ON THE STRUCTURE OF GEOMETRIES
WITH SPINOR-TYPE CONNEXION
M. c. Cullinan
A dissertation presented in partial
fulfilment of requirements for
the degree of
Doctor of Philosphy
universitj of New South Wal~s
C March 1975 .... .,,..-,·-,; ..,...... , I am indebted to Professor G. Szekeres for his constant help and guidance during the writing of this thesis.
I would also like to thank Ms. Helen Cook for her careful and expert typing, and Dr. H.A. Cohen for many stimulating conversations.
In addition it gives me great pleasure to acknowledge the continued support of my family and friends during the preparation of this thesis; without them this research could not have been completed. Abstract
The aim of this thesis is to present a synthesis, using standard techniques of algebra and of differential geometry, of some of the principal ideas and structures of the classical field theoretic description of spinor fields in curved space-time. A geometric model for spinor fields in curved space-time is described which is based upon properties of so-called tangent
Clifford algebras. The tangent Clifford algebra to a space-time manifold at a certain point is the quotient space of the algebra of covariant tensors at the point by a certain two-sided ideal, and is uniquely defined once the metric structure of the space-time manifold is given. Minimal left ideals of each tangent Clifford algebra are identified with spaces of four component spinors at the point. This model therefore features a direct method for synthesizing spinor fields from vector and tensor fields on a space time manifold.
Since a spanning set for each tangent Clifford algebra necessarily involves basis elements of four minimal left ideals, one is led to associate four four-dimensional spin-spaces with each point of a space-time manifold. The possibility of making transformations between these four spin spaces at each point offers a simple geometrical model for the class of unitary symmetry theories which relate to symmetries over a quartet of four dimensional spin spaces. Fields of higher spin are also readily accommodated in this model; as with the four-component spinor objects mentioned earlier, geometrical concomitants of (spinor-} fields of higher spin are constructed entirely from vector and tensor fields of various kinds.
iii IV
A theory of connexion is proposed for the spinor objects described above; this theory uses only standard methods of classical differential geometry, together with a certain amount of algebra, and leads, amongst other things, to the realization of a rather explicit relationship between the operation of taking the covariant differential of a tensor quantity on the one hand, and that of taking a 'Dirac'-type derivative (Q~( ) ,~> of associated spinor quantities on the other.
This theory is more general than existing models for spinor geometry as it embraces not only the usual theory of spinor connexion, but also a geometrical model for a certain class of generalized (Yang-Mills type) gauge fields in curved space-time. Of particular interest in this context are those space-time manifolds which do not admit a so-called spinor structure, for on these manifolds a geometry of spinors will generally involve connexion quantities which are direct geometrical concomitants of gauge fields such as the isotopic gauge field of Yang and
Mills.
At each stage in the development of this model, explicit calculations are given which serve to relate the various results to those of standard physical theory. In particular, a description of the so-called generalized
Dirac matrices which embodies both algebraic and differential properties is obtained from the general structure of the model. Like the other spinor objects with which this model deals, these matrices and their properties derive entirely from space-time properties of vector and tensor fields. CONTENTS
Abstract iii
Introduction vi
CHAPTER 1 The Algebra of Tangent Clifford Numbers 1
§1.1 The space-time manifold 2
§1.2 Tensors and tangent Clifford numbers 8
§1. 3 On minimal ideals of e; 22
§1.4 Frame transformations 35
CHAPTER 2 The Geometry of Tangent Clifford Numbers 53
§2,1 Linear connexion and covariant tensors 54
§2,2 Connexion and T,C, numbers; holonomic 62 c.onsiderations
§2,3 Connexion and T.C. numbers (continued); non-holonomic considerations 70
§2.4 Connexion and T,C, numbers (continued); spinor connexion 80
§2,5 On invariant distributions of minimal left ideals 88
CHAPTER 3 Physical Applications 108
§3,1 One-ideal distributions and four-spinor fields 109
§3,2 Multiple-ideal distributions; geometry and intemal symmetry 116
CHAPTER 4 Summary and Conclusions 128
Appendix Matrix representations of the r=ab etc. 132
Bibliography 137 Introduction
The literature of mathematical physics testifies to the existence of a close relationship between the development of our understanding of general relativity on the one hand and that of modern differential geometry on the other. The closeness of this relationship has stimulated research in both fields and has led, amongst other things, to our presently having a remarkably clear appreciation of the geometric foundations and content of relativity theory and of the theory of vector and tensor fields in curved space-time.
To date we have no understanding of the geometrical foundations of spinor field theory, nor of its geometrical content, that can compare with current geometrical models for vector and tensor field theory in terms of either depth or clarity. In fact our understanding of the genesis and geometrical roots of spinor field theory has progressed very little since
Dirac's original treatment of the quantum mechanics of the hydrogen atom and the well-known early work of INFELD and VAN DER WAERDEN [13].
Although the reasons for this are not entirely clear, one important contributing factor is surely that a full and satisfactory account of the geometrical foundations of spinor analysis has yet to be given; the geometrical framework necessary for a restructuring of the foundations of spinor field theory has quite simply not been available.
What would be the likely benefits, if any, accruing from a clearer understanding of the geometrical structures underlying spinor field theory, and what would one be entitled to expect of a definitive account of the geometry of spinor fields? An examination of the literature of orthodox spinor geometry, ranging from the by now classic paper of Infeld and Van der Waerden to more recent accounts such as that due to LUEHR and ROSENBAUM [16), reveals that the classical theory of spinor fields in curved-space time is predominantly composed of an inter-relation of ad hoe assumptions and constructions about which it is known that they are
(generally) consistent and that they seem to "work", but about which, it appears to this author at least, little else is known at a systematic level.
As a minimal requirement it would therefore be a desirable feature of any sensible geometrical model for spinor fields in curved space-time that it be able to provide a unifying perspective for at least some of the diverse assumptions and ideas to which we refer above, and a means of relating them to orthodox constructions of classical differential geometry.
At a more specific level it would be of considerable interest to know answers to a number of questions of a geometrical kind, questions for which solutions would seem to lie largely beyond the scope of orthodox ad hoe and axiomatic treatments of spinor theory. Typical of these questions are the following:
(i) What is the qeometrical/topoloqical content of classical
spinor field theory? Can a knowledqe of solutions of
Di l"ac' s equation, for example, tell us anything about the
underlying geometry and topology of space-time? How are
solutions of Dirac' s equation ( the equation of Rari ta and
Schwinger, equations for zero rest-mass fields) related
to the harmonic fields of classical geometry and to topoloqical
invariants of the space-time manifold?
(ii) What can a knowledge of space-time geometry and topology
tell us about the existence and properties of spinor fields?
For example, could a knowledge of space-time geometry and
topoloqy provide selection rules for solutions of wave Viii
equations representing fields of particular spin?
charge? iso-spin? And ultimately,
(iii) What is the geometric content of Lagrangian formulations
of field theory? Could the manipulation of field
Lagrangians as a means of obtaining self-consistent
field equations be replaced by an equivalent set of
manifestly geometrical postulates?
It is towards obtaining a basis for a possible clarification of some of these issues that this thesis is ultimately directed.
The model which we will describe and which will hopefully form this basis consists of a more or less systematic exploration of one central idea: that the key to a satisfactory geometrical understanding of the behaviour and properties of fields of spinor-type objects and of those issues raised above is contained in the notion of a suitably constructed local or - as we have called it - tangent, Clifford algebra. The idea of basing an account of the properties of spinor objects in curved space-time on the concept of a localized Clifford algebra structure is due to M.R!ESZ ([21]~ see also [20]). In his model Ri.esz identifies a "local Clifford algebra" with each point of space-time, and minimal left ideals of each local Clifford algebra with spaces of four component spinors at the point. By a "local
Clifford algebra" at a point P of a space-time manifold Jt one is evidently to understand an algebra generated by the actual tangent vector basis elements in the tangent plane to..,llg at :f>•J if in a suitable coordinate system around P these tangent vector basis elements
-+ are denoted (eA)P (A=~· 4) then each local Clifford algebra in Riesz
-+ model is to be thought.of as being generated by the (e\)p• the latter being subject to a constraint ix
'IEITLER ( [27] - [29]) has explored some flat space-time
applications of an idea of this kind, and in the first mentioned of these papers refers to the earlier use of a similar idea by workers such as Sommerfeld and Sauter.
The present essay is to be thought of as a developement and further clarification of the central idea of Riesz' model to which we refer above. However the model herein described differs from that of Riesz in a number of quite essential ways, the most important of which being in the mode of construction of each individual Clifford algebra. With us, each tangent Clifford algebra is a quotient space of the algebra of covarant tensors at a particular point by a certain two sided-ideal. The construction, which is based on a discussion due to CHEVALLEY [4], is of incisiveirnportance because it gives rise to a rather unusual and previously unremarked relation between tensor and spinor quantities, a relation which permits a developernent of algebraic and differential properties of spinor objects in terms of corresponding properties of underlying tensor fields. Our construction also imr;,lies that one does not have to "introduce" spinor-type fields into space-time as structures additional to tensor fields; being, as they are in our model, fundamentally derived from covariant tensor fields of various
ranks, spinor fields always exist on any space-time manifold, and one either takes them into account and studies their properties, or one
does not.
Other important differences between the model to be desc~ibed
and that propounded by Riesz include the way in which minimal left ideals X
of each local (tangent) a]qeb,ra are obtained; the manner in which transformation properties are assigned; the way in which inner products of spinor objects are, with us, related to inner products of underlying exterior forms; the way in which differential properties of our so-called tangent Clifford numbers are derived from differential properties of underlying exterior forms etc. In view of these differences, and the added fact that Riesz' model falls short of any real attempt to describe or assign differential properties to the aggregates of Clifford number objects with which it deals, it will be appreciated that virtually the only aspect that our model shares with that of Riesz is an adherence to the single central idea that was described earlier.
On the subject of differential properties of the spinor objects we will study,it is as well to mention an aspect of the geometry of spinor fields which is brought out by our analysis but which has not previously, to our knowledge, received any attention at all in spinor theory. This arises from the fact that the naturally ocurring spinor objects on an arbitrary space-time manifold are in fact sixteen dimensional rather than four dimensional. In the model presented here the differential properties of fields of spinor objects necessarily require for their full description two families of connexion coefficients.
One of these families corresponds to what in orthodox spinor theory is known as spinor connexion; the second family, the existence of which is reflected in the title of this thesis, has no counterpart at all in orthodox theory and is called by us ideal connexion, the epithet ''ideal'
referring to the fact that this kind of connexion describes a mixing of the constituent minimal left ideal components of a given tangent
Clifford number which takes place when the tangent Clifford number
undergoes parallel displacement in a general space-time. The properties
of ideal coefficients of connexion are affected by local curvature=related Xi
considerations as well as by such global issues as whether or not space time admits a spinor structure. It turns out that only when ideal coefficients of connexion have a certain simple form can our sixteen dimensional spinor objects be decomposed to yield the usual four dimensional ones. Therefore to suppose, for example, that one can generalize the Dirac differential operator for four component spinor fields in flat space-time to an arbitrary space-time manifold simply by including in this operator the usual spinor connexion coefficients
(i.e. by themselves) is to make a considerable and, as we shall show, generally unjustifiable simplification of the issues involved.
In what follows we will first of all (Chapters 1 and 2) describe the algebraic and geometric basis of our model; in order to avoid interruption to the developement, notes on physical applications of the model are held over to Chapter 3. Chapter 4 consists of a short summary of our deliberations, and concludes with some remarks on outstanding problems and possible areas for future research. Chapter 1 The Algebra of Tangent Clifford Numbers
Introduction
This chapter consists of four sections. In the first of these is to be found a brief summary of the standard tensor-algebraic ideas which underly our model. The second section introduces the notion of a tangent Clifford algebra and describes algebraic properties of constituent elements - our so-called tangent Clifford numbers.
Section three provides a construction of minimal left and right ideals of these tangent Clifford algebras. In section four is to be found a discussion of the representation and transformation properties of tangent Clifford numbers and a further elaboration of an account of inner products of these objects that was commenced in section two. l
§1.1 The Space-Time Manifold
In what follows, space-time will be represented by a manifoldc/& having the following properties:
(i) c)1, is an analytic manifold of dimension four i.e. a
Hausdorff topological space which has the property that each point P £ c...<-£ is contained in a connected open neighbourhood
11> C v"-1, homeomorphic to some open set ]) in the real affine four-space [ 4 , and on which is iI11Posed an analytic system of coordinates. An analytic system of coordinates onv"consists of an indexed family · { \U . cJI& ; i £ I} of open sets covering · 1
, and for each i £ I a homeomorphism
e . : \U . +]) . = e . cllJ. > c E4 1 1 1 1 1
such that if P £\.J.(')V .(:/ qi) with e.(P) = (x1 (P), •• ,x4cP)), 1 J 1 1 4 -1 1 4 e. (P) = (y (P) , • • ,Y (P)) = e. o e. Cx (P) , • • ,x (P)) then the J J 1 flmctions
A A 1 4 y = f(x, .. ,x) = ic!)
A=l, • • ,4, are analytic, in the sense that they are each expressible as a 4 convergent multiple power series in the variables x1 ,. • IX •
Furthenoore
(ii) this analytic system of coordinates onr,ftf, will always be assumed to be orientable, so that for all such intersecting open 3
sets V . , V., l. J
> o.
For each i e: I, the pair { qJ. , e. } will be called a local l. l. coordinate system (local c.s.) for\). or arolmdP e:V .• It will also l. l. be assumed that
(iii) cA is equipped with an indefinite Riemannian metric tensor of signature (-,-,-,+) - that is, with a symmetric covariant tensor field of degree two which is non-degenerate at each P e:J.tand which has signature (-,-,-,+). This metric tensor will be denoted g. Finally, it will be necessary to assume that
(iv) the analytic, orientable coordinate system on v'tiis also time-orientable, so that for all intersecting open sets \Ji, V j C ,;it, the flmctions /' defined above satisfy
The manifold cft having properties (i) -(iv) will be called the space time manifold.
Let us recapitulate some terminology relating to flmctions on manifolds (cf. KOBAYASHI and NOMIZU [ 15], p. 3 ) • Let,; be the algebra of analytic flmctions defined in a neighbourhood of P e:J'Zt
(a function on V cJLt to 1R is called analytic if with respect to a
local c.s. · {tU ,e}, f o e-1 is an analytic flmction on 1) = e (ll)) C E.4 to 'R. ) • A tangent vector to~ at P is a linear mapping ; P: ~; -+ IR with the property that for f, h £di;
-+ -+ -+ ~[fh] = ~[f]h(P) + f(P) up[h] (1.1.1)
-+1 -+2 An If two tangent vectors u P , up to c.1·~ at P have the same values for all argument functions: up[f]-+1 = up[f]~ for all f £.]';~ then we -+1 -+2 -+ M write up= up. A vector field u onv·6is an assignment of a tangent vector ,1'"P to each point P of~ • If f is an analytic -+ function defined in a neighbourhood of P £ Jl1, , then u[f] is a function defined by
-+ -+ (u [f]) (P) = up [f] •
. -+ -+ A vector field u is called analytic iff u[f] is analytic for all
-+l -+2 f £~;, p e: A . If u , u are vector fields such that -+1 -+2 -+1 -+2 u = u p p for all P £~, then we write u = u •
The set of tangent vectors to c),L at P formta vector space over l of dimension four which we will denote TP(J1,). For each point
P £ A contained in an open neighbourhood V with local c. s. · {IU, 8} , a natural basis in TP(~) is given by the vectors
. -+ a { (eA) P = ( ) A = 1, .• ,4} X =8(P) ax" -0
. 0 Where by = we mean that for f e:~p
-+ -1 eA [f] (P) = ( f o e <.!.>)x =S(P) • -0 -+ The function eA[f] will usually be abbreviated to f , so that ,\
f (1.1.2) = ,A
A The coefficients u (A = 1, •• ,4) appearing in the expansion -+ \-+ -+ u = u eA are called conponents of u in the local c.s.' {U ,e}; -+ u is analytic iff uA E~;, A = 1, •• , 4,P EJ,t, .
In the same notation, letf =· {p(s) E~; a~ s:; b} be a
differentiable path of class c1 lying in llJ C JI/, and passing through P, i.e. a c1 map of an open interval containing [a,b] EP-.. to 1U such that p (s ) = P, and represent in JI) = 8 ( ll.J) C [ 4 0 the points of 1? by equations
A A X (p (s)) = P (s) (1. 1. 3)
such that
X (p ( S )) = 8 (p ( S )) • - 0 0
-+ The operator xs : :J; Cs) -+ B. defined by
+ d + dx
obviously satisfies (1.1.1) and is called the tangent vector to \P
at x = p(s); the functions .9-.. x
linear f,mctional on tangent vectors at P. Let the value of the
If two such linear functionals h!, h; have the same 1-+ 2-+ values for all argument tangent vectors at P: [hp,~] = [hp'~]
for all;£ TP{J-1,), then we write h! = h;. A covariant vector field or one-form h on a subset l C J1, is an assignment of a one-form hp at P to each point P of q_ • If h1 , h 2 are one-forms such that 1 2 1 2 h p = h p for al 1 P £ lL , then we write h = h •
The set of real-valued linear f,mctionals on TP{J.f.) forms a
vector space of dimension four and constitutes the vector space
dual of TP{tft,); this space will be denoted TP{i,l,1,). For fixed -+ the map TP {c.4) -+ JR such that ~-+
linear functional on tangent vectors at P and therefore defines a
one-form at P which will be denoted dfp. There will be an
exception to this notation: the linear map TP cA ) -+ R such - •• = will for convenience be
= " = 1,.. ,4 and
'Thus
[w" ,e-+ ] = (1.1.5) µ
The four functionals {w\) p are linearly independent and therefore form a basis 1
A=l, •• ,4} (1.1.6) for T;(c.4) which we will call a natural basis for one-forms at p in the local c.s. nu ,e} .
Let f 1 , f 2 be two one-form fields on IL CA • The tensor product of f 1 , f 2 , denoted f 1 ® f 2 , defines a bilinear map from
, via the rule
= [fl ~ ] [ 2 -+2 p I p fp, '1p
-+l -+2 ,, '1p , '\, £ Tp (11"1, ) ; p £ L
The metric tensor g can be exhibited as a linear combination of bilinear functionals of this kind. If covariant components of gp are defined by writing
then with respect to the above basis for one-forms, g can be exhibited in the form
g =
Let i denote the determinant of [gAµ] and [g"µl the matrix of co-factors of gAµ in i ; then
Aµ g =
Aµ Under general coordinate transformations, the coefficients g transform like components of a contravariant tensor of degree two; we call
-+ g* = e 1J the contravariant metric tensor. g* determines a bilinear map
§1.2 Tensors and Tangent Clifford Numbers.
The following construction of the Clifford algebra over
T~(~) is based upon a discussion given by CHEVALLEY( [ L,\-l, p.38); the construction described by Chevalley has also been used by algebraic topologists in vector bundle theory (see for example
ATIYAH, BOTT and SHAPIRO [ \ ] ) but does not appear to have been used in a physical context.
