EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE Daniel Henry Gottlieb Abstract. Given a parameterized space of square matrices, the associated * *set of eigenvectors forms some kind of a structure over the parameter space. Wh* *en is that structure a vector bundle? When is there a vector field of eigenvec* *tors? We answer those questions in terms of three obstructions, using a Homotopy T* *heory approach. We illustrate our obstructions with five examples. One of tho* *se exam- ples gives rise to a 4 by 4 matrix representation of the Complex Quaterni* *ons. This representation shows the relationship of the Biquaternions with low dimen* *sional Lie groups and algebras, Electro-magnetism, and Relativity Theory. The eigens* *tructure of this representation is very interesting, and our choice of notation pr* *oduces impor- tant mathematical expressions found in those fields and in Quantum Mechan* *ics. In particular, we show that the Doppler shift factor is analogous to Berry's* * Phase. 1. Introduction This work was stimulated by the Gibbs Lecture of Sir Michael Berry given at the 2002 American Math. Soc. meeting in San Diego California. Berry's lecture discussed the discription of physical phenomina by means of slowly changing eig* *en- vectors of relevant linear operators, usually Hamiltonians of Quantum Mechanics. This work was advanced by several mathematical physicists, such as Barry Simon, under the name of Berry's Phase. The original papers are [Berry(1984)] and [Si- mon(1983)]. A multitude of similar phenomena are found in [Berry(1990)]. Berry's Phase can be thought of in terms of eigenbundles, or spectral bundle* *s as some mathematical physicists call them. These are vector bundles whose fibres a* *re spaces of eigenvectors associated to linear operators which are parameterized by the base space. There are two questions involving these spectral bundles. The first is: Wh* *en do they exist? The second is: What is a relevant connection to put on a spectral bundle which results in physical descriptions? The first question is topological, the second is more geometrical and of cou* *rse physical. We will approach the first question from a homotopy theoretical point* * of view. Spectral bundles are related to an area of Analysis concerned with spectr* *al projections. Mathematical physicists have incorporated some homotopy concepts, such as homotopy groups, in their study of spectral bundles, [Avron, Sadun, Seg* *ert, ______________ 1991 Mathematics Subject Classification. 57R45, 17B90, 15A63. Key words and phrases. exponential map, singularity, electromagnetism, energ* *y-momentum, vector bundles, Clifford Algebras, Doppler shift. Typeset by AM S-T* *EX 1 2 DANIEL HENRY GOTTLIEB Simon(1989)]. What we do here is study the existence of spectral bundles by mea* *ns of a commutative diagram. This will characterize when spectral bundles exist in terms of three obstructions, and will organize the many variants under which the existence problem can be posed. We illustrate the issues involved by giving a few simple examples and one so* *phis- ticated example. The sophisticated example consists of a set of 4x4 matrices wh* *ich are a representation of the biquaternions, that is the quaternions complexified* *. We denote the quaternions by H and the biquaternions by H C. The quaternions and biquaternions have been studied for over 150 years as a * *con- venient language for physics, [Gsponer, Hurni(2002)] The generalization of quat* *er- nions, called Clifford Algebras, has also been extensively studied by physicist* *s, especially by Dave Hestenes under the name of Geometric Algebra, [Hestenes, Sob- cyk(1987)]. Our particular 4-dimensional representation of the biquaternions naturally g* *ives rise to 4-dimensional representations of important low dimensional Lie groups a* *nd algebras. There is a conjugate representation also, and a öm dulus square map- ping", m , from these representations of the biquaternions gives well known rel* *a- tionships of low dimensional Lie groups, and electromagnetic energy-momentum tensors, as well as a cononical form of the eigenvectors of Lorentz transformat* *ions. This last feature allows us to see the Doppler shift factor as an analogue of B* *erry's phase. Finally in Section 8, we give two examples of probability distributions* * in Quantum Mechanics which can be expressed as inner products of eigenvectors . 2. Examples In this section we set up our basic point of view and illustrate with 4 exam* *ples. Let V be a vector space over the Real numbers R or the Complex numbers C. Consider the space Hom(V, V ) of linear maps from V to V . We assume that V is a finite dimensional space so that we can describe the topology of Hom(V, V ) sim* *ply. If a basis is chosen for the n-dimensional space V , then we have automatically chosen an isomorphism from Hom(V, V ) to Mn(K) where K stands for either the scalars R or C. Here Mn(K) denotes the space of n x n matrices with entries in * *K. This space is given the Euclidean topology of Kn2. Now let : B ! Hom(V, V ) be a continuous map where B is a topological space. We will call a field of linear operators (or matrices) on B. In the ph* *ysics literature, this is frequently called a system of linear operators parametrized* * by B. In Physics, B is usually an interval of the Real line and the parameter is freq* *uently thought of as time. Another variant is the field is over a parameter space B, a* *nd a physical process is represented by a path in the parameter space B. There is a trivial example where : B ! Mn(K) is the constant map which maps every point to the identity matrix I. In this case, any subbundle of the t* *rivial bundle is an eigenbundle. At the opposite extreme we give an example for which no eigenbundle exists. Let B be the rotation group in two dimensions, SO(2), and let be the inclusion map of SO(2) into the space of 2 x 2 matrices M2(R). Every rotation except for EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 3 the identity has imaginary eigenvectors and eigenvalues, hence there cannot be a real spectral bundle over SO(2). We will give four examples below which illustrate various issues which arise* * in the study of the existence of spectral bundles. Example 1 : Let B = R and let : R ! M2(R) be given by ` ' (t) = 10 t1 Every (t) has only one eigenvalue ~ = 1 with corresponding eigenspace spann* *ed by the vector (1, 0)T when t does not equal 0, and at t = 0, (0) = I so the eigenspace is all of R2. In this case the spectral line bundle exists and is tr* *ivial since there is a nonzero cross-section. For example, the map which takes t 7! (t, (1,* * 0)T ) is a cross-section. We