 5 17 18 respectively Itisamathematicaltheoremthatmagic discsexistforeveryP2 Fq withqapowerofaprime 1ThereadermayWnd itamusingtocheckvariousinstancesofthetheoremsofPapposandDesar gues see 15 6 Fig 15 14 2 Take q 2 so as to have enough points for a non degenerate conWguration Two examples Desargues for q 3 and Papposforq 5 usingthe discsofFig 16 1 are illustratedin Fig 16 2 The simplest case q 2 has particular interest from other direc tions 16 5 This plane with 7 points is called the Fano plane and it is depicted in Fig 16 3 the circle being counted as a straight line Although 16 3 Show this 16 4 Show how to construct new magic discs in the cases q 3 5 by starting at a particular marked point on one of the discs that I have given and then multiplying each of the angular distances from the other marked points by some Wxed integer Why does this work 16 5 The Wnite Weld F8has elements 0 1 e e2 e3 e4 e5 e6 where e7 1 and 1 1 0 show that either 1 there is an identity of the form ea eb ec 0 whenever a b and c are numbers on the background circle of Fig 16 1a which can line up with the three spots on the disc or else 2 the same holds but with e3in place of e i e e3a e3b e3c 0 360 16 2CHAPTER 16 2 2 3 3 1 1 0 0 6 6 5 5 4 4 7 8 9 10 11 12 2 3 1 0 6 5 4 7 8 9 10 11 12 13 14 1516 17 18 19 20 6 5 4 2 3 1 0 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 2324 25 26 27 28 29 30 a c b 2 1 4 6 2 1 4 10 3 1 5 2 12 5 1 2 7 4 d Fig 16 1 Magic discs for Wnite projective planes p2 fq q being a power of a prime The 1 q q2points are represented as successive numerals 0 1 2 q 1 q placed equidistantly around a background circle A freely rotating circular disc is attached with arrows labelling 1 q particular places the points of a line in p2 fq These are such that for each pair of distinct numerals there is exactly one disc setting so that arrows point at them Magic discs are shown for a q 2 b q 3 c q 4 22 and d q 5 7 28 23 222421 1 2 15 0 1 3 4 5 6 10 8 12 2 a b Fig 16 2Finite geometry versions of the theorems of Fig 5 14 a Pappos with q 5 and b Desargues with q 3 illustrated by respective use of the discs shown in Fig 16 1d and 16 1b 361 The ladder of infinity 16 2 20 6 3 54 1 Fig 16 3The Fano plane p2 f2 with 7 points and 7 lines the circle counting as a straight line numbered according to Fig 16 1a This provides the multipli cation table for the basis elements i0 i1 i2 i6of the octonion division algebra where the arrows provide the cyclic ordering that gives a sign its scope as a geometry is rather limited it plays an important role of a diVerent kind in providing the multiplication law for octonions see 11 2 15 4 The Fano plane has 7 points in it and each point is to be associated with one of the generating elements i0 i1 i2 i6of the octonion alge bra Each of these is to satisfy i2 r 1 To Wnd the product of two distinct generating elements we just Wnd the line in the Fano plane which joins the points representing them and then the remaining point on the line is the point representing the product up to a sign of these other two For this the simple picture of the Fano plane is not quite enough because the sign of the product needs to be determined also We can Wnd this sign by reverting to the description given by the disc depicted in Fig 16 1a or by using the equivalent arrow arrangements intrepreted cyclicly of Fig 16 3 Let us assign a cyclic ordering to the marked points on the disc say anticlockwise Then we have ixiy izif the cyclic ordering of ix iy izagrees with that assigned by the disc and ixiy izotherwise In particular we have i0i1 i3 i1i0 i0i2 i6 i1i6 i5 i4i2 i1 etc 16 6 Although there is a considerable elegance to these geometric and alge braic structures there seems to be little obvious contact with the workings of the physical world Perhaps this should not surprise us if we adopt the point of view expressed in Fig 1 3 in 1 4 For the mathematics that has any direct relevance to the physical laws that govern our universe is but a tiny part of the Platonic mathematical world as a whole or so it would seem as far as our present understanding has taken us It is possible that 16 6 Show that the associator a bc ab c is antisymmetrical in a b c when these are generating elements and deduce that this whence also a ab a2b holds for all elements Hint Make use of Fig 16 3 and the full symmetry of the Fano plane 362 16 2CHAPTER 16 as our knowledge deepens in the future important roles will be found for such elegant structures as Wnite geometries or for the algebra of octonions But as things stand the case has yet to be convincingly made in my opinion 3It seems that mathematical elegance alone is far from enough see also 34 9 This should teach us caution in our search for the under lying principles of the laws of the universe Let us drag ourselves back from such Xirtations with these appealing Wnite structures and return to the awesome mathematical richness that is inherent in the inWnite As a preliminary it should be pointed out that inWnite structures such as the