Let J'* denote the algebra of covariant tensors over T;(j,(,) with unit e~ement denoted lp (e:'J,;>, and let j P (g*) be the two- sided ideal of :J'; generated by {hp ® hp -g* (hp ,hp) - ~; hp E T; (J'f) } • f P (g*) consists of elements of the form fp ® (hp @ hp-g* (hp ,hp). lp) @
I I ..., * ,{,f _pi( fp; fp,fp £..; P , hp £ T;(v·~). Let -\op(g*) denote the quotient
algebra J ;/f P (g*), {;P (g*) the complexification of -C! (g*) and cr*
the canonical projection :J; + 4,cg*). Ring multiplication in
~ (g*) is defined by writing cr* (fp) cr* (f;) = cr* (fp ® f; ) , and
is clearly associative. cr* (lp) is the unit element of ~ (g*).
e; = ~p (g*) will be called the tangent Clifford algebra to~ at P; elements of t:,; will be called tangent Clifford numbers at
P (T.C. numbers at P), or simply T.C. numbers.
A field F of T.C. numbers defined on a subset IJ... of Jt is
a rule which assigns to each P £ l, a T.C. number FP £ -(;* p at P. If two such fields of T.C. numbers F1 , F2 defined on L have
the property that for each P £ L = , then we write
F1 = F2 • If f is a covariant tensor field on l CA i.e. a rule
which assigns to each P £ l a covariant tensor fp at P, then we will
write cr*(fp) = (cr*f)P. Hence if, for each P £ L
we will simply write F = cr*f. The set of T.C. numbers defined on
will be denoted { -G IL }.
Expressions for field$ of T.C. numbers in terms of local
coordinates are obtained as follows. Let {V,e} be a local c.s.
around P £ Jt and choose a natural basis for one-forms at P in
terms of the functionals (wA>p (1.1.6). Next, define
= cr*((wA)P) and(\.i 0 )P =
A w
cr*(l) = W0 (1.2.1) (o
The kemel of cr* is j P (g*) and hence the zero element of {,~ is
= (1.2.2)
In addition cr*(\',/) = cr*(l ® .)-) = cr*(if ® 1) = cr*(J)cr*(l) = A cr*(l)cr*(w), and hence
vf and (vf) 2 =
In a local c.s. { UJ ,e} a field of T.C. numbers on 'Vis therefore specified by linear combinations of the unit element W 0 and of the products
A wP
A < 4 p = (1. 2. 3)
(one verifies, incidentally, that -G~ is of (complex) dimension
0 sixteen). Note that the elements WA (also W ) are uniquely defined once space-time coordinates have been fixed~ therefore a field of T.C. numbers defined on Vis uniquely specified by the l(
coefficients appearing in its expansion in terms of the above products. However, rather than write T.C. nwnbers in terms of linear combinations of the products {1.2.3), it will prove convenient to adopt a special set of basis elements for e,P which we will now describe. Following notation and commutation relations established by GREEN [ If ] for an algebra of matrix valued functions, define the following totally skew-symmetric quantities:
w>..µ 1 {WA Wµ Wµ WA) = 2
WAµ\/ 1 {WA Wµv + Wµv WA) = 2
1 WAµ\/p {WA wµvp - wµvp WA) = 2
(A =I µ 7 \/ =I p; 1 =< A,µ,v,p =< 4) • {1.2.4)
The sixteen quantities
Q) {r.l) p { Aµ) <·wAµ\/) { Aµ\/p) {W p,.w ,W p, p,W p
{1 {1.2.5) in each GP {P £ \) ) , an arbitrary field of T.C. nwnbers F £ {-G IV } can always be expressed uniquely in the form f1. F = f + 1 f 0 3! AllV (1.2.6) (1.2.6) will be called an holonomic expansion for F, while the , (possibly complex) coefficients f 0 fA • • • will be called holonomic components of F with respect to the local c. s. { V , e} . With respect to a different local c.s., the same field of T.C. numbers is specified by holonomic components obtained in the following way. Suppose p e 'LJ i.nt>. (;,! ~) and that Fe: {,Cl 1l).U ll).}, and write J J. J -1 A A I 4 x = e. CP> , x • = e. CP> = e. 0 e . Cx) c= x -+ x • A = f (x , .. ,x )) J. - J J J. - 'A and also FP = F -1 = F Then xA-+ X e. (x) e~ 1 (x') J. - J - I A A I A ax w -+ w = etc., and hence from the linearity of cr*, ax0 i/-+ w'A= etc .•• therefore, on ll . n V . J. J 0 I A 1 I All F-+F(') = f' w + f' w + f' w 0 A 2 ! All 1 'AllV 1 + f' w + f w' AllVP 3! AllV 4! AllVP ':lXO" With f I ::: f f I = ~ f It is seen that under 0 0 , A I A ax a general coordinate transformations , holonomic components of a field of T.C. numbers transform like the components of totally skew tensors of various ranks. In particular, the coefficient of w' in the holonomic expansion of a field of T.C. numbers transforms like a scalar (coordinate-invariant) function, and will simply be called the scalar part of the field. If F e: { -GI V}, the scalar part of F will be denoted II F II . As a consequence of the preceeding discussion, the following notion is coordinate-invariant; a T.C. number FP £ ~; will be called regular if there exists an element FP-l £ t;, called will be called regular iff FP is regular for all P £ \1._ Note also that if holonornic components of a T.C. number at P are all real in one local c.s. arol.Uld P then they are real in any local c. S. arolm.d P, so that one can obviously also speak of the reality of a T.C. number in a coordinate-invariant sense. F p £ -{;* p will simply be called real if its holonornic components (with respect to some local c.s. ) are all real. F £ {"G I l.} will be called real iff FP is real for all P £ l. . Retuming to (1.2.2), it is to be emphasized that fields of T.C. numbers are equivalence classes of covariant tensor fields, under the equivalence fp = f;(mod Cjp(g*))) where fp,f; £J; and j P (g*) is the two sided ideal of :J; introduced earlier; the ). relationship between the elements W and fields of 4 x 4 matrix- valued functions which at each point of the space-time manifold satisfy a relation similar to (1.2.2) is discussed in §1.4. For the purpose of further clarifying the relationship between covariant tensor fields and fields of T.C. numbers, it is instructive to detail the effect of the projection cr* on covariant tensors of various ranks. (i) Let f[2] be a covariant tensor field of degree two defined on UC i}1, • In terms of a local c.s~ { tJ ,e} for V can be written in the form f [2] f w).. wµ, and = ;>..µ ® hence in the form a* maps this tensor field into a field of T.C. numbers i.e. into a field f g Aµ WO = Aµ We see that cr* maps the symmetric part of f[2 ] into a multiple 0 of the unit element W , an d th e sew-symmetrick . part o f f C2 ] into. (fAµWAµ). We remark that the first of these terms is that which would be obtained by applying a* to the tensor constructed by simp. 1 y con t racting. f[2 ] wi"th g;* t h e secon d term - wh" ic h d epen d s only upon the skew-symmetric part of f[2 ] - represents the part of f[2 ] which disappears when f[2 ] is contracted with g*. Similar results for covariant tensors of higher rank are easily obtained by A expressing p-fold products (1.2.3) of the W in terms of holonomic basis elements. Following GREEN [ II we can deduce for the . . . Aµ sew-symmetrick quantities W etc. (1.2.4) the properties 1 AV (WA wµ" - Wµv WA) gAµ w" - g wµ 2 = Aµ AV .!_ (WA wµvp + wµvp WA} g w"P + g wpµ + gAP wµ" 2 = (1.2. 7) hence using the definitions (1.2.4) we can deduce the decompositions = + g vp w Aµ + g APw µv g µpw Av { 1. 2. 8) Typically, from a covariant tensor field f[ 3] of degree three 3 de fined on 1U b y an expansion· f [ ] = f Aµv w A 'x'~ w µ ® w" one can construct a field of T.C. numbers if the coefficients f, have one or more symmetries, the number of /\µ\/ terms appearing on the right hand side of the expression will be reduced. *C (ii) Let Jp denote the complexification of jp* and let ~*C p c:J:cp denote the {complexified) Grassmann algebra of complex exterior multiforms over TP(J't).* In terms of a local c.s. { Q.J ,e} .,_*c around P, elements of J'p can be expressed in the form + .!,_ f, (P) {w\ Awµ Aw") + 1.f, (P) {wA Awµ Aw" A ~)p 3! /\µ\/ p 4 . /\µ\Ip (1.2.9) '" where 1 (w A 0 wµ [A~ µ] 2! w \;;,/ w and where f , f A, f ... are complex functions on \U C v<1., . It 0 ~µ A is easy to ve:; ..-ify that cr*(wA A wµ ) = WAµ , cr* (w A wµ A w'V) = W)..µv, cr* (w)., A J1 A wv A Wp) = WAµ'Vp , and hence that cr* ( f) = F = f W0 + f WA o A + (1. 2.10) (where cr * now means the canonical map: .rr*CcJ P -+ * Let cr* -Gp>. 3i b e th e restriction, . o f cr * toJ'pc:,;*C cr* is surjective, and injective also (its kemel is {O}). Hence it is invertible, and so there is a one to-one relationship between complex exterior forms and fields of T.C • .P * o-r:*C numbers over lJ The inverse map: ~P -+ u' P will be denoted *-1 (5~ Our next task will be to show that the metric tensor g* which defines inner products for covariant vectors and tensors can also be used to construct a simple non-degenerate hermitian inner product for fields of T.C. numbers. We have just shown that there exists a one-to-one relationship between exterior differential forms and T.C. numbers. It will be convenient to adopt a definition for the inner 11 product of T.C. numbers whereby the inner product of two T.C. numbers at P has the same value as whatever inner product we choose for the associated exterior forms at P. Our choice for the latter is the obvious one; if f 1 , f 2 are real exterior multiforms on (\Jee},{, which with respect to {llJ,e} have expansions i fi ), 1 fi ), A wµ+ 1 f i f i = fo • 1 + ), w + 2! ), µ w 3 ! ), µv (i = 1,2) then the (local) inner product of f 1 ,f2 at PE~ will be defined to be ( f 1 , f 2 ) = ( f 1 , f 2 )P with p p (1.2.11) Therefore if F 1 ,F 2 E { C, IlJ} are fields of T. c. numbers which with respect to { 1U ,e} have expansions 0 F. = fi W l. 0 (1.2.12) then the local inner product of F1 ,F2 at P will be written fl = defined as being given by the expression on the r.h.s. of (1.2.11): = (1.2.13) In the next section we will introduce systems of bases for {,*p that bear no simple relation to our holonomic basis elements (1.2.5). In adapting the inner product (1.2.11), (1.2.13) for use with these other systems of basis elements, it will be convenient to work with a slightly different form of (1.2.13) which is obtained in the following way. First of all we define a transposition operator~ for T.C. numbers as follows: (i) in its action upon holonomic basis elements,~ satisfies >.. >.. >.. >..1 (W 1 = W p W p-l w 1 < >.. < 4 p (ii) in its action upon general T.C. numbers,~ is linear, so that = (iii) W0 ~ = WO • (1.2.14) >.. ~ Then (W) = - w>..µv' and hence if F e: {G I l\J} is ,, given in {l),e} by (1.2.6), then 1 1 F"' = f W + f WA - o A -2: - -3: + In addition, for two such fields F1 ,F2 e: {' IL },(F1 F2)"' and hence One also has = Furthermore it can readily be verified that if real fields F1 ,F2 e: {~IV} are expressed with respect to { V ,e} by expansions (1.2.12), then = I... which is precisely the value of (1.2.11). (1.2.13) can therefore be written in the simple alternative form II FI\, F II (1.2. 15) = 1 2 p We can use this local inner product for real T.C. numbers to define an hermitian local inner product for non-real T.C. numbers as follows. necessarily real) fields of T.C. numbers; then F. can be written J in the form F . =A.+ iB. (j = 1. •• 3), (i2 = -1). Define J J J (1.2.16) where ( , ) is the real-valued operation on real T.C. numbers defined as in (1.2.15). Then (i) (ii) (iii) = A ( F 1 ,F 2 ) p 1.1 where* denotes complex conjugation. The verification of these properties is simplified by introducing a complex conjugation operator* for T.C. numbers; if FE {-GIL} is given by F =A+ iB where A and Bare both real fields, then the complex conjugate F* of Fis defined by This operation has the properties (i) (AF)* = A*F*, A E(, ; (ii) (F1 + F2 )* = (iii) = ( 1. 2. 16) can now F 1 * + F 2 *· ' be written in the simplified form = (1. 2 .17) Note that the operations~ and* conunute. ff Following an idea of M. RIESZ [l.l ] , minimal left ideals of GP will be identified with spaces of four-component spinors at P. (We recall that a left (right) ideal in an algebra Jt is a subset l (l> of elements of A such that (i) BEL => (A + B) Et ( (A + B) El. ) (ii) A E L I C E u4 => ( C A) E L ( A E l , C E.,4 => ( C A) Ei ). A minimal left (right) ideal of Ji is a left (right) ideal of~ which contains no other ideal but itself and the null ideal). Let be a basis for -(!~ such that the subset of elements {(Ym(n))P; 1 ~ m ~ 4} span a minimal left ideal of GP , (n = 1. •• 4); such a basis will be called a spinor frame for~~, while for arbitrary FP e: "t;p* the coefficients cf> appearing in the m(n) expansion = cf>m(n) (Ym(n)) P will be called spinor components of FP, The next sections are concerned with the construction of such a set of basis elements (Ym(n))P, and with the construction of a model for a spinor algebra based upon the above identification. §1.3 On minimal ideals of 1~p * Basis elements of the various minimal ideals of ,Cp can be specified in terms of certain multinomials in the generators of the algebra (in a A suitable local C.S. these are the (W )P, A= 1 •• 4). However in order that the specification of those basis elements be generally covariant, it is necessary to work with coordinate-independent linear combinations A of the (W >p and their products. For this purpose we introduce on each neighbourhood c.,,A't four fields of T.C. numbers, denoted G (m = 1. .4), le m which at each point satisfy the following conditions: = (1 < m, n < 4) p e: l (1. 3 .1) 13 For the moment the only other assumption that we will make concerning these fields is that for each value of m coefficients in cr*- 1 (G) m define analytic functions of local coordinates. Fields of T.C. numbers which at each point of some arbitrary neighbourhood, l say, satisfy a relation like (1.3.1) can be constructed as follows. Let jm (m = 1 .. 4) be indicators such that j 1 = j 2 = j 3 = i, j 1, and define j 2 (m 4) • 4 = e: m = m = 1. • Thus " Next, let {(gm)P e: T;; m = 1 .• 4} be an arbitrary but fixed orthonormal tetrad (of real co-vectors) at P e: l , normalised so that is chosen to be positive-timelike, so that if (g4)P expansion for (g4)P in terms of a natural basis for one-forms in TP(.,Af), A then g4T > O. Theorem: In order that (1.3.1) be satisfied it is necessary and sufficient that (Gm)P be expressible as -1 ~ (m = 1. .4) (1.3.2) where ~ e: ,P is some regular T.C. number at P. (The sufficiency of this condition will be clear, since Its necessity follows from a well known theorem of Pauli, as we shall show in §1.4). Hence fields of T.C. numbers which at each point of some arbitrary neighbourhood I_ C v4tsatisfy conditions like (1.3.1) can be constructed by assigning a set of (Gm)P (1.3.2) to each P e: l i.e. by specifying for each P e: L (a) a set of four tangent one-forms at P with properties as above, together with (b) some regular T.C. number,¾, say, and defining (Gm)P as in (1.3.2). Moreover, as a consequence of the above theorem, every set of such fields Gm on l can be synthesized in this way. We will now proceed with the construction of ideals of ( t. Following notation introduced by EDDINGTON ( [ 5 ] , p.106) for an algebra of matrix-valued functions, we start by defining symbols W0 , Gab, ~a (0; a< b ~ 5) as follows: w WO 0 = G G G (1 < m < 4) om = - mo = m - G = = So = - Gba = (1 < a < b < 5). We can deduce for these symbols the multiplication properties (G )2 w O = (at b, at c, b ~ c; no summation) 'lS and w if (abcdef) is an even permutation of = 0 (012345) , properties which are conveniently summarised as follows c~.f. SZEKERES [1-5], eqn. (5. 4)) : O O (1.3.4) For each P e: IL. , the sixteen quantities O are obviously linearly independent, and therefore provide an alternative basis for,; which we will denote {GP}. We will call this basis a non-holonomic basis, the epithet 'non-holonomic' referring to the fact that this basis essentially derives from the choice of a non holonomic basis in T;(c;t{,) (represented by the frame of the orthogonal A co-vectors { ( gm) P; m = 1 .•• 4 } ) • An arbitrary field of T.C. numbers F € {-GIL} therefore has a unique representation 1 F = w + (1. 3. 5) 0 2 where 0 and ab = - (0 , F, and the coefficients 0 ab non-holonomic components of F with respect to the field of bases {GI l.} = { {GP}, P EL } . Next we introduce symbols describing various special linear combinations of the Gab. DefineP (n) E {'GIL} (m= l. .. 4;n =+,-) m as follows: Cn> 1 p3 = 2 (Gl2 - nG34) Cn) i = CG - nw > p4 2 0 5 0 C n = + ' -) . (1.3.6) These symbols have the following properties: (i) P (1 < X < y < 3; n = +, -) , (1. 3. 7) where E is the alternating symbol on (1 2 3). (This property, as xyz well as those that follow, is verified directly using (1.3.3) and the definitions (1.3.6)), 11 (ii) P Cn>p Cn> + P Cn>p = P CinP Cl ~ x,y ~ 3;n = +, -) (1.3.8) and hence the Px(n) generate a Clifford algebra with unit element Ci nP 4 Cn >> , Cn = + , - > • (iii) P Cn>P C-n> = P (1 < x,y < 3; n = +, -). (1.3.9) = Writing E in the form e , we can summarize the above xyz oxyz 45 properties in the multiplication formulae. P Cn>p Cn> = i 6 P Cn>_ in<5 P Cn>_ i 6 P Cn>+ ie P P Cn>p C-n>= 0 (1 < m, n < 4; n = + , -). (1.3.10) m n = = We will also need the additional properties G p (n) = - E p (-n)G o4 m mm o4 G p (n) = P The quantities (Pm(n))P can be extended to a basis for the whole of ~ by introducing additional symbols Gm Cn) E {-G IlL}, Cm= 1 ••• 4;n = +,-),as follows: "lt G Cn) = .! G cw + nG ) = .!cw - nG )G rn 2 orn o 05 2 o o 5 orn (rn = 1 ••• 4; n = + , -) • (1. 3.12) These elements are related to the P (n) thus: rn . C-n) G (n) = inG P Cn) = - ine: P G • (1.3.13) rn o4 rn rn rn o 4 For each P e:L the sixteen quantities (G (n))r! (P (n)) Cm= 1. •• 4; rn rn P n = +,-) are readily seen to be linearly independent and therefore form a basis for G.;* . Using (1.3.6), (1.3.11), (1.3.13) and the alternative definition of the P (n) provided by rn P G Cn)G G G · l (W - ) orn on 2 o nGo5 ~rnnP4C-n)+i ~ P <-n)_i ~ P <-n) = - i nu nun4 rn nurn4 n i e P C-n) + omnp45 p (1 < rn, n ~ 4;n = + ,-) • (1.3.15) G G (n) = e: G (-n)G = inP G G (n) = - G (n) G = -nG Cn) o5 m rn o5 rn (1 < rn ~ 4; n = +, -) • (1. 3.16) We will also need to know how the P (n), G (n) multiply together. m m From (1.3.10), (1.3.12), (1.3.14) and (1.3.16) the following can be verified: P P Cn>G C-n> = - in G (G = -i o G <-n>+i o G <-n>_ i o G <-n>+ie G <-n> n mn 4 n n4 m n m4 n omnp45 p (1 < m, n < 4; n = +, -) • (1. 3.17) G = ino G (1 ~ m, n ~ 4; n = +, -) (1. 3.18) We will introduce a separate notation to handle certain special linear 1 (-) iP (-)) 1 (P (-) + iP (-)) x11 = 2 (P3 - 4 x12 = 2 1 2 1 (-) - iP (-)) 1 (P (-) + iP (-)) x21 = 2 (Pl 2 x22 = 2 3 4 i (-) i - -(G - iG (-)) = (G (-)+ iG (-)) x31 = 2 3 4 x32 2 1 2 i (-) iG (-)) i (G (-)+ iG (-)) x41 = 2 (Gl - 2 x42 = 2 3 4 / ... i i = - (G (+ )+ iG4 (+)) = (G (+) + iG (+)) xl3 2 3 xl4 2 1 2 i i = - (G (+) - iG2 (+)) = (G (+) - iG/+)) x23 2 1 x24 2 3 1 1 = (P (+ )+ iP (+)) = (P (+) + iP (+)) x33 2 3 4 X34 2 1 2 1 (+) 1 (+) = iP (+)) iP (+)) x43 2 (Pl - 2 X44 = 2 (P3 - 4 (1. 