regard this cross-section as a vector field of eigenvect* *ors. Example 2: Let B = R and let : R ! M2(R) be given by ` ' (t) = g1(t) f(t)1 where f(t) is a continuous real valued function which is greater than zero if t* * is positive and equal to zero if t is nonpositive; and g(t) has the opposite prope* *rty, for example g(t) = f(-t). In this example again, there is only one eigenvalue ~ = 1, but now the eigenspaces are spanned by (1, 0)T for t > 0 and (0, 1)T for t < 0,* * and at t = 0 the eigenspace is R2. Thus there is no continuous choice of eigenvect* *ors over R and so there is no eigenbundle. However, if we were willing to change the field slightly, by letting f(t) be zero in a small interval about 0, then we * *can connect up the (1, 0)T vector field continously with the (0, 1)T vector fields * *through eigenvectors in R2 near 0. So example 2 shows that degenerate eigenspaces are an obstruction to eigenbundles, but under some circumstances, a slight change in can eliminate the obstruction. Example 3: Let B = R3 and let : R3 ! M2(R) be given by ` ' (u, v, w) = uv wv Then (u, 0, u) has only one eigenvalue ~ = u and the associated eigenspace * *is the whole of R2. Off the line l given by {(u, 0, u)} however, has two distinc* *t real eigenvalues and the corresponding eigenspaces are one- dimensional and orthogon* *al, because (b) is a symmetric matrix. Let B0 = R3 - l. Then there are two spectral line-bundles over B0. But neither of them is a trivial line bundle. So there * *is no eigenvector field over B0. This is seen by moving around a loop which links the line l. The line bundle over the loop is not trivial, so it looks like a Mobius band. If we regard the* * map as mapping into M2(C), the eigenbundles over B0 are complex line bundles and must be trivial since complex line bundles are classified by the first Chern cl* *ass which lives in the second cohomology group with integer coefficients. Since B0* * is 4 DANIEL HENRY GOTTLIEB homotopically equivalent to the circle, the second cohomology group, and hence the Chern class, and hence the line bundle, must be trivial. This example was mentioned by M. V. Berry in [Berry(1990)] on Page 38, where he states that this phenomenon didn't seem to be widely known in matrix theory. The fourth example is more complex, and it is related to the quaternions H, the biquaternions H C, SL(2, C), SO(3, 1), so(3, 1), SU(2) and su(2) and other topics. Example 4: Let B = C3 and : C3 ! M4(C) so that (A1, A2, A3) is a matrix F such that 0 0 1 _____|_A1_____A2____A3___ F = B@ A1A | 0 -iA3 iA2 CA 2 | iA3 0 -iA1 A3 |-iA2 iA1 0 Or in block form, ` ~T ' F = 0~A x(A-iA~) where the notation x(-iA~) symbolizes the 3x3 matrix which operates on a column vector v to produce the cross product v x (-iA~). Let . represent the usual Euclidean inner product extended linearly to the c* *om- plex case. Thus ~A. ~A= A1A1 + A2A2 + A3A3. Then the eigenspace structure of (A~) depends on ~A. ~A. Case 1: A~.A~ 6= 0. In this case there are two nonzero eigenvalues, one the * *negative of the other (since the square of the eigenvalue equals A~ . ~A). Each eigenva* *lue corresponds to a two-dimensional eigenspace. Let B1 denote the set of all vecto* *rs A~ such that ~A. ~A6= 0. Then there are no eigenbundles for restricted to B1. Case 2: ~A. ~A = 0 and A~ 6= 0. In this case there is only one eigenvalue,* * 0, and it corresponds to a two-dimensional eigenspace. Let B2 denote the set of a* *ll vectors A~ such that A~. ~A= 0 and A~ 6= 0. Then there is an eigenbundle of ra* *nk two over B2. It splits as a Whitney sum of two trivial line bundles. So there* * are two linearly independent eigenvector fields over B2, and one of them consists o* *f real eigenvectors. Case 3: A~ = 0. In this case (~0) is the zero matrix, so every vector in C4* * is an eigenvector. The above assertions are proved in [Gottlieb(1998), (2001)]. See section 7 o* *f this paper. 3. Obstructions to the existence of eigenbundles We will show that the obstruction to the existence of spectral bundles over * *B for the field : B ! Hom(V, V ) consists of two crossections which must be constru* *cted EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 5 over B. A cross-section to a continuous map f : X ! Y is a map s : Y ! X so that the composition f O s is the identity map, 1Y , on Y . This means that we are a* *ble to choose in a continuous way one element in each fibre f-1 (y) of f. A cross-sect* *ion is a homeomorphism of Y to its image s(Y ) in X. Thus we may regard Y as a subspace s(Y ) of X. If the first two cross-sections, s1 and s2 exist, then the existence of a th* *ird, s3, gives an eigenvector field. Suppose we want to construct a spectral bundle whose fibres are k-dimensional eigenspaces over a field : B ! Hom(V, V ) where V is an n dimensional vector space. Then we first consider the product space BxKxGk,nxV . Here Gk,n = G(V ) is the Grassmannian space of k-planes in V . We define a subspace L3 of B x K x Gk,nx V as follows: L3 consists of all t* *he points (b, ~, W, ~v) in B x K x Gk,nx V so that ~ is an eigenvalue of (b), an* *d W is a k-dimensional eigenspace associated to ~, and ~vis an eigenvector in W . Now the projections B x K x Gk,nx V -ß3! B x K x Gk,n ß2-!B x K -ß1!B give rise to a sequence of mappings L3 -ß3!L2 -ß2!L1 -ß1!B where L2 := ß3(L3) and L1 := ß2(L2) are the images of the projections ß3 and ß2 respectively. That is: L2 and L1 are the subpaces of B x K x Gk,n and B x K consisting of the points (b, ~, W ) and (b, ~) respectively where ~ is an eigen* *value of (b), and W is a k-dimensional eigenspace associated to ~. Now the map ß3 : L3 ! L2 is a k-plane vector bundle. In fact it is a k-spect* *ral bundle with respect to the matrix field L2 ! Mn(K) defined by (b, ~, W ) 7! (b* *). Now this spectral bundle restricts to a subspace as a spectral bundle over the matrix field restricted to the subspace. So if s : B ! L2 is a cross-section t* *o the map ß1 O ß2 : L2 ! B, then the restriction of the spectral bundle over L2 to the spectral bundle over s(B) gives a spectral bundle ß3 : L03! s(B) over B for the matrix field . The above paragraphs give the notation and the proof for the following class* *ifi- cation theorem for spectral bundles: Theorem 3.1. The k-spectral bundles are in one to one correspondence with the cross-sections of the map ß1 O ß2 : L2 ! B It is convenient to break the cross-section s into two cross-sections: s1 : * *B ! L1, and s2 : s1(B) ! L02where L02denotes ß-12(s1(B)), the preimage of s1(B) contain* *ed in L2. Now the composition s2 O s1 is a cross-section to ß1 O ß2 : L2 ! B. On t* *he other hand, a cross-section s : B ! L2 induces the cross-section