真实之路 宇宙法则的完全指南 (彭罗斯)15纤维束和标准.pdf

15 Fibre bundles and gauge connections 15 1Some physical motivations for fibre bundles The machinery introduced in Chapters 14 and 15 is suYcient for the treatment of Einstein s general relativity and for the phase spaces of classical mechanics However a good deal of the modern theory of particle interactions depends upon a generalization of the speciWc notion of con nection or covariant derivative that was introduced in 14 3 this gener alization being referred to as a gauge connection Basically our original notion of covariant derivative was based upon what we mean by the parallel transport of a vector along some curve in our manifold M 14 2 Knowing parallel transport for vectors we can uniquely extend this to the transport of any tensor quantity 14 3 Now vectors and tensors are quantities that refer to the tangent spaces at points of M see 12 3 14 1 and Fig 12 6 But a gauge connection refers to parallel transport of certain quantities of particular physical interest that are best thought of as referring to some kind of space other than the tangent space at a point p in M but still to be thought of as being in a sense located at the point p To clarify a little what is needed here we recall from 12 3 8 that once we have a vector space here the space of tangent vectors at a point we can construct its dual space of covectors and all the various spaces of p q valent tensors Thus in a clear sense the spaces of p q tensors including the cotangent spaces covectors being 0 1 tensors are not anything new once we have the tangent spaces Tpat points p An almost similar remark would apply at least according to my own way of viewing things to the spaces of spinors at p see 11 3 Some others might try to take a diVerent attitude to spinors but these alternative perspectives on the matter will not be of concern for us here The spaces that we need for the gauge theories of particle interactions other than gravity are diVerent from these and so they are something new and it is best to think of them as referring to a kind of spatial dimension that is additional to those of ordinary space and time These extra spatial dimensions are frequently referred to as internal dimensions so that moving along in such an internal direction 325 does not actually carry us away from the spacetime point at which we are situated To make geometrical sense of this idea we need the notion of a bundle This is a perfectly precise mathematical notion and we shall be coming to it properly in 15 2 It had been found to be useful in pure mathematics1 long before physicists realized that some of the important notions that they had been previously using were actually to be understood in bundle terms In subsequent years theoretical physicists have become very famil iar with the required mathematical concepts and have incorporated them into their theories However in some modern theories these notions are presented in a modiWed form in relation to which spacetime itself is thought of as acquiring extra dimensions Indeed in many or most of the current attempts at Wnding a deeper framework for fundamental physics e g supergravity or string theory the very notion of spacetime is extended to higher dimensionality The internal dimensions then come about through the agency of these extra spatial dimensions where these extra spatial dimensions are put on an essentially equal footing with those of ordinary space and time The resulting spacetime thus acquires more dimensions than the standard four Ideas of this nature go back to about 1919 when Theodor Kaluza and Oskar Klein provided an extension of Einstein s general relativity in which the number of spacetime dimensions is increased from 4 to 5 The extra dimension enables Maxwell s superb theory of electromagnetism see 19 2 4 to be incorporated in a certain sense into a spacetime geometrical description However this 5th dimension has to be thought of as being curled up into a tiny loop so that we are not directly aware of it as an ordinary spatial dimension The analogy is often presented of a hosepipe see Fig 15 1 which is to represent a Kaluza Klein type modiWcation of a 1 dimensional universe When looked at on a large scale the hosepipe indeed looks 1 dimensional the dimension of its length But when examined more closely we Wnd that the hosepipe surface is actually 2 dimensional with the extra dimension looping tightly around on a much smaller scale than the length of the hosepipe This is to be taken as the direct analogy of how we would perceive only a 4 dimensional physical spacetime in a 5 dimensional Kaluza Klein total spacetime The Kaluza Klein 5 space is to be the direct analogue of the hosepipe 2 surface where the 4 spacetime that we actually perceive is the direct analogue of the basically 1 dimensional appearance of the hosepipe In many ways this is an appealing idea and it is certainly an ingenious one The proponents of the modern speculative physical theories such as supergravity and string theory that we shall encounter in Chapter 31 actually Wnd themselves driven to consider yet higher dimensional versions 326 15 1CHAPTER 15 Fig 15 1The analogy of a hosepipe Viewed on a large scale it appears 1 dimensional but when examined more minutely it is seen to be a 2 dimensional surface Likewise according to the Kaluza Klein idea there could be small extra spatial dimensions unobserved on an ordinary scale of the Kaluza Klein idea a total dimensionality of 26 11 and 10 having been among the most popular In such theories it is perceived that interactions other than electromagnetism can be included by use of the gauge connection idea that we shall be coming to shortly However it must be emphasized that the Kaluza Klein idea is still a speculative one The internal dimensions that the conventional current gauge theories of particle interactions depend upon are not to be thought of as being on a par with ordinary spacetime dimensions and therefore do not arise from a Kaluza