3.19) These sixteen linear combinations are obviously linearly independent and therefore provide an alternative non-holonornic basis in -G.;* , for each P EL They have the following particularly simple multiplication property: x_,.x = uLJ\. qn 1 < m, k,q,n ~ 4 (1.3.20) (this can be verified by enumeration of cases, using (1.3.10), (1.3.12), (1.3.14), (1.3.15), (1.3.17) and (1.3.18)). Let M be a 4 x 4 matrix =pq such that the entry at the p-th row and q-th column is 1 and all other entries are zero, (1 ~ p, q ~ 4). The product of any two such matrices can be written in the form ~mk ~qn = ~kq ~mn· The multiplication law (1.3.20) together with the correspondence (X ) P -+ M therefore mn =mn establishes an isomorphism of t;;, with the algebra of 4 x 4 matrices. Before discussing minimal ideals of -Gp* , it will be useful to summarize the linear relationship that exists between the basis elements Xmn(l ~ m, n ~ 4) and w0 and Gab (o of complex coefficients {emnab = - emnba; 0 {yabmn = - ybamn; 0 1 1 X = o W mn 2 4 mn o (0 < a < b < 5; 1 < m, n < 4) = = = X (O < a < b < 5) yabmn mn cw = X (summation) ). (1.3.21) 0 mm (Because the sets of indices (mn) and Cab) relate to different systems of labelling in --G; (and not just to different ranges within the same indexing system), we agree to underline those indices of each of these two kinds of coefficients that pertain to the X ). These mn coefficients, which are uniquely defined via (1.3.6), (1.3.12) and (1.3.19) are therefore subject to the restraints 1 = 2 Yabst yabmn = (1.3.22) A ,p* Returning to (1.3.20), leti,(qlp C '-'p be the linear vector space over the complex numbers spanned by { (Xpq)P e: -(;;; 1 < p ~ 4} , A (q = 1. •• 4) • The spaces I, (q) P define four minimal left ideals of Gp·* 11. We can verify this by invoking the vector-space structure of ;f,(q)P to deal with the additive requirement, and by using (1.3.20) in the following way to show that the multiplicative requirement is satisfied. * "' Let ¾>Ci O) £ -GP, F(q)P (~ 0) £ i(q)P be written KP = Kmn(Xmn)P, F = ( 1 ~ m, n , p ~ 4). Then for the product KPF(q)P we have = K (X )p Therefore the sub-space l (q) P C -e*p certainly defines a left ideal of -Gp·* Furthermore, this left ideal has the property of minimality, since if~ is allowed to assume any value in~,* all four basis elements (Xmq)P (m = 1 ••• 4) may occur in the product ~F(q)P (even if, for exanple, F() as written above is such that -G p* C ..p* Similarly, let (il (m) P "-' P be the vector s~ace over C. spanned by \ (Xrnk)P; k = 1. •. 4} , (m = 1. •• 4). The spaces -G*p = Naturally, the left and right decompositions of -GP* given here are not unique; basis elements of an arbitrary left deconposition, and those with which our spinor frame basis elements at P will be identified, are specified as follows. Let@ be a nonsingular complex matrix of degree four with entries and define cm- mn = em ( n ) , sixteen quantities {Y { ) ) e:-{; * by writing m n P P = {X ) mp p ep{n) (1 -< m, n < 4) {1.3.23 These sixteen quantities are linearly independent since det cm -:/ o. Let l(n) p C e; be the vector space over(C spanned by {{Ym{n))P;m = 1 ••• 4} , (n = 1 .•• 4); it is readily verified that the spaces .(.(n) P define four minimal left ideals of ,CP* , and hence that an alternative decomposition of "G;* into minimal left ideals is given by ,c*p (1.3.24) Similarly, for minimal right ideals: let _ be a nonsingular complex matrix of degree four with entries ( ~>- mn = ~ (m) n and define sixteen linearly independent quantities {Z(m)n)P by writing (1.3.25) Let{il(m)P C -GP* be the vector space over (m = 1 .•• 4); then the spaces ~(m)P define four minimal right ideals of -Gp* and * -Gp = ~(1) + ({(2) l(3) + F e{!P, F(I.) = FE is an element of £P. Consider a 4 x 4 matrix Q [cJ ] the entries of which are defined by E =~ (X )P. Since = mn nm mn E 2 = E implies n2 = Q and since Q necessarily has at least one non-zero entry, Q must have at least one (non-zero) eigenvector with unit eigenvalue. Let~= [~] be such an eigenvector of Q: Q~ =~;one readily m = ~ - verifies that the four T.C. numbers Ym = ~n(Xmn)P E = .•. = ~n(Xmn)P (m = 1.. 4) are linearly independent elements of l P and therefore form a basis of this space. It follows that for any minimal left ideal of 8p it is possible to choose basis elements in terms of linear combinations of the (Xmn)P like (1.3.23), and hence that (1.3.23) and (1.3.24) engender all possible decompositions of 'Gp into minimal left ideals. A similar statement can be made for minimal right ideals of ,P . From now on, by a left (right) decomposition of 'P we will always mean a decomposition of Gp into minimal left (right) ideals. Subscripts which pertain to minimal ideals will be referred to as ideal indices and will be enclosed in parentheses. Our summation convention will hold for repeated ideal indices. We complete this section by introducing some notation relating to the coefficients yabmn defined by (1.3.21). Let = - rba be 4 x 4 matricei with en tries yabmn (1 < a < b < 5). (1.3.26) (1.3.3) and (1.3.20) together imply the following multiplication rule: 1 r abr cd 2 eabcdef ef = = t 0 Values of the ~ab are given in an appendix. §1.4 Frame Transformations As foreshadowed at the end of §1.2, any set of basis elements for &t by which a left decomposition is specified will be called a spinor frame for t:t, or simply a spinor frame at P. In what follows it will always be assumed that we have at our disposal a set of fields X which satisfy mn a multiplication law like (1.3.20); at any point, P say, of .}t and for any left decomposition of -(;i it will therefore always be possible to choose a spinor frame for which constituent elements are expressible in terms of linear combinations of the form = (1 < m, n < 4) (1. 4.1) where 0 = [em(n)] is a non-singular 4 x 4 matrix. Henceforth individual spinor frames will always be assumed to have been chosen in this way. Let {YIU..}= {Ym(n) E {-C. IL }; 1 ;:;; m, n ;:;; 4} be a set of 16 fields of T.C. numbers on L C Jt obtained by assigning a spinor frame \(Ym(n))P; 1 ~ m, n ~ 4} to each P Ell..., and assume that these individual spinor frames can be chosen in such a way that the coefficients in cr*-l(Y ( )) define analytic functions of local coordinates. { Yll} '!- m n will be called a field of spinor frames over I. , while for arbitrary F E · { (, 11.} an expansion F = ~m(n) Ym(n) will be called a spinor frame expansion for F, or simply a spinor expansion for F. The coefficients ~m(n) appearing here will be called spinor components of F with respect to the spinor frames {Y IL}. With respect to a different field of spinor frames over L, the same field F is described by spinor components obtained in the following way. For each PEL, let E (P) and A(P) be non-singular complex matrices of degree four with entries [.:_]mn = cr(m) (n) and [A] = A respectively, having the = mn mn properties that cr(m) (n) and Arnn define analytic functions of local coordinab . -1 -1 -1 -1 in l (1 ~ m, n ~ 4) and write [E ] sm = cr (s) (m), [~ ]nt = \t Suppose that the spinor frames attached to points of L and which together constitute {Y I l} undergo a point transformation -1 -1 ; A sm(P) (Ys(t) )PO' (n) (t) (P) (1 < m, n < 4); (1.4.2) then the spinor components qi m(n) (P) of FP undergo a corresponding transformation qi m(n) (P)-+ qi 'm(n) (P) = Amp (P)qi p (q) (P) O' (q) (n) (P) or,in matrix notation, -+ -+ -+ cl> (P) -+ cl>' (P) = A (P) cl> (P) I: (P) • (1.4.3) == I.et {Y' I l} denote the field of spinor frames defined by {(Y~(n))P; 1 ~ m, n ~ 4, Pel.}; the transformation {Y l'l.. }-+ {Y'IU...} defined by (1.4.2) will be called a spinor frame transformation over l , and the transformation l -+ l '= A ]I: defined by (1.4.3) the == = =11;:a== transformation induced in spinor components of F by a spinor frame transformation over l . (For the moment we require of the independent matrices A , I: only that they be nonsingular, and that == = their entries be analytic functions; one sees that with respect to the action of these matrices on the Ym(n) which is defined by (1.4.2), I: (P) effectively determines a new left decornposition of ,; in terms of the old, while the choice of basis elements within each ideal of the new left decornposition is determined by A:P). Note that if we were to suppress all transformations on ideal indices - that is, were to restrict spinor frane transformations at P to those that preserve a certain left decomposition of -c; - a T.C. number at a single point P could be decornposed into a quartet of four ~-dimensional objects. (four-spinors). The question of the conditions under which such a decomposition might be carried out for a field of T.C. numbers requires for its answer a knowledge of the properties of a connexion for T.C. numbers, and will be dealt with in the next chapter. In §1.2 an expression for the inner product of two T.C. numbers was given in terms of their holonomic components. We will also need an expression for inner products of T.C. numbers in terms of spinor components; rather than working with indeterminate expressions like < Ym(p), Y (q) n ) , we introduce the notion of adjoint spinor components. Let F = ijJm(n) Ym(n) e: {'G ltL} be an arbitrary field of L.C. numbers; under a spinor frame transformation over the coefficients ( 1. 4. 4) undergo a transformation A A' -1 A A-1 Coefficients can now be expressed as 1 1 A = 2 ) 4 ( 1 = 4 trace (1. 4. 6) so that lly z II .! cS cS • m(p) (q) n P = 4 pq mn' (1.4.6)' then (1.2.17) can be written z )Cq, 2 Y ))JI = nclA(s)t (s)t m(n) mc n P A where the adjoint of F1 , denoted F1 , is defined by FA QllA Z (1. 4. 7) 1 = (n)m (n)m Such a set of basis elements can be constructed in the following way. -1 In the notation of ( 1. 4.1), let 0 denote the inverse of 0 -1 and write (0-l) for each P £ U... define = mn = 8 (m) n; -1 (1. 4. 8) = 8 ( ms) (Xsn ) p• (1.4.6) is obviously satisfied. Following the discussion of §1.3, the (Z(m)n)P determine a decomposition of~; into minimal right I ideals. Furthermore, (1.4.6) is invariant under spinor frame transformations at P provided that (1.4.2) is accompanied by a corresponding transformation of the (Z(m)n)P: (1.4.9) From now on, at each P e: L left and right decompositions of "G;* will always be assumed to have been chosen in such a way that the respective basis elements are related via (1.4.1), (1.4.8). Transformations like (1.4.9), made consequent on a spinor frame transformation at P, are the only ones to which basis elements of our right decomposition of-G;* will be subject. By starting from ~~sis elements for minimal left and right ideals specified by (1.4.1) and (1.4.8) respectively, the coordinated transformations (1.4.2) and (1.4.9) allow us to define a one-to-one correspondence between spinor and adjoint spinor frames at P. (1.4.7) will be called an adjoint spinor expansion for F1 • So far we have not allowed the underlying basis elements X mn to be transformed; spinor frame elements have been chosen in terms of certain specified linear combinations of the X , and mn spinor frame transformations have consisted of varying these linear combinations. the X for this purpose being considered fixed. mn This procedure will now be varied slightly; noting that for the basis elements Y' z• defined by (1.4.2) and (1.4.9) m(p)' (q) n respectively, the product Y~(p)z(q)n has the value 1 y•m(p) Z'(q) n = ~pqA- sm xstAnt we will now specify that a spinor frame transformation (1.4.2) at P e: L be always accompanied by the following transformation of the X mn -1 (X ) -+ (X' ) mn P mn P = A sm (P) (X s t)PA n t(P) 1 < m, n < 4. (1.4.10) Any basis for ~P* the elements of which can be obtained from the (Xmn) P by a transformation of this form will be called a non holonomic T.C. frame at P. The field of non-holonomic frames {(Xmn)P E~;* 1 ~ m, n ~ 4, Pell...} will be denoted {x j,l..} ; (1.4.10) then describes the transformation of elements of {X It..} induced by a spinor frame transformation over l . As a consequence of this discussion, the following multiplication rules, which summarise the relations that exist between the basis elements of spinor, adjoint-spinor and non-holonomic frames attached to points of l, are form invariant under spinor-frame transformations over X X = o X X Y = o Y mp qn pq mn mp q(n) pq m(n) Y Z = o X Z X = o Z m(p) (q)n pq mn (m)p qn pq (m)n (1 < m, n,p,q < 4) (1. 4 .11) If, with respect to {xl l} , F E {{, I\L } has an expansion of the form F = (1. 4.12) then under a spinor frame transformation overL, the coefficients in this expansion - the non-holonomic components of F with respect to { xl IL.} - undergo a transformation or in matrix notation (1.4.13)' q.:i. (1.4.12) will be called a non-holonomic expansion for F. Note that it is possible to write the r.h.s. of (1.4.10) in the form of a product of T.C. numbers; using (1.3.20) we simply -1 write X' We remark that (}.. - lx ) is mn = (A s tX s t)Xmn (A pqX pq ). pq pq p the inverse of the T.C. number (}..stxst)P, since and similarly for (}..stxst)P(}..;!xpq)P. Hence by writing L = }..stxst' -1 -1 L = Astxst' (1.4.10) can be written in the form -1 X -+ X' = L X L mn mn mn (1 < m, n < 4) (1.4.14) = = For convenience we specify that the coefficients yabmn' Smnab (1.3.21) be numerically invariant under all of the above transformations; it follows that since the Gab are expressible as linear combinations of the X (Gab= yab X ) , a transformation like mn mn mn (1.4.14) is necessarily accompanied by a siroilar transformation of the Gab: = (0 < a < b < 5) • (1.4.15) This transformation is clearly equivalent to (1.4.14), and for this reason the basis {GP}= {(Gmn)P; 1 ~ m, n ~ 4}, transforming in the fashion (1.4.16) consequent on a spinor frame transformation over l will be called a non-holonomic T. C. frame at P also, and I (1.4.15) the transformation of this frame induced by a spinor frame transformation over l . }._ We remark that holonomic basis elements such as the W are left unchanged by spinor-frame transformations over a coordinate neighbourhood V: we have already observed that once space-time coordinates have been chosen, the holonomic basis elements WA are uniquely defined via the projection o*, and one is therefore not free to make a transformation of these elements corresponding to (1.4.15). Returning to (1.3.1), one sees that this spinor frame invariance of the WA can be reconciled with the stated transformation properties of the Gab by interpreting (1.4.15) in terms of a change in the T. c. number ~ appearing in the specification (1.3.1) of G • From now on a spinor transformation om (1.4.2) over l will always be assumed to have been followed by a transformation of this "coefficient" T. c. number ~: -+ A I p = However, in spite of this spinor frame invariance of holonomic basis elements, if an element like WA is expanded in terms of the X , then under a spinor frame transformation over Q) the coefficients mn in this expansion - the non-holonomic corrg;>onents of WA with respect to {xlU} - will behave like non-holonomic components of any field of T.C. numbers. To be specific, suppose that with respect to {xl Q.)} WA has an expansion A = X • pq The set of coefficients { w A (P) £ C, , 1 ~ p, q ~ 4} define non pq holonomic corrg;>onents of (W A) P; under a spinor frame transformation (1.4.2), these components undergo a transformation w A(P) -+ w A'(P) = A (P) w A (P) A-l (P) mn mn rrg;> pq qn (1 < m, n < 4) or, in terms of matrices (1. 4.17) Note that not all of the sixty four coefficients w A (P) are pq independent. Substituting (1.4.16) into (1.2.2) and using (1.3.20) we obtain WA w µx + wµ i X pm mn pn pm mn pn non-holonomic components of (WA)P with respect to·{~} are therefore restrained by the conditions w A (P)w µ (P)+w µ (P)w A (P) pm mn pm mn which can be written in terms of matrices as (1. 4.18) The algebra of analytic matrix-valued functions over tU C J1, which at each point satisfy a relation like (1.4.18) has been studied by many authors, most recently by SCHMUTZER( [l?,], [l.lf-]), by LORD [lj ] and, as already noted, by GREEN ( [ 10] , [ II ] ) • In the present model these matrices appear merely as one of several convenient ways of handling non-holonomic COit\Ponents of underlying holonomic basis elements of -G;. Using the matrix elements wA (P) it is possible to write mn down non-holonomic c0It1Ponents of any T. C. number at P, once its holonomic COit\POnents are known. For exart1Ple, let fp= fA (\','\)PE:T;cA> be a covariant vector at P; with respect to {~}, Fp= cr*(fp) £ ~ can obviously be written FP = f,w A (P) (X )P. Hence non-holonomic I\ pq pq components of FP with respect to this frame can be described by A a matrix of functions fAn (P). One sees that the sequence of operations q* A Fp = fA (W )p is a formal statement of the process whereby the "spinor equivalent" of a "vector" fA is created by contracting the covariant index A with the index of a "connecting quantity" (in this case nA (P)). At the beginning of §1.3 it was stated that in order that the T.C. numbers (Gm)P satisfy (1.3.1) it is necessary and sufficient that there exist a T.C. number~ at P such that (Gm)P is expressible as (Gm)P = ~jmcr*((~m)P)Ap-l· By using the matrices A n (P) we can now prove the necessity of this condition for (1.3.1). From (1.4.18) and the properties of the coefficients gm). given earlier, the matrices n (P) = (1.4.19) m defined by ["u l X = J. cr * ("g ) =Jg . " ,W A = _, m pq pq m m m ml\ satisfy n (P)n (P) + n (P) n (P) = 20 r = m =n _ mn· (l (1.4.21) r on ~om = 2o mn I Therefore by a well known theorem of Pauli, there necessarily exists a non-singular matrix, which we will denote A(P), such that Q (P) = A (P)-lr A(P) In om == = = i.e. such that -1 = A (P) Q (P) A(P) • r om m by writing A_= [A(P)] (X ) .Then since Define ~ e: ~; --p mn mn P = (G >p = [r ] (X >p' the necessity of (1.3.2) for (1.3.1) m = om pq pq follows immediately. We conclude this section by demonstrating that in certain special cases, expressions for the local inner product of two fields of T. C. numbers in terms c:f their spinor COJ!\POnents (1. 4. 5) assume a form very similar to the standard "flat" space-time expression for the inner product of two four-cOJ!\POnent spinor fields. For this purpose we will need simplified expressions for the ( X ,x_ ) quantities mn -Kq and < Ym(n) , Yk (q/ ; we start by obtaining an expression for Gab *"' According to our local prescription, at any point P of a coordinate neighbourhood tlJ (G )P is expressible in the form om (1. 4.22) Using (1.2.14) and (1.4.22) and writing j e: j (m 1. •• 4) it is m* = mm = obvious that at each such P (G0 m)P "'* can be written (1.4.23) Next, assume that at each P e: 1LJ it is possible to choose the (G ) in such a way that the coefficient T.C. number AP satisfies om P = W, and that the spinor frame transformations over UJ 0 that we admit can be restricted in such a way that this condition is preserved (under spinor-frame transformations). Then = e: (G ) P and hence m om G e: G om *"' = m om . (1.4.24) Since G G ((G G ) i Gmn , it follows that om*"' om*"' = on om *"' = *"' G = mn*"' 1 < m, n < 4. and by a similar process of reasoning that = (1.4.25) Hence by introducing the notation = (1.4,24) and (1.4,25) can be summarised thus: G = e: G oa a oa 1 < a~ 5 Finally, by using G = * = = iGab we can oa*"' (GobG oa) i~a*"' *"' deduce that * '\, Gab = 0 < a < b < 5 (1.4.26) where to preserve uniformity of notation we have introduced = + 1. (1.4.27) Therefore if F E {'GI\\J} is specified by a non-holonomic 1 expansion F 4> w + 41 , complex = 0 0 2 F *"' = (1.4.28) However, in computing expressions like X it is convenient to mn*"' work with a slightly different form of this result. The following operation on non-holonomic components of T.C. numbers is suggested by an operation introduced by SZEDRES [ 151 for an algebra of matrix-valued functions. If, with respect to a certain field of non-holonomic frames, F E {G IlJ } is given by F = W + 0 0 ; defined by (1. 