ß2 O s =: s1, * *and the cross-section s2 is s O ß1 : s1(B) ! L02. The following diagram may be helpful in tracing the above notation in the th* *e- orem below. The horizontal arrows represent inclusion maps. 6 DANIEL HENRY GOTTLIEB L003 ----! L03 ----! L3 ----! B x K x Gk,nx V x ? ? ? s3?? ?yß3 ?yß3 ß3?y s2s1B ----! L02 ----! L2 ----! B x K x Gk,n x ? ? s2?? ?yß2 ß2?y s1B ----! L1 ----! B x K x ? s1?? ß1?y B ________ B Theorem 3.2. a) The set of s1 cross-sections is in one to one correspondence with the conti* *nuous functions ~ : B ! K so that every every ~(b) is an eigenvalue of (b) whose associated eigenspace has dimension k. b) The set of s2 cross-sections corresponds to the continuous selections of k-* *dimensional subspaces of eigenvectors with eigenvalues ~(b). c) The set of nowhere zero cross-sections s3 of the spectral bundle L003ß3-!s2* *s1(B) = B corresponds to the set of nowhere zero eigenvector fields for the eigenbu* *ndle. Proof. a) The cross-section s1(b) = (b, ~(b)) is continuous if and only if ~(b) is co* *ntinuous. b) s2(b) = (b, ~(b), Wb) where b 7! Wb picks out a k-dimensional subspace * *of eigenvectors with eigenvalue ~(b) contained in V , that is it is a function* * from B ! G(Vk). Now s2 is continuous if and only if the function B ! Gk(V ) is continuous. c) s3 is a cross-section to the vector bundle L003ß3-!s2s1(B) = B, so s3(b) is* * an eigenvector for (b). If s3(b) 6= 0 for all b in B, then the spectral bundl* *e has a trivial line bundle summand, or equivalently, a nonzero eigenvector field. Now let us consider L1 for complex spectral line bundles. This is the larg* *est of the possible L1's for a fixed . Every other L1 for higher dimensional comp* *lex spectral bundles, or for real spectral bundles associated to , must be a subs* *pace of the L1 for complex spectral line bundles. In those cases it is possible tha* *t there are no eigenvalues for (b) and hence there is no cross-section s1. Examples * *like the real rotation matrices SO(2) or the spectral 3-bundles of example 4 show t* *hat there is no s1 because ß1 is not onto. But for complex spectral line bundles,* * not only must ß1 be onto, but L1 is a topological branched covering of B, where we mean the following by topological branched covering: A space X which admits a continuous onto map p : X ! B such that all fibres are discrete and so that the path lifting property holds. That is for every x 2 X, and path oe in B startin* *g at oe(0) = p(x), there is a path __oein X so that oe = p O __oeand __oe(0) = x. Theorem 3.3. For complex line bundles, ß1 : L1 ! B is a topological branched covering of B. EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 7 Proof. Consider the mapping from B to the complex polynomials of degree n given by b 7! det(~I - (b)) This is a continuous map from b to the characteristic pol* *yno- mial of (b). The Fundamental Theorem of Algebra tells us that there are n roots of this polynomial counting multiplicities, for any point b. The roots are of c* *ourse, the eigenvalues of (b). I like to think of it using vector fields. Over each b* * in B xC is a fibre C. On each fibre there is a vertical vector field on C given by atta* *ching the vector pb(z) to z where pb is the characteristic polynomial for (b). Each * *zero has a positive vector field index, equal to the multiplicity of the correspondi* *ng root. The sum of the local indices adds up to a global index n for every fibre. The s* *et of the the zeros is L1. So every b is covered by at least one zero and at most n z* *eros. Hence ß1 is onto, and L1 consists of at most n connected components over B. As we move from one b to a nearby point, there are zeros in the new fibre close to where they were at b, because no zero can be annihilated by another since there* * are no nonpositive indices to cancil out. This gives L1 the branched covering struc* *ture. See [Gottlieb, Samaranayake(1994)] for a detailed discussion of the index of ve* *ctor fields. In the case of real matrices, the real characteristic polynomial det(~I - (* *b)) can be thought of as a vertical vector field on the fibres R. Again the zeros o* *f this vertical vector field on B x R gives us L1, but here it is not necessarily a br* *anched cover over B. The reason is that the zeros of the characteristic polynomial on * *the real line have indicial values of 1, -1 or 0 . The opposite signs and zero ind* *ices allow the zeros on the Real line to annihilate each other, so that there may no* *t be a nearby zero on a nearby fibre to continue the local covering of B by L1. The total index on each fibre R is 1 for odd order matrices and 0 for even o* *rder matrices, so the sum of the local indices of each zero add up to 0 in even dime* *nsions and 1 in odd dimensions. Thus, for odd dimensional matrix fields, there is alwa* *ys a zero of index 1 in each fibre, so ß1 is always onto in that case. For the ev* *en dimensional matrix field however, there is no guarantee of a zero in every fibr* *e, so ß1 may not be onto. The real matrix field may be considered as acting on a complex vector space.* * In this case, the zeros on the real line in C still have their indices of positive* * integers as well as their indices 1 or 0 on the Real line. In this case, a real zero's ann* *ihilation actually is given by a splitting of the zero into two complex conjugate zeros, * *which of course are off the Real line. Thus a real zero doesn't disappear, it splits * *into two conjugate zeros which leave the Real line in the Complex plane. Now we will reconsider our examples in light of the above considerations. Example 1 has only one eigenvalue for each b 2 R, so s1 exists. At each point b there is only one 1-dimensional eigenspace except at b = 1, where it is 2-dimensional. This potentially blocks the existence of s2, but it happens that* * we may choose a 1-dimensional eigenspace in the 2-dimensional eigenspace so that the choice of 1-dimensional eigenspaces is continuous. So s2 exists. There is* * an obvious eigenvector field, so s3 exists. It is worth remarking that given a ve* *ctor bundle over a contractible space such as R, the vector bundle must be trivial a* *nd there are always nonzero vector fields; or to say it another way, we can always* * split off a trivial line bundle. 