Klein type scheme It is a matter of interesting speculation whether it is sensible to regard the internal dimensions of current gauge theories as ultimately arising from this kind of Kaluza Klein type extended spacetime in any signiWcant sense 2I shall return to this matter later 31 4 Instead of regarding these internal dimensions as being part of a higher dimensional spacetime it will be more appropriate to think of them as providing us with what is called a Wbre bundle or simply a bundle over spacetime This is an important notion that is central to the modern gauge theories of particle interactions We imagine that above each point of spacetime is another space called a Wbre The Wbre consists of all the internal dimensions according to the physical picture referred to above But the bundle concept has much broader applications than this so it will be best if we do not necessarily tie ourselves to this kind of physical interpretation at least for the time being 327 Fibre bundles and gauge connections 15 1 15 2The mathematical idea of a bundle A bundle or Wbre bundle B is a manifold with some structure which is deWned in terms of two other manifolds M and V where M is called the base space which is spacetime itself in most physical applications and where V is called the Wbre the internal space in most physical applica tions The bundle B itself may be thought of as being completely made up of a whole family of Wbres V in fact it is constituted as an M s worth of Vs see Fig 15 2 The simplest kind of bundle is what is called a product space This would be a trivial or untwisted bundle but more interesting are the twisted bundles I shall be giving some examples of both of these in a moment It is important that the space V also have some symmetries For it is the presence of these symmetries that gives freedom for the twisting that makes the bundle concept interesting The group G of symmetries of V that we are interested in is called the group of the bundle B We often say that B is a G bundle over M In many situations V is taken to be a vector space in which case we call the bundle a vector bundle Then the group G is the general linear group of the relevant dimension or a subgroup of it see 13 3 6 10 We are not to think of M as being a part of B i e M is not inside B instead B is to be viewed as a separate space from M which we tend to regard as standing in some sense above the base space M There are many copies of the Wbre V in the bundle B one entire copy of V standing above each point of M The copies of the Wbres are all disjoint i e no two intersect and together they make up the entire bundle B The way to think of M in relation to B is as a factor space of the bundle B by the family of Wbres V That is to say each point of M corresponds precisely to a separate individual copy of V There is a continuous map from B down VVV V V V V M B Fig 15 2A bundle B with base space M and fibre V may be thought of as constituted as an M s worth of Vs The canonical projection from B down to M may be viewed as the collapsing of each fibre V down to a single point 328 15 2CHAPTER 15 to M called the canonical projection from B to M which collapses each entire Wbre V down to that particular point of M which it stands above See Fig 15 2 The product space of M with V trivial bundle of V over M is written M V The points of M V are the pairs of elements a b where a belongs to M and b belongs to V see Fig 15 3a We already saw the same idea applied to groups in 13 2 3A more general twisted bundle B over M resembles M V locally in the sense that the part of B that lies over any suYciently small open region of M is identical in structure with that part of M V lying over that same open region of M See Fig 15 3b But as we move around in M the Wbres above may twist around so that as a whole B is diVerent often topologically diVerent from M V The dimension of B is always the sum of the dimensions of M and V irrespect ive of the twisting 15 1 All this may well be confusing so get a better feeling for what a bundle is like let me give an example First take our space M to be a circle S1 and the Wbre V to be a 1 dimensional vector space which we can picture topologically as a copy of the real line R with the origin 0 marked Such bundle is called a real line bundle over S1 Now M V is a 2 dimensional cylinder see Fig 15 4a How can we construct a twisted bundle B over M M V M M B b a b a a b Fig 15 3 a The particular case of a trivial bundle which is the product space M V of M with V The points of M V can be interpreted as pairs of elements a b with a in M and b in V b The general twisted bundle B over M with Wbre V resembles M V locally i e the part of B over any suYciently small open region of M is identical to that part of M V over same region of M But the Wbres twist around so that B is globally not the same as M V 15 1 Explain why the dimension of M V is the sum of the dimensions of M and of V 329 Fibre bundles and gauge connections 15 2 Zero M S1 a b Fig 15 4To understand how this twisting can occur consider the case when M is a circle S1and the Wbre V is a 1 dimensional vector space i e a space modelled on R but where only the origin 0 is marked but no other value such as the identity element 1 a The trivial case M V which is here an ordinary 2 dimensional cylinder b In the twisted case we get a Mo bius strip as in Fig 12 15 with Wbre V We can take a Mo bius strip see Fig 15 4b and Fig 12 15 Let us see why this is a bundle locally the same as the cylinder We can produce an adequately local region of the base space S1by removing a point p from S1 This breaks the base circle into a simply connected4 segment5S1 p and the part of B lying above such a segment is just the same as the part of the cylinder standing above S1 p The diVerence between the Mo bius bundle B and the cylinder emerges only when we look at what lies above the entire S1 We can imagine S1to be pieced together out of two such patches namely S1 p and S1 q where p and q