4.29) (c.f. (1.4.28)). Hermitian conjugation has the following properties * ( *) * ( *) ). Fl + µ F2 (F F ) ( *) 1 2 (1. 4. 30) The second of these properties is verified by checking that for (*) all values of a, b, c, d, (GabGcd) = GcdGab. By inspection of the multiplication rules (1.3.4) one readily observes that leads to the required result. Expressions for hermitian conjugates of other non-holonomic basis elements are obtained as follows. From (1.4.28) and the definitions (1.3.6) and (1.3.12) one can easily verify that (P ( n) ) ( *) = - e: P (G (n)) (*) = G (-n) m m (1 < m ~ 4; n = +, -> (1.4.31) and hence from (1.3.19) that (X ) ( *) = X mn nm (1 < m, n < 4) • (1. 4. 32) Hence, if with respect to{xl V} , F e: {(; I U} has a non-holonomic *"-· expansion F = 't'mn.+- Xmn' then F can be written F G cp * X G (1.4.33) *"' = o4 nm mn o4 (this follows from (1.4.29), (1.4.30) and (1.4.32) ). Therefore if F.= cpi X e: {-GILi} are fields of T.C. numbers on n, (i = 1,2), J. mnmn V the local inner product of F1 and F2 can be expressed in terms of non-holonomic components as II G cp l*x G cp2 X II = o 4 nm mn o4 st st P S'O i.e. as = -41 trace er t lt r cI> 2) (1.4.34) o4 = = o4 = P where ~ 1 t denotes the (matrix) herrni tian conjugate of t 1. === = Note that the condition that (1.4.20) be equivalent to (1.4.21): = G A (*) G = o4--P o4 and that spinor frame transformations preserve this condition, is that the matrix A associated with a non-holonornic frame transformation should satisfy t -1 r o4 A r o4 = A • = = === where, as before denotes the matrix hermitian conjugate of A. Such matrices A have the form = A = exp (i :\ I + (1. 4. 35) = where the coefficients :\, :\ab = - Aba are real (0 We can use (1.4.34) to obtain a similar result for the expression of inner products of T.C. numbers in terms of their spinor components. First of all we recall that at any (fixed) P e: ,U , basis elements of a given left decomposition of -e;; are related to the (Xrnn)P via an expansion (Ym(n))P = (Xmp)P ep(n) where the matrix of coefficients 0 = [6p(n)] is non-singular. If = Fi e: ce I \J} (i = 1,2) have spinor expansions Fi = /m(n) ym(n) , the local inner product of F1 and F2 at P e: \U can be written I * * 2 = IIG X G X II o4 ~ m(n) pm ep(n) o4 ~ s(t) sq ep(t) P 1 + 1t + 2 (1.4.36) i.e. = - trace (0 r et h\ r ) 4 - o4 '+' o4 P =a:= = = - Next, suppose that at each such P £ QJ it is possible to restrict attention to those spinor frames having the property that the associated matrix of coefficients 0 (which relates the (Ym(n))P and the (Xrnn)P, as described above) satisfies ce r 0 t> = (1. 4. 37) = = o4 - P - where 3 is some fixed, position-independent (non-singular) 4 x 4 matrix. Suppose also that ideal transformations at each such P can be restricted in such a way that this condition is preserved (under spinor frame transformations). Then (1.4.36) can be written = ¾trace(= ~l (1. 4. 38) The r.h.s. of this expression now depends upon P only through the position dependence of the spinor components of the fields F1 , F2 . The condition that (1.4.38) be form invariant under spinor frame transformations is simply that the matrix E associated with the = ideal component of such a transformation (cf. (1.4.3)) satisfy = -;:t = As an exatr1?le of a system of matrices satisfying the above constraints we have the following. Suppose it is possible to choose a left decomposition of each tangent Clifford algebra in such a way that at any point of a coordinate neighbourhood {Uthe matrix 0 :=. which appears above is expressible in the form 0 0 (P) 1 E (P) (I - if ) (1. 4. 39) = 45 - fi = = 0 for some (non-singular, position dependent) matrix E (P) which satisfies 0 E CP> = r05 == = (1.4.37) and (1.4.39) together imply = Therefore in this case (1.4.36), and hence (1.4.38), can be written = -41 trace er ~ 1t (1.4. 40) = oS - Properties of an inner product of this kind have been investigated by Szekeres [ lb] Chapter 2 The Geometry of Tangent Clifford Numbers Introduction This chapter consists of five sections. In the first of these a summary is given of the standard geometrical treatment of differential properties of tensor fields. In section two it is shown that this treatment of the differential properties of tensor fields provides a remarkably natural basis for a geometrical treatment of the differential properties of T.C. numbers when the latter are described in terms of holonomic components. In sections three and four we obtain a description of differential properties of T.C. numbers in terms of non-holonomic bases by introducing two families of connexion coefficients - our so-called non-holonomic and ideal coefficients of spinor connexion. As foreshadowed in the introduction, the first of these families of connexion coefficients corresponds to what is usually known as spinor connexion, while the second family - our so-called ideal coefficients of spinor connexion - has no counterpart at all in orthodox spinor analysis and is introduced in order to describe a mixing of the constituent minimal left ideal components of an arbitrary T.c. number which takes place when the T.C. number undergoes parallel displacements. One rather interesting facet of this analysis is our deduction of a previously unremarked relationship between the operation of taking the covariant differential of a tensorial quatity on the one hand and that of taking a 'Dirac-type' derivative n~c ) ,~ of associated spinor quantities on the other. Section five is devoted to a discussion of the issues involved in constructing fields of four-component spinor entities from fields of T.C. numbers. 5"3 §2.1 Linear connexion and covariant tensors Let PO , P 1 be points of c),£ ; let V C ~ be a connected open neighbourhood containing P0 and P1 with local c.s. {UJ, e}, and let .le = { p ( s) £ V; a ~ s ~ b} be a differentiable curve of class C' lying in\) and passing through P0 and P1 such that 4 p(O) = P0 , p(l) = P1 • :Represent in 'lt) = e('U) c [ the points of i_) by equations x >. (p(s)) = p >. (s) a ~ s < b. ). ). Leth = { h (s) = h). (p(s)) (w )p(s) = h). (s)w (s); a ~ s < b } be a covector field defined along P . A parallel displacement of s I * IA a covector alongp is a non-singular linear mapping Ps :Tp(s') (v·b) + T;(s) s' s 11 "f s • lp s' s' p = identity. s • (a< s, s', s" < b) (2 .1.1) The possibility of defining a parallel displacement operator along E for covectors derives from the existence and properties of so-called horizontal paths in the principle bundle of linear frames over\\); _(c.f. KOBAYASHI and NOMIZU (15], p.68). Let f: 1 [h (s')] denote the value of s' * u Ps on h(s') e: Tp(s') ((ll,); noting that for any q (a - s < q < b - s), P:q [h (s+q)] e: T; (s) (e/4) , we define D lim ! {i s+ q [h ( s+ q) ] - h ( s) } ds h(s) = q s q-+O (c.f. KOBAYASHI andNOMIZU, op.cit., p.114). ~s h(s) is called the intrinsic derivative of h(s) with respect to the tangent vector + ; to at p(t). his said to be parallel along p iff s 1P D ds h(s) =Oat all points of ~ • It is easy to verify from the linearity of ff').It' SS I that D d /\ ~ ds h(s) = as hi\ (s) )w (s) + hi\ (s) ds (2.1.2) . D ;>,.( ) Since ds w s e: T;(s) ctfl) and since the w;>,.(s) form a complete D set in T;(s) (J/,), it is possible to write ds ;>,. D = - w (p (s)) wcf> (s) ds cl> ( 1 < ;>,. < 4) , (2.1.3) which serves to define the real coefficients wcf>A(p(s)). (These coefficients, which depend upon the path f as well as upon the parameter values, evidently completely determine the effect of the mapping ,p:' on basis elements of T;(s)(J1. ). The minus sign is included for convenience). The condition that h be parallel along P can now be written or, since the basis covectors w;>,.(s) are linearly independent, d = o. (2.1.4) ds The coefficients wA0 (p(s)) are said to define a linear connexion along f iff their dependence on the path 'P is of the form a 0 dx wA (p(s)) = r A (P) ds (p Cs)) (2 .1.5) P = p(s); a the coefficients rA, which are functions only of the point P, are called coefficients of linear connexion over lJ with respect to the local C. s. { V , e} . For coefficients wA0 (p(s)) having dependence of the form (2.1.5), (2.1.4) has a set of C' solutions that assume prescribed values when s = O. Hence for such a set of coefficients, the mapping 1Dss'J[ defines an isomorphism Tp(s)* (J.1,), an isomorphism which is easily seen to be independent of the parametrization along f. Rules for the construction of absolute differentials of other tensors are summarized thus: D d f .1 f ds 0 = ds 0 f E Jo op p D A A dx w WO ds = rcr D -+ a -+ dx e ds eA = r A a ds .£_ t + a ds 1 (2. 1. 6) In particular, if f is an arbitrary exterior multiform defined at each P e QJ by an expansion of the form (1.2.9), write dxcp ,r, ~f = f etc. (c.f. (1.1.4)); then along .._, ds o 0' et> ds w µ (2 .1. 7) where let> denotes covariant differentiation with respect to the connexion , so that p = 1. .• 4 (2.1.8) Such a multiform is called parallel along p iff ~s f = 0 at all points off and integrable iff fo,cp = 0 and the equations A cp 0 are integrable (p 1 ••• 4) • p I = = Lett be an arbitrary tensor field defined over the whole of the neighbourhood 1lJ , and in the notation of ( 1. 1. 5) , write Dt dx4> = ds = 04> t ds ax4> ds ci4>)x=p(s) is the tangent vector tof at x = p(s)). Define a T; Ccft) - valued linear functional ( ~l ® D4> t) P on TP C/'Z) by writing -+ 0 at x is therefore s ] = !?!). That the values of two = ••• ds such functionals (w4>{8)od>tl)P, (w4> 0 D4>t2)P coincide for every TP(r/t) is expressed by writing (~~ ® D4>tl)P = The tensorial one-form defined on\\) by D t = is called the absolute or covariant differential oft. Let t 1 ,t2 be arbitrary fields of tensors defined on L C JI{, and let '2 be an arbitrary C' path lying in l) CL ; then = for every such path P * ot1 = Dt2 at all points of V., where ~s denotes absolute differentiation along f . Thus absolute derivatives and covariant differentials provide equivalent descriptions of the differential properties of tensor fields. Rules for the construction of covariant differentials of arbitrary tensors follOW" immediately from (2.1.6); one has D(f .1) w"A@ Dcj, (f • 1) = w Dw"A "A - r "A w 0 w0 = w ®o -+ (J -+ -+ e D e"A = w ®0<1> e"A = r "A Wcj, ® (J a,13 e: CC. • (2.1.9) In particular, if, with respect to a local C. S. {t\) , 0} , an arbitrary exterior multiform f is defined at each P. e: 0 by an expansion (1.2.9), then Df = (2.1.10) :Returning to (2.1.5), the coefficients r~ will be said to define a metric (linear) connexion iff within each coordinate neighbourhood of J1, they satisfy ."Aµ g , O; (2.1.11) connexion coefficients of this kind may be written uniquely in the form = { "A } + (2.1.12) (J A where { 0 ~} denote the Christoffel symbols constructed from { A } - g } a~ a~,P and where the coefficients TA satisfy a~ = 0 (2.1.13) and define components of the torsion tensor. Next we recapitulate some terminology relating to differential forms (cf. GOLDBERG [ G} ] , p. \2 ). Let f(p) be an arbitrary exterior p-form over l C J'1, defined in a local c. s. { V CL ,0} by a local expansion f (p) = 1 (p = 1. •• 4) p! f .1 (p = 0) 0 The adjoint of f(p), denoted *f(p), is an exterior (4-p)-form which has a local expansion 1 f A1f.\ Apµ Ap+l µ4 E g •• g PW A •• A w (4-p) ·' I\'1 ••• 'l\p µ 1" . " µ p µp+ 1. . µ· 4 (p = 1, 2, 3) 1/ 4 f *l fo<-1,> 2 wl A ••• Aw (p 0) 0 = = Al 1\ A4 µ4 f E µ g g .1 ( p = 4) A1···A4 µl ••• 4 "' 1/ where E = (-.L) 2sig(A ••• A )is the alternating tensor A1···A4 .d' 1 4 density on Al •• A4 and where 1 = det (gAµ). Note that for two such p-forms f (p) f (p) 1 ' 2 f (p) A * f (p) 1 2 1/ = (-i,> 2 ( fl (p) ,f2 (p) ) (2.1.14) (cf. (1.2.11)). For the remainder of this section it will be rA supposed that the µcp are coefficients of a rretric connexion, i.e. satisfy (2.1.11). The exterior differential. of f(p), denoted df(p), is a (p+-1)-form defined on V by a local expansion A Aw P p = 1,2,3 dfo .1 = f o,cp w 0 p = 4 (2.1.15) (where df is the differential off), while the exterior co- o 0 differential of f(p), denoted cSf(p), is a (p-1)-form defined on lJ by *d*f(p) 1 q>Al A2 A (p-l)! g fA 1 ••• AJ (2. 1. 16) A p-form will be called closed iff df(p) = O, and co-closed iff of(p) = o. A p-form will be called pseudo-harmonic iff it satisfies (do + od) f(p) = O (p = o ••• 4). (A a-form f is 0 therefore pseudo-harmonic iff odf0 = gAµfolAIµ = 0). Finally, a multiform f will be called closed(, co-closed, pseudo-harmonic,) iff it consists of a sum of p-forms each of which is closed (, co-closed, pseudo-harmonic). § 2. 2 Connexion and T. C. numbers; holonomic considerations In §2.1 a linear connexion was regarded as defining an isomorphism between cotangent spaces toJ1,attached to various points of the path P . Since this isomorphism induces isomorphisms of the corresponding spaces of tensors of arbitrary rank, it is apparent that a linear connexion over V also defines an isomorphism between tangent Clifford algebra structures attached to the different points off • That this is so can be demonstrated directly by observing that the operator r:· that was introduced to describe the parallel displacement along f of cotangent vectors can also be used to construct a parallel displacement operator * for T.C. numbers. Let cP et:*p denote the image of Tp * (,)t) *-1 *-1 * under o*, and let denote the restriction of (']°.Ji to \1 cP (cf. example (ii) , §1.2). Define 0 (a< s < b, a - s < q < b-s) (2. 2 .1) (where 0 means COir\POSition). This operator has the following properties: .... (i) p :+q is a non-singular linear map from c;(s+q) to c;(s); A A therefore if {H (s) = a* (h (s)) = hA (p (s)) (W ) p (s) = hA (s) W (s); a < s < b} is a field of T.C. numbers along~ derived from a A covector field h(s), then P:+q [H(s+q)"] £ c;(s); and secondly " II. (ii) -ps' 'It) s" f" s" s o.C s' = s A A. s' s identity 'P s o "P s' = ( cf. (2. 1. 1)) • By defining A D H (s) = lim l { ~" s+ q [H ( s+ q) ] - H ( s) } q s ds q-+O H(s) £ c;(s) it is apparent that one can make a formal identification ,.. D "D D cr*(h(s)) = a* - h(s) ds H(s) = ds ds and hence deduce that and in particular that D w\s) = - w A (p(s)) wcr (s). (2.2.2) ds a More generally, define the effect of 11\s+q" on an , Jc:.s A arbitrary k-fold product of the (W) ( ) and on the product p s+q k * of k arbitrary elements H' (s+q) , •• ,H (s+q) £ C ( ) in such p s+q a way that A m . m ,._ P s+q [ II HJ (s+q)] = II P s+q [Hj (s+q)] s j=l j=l s II. m . n . m " "f s+q [a II HJ (s+q)+ S II HJ (s+q) ] = a II ]l> s+q [Hj (s+q)] s j=l j=l j=l s n ,.. m s+q j +(3 j~l Jt. s [H (s+q)] a < s < b, a - s < q < b - s, 1 ~ m ~ k; (2.2.3) These rules determine uniquely the effect of the operator]? :+q on general ele~ents of * Next,,define the absolute derivative of an arbitrary eP (s+q )" T.C. number Fp(s) E'(!:(s) by writing D l l\s+ -d F(s) = lim - {p q[F(s+q)] - F(s)} (2.2.4) s q+O q s It is readily verified, for example, that if F1 (s), F2 (s) E e;(s), then D 1 2 1 D 2 ( ds F ( s)) F ( s) + F ( s) ( ds F ( s)) • D Applying the operator ds to both sides of the equation W0 = ! gAJ.I (w¾µ + w¾A) it follows that Henceforth the coefficients r~, will always be assumed to be coefficients of a metric connexion; therefore D W = 0. (2.2.5) ds o Suppose F E {'GIL} is specified on \) C l by a local expansion F f W + f WA l f,µWAµ + l f W).µv = o o A + 2 ! /\ 3 ! Aµv {2.2.6) then along the path ~ C l) D dx ds F{p{s)) = ds {2. 2. 7) Dds F{p{s)) will be called the absolute derivative of F{p{s)) e: "p{s)* -+ with respect to the tangent vector *s tot at p{s). Note that as D a consequence of {2.2.4), ds F(p{s)) has the same value as the D *-1 image under cr* of cr F {p { s) ) ) : ds "Ji { D F(p(s)) = cr* {~ (cr*-l (F{p(s)) ))) ds ds ~ {2.2.8) *-1 Hence F e: { e I1L. } will be called parallel along ! iff cr~ F{p{s)) is parallel along~ • Related to the absolute derivative of a field of T.C. * numbers is its so-called Clifford differential. If t e: is p 0 p a covariant tensor of rank r, then Dt is a covariant tensor of p rank {r+l). From each of these covariant tensors can be constructed a T. c. number; the T. C. number obtained from Dt {i.e. cr*{Dt ) ) p p will be called the Clifford differential of o*t at P. p *-1 More formally, suppose F e: {(:I L}, write fp = o~ Fp and define (2.2.9) The field of T.C. numbers DF thus defined will be called the Clifford differential of F. In terms of a local C. S. {U, e} around P e: l. , DF has a local expansion DF = F. (2. 2 .10) It is easy to write down the Clifford differential of an arbitrary field of T.C. numbers; if, with respect to our local c.s., F e: {'GIL} has an expansion (2.2.6) then in the same coordinate system DF" is expressible as (2. 2 .11) Note that in order that DF vanish on lJ , it is not necessary that D~F should itself vanish. Consider the second term in (2.2.11); (DF) (l) = W~ WAfA,~. Since (2. 2 .12) this term vanishes provided both g~A fAI~ = 0 and (fAI~ - f~IA) = O. Therefore in order that (DF) (l) vanish, it is sufficient that (a;-lF) (l) = fA wA be both closed and co-closed. More generally, from (1.2.7) and (1.2.8) one has (2.2.13) summarised as while by constructing all permutations on (AµVp) of the product and summing with appropriate signs, one readily verifies that (2.2.14) The last equation may also be written as = Substituting (2.2.13) and (2.2.14) into (2.2.ll)yields 0~ 0~ DF = g f O I ~ wO + ( f O, A - g f ol ~ (2. 2 .15) Therefore in order that DF vanish, it is necessary and sufficient that go~ f I = 0 0 ~ 0~ = f g fAol~ o,A 1 0~ g = 2! fAµol~ - f[AIµ] = 0 = (2.2.16) These conditions may be given succinct expression in terms of the operators d and O • write *-1 f (o) 4 f (q) 0'3i F = + E with 1 A f(q) Al = !... f w A ... Aw q Then DF may be written q! A1 ••• Aq /. .. + cr* (2df ( l) + cr* (4df< 3>). (2 .2 .17) Hence DF vanishes iff = 0 = of <3> = - 2df(l) = 0 = - 4df< 3> (2.2.18) (cf. (2.2.16)). It is natural to enquire what exterior forms satisfy constraints of this kind. Obviously, in order that (2.2.18) be satisfied, it is necessary that the f(q) should each be pseudo-harmonic ((do+ od)f(q) = o, q = 1,2,3; dof(4) = odf(o) = O), while it is sufficient that the f(q) should (q') each be both closed and co-closed. When f = f · is homogeneous of degree q', this sufficient condition is seen to be also a necessary one. Hence Theorem: Suppose F e: { e IL} is the image under cr* of an homogeneous p-form; then the Clifford differential of *-1 F vanishes iff cr "J, Fis both closed and co-closed. A field F of T. C. numbers will be called closed (, co-closed, pseudo ' *-1 harmonic) iff cr3i Fis closed (,co-closed, pseudo-harmonic). Note also that if the Clifford differential of F vanishes, then holonornic components of F satisfy = 0 (2.2.19) In the following sections we will introduce a number of different expressions for both absolute derivatives and Clifford differentials of fields of T.C. numbers; these different expressions correspond to the different systems of bases that we have introduced in each tangent Clifford algebra. It is clear that once absolute derivatives of a T.C. number are specified for every C' path lying in a given coordinate neighbourhood, the Clifford differential of the T. c. number is then uniquely determined. However, the converse statement - that once the Clifford differential of a field of T.C. numbers is known, absolute derivatives are also known for every C' path - is not true in general. §2.3 Connexion and T,C. numbers (continued); non-holonornic considerations The rules given in the previous section for the construction of absolute derivatives and Clifford differentials of fields of !.C. numbers can also be used to obtain a description of the differential properties of fields of T.c. numbers in terms of the various systems of non-holonomic basis elements. It is convenient to begin by looking at some of the differential properties of fields of non-holonornic frame elements. Let ...,, {X; 1IJ } be a field of non-holonomic T.c. frames over soma coordinate neighbourhood U C ~ and let {X; = {(X } ( } 1 < m, n < 4, p(s} f} mn p s e: e:'E } denote the subset of elements of {X; \) } attached to points of the path '! C U Suppose that {x, U} is chosen in such a way D depends on! only through the components of the that -ds (Xmn } p ( s } -+ tangent vector field and that this dependence on the x,s -+ components of x is linear and of the first order. Then since " s -D (X } e: G* there exist 256 x 4 (possibly comnlex} ds mn p(s} p(s} ··r coefficients K ~(P} (1 < m, n, u, v < 4; 1 < 4> < 4) such that mnuv'I' ax4> (2. 