8 DANIEL HENRY GOTTLIEB Example 2 is the same as Example 1, except that it is impossible to choose a 1-dimensional subspace at b = 0 in such a way to make a continuous selection of* * 1- dimensional eigenbundles. Hence s2 does not exist. The possibility was mentioned of altering slightly to eliminate this obstruction to s2 existing. For 1-dime* *nsional B's such as a line interval or a circle, this can always be done. Of course, s* *ince Hom(V, V ) is contractible,we can always homotopy to a constant and obtain a new s2, but this is too large a change for most purposes. There are homotopy obstructions to changing so as to eliminate the obstruc* *tion to s2. Suppose that D is the unit disk in the plane. Let B = D, and let (b) be a symmetric matrix of order 2 with eigenvalues 1 when b 2 S1, where S1 is the boundary of D. Suppose that the +1 eigenvectors are pointing orthogonally outsi* *de of D. The it is impossible to extend over D with values symmetric matrices su* *ch that every matrix has no 2-dimensional eigenspace. This follows since the outwa* *rd pointing eigenvector field cannot be extended to a nonzero vector field over D,* * since such a vector field has index = 1. Since every symmetric matrix has a two frame of eigenvectors whenever the two eigenvalues are distinct, such an extension of would give rise to a a nonzero vector field. Contradiction. Example 3 exhibits some homotopy type features. Recall ` ' (u, v, w) = uv wv Since the matrices are symmetric, the eigenvalues are real and we can find cont* *inu- ous eigenvalue functions on B = R3. Hence s1's exist. On the other hand, s2 does not exist. We know that if an s2 existed, there would be a eigenbundle over R3, which is contractible. Hence it would be a trivial line bundle. But we know t* *hat on a circle linking l, the restriction line bundle is not trivial. So that con* *tradicts the triviality of a bundle over R3. If we consider the question over B0 = R3 -* * l, we have eliminated degenerate eigenspaces, every eigenspace is 1-dimensional, so we can choose a continuous selection of eigenspaces, so s2 exists, and we have a spectral line bundle over B0. But it is not a trivial bundle. Now real line bun* *dles are classified by their Stiefel-Whitney class w1, which lives in the first coho* *mology group of B0 with Z2 coefficients, H1(B0, Z2). Now B0 is homotopy equivalent to S1, and so there is only one nonzero w1 2 H1(B0, Z2) = Z2. If we consider the same field acting on a complex two-dimensional vector spa* *ce, we again get a spectral line bundle over B0, but this time the bundle is trivia* *l in that is there is a nonzero eigenvector field, but it is not completely real. A * *complex line bundle is classified by its Chern class c1 2 H2(B0, Z), the two-dimensional cohomology group with integer coefficients. Since B0 is homotopy equivalent to a circle, the two-dimensional cohomology must be zero and hence c1 = 0, so the bundle is trivial. Example 4 has the property that every eigenspace has complex dimension 2 except for the 0 matrix. If we remove the 0 matrix from consideration, we see t* *hat if s1 exists, then s2 would exist and we would have an eigen 2-bundle. If we re* *strict to Case 2, the set B2 of all vectors ~Asuch that ~A. ~A= 0 and ~A6= 0, we get s* *1 since the only eigenvalue is 0. Hence in this case there exists an eigenbundle of ra* *nk 2 over B2. Let us write ~A:= ~E+ iB~ where ~E and ~B are real vectors. In this ca* *se, EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 9 where ~A. ~A= 0, we have E = B and ~E. ~B= 0. We may describe the eigenspace by means of two linearly independent eigenvectors: E~+ iB~ and E2u + ~Ex ~Bwhere u = (1, 0, 0, 0). Here we are regarding the 3-vectors as living in the space or* *thogonal to u. These eigenvectors each give rise to an eigenvector field which shows tha* *t over B2 the eigenbundle of rank 2 splits as a Whitney sum of two trivial spectral li* *ne bundles. In Case 1 of Example 4, where B1 is the set of vectors ~A such that ~A. ~A6=* * 0, we see that s1 does not exist. In this case each matrix has two distinct eigenv* *ector spaces. Recall that for complex line bundles, Theorem 3.3 states that ß1 : L1 !* * B is a branched covering of B. If we restrict ourselves to matrices so that every eigenvalue is distinct, then the branching part of the branched covering is eli* *minated and we have a covering. Each connected component of the covering space is a connected covering space. A cross-section s1 exists if and only if there is a c* *onnected component which is homeomorphic to B, that is, if and only if there exists a one to one covering of B. In situation at hand, the eigenvalues are are not distinc* *t, but there are only two of them, one being the negative of the other. This gives ris* *e to a two to one covering of B1. Hence s1 does not exist. In this case, if we move around a closed curve in B1 which loops B2 one time, we arrive at the same matrix, but the eigenspace has been transported to the eigenspace corresponding to the opposite eigenvalue. This is a subtle effect w* *hen encountered without the aid of the double covering point of view. We will add one more example to our list of four examples. This will actual* *ly be an extension of Example 4, and is a faithful 4-dimensional representation of* * the Biquaternions H C. Example 5: Consider the set I + S of all 4 x 4 matrices of the form aI + F where a is any complex number and I is the identity matrix and F is any matrix from Example 4. That is F 2 S and so has the form ` ~T ' F = 0~A x(A-iA~) Here B = C4, and (A0, A1, A2, A3) = A0I + F . That is: 0 A0 1 _____|_A1_____A2____A3___ (A0, A1, A2, A3) = B@ A1A | A0 -iA3 iA2 CA 2 | iA3 A0 -iA1 A3 |-iA2 iA1 A0 Let < , > represent the usual Minkowskian inner product extended linearly to* * the complex case. Thus, if A := (A0, A1, A2, A3) =: (A0, ~A), then = -A0A0 + A1A1 + A2A2 + A3A3 = -A0A0 + ~A. ~A. Then the eigenspace structure of (A) depends on . Case 1: 6= 0 and ~A6= 0. In this case there are two nonzero eigenvalu* *es when A~ . ~A 6= 0. Each eigenvalue corresponds to a two-dimensional eigenspace. Let B1 denote the set of all vectors A such that 6= 0. Then there are no eigenbundles for restricted to B1. 10 DANIEL HENRY GOTTLIEB Case 2: = 0 and ~A6= 0. In this case there is one or two eigenvalues,* * but one of them is equal to 0, and it corresponds to a two-dimensional eigenspace. * *Let B2 denote the set of all vectors A such that = 0 and ~A6= 0. Then there * *is an eigenbundle of rank two over B2. It splits as a Whitney sum of two trivial l* *ine bundles. So there are two linearly independent eigenvector fields over B2, and * *one of them consists of real eigenvectors. Case 3: A~ = 0. In this case (A) is a diagonal matrix, so every vector in C* *4 is an eigenvector. We note that the cases of Example 5 seems to be very similar to the cases of Example 4, but now the eigenvalues are not each other's negatives, and in Case 2 there are one or two eigenvalues. But one of them is always zero, so s1 exists * *in that case since the eigenvalue map is the constant zero. But then the nonzero eigenv* *alue also must form an eigenfunction over B2, and so there is another spectral 2-bun* *dle over B2. Over the region where A0 = 0, this second spectral 2-bundle is identic* *al with the first. 