totality of natural numbers N might be part of some mathematical formalism aimed at a description of reality whereas it is not intended that these inWnite structures have direct physical interpretation as inWnite or inWnitesimal physical entities For example some attempts have been made to develop a scheme in which discreteness and indeed Wniteness appears at the smallest level while there is still the potential for describing indeWnitely or even inWnitely large structures This applies in particular to some old ideas of my own for building up space in a Wnite way using the theory of spin networks which I shall describe brieXy in 32 6 and which depends upon the fact that according to standard quantum mechanics the measure of spin of an object is given by a natural number multiple of a certain Wxed quantity 1 2 h Indeed as I mentioned in 3 3 in the early days of quantum mechanics there was a great hope not realized by future developments that quantum theory was leading physics to a picture of the world in which there is actually discrete ness at the tiniest levels In the successful theories of our present day as things have turned out we take spacetime as a continuum even when quantum concepts are involved and ideas that involve small scale space time discreteness must be regarded as unconventional 33 1 The con tinuum still features in an essential way even in those theories which attempt to apply the ideas of quantum mechanics to the very structure of space and time This applies in particular to the Ashtekar Rovelli Smolin Jacobson theory of loop variables in which discrete combinator ial ideas such as those of knot and link theory actually play key roles and where spin networks also enter into the basic structure We shall be seeing something of this remarkable scheme in Chapter 32 and in 33 1 we shall briefly encounter some other ideas relating to discrete spacetime Thus it appears for the time being at least that we need to take the use of the inWnite seriously particularly in its role in the mathematical descrip tion of the physical continuum But what kind of inWnity is it that we are requiring here In 3 2 I brieXy described the Dedekind cut method of constructing the real number system in terms of inWnite sets of rational numbers In fact this is an enormous step involving a notion of inWnity 363 The ladder of infinity 16 2 that greatly surpasses that which is involved with the rational numbers themselves It will have some signiWcance for us to address this issue here In fact as the great Danish Russian German mathematician Georg Cantor showed in 1874 as part of a theory that he continued to develop until 1895 there are diVerent sizes of inWnity The inWnitude of natural numbers is actually the smallest of these and diVerent inWnities continue unendingly to larger and larger scales Let us try to catch a glimpse of Cantor s ground breaking and fundamental ideas 16 3Different sizes of inWnity The Wrst key ingredient in Cantor s revolution is the idea of a one to one 1 1correspondence 4Wesaythat twosetshave thesame cardinality which means inordinarylanguage thattheyhavethe samenumberofelements if it is possible to set up a correspondence between the elements of one set and the elements of the other set one to one so that there are no elements of either set that fail to take part in the correspondence It is clear that this procedure gives the right answer same number of elements for Wnite sets i e sets with a Wnite number 1 2 3 4 of members or even 0 elements where in that case we require the correspondence to be vacuous But in the case of inWnite sets there is a novel feature already noticed by 1638 by the great physicist and astronomer Galileo Galilei 5that an inWnite set has the same cardinality as some of its proper subsets where proper means other than the whole set Let us see this in the case of the set N of natural numbers N 0 1 2 3 4 5 If we remove 0 from this set 6we Wnd a new set N 0 which clearly has the same cardinality as N because we can set up the 1 1 correspondence in which the element r in N is made to correspond with the element r 1 in N 0 Alternatively we can take Galileo s example and see that the set of square numbers 0 1 4 9 16 25 must also have the same cardinality as N despite the fact that in a well deWned sense the square numbers constitute a vanishingly small proportion of the natural numbers as a whole We can also see that the cardinality of the set Z of all the integers is again of this same cardinality This can be seen if we consider the ordering of Z given by 0 1 1 2 2 3 3 4 4 whichwecansimplypairoVwiththeelements 0 1 2 3 4 5 6 7 8 of the set N More striking is the fact that the cardinality of the rational numbers is again the same as the cardinality of N There are many ways of 364 16 3CHAPTER 16 seeing this directly 16 7 16 8 but rather than demonstrating this in detail here let us see how this particular example falls into the general framework of Cantor s wonderful theory of inWnite cardinal numbers First what is a car