3 .1) = {K.mnuv'I' ~(P} (X uv)p} p (} s ds Note that not all of these coefficients K ••• 4> are independent; by taking the absolute derivative of each side of the equation X X = o X according to the rules outlined in the previous mp qn pq mn section and using (2.3.1) to eliminate derivative terms, one readily verifies that (2.3.1) is expressible as .!?.,_ X l (:K o + .K o }X ds mn = 4 mquq4, vn qnqv4> mu uv Furthermore, by contracting this expression with o and using mn D W = 0 we deduce that ds 0 K + K = 0 (2.3.2) mqnq4> qnqm4> 71 1 and hence by writing Kmn~ = K that (2.3.1) is expressible 4 nqmq~ in the form D X O - K O ) X (2.3.3) ds mn = (Kum~ vn nv~ um uv where, as a consequence of (2.3.2) K = o. (2.3.3) I mm~ Suppose F £ {f:IUJ} (2.2.6) is given a non-holonomic F = ~ X ; then the absolute derivative of F at expansion mn mn -+ 0X £ T dxo Q_F={"' · +K "'_,.. K} X (2.3.4) ds 'f'n,n, a mqa 'f' qn 'f'mq qno mn ds while the Clifford differential of F can be written DF = WO {"' + K ~ "' K } X (2.3.5) 'f'mn,o mqo qn - 'f'mq qncr mn Non-holonomic components of F will be called integrable over UJ iff the differential equation ~qn - ~mq Kqncr = 0 is integrable. By writing this last equation in matrix form 13 = one readily verifies that in order that non-holonomic components of F be integrable it is necessary that = 0 where (2.3.6) Naturally, as a consequence of the way in which the coefficients Kmn "consistency" theorems. Let J)F denote the (T. C. number) 4> coefficient of : on the r.h.s. of (2.3.4) and, as in the previous section, DF that on the r.h.s. of (2.2.7). Then: Theorem 1: A n.s.c. that (2.2.7) and (2.3.4) be consistent for arbitrary C' paths passing through any (fixed) P e: Q.J is that = ( 2. 3. 7) Proof: The sufficiency of this condition for the consistency of (2.3.4) and ( 2. 2. 7) is obvious, as is the necessity since in ,. -+ general 0 (Q. E. D. ) • neither (Dl)p(s) nor (l)l)p(s) depend upon X s • .., .. Furthermore Theorem II: A n.s.c. that (2.3.7) hold at each P £\) and for arbitrary F £ {,G IV} is that the matrix-valued functions QA on UJ defined at P £ \J by a local expansion (WA) P = w A(P)(X ) P (with r?· = [w Al) mn mn = mn satisfy the partial differential equations (2. 3. 8) Proof: That (2.3.8) constitutes a necessary condition for (2.3.7) follows directly by substituting WA for Fon the l.h.s. of (2.3. 7) and (w AX ) for Fon the r.h.s.. One readily verifies mn mn that A. (w A + K w A - W K ) X mn ' mp pn mp pn mn which leads immediately to (2.3.8). That (2.3.8) also constitutes a sufficient condition for (2.3.7) is proved thus. Assume that the relationship between holonomic components (with respect to some local c. s. { QJ , e} and non-holonomic components (with respect to some field {X;U)} of non-holonomic T.c. frames) of fields of i. c. numbers over IU C v<-f, is described by a set of 4 x 4 matrix functions =QA which satisfy (1.4.18) and (2.3.8). A1 ••• A Al ... A Define (4 x 4)-matrix functions S'2 P = Lio p] (p = 2,3,4) = mn on lJ by writing Al ••• Al".• Ap = w cw mn (P)(Xmn)P , p £ 1U p = 1 ••• 4. (2.3.9) 1S Then 1 n }i.µ = - ( n;>.. nµ nµn;>.. 2 - = = = =- 1 ,-lµ" rlnµ" + nµ"s::l ) = -2 ====== n;>..µvp 1 ;>.. µvp = - ( nµ"pn;>.. ) 2 n n = == = = (1 ~ ;>..,µ,v,p < 4). (2.3.10) (c.f. (1.2.4)); hence with-respect to {X; V}, 4 l ;>.. 1 ••• ;>.. F = (f W + l - f W P) E {'G IV} can be given a o o p=l p! ;>.. 1 ••• ;>..p non-holonomic expansion F = X with t = [mn] given by mn mn .. 4 l ;>.. 1 ••• ;>..P ~ = f I + l -p! f ;>.. ••• ;>.. _n o= p= 1 1 p - Now by differentiating each of the equations in (2.3.10) partially with respect to x and using (2.3.18) one easily verifies that (2. 3.11) and hence that the invariant derivative S, ,l = ( can be written + etc. • • ] x mn mn + etc. • • • J X , mn mn which is the same as the local expression for D~F. Hence (2.3.8) implies (2. 3. 7) • (Q. E. D. ) • For unrestricted coefficients r~~ ,K~, it will generally = be impossible to find matrices nA satisfying (2.3.8), and in order that solutions of these equations exist it is necessary that certain conditions of integrability be satisfied. By differentiation of (2.3.8) we have nA A _na A nA - -R + [ K~ 1 = 0 - I ~re = a~e 0,n ,e,~ = = = where A rA - A rA r" - rA r" Ra~e = ae,~ r a~' e + "~ ae ve a~ A are components of the curvature tensor formed from the r0 ~, and where K~e is as defined in (2.3.6). It follows that any solution of (2.3.8) must also satisfy (2.3.12) A necessary and sufficient condition for the existence of a simultaneous solution of (2.3.8) and (2.3.12) is obtained as follows. By repeated partial differentiation of (2.3.12) and the use of (2.3.8) we arrive at the following sequence of conditions T1 that must be satisfied by solutions of (2.3.8), (2.3.12): [ r/' K ] = '_Jn,IIP (2.3.13) where RA I denotes t h erth covariant... derivative o f"-R crµv P 1 •• Pr crµv with respect to the denotes the invariant linear combination = K - rcr K - rcr K +[ K, 5ivl IP :)l\J, p µp O'\J vp :J.10' _p 5iv1 (2.3.14) and where K = ( K ) r = 2 .. (2. 3 .15) µv I IP 1 •• Pr µv 11 P1 •• P r-1 I I Pr denotes the invariant expression + r = 2 •• (2. 3 .16) Then Theorem: In order than (2.3.8) admit a solution it is necessary and sufficient that there exist a positive integer N < 64 such that both of the following conditions are satisfied: (i) equations of the first N sets of (2. 3.12) and (2. 3.13) are consistent at all points ofV; (ii) equations of the (n+l)th set are satisfied as a consequence of the first N sets being satisfied. Furthermore, if p is the number of indendent equations in the first N sets of (2.3.12), (2.3.13), the solution of (2.3.8) involves 64-p arbitrary constants. This theorem is a special case of a theorem on partial differential equations due to EISENHART ( [6], p. 17). Its proof is omitted. An equation of the form (2.3.8) is often postulated as a starting point for discussions of spinor connexion and of the differential properties of fields of so-called Dirac matrices in curved space-time (see, for example, GREEN ([10], [11]); see also BRILL and WHEELER [2], LICHNEROWICZ [17], SCHMUTZER ( [23], [24]) and more recently BUCHDAHL [3]). Wheeler (op.cit) refers to "general solutions" due to KLEIN ([14]) of an equation like (2.3.8), but I have not been able to verify this reference. In the present model, (2.3.8) and the associated conditions (2.3.12) function only as consistency conditions serving to relate expressions for absolute derivatives (and also Clifford differentials) with respect to fields of holonomic and non-holonomic frames. In this regard it will always be assumed that the integrability conditions (2.3.12) are satisfied, and hence that non-holonomic T.C. frames are distributed over each coordinate neighbourhood in accordance with solutions of (2.3.8). As a simple application of the results of this section we have the following Theorem: In order that the Clifford differential of F X e: { = ~ mn mn ,e I\) } vanish it is necessary and sufficient that non-holonomic components of F satisfy <~,~ + [~ ~, !]) = 0 (2. 3. 17) (Proof omitted). Attention is drawn to the appearance here of a Dirac-type matrix differential operator n~< ) ,~· The next section is devoted to a discussion of differential properties of fields of T.C. numbers in terms of spinor frame elements. 10 §2.4 Connexion and T.C. numbers (continued); spinor connexion A description of the differential properties of fields of T.C. numbers in tenns of spinor frame elements may be obtained by means of what is essentially a repetition of the argument of the preceeding section. Let {Y; V} be a field of spinor frames over the neighbourhood 'V C ~ and let {Y;'2} = {(Ym(n))p(s) e: Gp(s);* 1 < m, n < 4; p(s) e:'f } denote the subset of elements of {Y;U,} attached to points of EC V . We will assume that {Y; IU} can be chosen in such a way that for the spinor frames attached to the points of? , Dd (Y ( ) ) ( ) depends s m n p s -+ on only through the components of the tangent vector field~, and that f s -+ this dependence on the components of xs is linear and of the first order. Then ••• there exist 256 x 4 (possibly complex) coefficients vm(n)u(v)~ (1 ~ m, n, u, v ~ 4; 1 < ~ < 4) such that dx~ ds (2.4.1) As in the corresponding argument of the previous section, not all of the coefficients vm(n)u(v)~ defined in this way are independent; by taking the absolute derivative of each side of the equation Xmu Yv(n) = ouvym(n) according to the rules described in §2.2, and using (2.3.3) and (2.4.1) to eliminate derivative terms, one readily verifies that (2.4.1) is expressible as 1 dx$ V (2.4.2) 1 and hence, by writing 4 "s(v)s(n)4> (P) =v(n)(v)$(P), as 12_ Y = (Kum,1, Ovn + V O )Y ds m(n) "' (n) (v) $ mu u(v) (2.4.3) The coefficients ~um$' "en) (v)$ defined by (2.3.3) and (2.4.3) respectively are functions only of the point P and will be called spinor coefficients of connexion over V with respect to the field of spin or frames {Y; U)}. With respect to a different field of spinor frames over 1l), the same connexion structure is specified by spinor coefficients obtained in the following way. Suppose that basis elements of {Y;11)} undergo a point transformation given by (1.4.2) and that under such a transformation (2.4.3) is form invariant i.e. is transformed into $ D Y' = { K ' o + v ' By substituting into this expression the value of Y~(n) given by (1.4.2) and expanding the left hand side, one readily discovers that appropriate transformation laws for spinor coefficients of connexion are given by + ' = - A A-l + A A-l Kum~ us,$ sm us l or in matrix notation -1 + K K' = - A,cf> A + A K -1 + I: I: I: -1 N I: (2.4.5) Ncf> N' = + =cf> = ='cf> = =4> = The coefficients kumcf> defined by (2.3.3) and transforming in the fashion (2.4.4) consequent on a spinor frame transformation over U will be called non-holonomic coefficients of spinor connexion, while the coefficients v ••• will be called ideal coefficients (n) (q) cf> of spinor connexion. Suppose that with respect to {Y;IU} Fe: {-GIIU} (2.2.6) is given a spinor frame expansion F = ~m(n)Ym(n); then in addition to the expressions given in §2.2 and §2.3, the absolute derivative of F along f can also evidently be written axcf> ds (2.4.6) while a similar alternative expression also exists for the Clifford differential of F: (2.4.7) For convenience the coefficient of w in this last expression will be denotedcf) F, so that DF = (2. 4. 8) will denote the expression of DF in terms of spinor coefficients of connexion. Spinor coefficients of F will be called integrable over UJ iff the differential equations 1/lm(n) , + Kms 1/ls (n) + 1/lm(t) v (t) (n) = 0 are integrable. In terms of matrices this last equation can be written -+ + qi N = 0 == hence in order that spinor components of F be integrable it is necessary that = 0 where (2.4.9) We will call KA and N the non-holonomic and ideal components =µ Aµ of spinor curvature respectively. Our next task will be to relate the connexion structure described here to those introduced in previous sections. The results developed here are very similar in content to those given in 2.3, and proofs will be omitted. We will assume throughout that the N independent consistency conditions of (2.3.12) and (2.3.13) are satisfied, so that non-holonomic frames are distributed over a typical coordinate neighbourhood in accordance with solutions of (2.3.8). The consistency of our holonomic and spinor frame descriptions of the differential properties of fields of T.C. numbers is formulated as follows: Theorem I: A n.s.c. that (2.2.7), (2.4.3) be consistent for arbitrary C' paths passing through PE '1,J is that = (2. 4.10) Theorem II: A n.s.c. that (2.4.10) hold at each PE 'tl.J and for arbitrary F E {~I U} is that the matrix-valued functions 0A on U defined at P E 1LJ by a local A = 6m(n) (P) (Ym(n)) P (with [~A]mn = em(n)A) satisfy the partial d.e. = o. (2.4.11) With regard to proof of the sufficiency of this condition, note that from the 0 A can be derived spinor expansions for the remaining Al ... ")..s holonomic frame elements W (s = 2,~,4). Write Al •.• A [Al ... ). 1 A ] w s = W s- W s , and in each term of this product expand A the last term w s in terms of spinor frame elements, and the Remaining terms w\···\-1 in terms of non-holonomic frame A1 ••• A elements. Then it is easy to verify that W s has a spinor- frame expansion Al •.• A Al ..• \ V s y = em(n) m(n) with A ] Al •.• A [Al .•. As-1 s 0 s = n 0 Al •• ·\ = ) ; furthermore, as a consequence em(n) \· •• A of (2.3.8) and 2.4.11), the matrices~ s satisfy the partial differential equation A Al •.. As-lo r s 0 a = o. (2. 4. 12) In what follows it will always be assumed that the integrability conditions for (2.4.11), viz. = + (2.4.13) (t:.{.(2.3.12)) are satisfied, and hence that spinor frames are distributed over a typical coordinate neighbourhood in accordance with solutions of (2.4.11). As a consequence of the preceeding discussion of consistency we have the following results for spinor components of T.C. numbers: Theorem: In order that the Clifford differential of F = -+ -+ + + = 0 (2.4.14) (Proof omitted). Also Theorem: Suppose F = of an homogeneous p-form. In order that spinor components of F satisfy (2.4.14), it is necessary and sufficient that F be both closed and co-closed. To conclude this section we will show how our spinor coefficients of connexion are used in a description of differential properties of fields of T.C. numbers in terms of adjoint spinor frame elements. Of the available methods for defining absolute derivatives and Clifford type differentials of adjoint spinor-frame elements we will need only the following one. In § 1. 4 we defined the adjoint FA of F e: { G I O} by .... -+ A II p"'* II writing F = Ym(n) z(n)m" The adjoint of DF = w II z (nm) ( :i' ,,_F) "'* y In particular, if we write II Y "'*y II = cr then p (q) s (t) p* (q) *s (t) A A * * o )cr z w Now as a conseq \Ence of the manner in which inner prod u::ts of TS::. n unbers were de fined, the local inner prod u::t < F1 ,F2>'P of two 1 2 arbitrary T.C. numbers is preserved when FP, FP are parallel displaced along any fixed curve passing through P. In terms of spinor components, that this is so is expressed by writing cr -cr {K * + v m*(n)*p(q) ,~ s*(t)*u(v) sm~ ont opu o qv (n) (t)~*o ms opu oqv + K O O O + V O O O } = 0. up~ ms nt qv (q) (v) ~ ms nt pu A (cf. (2 .1.11)). It follows that by writing W(s)t = W* a p(q) p*(q)*t(S) I A A we can express (DF) in the form A A { A A A } (DF) = W(q)p,~ - W(q)s Ksp~ - V\q) (s)~ W(s)p (2. 4.15) AA A (cf. (2.4.7)). We will write this expression as D F, so that A A (DF) = (2.4.16) AA A A D F will be called the adjoint Clifford differential of F. Note that if DF = 0 then DAFA = O; hence the following Corollary: If F e: {G IU} has vanishing Clifford differential, then adjoint spinor components of F satisfy + -+ -+ ( and let ~ C ~Po -e*p be some arbitrary but fixed minimal left ideal of 0 Theorem: Parallel displacement of T. C. numbers from e;, 0 along ~ maps lp into a minimal left ideal of 0 AP Proof: Let po denote the operator for parall~l (i) ~ + E: AQ , BQ e; et O (AO BO) r:!o 0 0 ··o ··o ·-o "O (ii) and having shown both of these then also that (iii) f! contains no other ideal but itself and the null ideal. Qo Since ~ u"'o* is invertible (its inverse ··o "p0 is leQ), it is bijective; is the image of r/ 0 under 0 ·-o I\ Q ppo· Properties (i) and (ii) will now be seen to follow immediately, 0 since £P is it self a minimal left ideal. Therefore J O is 0 * -o certainly a left ideal of eO • Furthermore, suppose this left ·o ideal contains an ideal, J; say, which is itself a minimal ideal -o ~ Qo Note that for a given lp the minimal left ideal of G; 0 0 into which ip is mapped by parallel displacement is obviously 0 dependent on the particular path along which the displacement takes place; if a path between PO and Q0 other than 1P is used, the image of ~P,I in .Pu * under parallel displacement will generally o Qo differ from JQ. In fact it is in general not possible to assign ·o to each point of the space-time manifold a single minimal left or right ideal of T.C. numbers in such a way that parallel displacements along arbitrary paths of ./C( map the collection into itself. Nor, for that matter, is it generally possible to assign to each point of space-time any fixed set of (<4) minimal left or right ideals in such a way that the collection over the whole ofJ( of all such sets has this translation-invariant property. However, suppose that it ~ possible to assign to each point of P of some domain l. C Jt. a subspace ,l by basis elements of q (<4) minimal left ideals of -GP* such that the collection 1.,.(q) = U 1./t is invariant under parallel displacements, Be:~ that is, is.,•mapped into itself in a nonsingular way when constituent elements are parallel displaced along paths of L· Such a collection ,/..(q) will be called an invariant distribution of q minimal left ideals on l, whilel, will be said to admit an invariant distribution of q minimal le ft ideals. Suppose l C J1, admits a distribution {, (q) of this kind. For .1(q)c p * each P e: l let "° PC u P denote the orthogonal complement of {}t and write l..,(q) c = U £ Ct c. {. (q) c will be called the complement 1 C ) . Pe:\_ of ,1.., q. Let Fe: {tlL} be an arbitrary field of T.C. numbers on Then for each p we can write F uniquely in the form l• e: L p Cl.> ( l.C,) F + with Fp CL> e: FCL) will be called B = Fp Fp ' J}tetc. !