4. Biquaternions The set of matrices of Example 5 0 A0 1 _____|_A1_____A2____A3___ (A0, A1, A2, A3) = B@ A1A | A0 -iA3 iA2 CA 2 | iA3 A0 -iA1 A3 |-iA2 iA1 A0 is a representation of the biquaternions. Obviously it is isomorphic to C4 as a vector space. We will list a basis be* *low which will reveal the relationship of the matrices and the biquaternions. Let x denote the matrix above in which A1 = 1 and the other Ai = 0. That is 0 1 0__|_1_0___0__ x = (0, 1, 0, 0) = B@ 10 | 0 0 0 CA | 0 0 -i 0 | 0 i 0 In the same way we define matrices y := (0, 0, 1, 0) z := (0, 0, 0, 1) I = (1, 0, 0, 0), the identity matrix of order 4. Now xy = iz and x2 = y2 = z2 = I and xy = -yx. Then the basis {ix, iy, iz,* * I} obviously has the relations defining the biquaternions. There is another representation of the biquaternions in which the traceless * *ma- trices are given by ` ' A~T F = 0~A x(iA~) EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 11 These matrices differ from the previous set in Example 4_by_changing the -i to * *+i. If we denote the set of matrices of Example 4 by S, let S denote the set of mat* *rices of the form F above. Now let {X, Y, Z, I} be the complex conjugates of {x, y, z, I} respect* *ively. These new elements satisfy XY = -iZ and X2 = Y 2 = Z2 = I and XY = -Y X. So the basis {-iX, _-iY,_ -iZ, I} obviously has the relations defining the biquaternions for I S. __ __ __ Now it happens that any F 2 S commutes with any G 2 S. That_is F G = G F for F, G 2 S. This gives rise to a pairing (I S) (I S) ! M4(C) given by A B 7! AB where the product AB is in the space of 4 x 4 complex matrices. This pairing is an isomorphism of rings. This can be seen by observing that the following set of sixteen matrices forms a basis of M4(C): Theorem 4.1. The set of sixteen matrices I, xX, yY, zZ, a) x,y, X,Y, yZ,xZ,zY,zX, z, Z, xY, yX forms a basis for M4(C), the vector space of 4 x 4 complex matrices. b) The square of each of the matrices in the basis is I. c) Each matrix is Hermitian, so real linear combinations of the basis are the 4 x 4 Hermitian matrices. d) Every matrix has zero trace except for I. Proof. Theorem 3.3 of [Gottlieb(2001)]. It is easy to calculate any 4 x 4 matrix in terms of this basis using MATLAB. Below I produce a matrix whose first column is x written as a column vector of length 16. (this is done by x(:), which counts from 1 down the first column and then down the next column until you arrive at the 4 x 4 term which is the last number of the vector). The remaining columns are given in the order as shown below in the definition of Total. (4.1) Total = [x(:) X(:) y(:) Y(:) z(:) Z(:) xY(:) yX(:) yZ(:) zY(:) zX(:) xZ(:) xX(:) yY(:) zZ(:) I(:)]; Now any 4 x 4 matrix M can be converted into a vector M(:). The command Total \ M(:) gives the vector of coefficients which when multiplied with the ba* *sis in the order found in Total above will give the linear combination of M in term* *s of the basis. Now since M4(C) is the complex Clifford algebra C`(4), there must be generat* *ors 12 DANIEL HENRY GOTTLIEB ff0, ff1, ff2, ff3 so that ffiffj + ffjffi = ffiijI. One such set of ff's is gi* *ven by ff0= x ff1= y ff2= zX ff3= zY. Theorem 4.2. Let F, G 2 S satisfy F u = ~Aand Gu = ~B. Then a) F G + GF = (A~. ~B)I __ __ b) F G = G F c) [F, G]u = 2iA~x ~B d) eF = cosh(~F )I + sinh(~F_)~FF where ~F is an eigenvalue of F . Proof. Corollary 4.7, Theorem 4.8, Corollary 4.4, and Theorem 8.5 of [Gottlieb(* *1998)] respectively. Now every nonsingular matrix A 2 M4(C) gives rise to an inner automorphism of M4(C) given by B 7! A-1 BA. These maps transform the basis into a new basis with the same algebraic properies, but the form of the representative matrices can be quite different. We will end this section discussing what distinguishes* * our representation from the other representations. __ The matrices of S (or S ) are skew symmetric with respect to the Minkowski metric - + ++. That is equivalent to the property F T = -jF j where F 2 S, F T is the transpose of F and j = the diagonal matrix with -1, 1, 1, 1 down the main diagonal. A popular set of matrices are the skew symmetric matrices with respec* *t to the Euclidean metric. They satisfy A = -AT . Now j1=2F j-1=2 is a skew symmetric matrix if j1=2 and j-1=2 equal the diagonal matrix i, 1, 1, 1 respectively. He* *nce if ` ~T ' (4.2) F = 0___|A________~ A | i(xA~) then ` ~T ' (4.3) j1=2F j-1=2 = -i 0___|_-A______~ A | (xA~) __ Thus M4(C) is the tensor product (I + j1=2Sj-1=2 ) (I + j1=2S j-1=2 ). This means that the transformed S matrices still have squares equal to a multiple of* * the identity, and it satisfies_the same exponential equation as in Theorem 4.2d. An* *d the transformed S and S still commute, but they are no longer the complex conjugate of each other. It is this property which gives our representation its distinct* *ive advantage, because the öm dulus squared map" is a multiplicative homomorphism on S. The matrices of the form ` ~T ' (4.4) F = 0___|A____~ A |xC~ EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 13 are the skew_symmetric matrices with respect to the Minkowski inner product. So S and S are skew symmetric matrices with respect to the Minkowski inner product. The only skew symmetric matrices with respect to the Minkowski inner__ product whose squares are multiples of I are precisely the matrices of S and S* * . [Gottlieb(1998)], see Theorem 4.5 . __ Now note that_if_F 2 S, then both the complex conjugate F and the_transpos* *e_ F T are both in S . Thus the pseudo automorphisms conjugation : A 7! A , which is antilinear in that it changes the sign of i, and transpose_: A 7! AT , which reverses the order of multiplication,_interchange S and S . In terms of our b* *asis, ax + by + cz 7! __aX + bY + _cZ under conjugation and ax + by + cz 7! aX + bY + cZ under transposition. The composition of conjugation_and transposition yiel* *ds the Hermitian conjugate y : ax + by_+ cz 7! __ax + by + _cz which is an antili* *near isomorphism which preserves S and S. __ On the other hand, S and S are interchanged by the inner automorphism A 7! jAj. That follows since jF j = -F T when F 2 S. In terms of our basis, ax + by* * + cz 7! -aX - bY - cZ. 5. The Modulus squared map We define the modulus squared map and list several of its properties in this section. Definition. The modulus squared_map_is a multiplicative homomorphism m : (I + S) ! M4(R) given by A 7! m(A) = A A. Its image m(I + S) is denoted by M. To show that this definition is well-defined, we must show that its image i* *s in the set of real matrices; and that it preserves matrix multiplication. The fol* *lowing lemma does that. Lemma 5.1. Suppose A and B square matrices. Then __ __ a) A A is a real matrix if and only if A and A commute. b) m(AB) = m(A)m(B) Proof. a) A_matrix_is_real_if_and only if_it is equal to its own complex conjugate._ * *Now AA = A A = AA since A and A commute. Conversely,_suppose AA is real. Now A = C + iD where C and_D_are real. So AA = (C + iD)(C - iD) = C2 - D2_+ i[D,_C]. Since AA is real, the commutator [D, C] = 0. This impli* *es that AA = A A. b) First of all , note_that_I_+ S is closed under multiplication. See Lemma 6* *.2. Then m(AB) = ABA B = AA BB = m(A)m(B). We will call m the modulus squared map in analogy with the complex absolute value squared of a complex number. Now m has many striking properties. The following are the most interesting. 