are two distinct points of S1 then we can piece the whole of B together out of two corresponding patches each of which is a trivial bundle over one of the individual patches of S1 It is in the gluing together of these two trivial bundle patches that the twist in the Mo bius bundle arises Fig 15 5 Indeed it becomes particularly clear that it is a Mo bius strip that arises with just a simple twist if we reduce the size of our patches of S1 as indicated in Fig 15 5b this reduction making no diVerence to the struc ture of B It is important to realize that the possibility of this twist results from a particular symmetry that the Wbre V possess namely the one which re verses the sign of the elements of the 1 dimensional vector space V This is y 7 y for each y in V This operation preserves the structure of V as a vector space We should note that this operation is not actually a symmetry of the real number system R In fact R itself possesses no symmetries at all The number 1 is certainly diVerent from 1 for example and x 7 x is not a symmetry of R not preserving the 330 15 2CHAPTER 15 a b Fig 15 5 a We can produce an adequately local simply connected region of the base S1by removing a point p from it the part of the bundle above S1 p being just a product The same applies to the part of B above S1 q where q is a diVerent point of S1 We get a cylinder if we can match the two parts of B directly but we get the Mo bius bundle as illustrated above if we apply an up down reflection a symmetry of V to one of the two matched portions b The resulting Mo bius strip is little more obvious if we reduce the size of the two parts of S1so that there are only small regions of overlap multiplicative structure of R 15 2 It is for this reason that V is taken as a 1 dimensionalrealvectorspaceratherthanjustasthereallineRitself We sometimes say that V is modelled on the real line We shall be seeing shortlyhowother Wbre symmetriesprovideopportunitiesforotherkindsof twist 15 3Cross sections of bundles One way that we can characterize the diVerence between the cylinder and the Mo bius bundle is in terms of what are called cross sections or simply 15 2 Explain this 331 Fibre bundles and gauge connections 15 3 sections of a bundle Geometrically we think of a cross section of a bundle B over M as a continuous image of M in B which meets each individual Wbre in a single point see Fig 15 6a We call this a lift of the base space M into the bundle Note that if we apply the map that lifts M to a cross section of B and then follow this with the canonical projection we just get the identity map from M to itself that is to say each point of M is just mapped back to itself For a trivial bundle M V the cross sections can be interpreted simply as the continuous functions on the base space M which take values in the space V i e they are continuous maps from M to V Thus a cross section of M V assigns 6in a continuous way a point of V to each point of M This is like the ordinary idea of the graph of a function illustrated in Fig 15 6b More generally for a twisted bundle B any cross section of B deWnes a notion of twisted function that is more general than the ordinary idea of a function Let us return to our particular example in 15 2 above In the case of the cylinder product bundle M V our cross sections can be represented simply as curves that loop once around the cylinder intersecting each Wbre just once Fig 15 7a Since the bundle is just a product space we can consistently think of each Wbre as being just a copy of the real line and we can thus consistently assign real number coordinates to the Wbres The coordinate value 0 on each Wbre traces out the zero section of marked points that represent the zeros of the vector spaces V A general cross section provides a continuous real valued function on the circle the height above the zero section being the value of the function at eachpoint of the circle Clearly there are many cross sections that do not B M a b Fig 15 6 a A cross section or section of a bundle B is a continuous image of M in B which meets each individual Wbre in single point b This generalizes the ordinary idea of the graph of a function 332 15 3CHAPTER 15 Zero a b Fig 15 7A cross section of a line bundle over S1is a loop that goes once around intersecting each Wbre just once a Cylinder there are sections that nowhere intersect the zero section b Mo bius bundle every section intersects the zero section intersect the zero section non vanishing functions on S1 For example we can choose a section of the cylinder that is parallel to the zero section but not coincident with it This represents a constant non zero function on the circle However when we consider the Mo bius bundle B we Wnd that things are very diVerent The reader should not Wnd it hard to accept that now every cross section of B must intersect the zero section Fig 15 7b The notion of zero section still applies since V is a vector space with its zero marked This qualitative diVerence from the previous case makes it clear that B must be topologically distinct from M V To be a bit more speciWc we can begin to assign real number coordinates to the various Wbres V just as before but we need to adopt a convention that at some point of the circle the sign has to be Xipped x 7 x so that a cross section of B corresponds to a real valued function on the circle that would be continuous except that it changes sign when the circle is circumnavi gated Any such cross section must take the value zero somewhere 15 3 In this example the nature of the family of cross sections is suYcient to distinguish the Mo bius bundle from the cylinder An examination of the family of cross sections often leads to a useful way of distinguishing various diVerent bundles over the same base space M The distinction between the Mo bius bundle and the product space cylinder is a little less extreme than in the case of certain other examples of bundles however Sometimes a bundle has no cross sections at all Let us con