(q); (£) the component of F in any field F for which Fp = Fp at each P e: l will be called a q-minimal left ideal distribution of . . .1(\1 . T.C. n umb ers on l associated with a:, or, for simplicity, a q-ideal distribution of T.C. numbers on l. (Hence •• one-ideal distribution, two-id~al distribution etc •• ) ~, Conditions for the existence of invariant distributions of minimal left ideals of the various degrees may be obtained by analysis of the block structure of matrices associated with ideal coefficients of connexion in the various cases. We start by considering invariant distributions of a single minimal left ideal. Suppose l C J'( admits an invariant distribution £(l) of one minimal left ideal. Then without loss of generality we may choose a field of spinor frames for \L in such a way that at each P, L<;>is spanned by {(Ym(l)) P ; m = 1--4}. As before we will assume that spinor frames on l can also be chosen in such a way that absolute derivatives of frame elements along any path, ! say, of \. depend on , only through components of the vector field tangent to ! and that this dependence is linear and of the first order. Then in the notation of the previous sections, in order that /.}1) form an invariant distribution of one minimal left ideal on l it is necessary that for any C' path ! C l there should exist coefficients D amn~ (1~m, n~4; 1,~(4) such that ds (Ym(l)) is expressible as p(s) D = a Y dx~ i.e. such that ds (Ym(l)) mncji n(l) - P (s) ds = a o mu~ vl (cf. (2.4.3)). Now by contracting this expression with o , using mu Kmm~ = 0 ((2.3.31) and writing 2ia~ = amm~ (summation; i 2 = -1) we see that in order that l admit an invariant distribution of one minimal left ideal it is necessary that there exist a local c.s. and a field of spinor frames with respect to which (2.5.1) at each P e:U... Now for the remaining elements Ym(a) (a = 2,3,4) of our spinor frame field on l , D + ds a= 2,3,4. (2.5.2) Since this expression contains terms in Ym(l) (viz. v(a) (l)~Ym(l)), if F = (Y ) e: * P ~ m(a) m(a) P -C P (summation over a= 2,3,4) is parallel 0 0 0 displaced along an arbitrary path to Q0 e: L,F0 will have the general ··o CL> form FQ = ~m(n) (Ym(n))Q. consider the component F0 of F0 0 0 -o --o in -~'Q . This is non-zero unless the v(a) (l)~ vanish (a= 2,3,4) at 0 all points of the path along which the displacement was made. When FQ (£) is parallel displaced back along the same pa.th to~* , it is 0 0 mapped into the zero element. This contradicts the requirement that ~ Qo JI (1) the restriction of ~P to ~Q be non-singular. Hence in order 0 0 that l admit an invariant distribution of one minimal left ideal it is also necessary that v(a) (l)~ = 0 (a= 2,374) at all points of l It is in fact not hard to verify that this last condition together with (2.5.1) also constitute a sufficient condition for the existence of l (l). Hence Theorem: In order that l C ,ft, admit an invariant distribution of one minimal left ideal it is necessary and sufficient there exist a local c.s. and a field of spinor frames with respect to which, at each P e: l, t " ( 1) (n) cf> (P) = "(n) (1) cf> (P) = 2 acf> (P} onl 4> = 1••41 n = 1••41 PE 0.. (2..5.3) for some act>. It is not hard to see that when (2.5.3) prevails f..(l}c forms ., ( 1) • an invariant distribution in much the same way as does a. Since' a b asis' in. eaeh .I(ck.p l} C can b e ehosen in. t erms o f e 1emen t s th a t ,l(l}c span three minimal left ideals of G;, "J... is clearly an invariant distribution of three minimal left ideals. Analogous results can obviously be obtained for invariant distributions of two minimal left ideals, it will be readily verified that in order that LcJ.! admit an invariant distribution of two minimal left ideals it is necessary and sufficient that there exist a local c.s. and a field of spinor frames with respect to which, at each PE l, matrices associated with ideal coefficients of connexion each (simultaneously} have a general block structure (0 1°> with non-zero coefficients in the top-left and bottom-right 2 x 2 blocks only. Further information about the specific conditions in which these different block structures arise is abtained by considering solutions of (2.3.8} and (2.4.12}, viz. + ((2. 3. 8}} 0 A + + 0 AN = 0 ((2. 4.11)) .. ,cf> ·= cf> Let 0 be a (non-singular) 4 x 4 matrix defined by 1 0 = + 2 (2.5.4) = == It is readily verified that .if &land 0 A satisfy (2.3.8) and (2.4.11) respectively then 0 satisfies (2.5.6) + ~cf> 0 = 0 + 0 = 0 (2 .5. 7) = ~cf>9 Conversely, by using the alternative definition of 0 provided by Wo = em(n) Ym(n) ( 0 = rem(n)]) and assuming that (2.5.6) and = (2.5.7) (together with (2.3.8)) are satisfied it is easily shown that 0 A = QA 0 satisfies (2. 4.11). Hence in order that (2. 4.11) be satisfied it is necessary and sufficient that 0 (2.5.4) satisfy (2.5.6) and 2.5. 7). Now by examining the commutators of the QA with remaining matrices of the ring it will be readily seen that the integrability condition for (2.3.8) 1 viz. A ,.,a ((2.3.8)) R crcf,9 : = , can be written as (2.5.8) en,...,=.!. enµ gv - r? il)), and hance that whenever (2.5. 7) is = 2 = = = = assumed satisfied, (2.5.7) can be expressed in the form 0 (2.5.9) == == This shows that values of the ideal curvature matrices fully determinable in terms of the independent quantities their derivatives, the matricies nA and the matrix 0 which describes a spinor expansion of the unit. When the rA have µ ~ are needed in a description of the ~<1> 0 • Equations (2.5.6) and (2.5.9) are of considerable importance in this model, for in principle they contain all the information we need to determine the block structure of the ~· In order to determine this block structure in a specific situation i.e. given. prescri. b e d va 1 ues o f th e quantities~,,· · ,..,A rAµ However equations (2.5.6) and (2.5.9) can also be used to obtain conclusions of a more general kind concerning the block structure of the ~· Consider the simplest - and also the most important-case, that in which the relationship between holonomic and non-holonomic components of fields of T.C. numbers is expressed by the existence of a solution of (2.3.8) of the form r? = j gAµg r , where the indicators j which appear he.re are = m mµ _om m as introduced at the beginning of §1.3 and where the fields g mµ are components of a tetrad field on space-time which at each point has a spatial and temporal orientation the same as that of the tetrad introduced at the beginning of §1.3. It will be readily verified that in order that (2.3.8) yield a solution of this kind it is necessary and sufficient that non-holonomic coefficients of spinor connexion should have the form i . . AV C rcr > r ~~ = 2 Jm Jn g gnv gmA,~ - gmcr A~ Jt1ll. Now by inspecting representations listed in an appendix it is apparent that in this situation the matrices ;;.A and~~ already have a block structure of the form(-::+£), with non-zero coefficients in the top-left and bottom- o right 2 x 2 blocks only. Referring to (2.5.6) it is clear that in such a model the existence of a two-ideal distribution connexion structure may be assumed provided the matrix 0 has this same block structure. A similar conclusion can be drawn for models in which the connection between holonomic and non-holonomic components of T.C. numbers is expressed by the existence of a solution of (2.3.8) of C+H r + (-)). the form rl = j gm _om jm gm ~ms-' for by inspecting values of r ab = m = which are listed in an appendix it will be readily seen that such A A n have the same block structure as then given above. However models such as that described by GREEN [11], when translated into the present formalism, seem to necessitate our considering a relationship between holonomic and non-holonomic ~omponentsin which the n}.. }.. 1 }.. : are quite general linear combinations of the rab: n = 2 gab ~ab· In this case the matrices~+ cannot be expected to have any block structure other than full 4 x 4 form , in any frame. In such a model there are no T.C. number structures analogous to four component spinor fields, the simplest geometrical spinor quantities being sixteen component ones. We shall have more to say about these and related matters in the next chapter. It is also possible to give a sufficient condition for the existence of invariant distributions of one minimal left ideal in terms of the existence of so-called spinor structures on space-time. Before formally stating our result it may be useful to recapitulate some of the more important fractures of spinor structures. Following the definition given by GEROCH ([7]; see also [8]), a spinor structure on .,.CZ is a principal fibre bundle (cf. KOBAYASHI and NOMIZU [15], p. 50), flJ say, with group Spin (4) (the spinor covering of the restricted (i.e. time preserving, spatial-parity preserving) Lorentz group L), along with a 2:1 0 mapping+ from J to the principal bundle O of orthonormal frames (which has as group L ) on cl1f , such that 0 (i) + maps each fibre of J into a single fibre of(; (so that with each element of ef P, the fibre of f!! at P, is associated a single orthonormal frame at P, for each P Ec..A'l), and such that (ii) + commutes with group operations. That is, for each A E Spin (4) ,+ 0 A = µ(A).+ whereµ: Spin (4) ~ L0 is the covering map of the restricted Lorentz group. That a spinor structure exists on Jt (in the sense of Geroch) implies that if P is any point of tfZ and Y is a closed curve lying in C)P (the fibre of e> over P) which cannot be contracted to a point i.e. which corresponds to a rotation of an orthonormal tetrad at P through 2Il, then it is not possible to contract t to a point by deforming it throughout the entire bundle eJ. In other words O must have the same connectivity properties as L0 • On the other hand, that J'1, does not admit a spinor structure means that at some P e:: v1I/, there exists a curve t C op which cannot be contracted to a point in op but which can be so contracted by continuous deformation over e}. In terms of four-component spinor entities, that a spinor structure exists on ./It implies not only that it is possible to define fields of four-component entities which are mapped into themselves under parallel displacement along paths in.flt; it also means that -+ if 4> is such a field on cA then for each P e:: J-1 the fundamental -+ -+ + ambiguity in the sign of 4>P is preserved when 4>P is parallel displaced around any closed curve lying in A which passes through P. (In this context 1 is to be considered a cross-section of a fibre bundle with fibre ([, 4 and group Spin ( 4) associated with .K , while "parallel displacement" means with respect to the natural connexion structure for these objects which derives from the connexion on f!J. See KOBAYASHI and NOMIZU (op.cit) pp. 87, 114). It will be evident from this second statement that in order that space-time admit a spinor structure it is necessary that it also admit an invariant distribution of one minimal left ideal. There is some interest in proving this result formally, for we will further discover that the existence of a spinor structure is not only a sufficient condition for the existence of an invariant distribution of minimal left ideals on J't; it is in fact also a sufficient condition for the existence of four independent invariant distributions of single minimal left ideals. To establish this we will show first of all that for any minimal left ideal, say, of GP* there exists a certain equivalence class of bases for £ (i) P, which we will denote (S (it, the elements of which bear a 1:1 correspondence with fibres belonging to c!p· We will then use the natural connexion structure on ~ to define an absolute differentiation operator for elements of these equivalence classes which maps each £ (i) P into itself, for each P i:: .,.CZ, and to hence show that for each left ideal in an arbitrary left decomposition G; = aci> + !c2> + l< 3> + L <4 > the relevant idea1 connexion p p p p oefficients have a "one-ideal" structure. We begin by introducing the spaces f> (!) and showing that at each P i::rft there exists a 2:1 correspondence between ( i) elements of f> P and orthonormal tetrads at P. For each P i:: ~ 4 let (W0 ) P = r (E. ) be an arbitrary but fixed parti,tion of i=l 1 P the unit into primitive idemponents (so that (E.) (E.) = 0 .. (E.) , 1,i, j~4) 1 p J p 1J 1 p and suppose the minimal left and minimal right ideals generated by (Ei) Pare deno'!led £(i)P and~(i)P respectively. In what follows it will be convenient to omit the point designation of elements of ,e;;. Let {gp} = {(gm) i:: T; (..,cg); m-1··4} be an arbitrary but fixed p orthonormal tetrad at P, chosen in such a way that y ., ., g((gm) p' (gn)P) = i::m omn wth (g4>, > 0 (see our remarks at the y V ~ beginning of §1. 3), define G = cr* ( (g ) ) (m=l· • 4) and let X m mp mn (l~m,n~4) ,be 16 T.C. numbers at P which have a multiplication law roo V of the form (1.3.20) and which are constructed from the G according m to the procedure described in §1.3 • Consider bases . { (i) ,II } p y (i) Ym € ~(i)P; m=l··4 of J.,(i)P for which m is expressible in the form y y (i) = X .T m mi m=l • • 4 (2.5.10) * where Tisa nonsingular element of Gp" It will be readily verified that in order that elements of this form belong to l(i)P it is necessary and sufficient that T satisfy -1 y T X .. T = E, 11 1 (no summation). (2. 5.11) (For the purpose of showing this it may be useful to write the Y (i) thus: m = 4 v -lv y (i) E X t) T) (T X.. T) m s,t=l S 11 (no summation) • (2.5.12) 4 ,, In this regard note that X E Xst)T satisfies s,t=l x2 = X = lx X EXm 4 E X E = .!, E m m 4 m (m=l··4; no summation). (2.5.13) We will have occasion to refer to these properties at a later stage). Before proceeding, it will prove useful to have atour 1isposal the following lemmas. fO( ., Lemma 1. For a given set of X and a given partition of the mn 4 unit W = E E. into primitive idempotents, there 0 i=l 1 exists one and only one solution of -1 V A X .. A= E. 11 1 (i=1··4; no summation) (2.5.14) that also satisfies 'I/ V V X,. A X,, = X,. 11 11 11 (i=1··4; no summation). (2.5.15) Proof: Suppose there exists a nonsingular B(#A) which satisfies ., -1" -1 V -1 B X .. B = x .. so (BA ) X. . = X .. (BA ) • 11 11 11 11 -1 4 ., -1 ., Therefore BA = r xi1 (BA ) xii" It follows that BA-l must necessarily i=l -1 4 ., have the form BA = r ei xii (Si# o, i=1··4) and hence that if Band i=l 4 ., and A both satisfy (2.5.14 then B-,( E S1. X. . )A. Next, suppose that i=l 11 ., ., ., ., .., .., X.,BX, .=X .. (i=1••4). Then S . X,. A X,. = X. . (i=l• • • 4) , so S. = 1 (all i) 11 11 11 1 11 11 11 1 Therefore B = A (Q.E.D~). Lemma 2. For a given partition of the unit W = EE. into primitive 0 1 idempotents there exists a 1:1 correspondence between orthonormal tetrads at P which have the spatial and ternporal orientation described above and solutions of s2 = S SES = m .;s4 E SE = .!, E mm 4m (m=l· •4; no summation). (2.5.16) (Ol Proof (a) Let {(g) £ TP* (,It); m=1••4} denote any orthonormal rn F tetrad at P having the spatial and temporal orientation described above , define G = cr*((g) , and let X (l~rn,n~4)denote 16 T.C. rn rn P mn numbers at P which have a multiplication law of the form (l.3.20) and which are constructed from the G according to the procedure rn described in §1.3 (these X are thus uniquely defined in terms of mn 0 0-l O O the (g) ). Let T denote the unique solution of T X.. T = E. , X •• T X .. = X •• rn P 11 1 11 11 11 o~, 4 0 (i=l""4) and define S = T r Xst) T. By using (2. 5 .13) ·one readily s,t=l verifies that the T.C. number S thus defined satisfies (2.5.16) (b) Suppose S £ * satisfies (2.5.16) and write Smn = 4E SE • ,ep rn n Then S s = 8 (l~rn,s,t,n, ~ 4) and s = w (summation} • ms tn st smn mm 0 0 Write G = (l~rn~4) (where the matrices r [r ] orn ornpq spq = t =Orn =Orn pq t ornpq 0 0 0 0 are as listed in an appendix). Then G G + G G = 28 w orn on on .Orn mn o . Therefore by virtue of a result described in §1.3 there exists a non-singular T.C. number, B say, at P which has the property that *-1 _, t B ·· S B) form an orthonormal tetrad at P with the a~ am pq pq required spatial and temporal orientation. Since this T.C. number Bis determined to within a scalar multiple of the unit element, the tetrad thus defined is unique. (Q.E.D). To retum to our main discussion, it follows from the first of these results that there exists one and only one T.C • ., ~ ., number satisfying (2.5.11) that also satisfies x .. TX .. = X.. 11 11 ., 11 (i=l·•4; no summation). Let this T.C. number be denoted T. Further, for the particular tetrad {g} that we have introduced, let the p corresponding (uniquely determined} solution of (2.5.13) be denoted ., (i) ~ S, and let(B P denote the set of all bases for a,(i)P for which (03 constituent Y (i) can be written m .., .,. y (i) = ;\ TE SE. m mn m l. A= [;\ ] £ Spin (4). (2.5.17). mn By comparison with (2.5.10) one sees that there are precisely two elements of (B ii) for which constituent Ym (i) can be written in the form (2.5.10) where T satisfies (2.5.11) (these have " ,I , ( i) V ( i) T = T and T = - T respectively, and give Y = + Y where m m ,/ (i) V Y =TE SE.)." Hence to the tetrad {g} of orthonormal m m l. p tJ (i) cove ctors at P there correspond exactly two elements of IJp , for each value of i. Next it is necessary to show that with any other tetrad at P (having the same relative orientation as {gp}) can be associated exactly two elements of 8P (i) , for each i, and hence that the definition of the ep (i) is essentially independent of the particular y frame {gp} that was introduced at P. Let {gp}' = {(gm)i £ TP* (""'); m=1••4} be any orthonormal tetrad at P having the same relative orientation such that (g)' = p (g) (m=l··4). Suppose A= [;\ ] £ Spin (4) m P mn n P = mn is defined by writing pmn r =Ar A-l (A is thus defined only om =on == = == to within a+ sign). £ .P Let xmn ' \.Jp* (l~m,n,4)be 16 T.C. numbers at P which have a multiplication law of the form (1.3.20) and which are constructed from GI = o* ((g) ') according to the procedure m mp described in §1.3, and choose a basis for ,£(i)P in terms of elements (i)' (i)' 1 Y such that Y = X ~ T' where T' satisfies T1• X. ~ T' = E .• m m nu 1.1. 1. -lV Since we can write X' = L X L (l~m,n~4; L nm mn = ).st Xst) it follows ,/ V .., that T'necessarily has the form T' = L-l (L 0 X ) T where T IS n nn fot- AS defined previously and where the cr. are non-zero 1 {i)' -1 {i) ' complex numbers; hence Y = L cr.Y • Therefore from {gp} m 1m can be obtained two elements of ~·(i), corresponding to the values cr. = 1 and cr. = -1, for each value of i. It follows that from any 1 1 orthonormal tetrad at P can be constructed exactly two elements of f, P {i) {i=l • • 4) • To show that with each element of (8P {i) can be associated a unique tetrad at P we proceed thus. Suppose {Ym{i) e: !