14 DANIEL HENRY GOTTLIEB Theorem 5.2. The set M is homeomorphic to the cone over the projective space P C3 Proof. As a vector space (I +S) is isomorphic to C4. The modulus map m has fibr* *es S1 over all points of M (except for 0) since m(F ) = m(ffF ) when ff is a compl* *ex number of unit modulus. Then m can easily be seen to be an identification map, and the identification of C4 by identifying any vector to its multiple by a sca* *lar with the same modulus is the cone over P C3 with 0 as the vertex of the cone. Corollary 5.3. The image of m restricted to the unit 7-sphere in I + S is the complex projective space P C3 The Lorentz group is the set of linear transformations L on Minkowski space which preserves the Minkowski metric, that is < Lu, Lv >=< u, v >. It has four connected components. The component containing the identity is called the proper Lorentz group and is denoted by SO+ (3, 1). The complex Lorentz group is the set of linear transformations on complexi- fied Minkowski space R3,1 C which preserve in Minkowski metric. The complex Lorentz group, L(C), has two connected components. It plays a role in physics, [Wightman(2000)]. The identity component of the complex Lorentz group intersects I + S in a subgroup, which I will call the biquaternion Lorentz_group. Similarly, the iden* *tity component of the Lorentz group intersects I + S in a subgroup which is isomorph* *ic to the other by compex conjugation. The other complex Lorentz group component is disjoint from both biquaternions. Theorem 5.4. The image of m restricted to the biquaternion Lorentz group, which consists of the set {aI+F | a2I-~2 = 1}, is the real proper Lorentz group SO+ (* *3, 1). Corollary 5.5. The Lorentz Group SO+ (3, 1) is exponential, that is it has a s* *ur- jective exponential map from so+ (3, 1) . We will prove Theorem 5.4 and Corollary 5.5 in the next section. Corollary 5.5 was proved in [Nishikawa (1983)]. In fact Nishikawa shows that SO(n, 1) is exponential. Theorem 5.6. m(S) = The set of electromagnetic energy-momentum tensors. proof. Suppose F 2 S. Then F u = E + iB , and if we imagined E and B as electri* *c_ and magnetic vectors, then the corresponding electro-magnetic tensor T = 1_2F F. See Proposition 5.1 with Definition 3.8 in [Gottlieb(1998)] . See [Parrott (19* *87)] for a mathematical account of electro-magnetic energy-momentum tensors. Theorem 5.7. m(S3) = SO(3) where S3 is the unit 3-sphere, that is the real unit quaternions. proof. The real unit quaternions are represented by {aI + bix + ciy + diz) where x, y, z are the basis matrices of section 4, and a, b, c, d satisfy a2 + b2 + c* *2 + d2 = 1 EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 15 and are real numbers. If we multiply {aI + bix + ciy + diz) by a unit modulus complex number, the element remains in the real quaternions if and only if the number is 1. Thus m is a 2-1 covering map, so its image must be SO(3). The real unit quaternions_S3_acts on the right of unit biquaternions S7 = {a* *I + bx + cy + dz|a__a+ bb + c_c+ dd = 1}. The quotient map is the famous Hopf fibra* *tion S3 ! S7 ! S4. Now m : S7 ! CP 3 is a principal S1-fibre bundle and is an equivariant map from the free S3 action on S7 to the induced SO(3) action on CP 3. The action of SO(3) on CP 3is not free. Consider the set of matrices in 1+S of the form {aI +F | a2 = ~2}, where ~ i* *s the eigenvalue of F . These matrices are those aI + F such that (aI + F )(aI - F ) * *= 0. In biquaternion jargon, these are called nullquats or singular quaternions. Si* *nce F (~I +F ) = ~(~I +F ), we see that the image of ~I +F consists of the eigenvec* *tors of F corresponding to the eigenvalue ~. The fact that (~I + F )(~I - F ) = 0 im* *plies that the kernel of F (~I + F ) consists of the eigenvalues of F corresponding t* *o -~. Thus ~I +F has rank two. But it is not a spectral projection unless ~ = 1=2. Wh* *en ~ = 0 we have the null matrices N such that N2 = 0. Here the eigenvector space is both the image and the kernel of N. So N cannot be made into a projection by scalar multiplication. However, N does map C4 onto the subspace of eigenvectors of N. Theorem 5.8. The image of a nullquat under m is a linear transformation from R4 to a real null 1-dimensional subspace of eigenvectors of the nullquat. proof. See Theorem 6.7c in [Gottlieb(1998)]. 6. The Exponential Map In this section we show that the exponential map for the proper Lorentz group is surjective using novel methods. In order to discuss eigenvector spaces and exponential maps more fully, we w* *ill change our notation to emphasize the real matrices. We shall follow the notation of [Gottlieb(1998) and (2001)]. Let F 2 S now be denoted by cF where ` T ' cF := 0A x A(-iA) where A = E + iB Then cF := F - iF *where F now denotes the real part of cF and -F *is the imaginary part. Thus ` T ' ` T ' F = 0E ExB and F *= -0B -BxE . Similarly we define _cF := F + iF *. Now F is a linear transformation on R4 which is skew symmetric with respect * *to the Minkowski metric, and cF will be called its complexification . We may regar* *d F 16 DANIEL HENRY GOTTLIEB as a 1-1 tensor corresponding to a two-form ^F. Then F *corresponds to the Hodge dual *F^. If we apply the modulus squared map to cF , we get _cF cF := 2TF where TF has the form of a multiple of the energy-momentum tensor of the electromagne* *tic field two-form F^ corresponding to F . On the other hand we may regard F as an element of the Lie algebra so(3, 1). Theorem 6.1. The exponential map Exp: so(3, 1) ! SO(3, 1)+ given by F 7! eF is onto. That is, for every proper Lorentz transformation L, there exists an F* * 2 so(3, 1) so that L = eF . To prove the above theorem, we need to consider the