{i) m=1··4} e:03P {i); then there exists a matrix A= [A ] e: Spin {4) such that Y 1i) = A TE SE. = mn m mnn1 where S satisfies {2.5.16) and where A= T" satisfies {2.5.14), V {2.5.15). {In these last constraints, the X,. are now to be considered 11 constructed from the tetrad associated with S) . Let Z m{i) e: ft {i) P . h . f {i) {i) ~ {m= 1 •• 4) b e f our T.C. n umb ers wh 1c sat1s y Z Y =u E., m n mn 1 {i=1·•4). The Z {i) are uniquely defined by these equations; m in fact by virue of the properties of "sand T., listed earlier, it is readily verified that the solution of these equations is {i) " ,,_1 -1 -1 -1 -z = E,SE T A , where A = [A ] • Next, define m 1 n nm nm = nm s = y {i) z {i). It is readily verified that s satisfies each p,q=l p q of the constraints listed in {2.5.16). As a consequence of the second of the lemmas given above we can therefore associate with the Y {i) a unique tetrad of co-vectors at P, for each value of i m and at each P e: .,If • It follows that there exists a 2: 1 correspondence between elements of A connexion for T.C. numbers can be obtained from the connexion on the spinor structure e/ in the following way. First (i} we remark that elements of(BP can obviously be put into a 1:1 correspondence with elements of Spin (4), for each value of i and at each P e: .}(. Therefore for each value of i there exists a 1:1 correspondence between elements of fop (i} and individual elements of the fibre J P of ff over P. Let ::k- be a fibre bundle with 'fibre C 4 and group Spin (4) associated with{!, ~P the fibre of ~ over P e:,J,t. (Properties of vector bundles associated with a given principal fibre bundle are described by KOBAYASHI and NOMIZU (15], p 54; see in particular the account given in this reference of the manner in which connexions on associated bundles are defined (pp. 87,113}}. As a consequence of the correspondence between elements (i} of (!f P and frames of (8P there exists a 1:1 correspondence between elements of :fi.P and T.C. numbers at P which are contained wholly within !,.(i} P, for each value of i and at each P e: It. S uppose th e mapcJlp,r,., + iv(i}P-' 1s· deno tan·th e , us n.ctL+-'·&R"p ~(i}P : hp + HP (i} = n(i} (hp}. Let le denote an arbitrary but fixed path in cl( which passes through P, suppose ! has parameter s and let the absolute differentiation operator along this path for elements of ~P be denoted ~s (see KOBAYASHI and NOMIZU (op.cit}, p. 114}. Since this operator maps :J'.f.p into itself and since we can always choose the " differential operator i for T.C. numbers at Pin such a way that D Il(i}-1 O ds ' A it follows that ~s maps each £(i}P into itself, and hence that for each minimal left ideal of GP* ideal coefficients of connexion have a "one-ideal" form. Therefore there must necessarily exist a I ol, field of spinor frames with respect to which ideal coefficients of connexion can be represented by a diagonal matrix. Hence Theorem: In order that .}1, adJ11it four invariant distributions of single minimal left ideals it is sufficient that ,.,It admit a spinor structu.re. By using methods from algebraic topology, GEROCH [8] has obtained a short list of some of the more common solutions of Einstein's equations which have a spinor structure. This list includes the Schwarzschild, Robertson-Walker, GcSdel and fluid-ball solutions. It follows from our analysis that the space-times represented by these solutions also admit invariant distributions of one minimal left ideal. There has not been an opportunity to substantiate these findings usinq the methods described earlier in this section. To summarise the results of this section, there are a number of different possible invariant invariant T.C. number sub- structures that may exist on space-time, including invariant distributions of one, two and three minimal left ideals; there are also invariant distributions that are closely related to spinor structures. Of the existence condibans that can be given for these various distributions, those that pertain to the last mentioned are the most restrictive, since they involve global topological properties of space-time as well as local "curvature-related" ones. For manifolds which do not admit a spinor structure, ideal coefficients of connexion may have either of the "one-ideal" or ittwo-ideal" forms described earlier, depending on properties of the matrices A n and of the curvature tensor, but will generally involve non-zero entries in all non-diagonal positions. In such manifolds it is thus not generally possible to form fields of four component (single left ideal) T.C. number entities which are mapped into themselves under parallel displacement. Chapter 3 Physical Applications Introduction. This chapter consists of two sections and is devoted to a discussion of applications of our model in physical theory. In the first section it is shown that with certain simplifying assumptions the column matrices associated with one-ideal distributions of T.C. numbers can be shown to have a number of properties which are normally associated with four-component spinor fields. in the classical theory of spinor fields. By virtue of these properties, one-ideal distributions of T.C. numbers are distinguished as the geometrical concomitants of four-spinor fields. In particular, one-ideal distributions of T.C nmrhers having vanishing Clifford differential appear as concomitants of four-spinor fields which satisfy the zero rest-mass Dirac equation. When viewed in this way it is clear that the usual curved space-time classical Dirac equation comes about only when the existence of an invariant distribution of one minimal left ideal is assumed. This is a quite strong assumption, and. is evidently not satisfied in the most general situation. In the first section we will also show how fields which describe particles of higher spin may be respresented in terms of T.C. number-type fields. In section two it is shown that our model not only supplies a geometrical model for four-component spinor fields, it also embodies a geometrical model for the spinor fields which appear in the class of unitary symmetry theories that pertain to symmetries over a quartet::of four 4-dimensional spin spaces. This discussion centres mainly upon an exhibition of the isotopic gauge theory of Yang and Mills in terms of the geometry of T.C. numbers. (01 §3.l One-ideal distributions and four-spinor fields. We will assume throughout this section that the space-time manifold admits an invariant distribution of one minimal left ideal, i.e. that there exists a rule which assigns to each P E J1, a minimal left ideal I, p C '(; p* such that the collection J. = u I., PEA p is invariant under parallel displacements. It will be recalled that a one-ideal distribution of T.C. numbers on .f1. associated with an invariant distribution ! of minimal left- ideals is a rule which assigns to each P E c)1. an element of/. P. Let F (I.) be such a one-ideal distribution, and for each P E..)( let { (Ym(l) )P E ,GP; m = 1. .4} be a basis for LP. Then with respect to this choice of frames F ( I. ) may be written As foreshadowed in the introduction to this chapter, with certain + simplifying assumptions the column vector ~ = [~] of spinor components m of F defined by this expansion can be readily shown to have a number of properties normally associated with classical four-spinor fields. Let us write down these properties. (i) Behaviour of under a basis change in each 1,P. Suppose the following basis change is made in each -£ P + ,-1 (P) ( ) = l\sm Ys (1) P m = 1. .4 no + Then according to (1.4.3) the column vector t(P) undergoes a contragradient transformation + + I + t (P) + ·t (P} = At(P) +A while for the matrix t (P) of spinor components of the adjoint (i) A (£) F of F , +A 1 +A -1 -;A (P) + t (P) .::t (P) A (P) (cf. ( 1. 4. 4)). When basis changes in each f. P are restricted to representations of the spinor covering of the Lorentz group, these transformation properties are clearly identical with those normally associated with spinor fields. + (ii) Metric properties of t Let F (') F (L} be one-ideal distributions of T.C. numbers with 1 , 2 + + spinor components t 1 , t 2 respectively. From (1.4.5) the local inner -(.() . (() (.() product of F 1 {t) and F2 can be written ( F1 ,F2 ~ = +A + t 1 (P). ~2 (P). Under the assumptions that led to the simplified form (1.4.41), this expression becomes ( F Cl) F (~ ) 1 : *"' (P) f °:t (P) 1 ' 2 P: -.,. -'¥1 • _o4 .,..2 (3. 1. 1) where ~04 is a 4 x 4 matrix with constant entries the value of which is given in an appendix, and where* and"' now denote the operations of complex conjugation and (column+ row) vector transposition respectively. Ill The similarity of (3.1.1) to standard flat space-time four-spinor expressions is clear (c.f. ROMAN [21] , pp.133 (eq. 2.97), 135 (eq. 2.102)) -Ii (iii) Differential properties of I From (2.4.6), (2.5.3) when FP et) undergoes an infinitesimal parallel displacement along some C' path ! from P = p (s) to P' = p(s + ds) (c f. notation of §2.4), spinor components of F(/.) ? + + undergo a consequent change l + !• = ~ + 8~ with ds. (3.1.2) The Clifford differential of F (I.) can therefore be written (3.1.3) In particular when the Clifford differential of F ( l.) vanishes, spinor components of F( £) must necessarily satisfy (3. 1. 4) The similarity of these expressions to those of orthodox spinor analysis will be obvious. The column matrix-valued functions associated with spinor components of one-ideal distributions thus have all of the properties by which the four component fields of classical spinor field Ill.. theory are normally distinguished, and may therefore be legitimately interpreted as geometric concomitants of four-spinor fields. The column matrix-valued functions associated with spinor conponents of one-ideal distributions will be called spinor distributions., while (3.1.4) will be interpreted as a zero rest-mass curved space- + time variant of Dirac's equation for the distribution~. It is worth re-iterating that in writing down (3.1.3) we have assumed the existence of an invariant distribution of one minimal left ideal • Therefore from a geometrical point of view the classical (curved space-time) Dirac equation for a four component field only arises when the existence of an invariant distribution of one minimal left ideal is assumed. On an arbitrary space-time manifold one is certainly not entitled to make an assumption of this kind. On such a manifold the naturally ocurring spinor objects are thus sixteen dimensional rather than four dimensional. To conclude this section we will show how fields representing particles of higher spin (~ 3/2) are accommodated in this model. We recall that in flat space-time a free particle of higher spins is described by a wave function which satisfies some wave equation together with a number of supplementary conditions, so that the wave function has exactly 2(s+l) fermion degrees or (2s+l) boson degrees of freedom. Boson fields are describable in terms of purely tensorial quantities of various kinds, and there is obviously no difficulty incorporating these into our model. The most satisfactory way of accommodating fermion fields of higher spin in our model is through a curved space-time generalization of a scheme originally due to Rarita and Schwinger (see ROMAN [22], p. 159). In this formalism a fermion of spins () 3/2; sis thus an odd integral multiple of¼.) is describable in terms of a spinorial quantity which has s - ~ lll world vector indices and one spinor (column) index and which satisfies some wave equation together with suitable supplementary conditons (cf. ROMAN [op.cit]p. 159). As we will now demonstrate, spinor quantities which have this same configuration of indices in an arbitrarily curved space-time arise quite naturally in the context of Clifford number models for classical fields. In the construction of tangen~ Clifford number that was given at the beginning of §1.2 we considered the quotient space of the alga,·ra of covariant tensors at each point of space-time by a certain two-sided ideal of tensors at the point. The effect on covariant tensors of the projection cr* was, essentiallv, to factor out by contraction the symmetric part of every pair of covariant indices whilst preserving skew-symmetrized "remainders" at each stage of the factoring. We will now show that geometric spinor quantities having the same formal structure and properties as generalized (i.e. curved space-time) Rarita-Schwinger functions can be obtained in this model simply by extending the domain of cr* in a certain way to include contra-variant and mixed tensor quantities as well as covariant ones. Let Jp denote the algebra of (contravariant) tensors over TP(Jt), 'JP* the algebra of covariant tensors, and consider the tensor * l * * IP* i * product :Jpt3( Jp/ J P) = JP 0 t:P, where J = J P (g ) is the two-sided ideal of jp* described in §1.2. Since the projection cr*: jp* + Jp/ * j*P = bpp* is linear, the linear map t Qt f + t e cr*f is uniquely defined (cf. KOBAYASHI and NOMIZU [15], p. 18) and, without loss of lllf- generality, may be denoted a*. Elements of ~ Gr> (P* will be called generalized tangent Clifford numbers (G.T.C. numbers) at P. A field of G.T.C. numbers on Lc,M is a rule which assigns to each P e:l. a G.T.C. number at P. Properties of T.C. numbers outlined in the previous chapters can all be readily generalized to yield analogous results for G.T.C. numbers. However for our present purposes it will suffice to summarise these properties in one or two general assertions. (i) Spinor components of G.T.C. numbers can be defined, and give rise to quantities having one non-holonomic index, one ideal index and a number of contravariant world-vector indices. For example, each mixed (1,1) tensor has associated with it an element of Tp(c11,)II 8 "p* such that if f = fA µeA-+ (iw,µ then A -+ µ F = cr*f = f µ eA ® w has a spinor expansion of the fnm (3.1.5) (ii) Differential properties of fields of G.T.C. number can be assigned in such a way that notions such as absolute derivative and Clifford differential carry over to G.T.C. numbers in a natural way. In particular, the connexion structure for T.C. numbers carries over to G.T.C. numbers in such a way that whenever space-time admits an invariant distribution of one minimal left ideal, each US G.T.C. number has associ,ated with it a distribution the conponents of which bear one spinor :index and a number of world vector indices. In such a situation, the field F (3.1.5), for exanple, has associated with it a distribution + = ~ e, e y m A m the Clifford differential of which can be written as (3.1.6) A full account of the formal developement of these and similar properties is rather lengthy and has been omitted. Quantities such as +A the column~ which appears in (3.1.6) will be interpreted as the geometric structures underlying generalized Rarita-Schwinger functions. To summarise this discussion, by extending the domain of the canonical map a* to include arbitrary mixed tensors as well as purely covariant ones, one can construct fields of conposite tensor - T.C. number quantities having properties which make them suitable for interpretation as the geometric structures underlying higher spin wave functions. §3.2. Multiple-ideal distributions;geometry and internal symmetry Having identified our so-called spinor distributions as the geometric correlates of four COl'l!Ponent spinor fields, the question naturally arises as to how one is to interpret multiple-ideal distributions of T.C. numbers and the spinor components of general fields of T.C. numbers. In this regard it is obviously sil'l!Plest to look for an interpretation of the constructions of our model in terms of existing physical-theoretic structures. Of the possible ways in which a physical interpretation of multiple-ideal distributions might be apprached the most straight forward would seem to be one in which these distributions are interpreted in terms of the multiple spin-space quantities encountered in the various theories of unitary symmetry. Indeed, in the context of a discussion of Clifford algebras in flat space-time, TEITLER [29] has discussed a representation of theories of unitary symmetry in terms of transformations between minimal left ideals of a vector Clifford algebra, and inspite of the many ways in which the model described here differs from that of Teitler, there seems to be no reason why such an approach should not be useful in our model. However, it is hardty necessary to point out that we are starting here from assumptions rather different from those of Teitler. As already remarked, the naturally occuring spinor objects in our model are actually sixteen component ones; in a general space-time there is quite simply no a priori reason for assuming the existence of the reduced four-col'l!Ponent structures we have called spinor distributions. Further, transformations mixing up minimal left ideals are admitted in our model not merely as an ad hoe generalization of a sil'l!Pler model; they are an integral and necessary part of its essential geometric structure. In this regard the model for unitary symmetry developed here is rather different from that propounded by Teitler. 111 By way of illustrating how multiple-ideal distributions of T.C. numbers are interpreted in terms of the multiple spin-space structures encountered in physics, we will now sketch a geometric model for the isotopic gauge field theory of Yang and Mills (YANG and MILLS [30]). Geometric models for more complex theories of internal symmetry, up to and including those which relate to symmetries over four (four-dimensional) spin spaces, may be obtained in an analogous fashion, but will not be dealt with here. First we recapitulate some of the more important features of the Yang-Mills treatment of isotopic gauge invariance. According to the theory of isotopic spin, the closeness of proton and neutron rest-masses and the approximate equality of certain proton-proton and proton-neutron interactions can be accounted for by assuming that proton and neutron correspond to different states of the same particle. This underlying particle is described by a 'two-component' super-wave function, one (four-spinor) component of which can be chosen to represent a proton wave function and the other a neutron wave function. In order that a theory of nucleon-nucleon interaction conform to the requirement that, in the absence of electro-magnetic interaction, it should equally well describe both p-n and p-p interaction, it is sufficient that it be covariant with respect to the mixing of proton and neutron parts of this super wave function: -+ + I '¥ -+- '¥ = ( 3. l.1) -+ (In this equation '¥ is a 2 x 1 column matrix, the entries of which may be thought of as (transposed) column four-spinors). It is usual to restrict such transformations to representations of the unitary unimodular group ,,, in two dimensions; in addition, the matrices which represent these transformations, such as Sin (3.1.1), are in the first instance constant functions of (space-time) position. The invariance of a physical theory under transformations like (3.~.1) is usually called isotopic gauge invariance of the first kind. According to Yang and Mills the localised field concept that underlies usual physical theories demands a rather wider kind of invariance than that described above; not only should a theory of nucleon-nucleon interaction admit transformations of the 'compound-nucleon' super wave function like (3.l.l), a satisfactory theory should also be covariant under transformations like (3.t.1) even when isotopic spin transformation matrices are quite arbitrary functions of position. A stipulation of this kind of invariance(which is called isotopic gauge invariance of the second kind) in a theory leads to the requirement that + partial derivatives of 1¥ should appear in the theory only in a + combination like (I - i E B) ~ where the B (4> = 1 •• 4) are 2 x 2 matrices which Wlder a change of isotopic gauge + + 1 4 +, 1 4 1 4 -1. 