complexification so(3, 1* *) C operating on R3,1 C. This last is isomorphic to C4 and has an inner product whi* *ch is of the type - + ++ on R3,1 and extends to the complex vectors by = <~v, iw~> = i<~v, ~w>. See [Gottlieb(2001), Section 2] for more details. Now let c : so(3, 1) ! so(3, 1) C given by cF = F - iF *. The image of c, denoted S, is a three-dimensional complex vector space. The set of operators of the form aI + bcF will be denoted by I + S. Note that I + S is a vector space isomorphic to R3,1 C, and that I +S is closed under multiplication, as the foll* *owing lemma shows. Lemma 6.2. Let F and G 2 S denote cF and cG. Then (aI + bF )(ffI + fiG) = (aff + bfi)I + (bffF + afiG + bfi_2[F, G]) Now we say that L 2 I+S is a biquaternion Lorentz transformation if * * = . Any biquaternion Lorentz transformation L must have the form L = aI +bF* * , where F 2 S, such that a2 - b2~2F= 1. That is, L-1 = aI - bF . Theorem 6.3. Every complex Lorentz transformation L is an exponential, that is L = eF for some F 2 S, except for L = -I + N where N 2 S is null, that is N2 = 0. Proof. Recall [Gottlieb(1998), Theorem 8.5] where F 2 S that (**) eF = cosh(~F )I + sinh(~F_)_~F F Now L = aI +H where H 2 S and a2-~2H = 1. So the first obstruction to showing that L is an exponential is solving the equation cosh(~) = a. We shall show bel* *ow that such a ~ always exists. Next, if sinh(~)_~6= 0, then ` ' L = aI + H = cosh(~)I + sinh~_~ __~___sinh~H=: cosh(~)I + sinh~_~D = eD Hence L may not be an exponential if sinh(~)_~= 0. EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 17 Now sinh_~_~= 0 exactly when ~ = ßni for n a non-zero integer. (Note that sinh(0)_= 1). Then 0 a = cosh(~) = cosh(ßni) = cos(ßn) = (-1)n. If n is even, then L = I + N = eN where N must be null. If n is odd, then a = (-1)n = -1, so L = -I + N where N must be null or zer* *o. Now eB = -I where B 2 S has eigenvalue (2k + 1)ßi. But -I + N = -e-N cannot be an exponential, because it has a real eigenvector with negative eigenvalue.* * This proves Theorem 6.3 except for the following lemma. Lemma 6.4. a) cosh(~) = a always has a solution over the complex numbers. b) sinh(~) = 0 if and only if ~ = ßni. ~ - e-~ Proof. First we show b). Now sinh(~) = e________2= 0. Thus e2~ = 1, hence 2~ = 2ßni so ~ = ßni. ~ + e-~ Next we show a). Now cosh(~) = e________2= a. Hence (e~)2 - 2ae~ + 1 = 0 p ________2 p ______ Hence e~ = 2a_____4a__-_4_2= a a2 - 1. p ______ Now e~ = b has a solution for all b except b = 0. But a a2 - 1cannot eq* *ual zero, hence we have shown there is a solution for each a. Proof of Theorem 6.1. We show the exponential map is onto SO(3, 1)+ by show- ing the products of two exponentials1is an1exponential._ That is eF eG = eD * *for F, G, D 2 so(3, 1). Now eF = e _2cFe _2cFwhere _cF = F + iF_*. This follo* *ws since cF and1_cF1 commute.1_Also1for_this1reason,1ecF and ecG commute. Thus eF eG = e _2cFe _2cGe _2cFe _2cG. Now e _2cFe _2cGis a complex Lorentz trans* *formation in I + S. So either it is an exponential ecD , or_it has the form -I + cN = -e* *cN by Theorem 6.3. Now Theorem_6.3 also holds for I + S. Hence we have eF eG = e2D or eF eG = (-ecN )(-ecN ) = e2N . Corollary 6.5. The exponential map Exp : so(3, 1) C ! SO(R3,1 C) is not onto. If N 2 so(3, 1) is null, then -eN is not an exponential even though -ec* *N is an exponential. Proof. As explained`in [Gottlieb(2001)],'we can extend duality F *to skew symm* *et- ~E ric matrices 0~E xB~ where ~Eand ~Bare complex vectors. Then cF = F - iF * and _cF = F + iF * satisfy the same properties as in the complexification of * *the real case. Now consider_eF eG where_F , G 2 S. Then cF = 1_2cF + 1_2_cF ,* * so eF eG = e 1_2cFe 1_2cFe 1_2cGe 1_2cG. Now cF = cA for some A 2 so(3, 1), and _* *cF = _cA0 for 18 DANIEL HENRY GOTTLIEB A02 so(3, 1), hence _cA0cB _cB0 cA cB _cA0_cB0 eF eG = ecAe e e = (e e )(e e ), _ and so ecAecB_ equals either ecD or -ecN . But (-I)ecN = e(2n+1)ßicEecN = e(2n+i)ßicE+cN where E has eigenvalue equal to 1. So in both cases ecAecB is an exponential. _ _ Now -ecN is an exponential since -ecN = eßicEecN = eßicE+cN where E has eigenvalue ~cE = 1. On the other hand -eN , where N is the real part of a null* * cN, cannot be an exponential, since if -eN = eF , then s, the unique eigenvector * *for eN , applied to this equation gives -s = eF s = e~F s, so ~F = (2n + 1)ßi for * *some n. Thus F has another linear independent null eigenvector, which contradicts -* *eN having only one. 7. Eigenvectors In this section we give explicit formulas for the eigenvectors and eigenval* *ues of proper Lorentz transformations and their Lie algebra. We show the Doppler shift factor arises as a kind of Berry's phase. Theorem 7.1. Let F 2 so(3, 1) and let ~F be an eigenvalue of F and ~T be an eigenvalue of TF . The eigenvalue of cF is ~cF = ~F - i~F* and r _______________________22 a) ~T = ( E__-_B___2)2 + (E . B)2 r ________________22 q ______________ b) ~F = ~T + (E__-_B_)__2, ~F* = ~T - (E2-B2)_2. proof. This is Theorem 5.4 of [Gottlieb(1998)]. Now the image of ~cF I +cF is the 2-dimensional space of eigenvectors of cF* * with eigenvalue_~cF . The image of ~_cFI +_cF is the 2-dimensional space of eigenva* *lues of cF . Note that this is the complex conjugate of the eigenspace of ~cF I +cF . * *Now let u be a vector of length -1 in the Minkowski metric, an observer in relativity * *theory. Then s := (~cF I + cF )(~_cFI + _cF )u is in both eigenspaces, since the opera* *tors commute. And s is a real vector since u is. So s is not only an eigenvector * *for cF and _cF , but also for the real part F and the imaginary part F *, and he* *nce for the stress-energy tensor TF and the Lorentz transformation eF . See secti* *on 5, [Gottlieb(1998)]. Theorem 7.2. The eigenvector s := (~cF I + cF )(~_cFI + _cF )u for F 2 so(3, * *1) with E = F u and B = -F *u satisfies the following equation: E2 + B2 (7.1) s = 2 (~T + _________2)u + E x B + ~F E - ~F* B . proof. This is Corollary 6.8 of [Gottlieb(1998)]. EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 19 Corollary 7.3. For a null N 2 so(3, 1), the eigenvector is E2 + B2 (7.2) s = 2 ( _________2)u + E x B proof. Now N null is the real part of the null cN. So ~cN = ~N - i~N* = 0. Hence ~N = ~N* = ~T = 0. Then plug this into Theorem 7.2. Since there are at most two eigenvalues ~cF , one the negative of the other,* * and since the null matrices have only one eigenvalue, 0, we see from the above resu* *lts that there are two null real eigenvector spaces for the generic case and one nu* *ll real eigenvector space for a null matrix. Now we can use the above formulas to give us something like a connection on the eigenbundles of a field of F 2 so(3, 1) on Minkowski space-time. And we can consider what occurs as we move around a closed time-like circuit in space-time, that is, two time-like paths starting with the same velocity at time 0 and endi* *ng at the same point at some positive time. Then the eigenvectors formulas will progr* *ess according to the formulas until they meet at a future time where they lie in the same 1-dimensional space, but they differ by a factor. We can calculate that fa* *ctor. It only depends upon the tangent velocities u and u0 at the point of intersecti* *on and the factor is real This differs from Berry's phase, in which the factor is * *complex and usually depends upon the history of the paths, yet it has the same feel to * *it. We follow Scholium 8.2 of [Gottlieb(1998)] Let su be an eigenvector of F corresponding to ~F as seen by an observer u. Suppose (7.3) u0= ____1____p_(u + w) 1 - w2 is another observer. Then u0 sees a different eigenvector su0. But su0 must b* *e a multiple of su since they are eigenvectors. So the question is, what is the mul* *tiple in terms of E, B and w? The answer is: Theorem 7.4. 2 3 * B . w 7 (7.4) su0 = ____1____p_641 + -(E_x_B)_._w_+_~F_E_._w_-_~F________225su. 1 - w2 ~ E__+_B___ T + 2 Proof. Define (7.5) '(v) = where s- is an eigenvector corresponding to -~F . Then ' is a linear map whose image is the span of su and whose kernel is the space of vectors orthogonal to * *s- . Now '(u) = su. 20 DANIEL HENRY GOTTLIEB ____ _ Now := (~cF I + cF ) O (~cF I + cF ) has the_same_properties and let (u) * *:= su. Then = '. Let s- = - (u) = (-~cF I + cF ) O (-~cF I + _cF )u. Now ` 2 + B2 ' (7.6) su = 2 ~T u + E________2u + E x B + ~F E - ~F* B from (7.2) and s- is the same with the signs changed on ~F and ~F* : ` 2 2 ' (7.7) s- = 2 ~T u + E__+_B___2u + E x B - ~F E + ~F* B 0, s- > Now su0 = '(u0) = ` ' (7.8) su0 = ____1____p_1 + _su. 1 - w2 Now ` 2 + B2 ' (7.9) = -2 ~T + E________2 using (7.7). Then using (7.7) to calculate and substituting this into * *(7.8) we obtain (7.4). Now (7.4) holds for all F 2 so(3, 1). If we restrict to null F we should see* * (7.4) reduce to a simpler form. In the null case ~F = ~F* = 0 and E = B. So equation (7.4) reduces to ` ' (7.10) su0 = ____1____p_1 - w . (E_x_B)_2 su. 1 - w2 E Now w . (E_x_B)_E2is the component along the E x B direction. If we assume that w = wr, that is w is pointing in the radial direction, then r _______ (7.11) su0 = 1_-_wr_1s+uw. r _______ r Here 1_-_wr_1i+swthe Doppler shift ratio. This suggests that null F propagat* *e along r null geodesics by parallel translation. __ Now the fact that I + S and I + S commute leads to a richer situation in ana* *logy to Berry's phase considerations. If V is_a 2-dimensional eigenspace for F 2 I +* * S, then it is invariant under any G 2 I +S . In fact, any null 2-dimensional subsp* *ace of complexified_Minkowski space is either an_eigenspace of an F 2 S or an eigenspa* *ce of an F 2 S . The action of x, y, z on V is an irreducible action of the spin* * Lie algebra, and the action of X, Y, Z on V is also an irreducible action of the s* *pin Lie algebra on V . The particular basis of the actions have a sign difference w* *hich [Ryder(1988)] calls left and right spin 1/2 actions. _ Now, for example, the nullquat (~cF I + cF ) composed with ecG and applied to a vector u must be an eigenvector of cF. So if these three quantities are varie* *d, one gets a formula giving the progression of an eigenvector of cF . EIGENBUNDLES, QUATERNIONS, AND BERRY'S PHASE 21 8 Physical examples of eigenvectors and quantum probability We will point out two examples of inner products of eigenvectors of F in M which give probabilities underlying two important cases in [Sudbery (1986)]: Pa* *ge 200, equation (5.84) which gives the probability of spin along an axis at angle* * ` from the spin direction of the particle. In this case the probability of spin +* *1=2 is equal to the Minkowski innerproduct - 1_2 = sin2(`=2) where u is an observer, i.e. = -1, and v and w are unit vector in the r* *est space of u pointing along the direction of spin of the particle and the directi* *on of the measurement, usually the gradient of a pure B field. Note both u+v and u+w are both null vectors, and hence possible eigenvectors of some operators in M. The other example is on P. 273, equation (6.121) of [Sudbery (1986)]. Here t* *he distribution of electrons with specific velocity v is given by 1 - v cos(`), wh* *ere the electrons decay from a Cobalt 60 atom in a strong magnetic field B. Here ` is t* *he angle between the magnetic field B and the velocity of the electron v. If we le* *t u represent the center of mass observer u and u0 = ___1____p_(1-v2)(u + v) repres* *ent the 4- velocity of the electron and u+ 1_BB be the normalised eigenvector of F represe* *nting the pure B field, then p ________ 1 -< (1 - v2)u0, u + __BB> equals this distribution. References J. E. Avron, L. Sadun, J. Segert, and B. Simon(1989), Chern numbers, quaternion* *s, and Barry's phases in Fermi systems, Commun. Math. Phys, 124, 595 - 627. Michael V. Berry(1984), Quantal phase factors accompanying adiabatic changes, P* *roc. Royal Soc. Lond. A 392, 45-57. Michael V. Berry(1990), Anticipations of Geometric Phase, Physics Today Decembe* *r (1990), 34-40. Daniel H. Gottlieb(1998), Skew Symmetric Bundle Maps, Contemporary Mathematics * *220, 117 - 141. Daniel H. Gottlieb(2001), Fields of Lorentz transformations on Space-Time, Topo* *logy and its Applications, 116, 102 - 122. Daniel H. Gottlieb and Geetha Samaranayake(1994), Index of Discontinuous Vector* * Fields, New York Journal of Mathematics 1, 130-148.. Andre Gsponer and Jean-Pierre Hurni(2002), The Physical Heritage of Sir W.R. Ha* *milton, In- dependent Scientific Research Institute report number ISRI-94-04 (arXiv:math* *-ph/0201058). David Hestenese and Garret Sobcyk(1987), Clifford Algebra to Geometric Calculus* *: A Unified Language for Mathematics and Physics, Kluwer Academic Publishing, Amsterdam. Mitsuru Nishikawa(1983), On the exponential map of the group O(p, q)0, Memoirs * *of the Faculty of Science, Kyushu Univ. 37, ser. A, 63-69. 22 DANIEL HENRY GOTTLIEB Stephen Parrott(1987), Relativistic Electrodynamics and Differential Geometry, * *Springer-Verlag, New York. L. Ryder(1988), Quantum Field Theory, Cambridge University, Cambridge. Barry Simon(1983), Holonomy, the quantum adiabatic theorem, and Berry's phase, * *Physical Re- view Letters 51, 2167-2170. Anthony Sudbery(1986), Quantum Mechanics and the Particles of Nature: An Outlin* *e for Math- ematicians, Academic Press, New York. Math. Dept., Purdue University, West Lafayette, Indiana E-mail address: gottlieb@math.purdue.edu