1 4 1¥ (x , •• ,x ) + 1¥ (x , •• ,x) = S(x , .• ,x) f(x , •• ,x) (3.l.2) undergo a corresponding transformation 1 4 as = B (3.l. 3) In this formalism the field ~4> as introduced ensures the covariance of + expressions containing ', under transformations like (3.1.2). It constitutes a vector potential for a gauge-invariant field strength quantity = + ie: which is used in the construction of a Lagrangian density for the Bcj, field. It will prove useful to list some intermediate quantities introduced by Yang and Mills. These are vector and tensor triad components, denoted bxA (A= 1 •• 4; x = 1,2,3) and b, XAµ (1 = 2 b T Bcj, XA = X and 2 b T BA = XAµ x' =µ = (3.1..4) where the T are 2 x 2 isotopic spin 'angular momentum' matrices = X (numerically these may be taken to have the same values as the Pauli matrices). The matrices B1 , B2 , B3 and Tx are taken to be hermitian, and B4 skew- ======hermitian. Hence the coefficients bxA are real for all x when A= 1,2,3 and imaginary for all x when A = 4. Next, consider the following geometrical situation. Suppose that the space-time manifold admits an invariant distribution of two minimal left ideals (but not a one-ideal distributiort and let this distribution be ~ (2) denoted ~ • If as basis in each £ ; 2> we use 1"10 {(Ym(u))P; m = 1 •• 4, u = 1,2} then matrices associated with ideal coefficients of connexion necessarily have a block structure like with non-zero coefficients in the "top-left" and "bottom-right" 2 x 2 blocks only. Furthermore, of the non-zero coefficients in these matrices, only those lying in the top left block contribute to .I (2) expressions for Clifford differentials of elements of J... (this is because spinor components of elements of £ <2 > can be represented by 4 x 4 matrices with zeroes in the last two columns). A glance at representations listed in an appendix indicates that the linear vector space of 4 x 4 matrices which have non-zero entries only in the top-left II {-) 2 x 2 block is spanned by matrices Cm= 1.. 4) which have m (constant) entries defined by Pm(n) = II (n) X (cf. (1.3.9) and the m pq pq appendix). Hence the top-left 2 x 2 block of each of the ideal connexion matrices N4> can be written {-) " II(-) N 4> = mcp = m (3.l..5) The corresponding component of the ideal curvative matrix is = " II<-> m\µ = m l In order to change the mode of description of these quantities to that of 2 x 2 matrices, we introduce four 2 x 2 matrices T :the entries of which ...m are defined by 111 = 21 [JI (-)] = X UV 1 < X < 3 = [T4] = 21 [ - 1' rr<-> J = 0 = UV = 4 UV UV (1 < u, V ~ 2) ( 3 • .1.. 7) Since, by virtue of the multiplication rule for the II(n) given in an n appendix 1 + T T = o I J X xy = T T = .,.Y X (we have written T = I, the unit 2 x 2 matrix) , the T are simply 4 X = == = a particular representation of the Pauli matrices. Therefore if,as the 2 x 2 (-) equivalent of ~ , we write = 2v "' T (3.2.8) m"' =m then the Clifford differential of F = Y e: ,J <2> becomes m(u) m(u) "J.. -+ -+ -+ = [4> n + K 4> + 4> N ( 2 )] Y ='v =o - -=a mu m(u) (3.l.9) -+ (n.b. 4> is 4 x 2; summation over the ideal index extends only over the values 1,2). The ideal curvature quantity that corresponds to (3.1.8). viz. Ill. - N (2) _J.l , A can also be written ( 3. 1. .10) Note also that if under a (non-singula~ ideal transformation in each 4,.<2>p -+ -+ = == == (3.l.ll) then also N (2) (2) I -1 +E-1 N (2) -+ N = - E E E ~ = = , 4> = _. and N (2) (2) I I -1 N(2) -+ NA = E Aµ =µ = Aµ = (3.1.12) Comparing the situation just described with the isotopic spin theory of Yang and Mills as outlined earlier we see that with regard to mathematical content the two descriptions would seem to be equivalent provided that if -+ (i) the iso-spinor 1¥ that appears in (3.l.2) and the matrix of spinor -+ = components~ that appears in (3.l.11) are equated thus: -+ = 1¥ ( 3. 2. 13) or in some similar manner (as before, t means matrix hermitian conjugation) then J (1'3 (ii) whenever an ideal transformation is made in each£';> , a corresponding change is made in axes of the iso-spin space attached to the same point, and in such a way that if (3.l..13) then ( cf. ( 3. 2 . 3) and ( 3. 2.. 12)) "'t 1 4 -1 1 4 L (X I•• 1X ) = S (x , .. ,x ); (3. 2.14) also (iii) the coefficients that describe spinor connexion for elements of t 2 > and those that define isotopic gauge fields are related, and insuchawaythatif (3.l..13) then (cf. (3.l..3), (3.2.12)) for N (2) (2)t N(2) = N (2)t cj, = 1,2,3 = - N = i e Bcj, I while cj, _cj, =T =T = i e B (T = 4) with = o, i.e. - _T \) 4cj, \) * = \) = b xcj, - xcj, xcj, cj, = 1,2, 3 \) * = \) = b XT XT XT (T = 4) ( 3. 1. 15) 4 (n.b. with Yang and Mills, x = it) • Note that if we assume that = 0 ( 3. ,.. 16) (and a direct comparison of expressions like (3.l.9) with the differential operator of Yang and Mills would seem to require this), and hence that = O, then by virtue of (2.3.12) RA = O and hence (cf. (2.4.12)) µvp = O. As a consequence, in order that our model successfully mirror the constructions of Yang and Mills, it is necessary that space time be curved. Now with regard to the orthodox view that the domain of isotopic spin theory is flat space-time, we remark that GEROCH [ 8 ] has shown that when space-time is flat it necessarily admits a spinor structure; the geometry of a Yang-Mills isospinor field in flat space-time is therefore the geometry of a direct sum of two ordinary spinor wave functions. Expressed in slightly different terms, in the parallel transport of isospinors in flat space-time there is no "mixing" of the constituent proton-neutron isospin states; in such a space-time a proton wave function is in fact everywhere a proton wave-function, a neutron wave-function everywhere descriptive of a neutron-like state. Implications of this kind are at variance with the stated intention of Yang and Mills of localising the concept of isotopic spin, and would seem to imply that the true domain of isotopic spin theory is in fact curved rather than flat space time, in which case (3.2.16) is not relevant. It follows from these remarks and from the preceeding analysis that two-ideal distributions of T.C. numbers may be legitimately viewed as the geometrical structures underlying isospinor fields in space-time. This mode of representing isotopic spin leads to several rather interesting conclusions when combined with our earlier observation about the relation between one-ideal distributions and fields of four-spinor fields; we can, for example, infer from the previous discussion and from the results obtained in §2.5 that (i) only certain kinds of space-time manifold admit isotopic gauge fields with non-trivial field strength components, and also that (ii) an assumption of the existence of non-trivial isotopic gauge l'l.S fields carries hitherto unrecognized information of a geometrical kind about the space-time manifold on which the fields are defined. From the foregoing analysis it is clear that any of the theories of unitary symmetry that pertain to symmetries over a number (~ 4) of spin spaces may be modelled in terms of the geometry of invariant ideal distributions of T.C. numbers, and hence that conclusions analogous to the above may be drawn for any of these more general theories. A geometrical model for non-trivial SU(3) gauge fields in terms of three-ideal distributions, for example, will generally require that the space-time domain be one of those manifolds for which a three-ideal connexion structure like (2.5.3) is indicated. A model of this kind is currently being investigated by LYNCH [ 11]. From its earli·est developments, classical spinor field theory and the theory of elementary particles have been based on the primacy of two and four component spinor fields. More complex theories of particle properties such as the various theories of unitary symmetry have necessitated ad hoe increases in the dimensionality of spin-space from four to eight, twelve etc. While these models have obviously had a great deal of success from the point of view of the construction of a phenominological account of particle properties, they leave orthodox classical geometrical models for two and four component fields in a somewhat uncertain position. From the point of view of orthodox spinor geometry it is hard to see what kind of (geometrical) argument could justify or lead one to attaching two, three or four four dimensional spin spaces to each point of space-time, rather than just one. l'lb In the present model, however, the spinor objects with which one deals are in the first instance sixteen-dimensional; the problem is therefore not so much of understanding now the dimension of spin-space at a given point might be increased, but is rather one of understanding the geometrical and topological conditions under which our sixteen component fields might be decomposed into direct sums of invariant distributions of lower dimension, and the meaning of these geometrical conditions in physical terms. We complete this section with a remark concerning the place of the.electromagnetic field in our model. As already noted, the matrix associated with spinor coefficients of connexion is fundamentally traceless (a field of spinor frames can always be found for which trace i (K) = O; the term a. I which appears in (3.1.3), for example, ~ 2 = arises from the (1,1) entry of the ideal connexion matrix~~' and is not to be considered a part of K~). Written either on the left or in its original form on the right, this term describes how the relative phase and magnitude of a spinor distribution change from point to point of)f; the field-theoretic function of its pure imaginary part is therefore the same as that of a vector potential associated with a classical electromagnetic field, and without further ado we will refer to the tensor field having local components f,µ = Re (a - a ,) simply as the A A,µ µ,A electromagnetic field. In expressions for spinor components of Clifford differentials i of arbitrary T.C. numbers the term 2 a~:_ must obviously be written in its original form operating from the right. Notice however that ~~A is itself necessarily trace-less (cf. (2.5 )); as a result the pure imaginary parts of remaining diagonal elements of :~A must necessarily include at least one non-zero quantity having a sign opposite to that of 11.7 (ifA~). The net effect of this is that different ideal components of a given T.c. number will generally "see" different electromagnetic fields; some of them may even "see" no electromagnetic field at all, in spite of the fact that a~~ O. In physical terms this situation is described by saying that the various minimal ideal components of a given T.C. number will generally be coupled to a specified (physicial) electromagnetic field in different ways and are therefore to be considered as being descriptive of particles having different charges. Such a state of affairs is clearly in general accordance with our earlier depiction of isotopic spin in terms of two-ideal distributions of T.C. numbers; the two components of a two-ideal distribution of T.C. numbers can represent different charge states simply because one of them "sees" an electromagnetic field and the other does not. A model for the internal gravitational structure of a classical charged particle which incorporates this representation of the electro magnetic field is currently under investigation by LYNCH [19]. Chapter IV Summary and Conclusions It may be of assistance to the reader if we preface some remarks concerning the further developement of our model with a brief summary of its main features as developed in the preceeding chapters. We commenced §1.2 with a construction of the tangent * Clifford algebra ep to ..)1 at Pas a quotient space of the space of covariant tensors at P under a certain equivalence relation; this construction leads to an explicit relationship between covariant vectors and tensors on the one hand and our so-called tangent Clifford numbers on the other. In particular it leads to a one-to-one reciprocal (bi-jective) correspondence between exterior differential forms and fields of T.C. numbers. As the first of a number of applications of this result that were made, we were able to define the local inner product of two real T.C. numbers as having the same value as the inner product of underlying exterior forms. Next we constructed basis elements for minimal left and right ideals ofCP* and introduced the concepts of spinor frame, non-holonomic frame and adjoint spinor frame. Whenever a spinor (, non-holonomic, adjoint spinor) frame for C,p* has been chosen, any T.C. number at P can be given a spinor (, non-holonomic, adjoint spinor) expansion 9 thus giving rise to a set of spinor (, non-holonomic, adjoint spinor) components of the T.C. number. There are thus four possible modes of description of T.C. numbers: the holonomic, spinor, non-holonomic and adjoint spinor modes. Implicit throughout the whole of this essay is an assumption that these different modes provide fully equivalent descriptions of the properties of T.C. numbers. l'l'f In §1.4 we showed how the various T.C. number coefficients behave under spinor frame transformations. We also obtained an expression for the local inner product of two T.C. numbers in terms of their spinor components and were able to demonstrate that with certain simplifying assumptions spinor expressions for inner products can be written in a form very similar to standard expressions for inner products of classical (flat space-time) four-spinor fields. We were also able to demonstrate that the relation between our holonomic and non-holonomic basis elements is fully describable in terms of fields of 4 x 4 matrices which are identifiable as curved space-time variants of the classical Dirac matrices. In the second chapter we discussed differential properties of fields of T.C. numbers; we started by summarizing the standard treatment of differential properties of exterior forms according to classical differential geometry, and then showed that these standard methods for exterior forms, together with the bijective correspondence between forms and T.C. numbers about which we have already remarked, were sufficient to allow us to define Clifford number counterparts of tensorial notions like 'absolute derivative' and 'covariant differential'. Next we showed how differential properties of fields of T.C. numbers are described when non-holonomic or spinor basis elements are used in each tangent Clifford algebra in the place of holonmmic frame elements. The consistency of these descri~tions with the earlier one is expressed by a number of consistency conditions which are given in §2.3 and §2.4. One of these conditions is already familiar to us as the so-called 'fundamental equation' of curved space-time spinor theory •viz: + + = 0 In §1.5 we examined the different relationships that are possible between minimal ideals of T.C. numbers at different points of space-time. We concluded that a number of invariant Clifford algebra sub-structures are possible; our invariant distributions of a single minimal left ideal, of two minimal left ideals and of three minimal left ideals. Necessary conditions for the existence of these different distributions were stated in terms of a common reduced block structure of the matrices 1 g Ra nJJ" (1 ~ ~ < e .::. 4). 2 aµ v~e= ' We were also able to give a sufficient condition for the existence of invariant distributions of a single minimal left ideal in terms of global spinor structures on j//, • With regard to further work on this model, the most pressing mathematical problems would seem to lie in the general area of further clarifying the issues raised in §2.5. For example, we are not as yet able to provide conditions for the existence of our various invariant distributions that are both necessary and sufficient. As a first step towards obtaining these conditions it would obviously be of great interest to have a list of concrete examples of manifolds on which the different invariant ideal distributions arise. Clarification is also needed of some of the basic questions relating to an interpretation of the constructions of our model in terms of orthodox field theory. For example, in §3.l we found that the column matrix functions associated with spinor components of one ideal distributions of T.C. numbersshare a number of important properties with the four-component fields of classical spinor analysis and may therefore be legitimately interpreted as geometrical concomitants of classical four-spinor fields. Since any covariant tensor field (of any rank) gives rise to a field of T.C. numbers,it follows that when space-time admits an invariant distribution of minimal left ideals every covariant tensor field has associated with it a spinor distribution ( i.e. a classical four-spinor field). The question then arises as to how these extraordinary spinor fields associated with covariant tensor fields are to be interpreted in terms of orthodox classical spinor field theory, and in particular how this kind of relationship between tensor and spinor quantities is to be reconciled with the orthodox theory of interacting classical fields. Although some progress has been made towards resolving these issues we are not at the moment able to provide results that could be considered conclusive. ('32. Appendix. Matrix Representations of the ~ab etc. Matrix representations of the coefficients yabmn that appear in (1.3.21), (1.3.26) are readily calculated using (1.3.6), (1.3.13) and (+) (-) ( 1. 3.19) • For example, from (1.3.6) G23 = P 1 + P1 = x 34+x43+x21+x12 i.e. G = + l + + ) X which gives 23 (o m1 no m 2 n m no n m2 o o 3o 4 o 4o 3 mn . 1 Similarly .::.2 3 = 1 . . 1 . . 1 . (i) G = G (+) + G (-) implies om m m -i . • • -1 .-i 1 = ~02 = . i . . 1 . i -1 • . • • -1. • . . -1 • = i .-1 i . -1 • . -i .-1 while (ii) G = i(G (-) G (+)) gives mS' m m . .-1 i .-1 .-i ~15 = .-1 . i . . -1 . -i . . -1 i i -1 ~45 = -1 -i • -1 . -i In addition (iii) G = e: (P (+)+P (-)), (x,y,z = 1,2,3) gives xy xyz z z = p (+)+P (-) 1 1 thus 1 i r -i -1 31 = 1 . -i • -1 i 1 1 while 1 1 (-) ( +) (iv) = p - p (x = 1,2,3) yields X X 1 i -i r 14 = 1 ~24 = -1 i -1 . -i 1 -1 . -1 1 . ( (+) + p (-) For the last of the Gab, G05 = -1 P 4 4 implies 1 = 1 . -1 • -1 1 0 = -i (P (-) - P (+)) yields II I" = 1 while w 4 4 1 1 which checks. For completeness we will also list matrix representations of the P (n) G (n) with respect to the X For example, from (i), (ii) m ' m ~ Gm(+) = i (G 0 m + iGm5). Hence if we write G (n) = m P Cn> m = then the following representations are readily verified IT ( +) = IT (+) = =l 2 1 . -i 1 i IT (+) IT (+) _3 = 4 = 1 . -i . -1 . -i 1 i (-) = = ::i 1 -i 1 i II (-) -1 i = =4 = -i -1 r <+> r <+> = i = 1 =• =2 -i -1 r <+> r <+> = i = -1 3 4 r (-) r <-> = --'l 2 = i 1 i -1 r <-> = r <-> = = 3 =4 i -1 -i -1 Using (1.3.10) and (1.3.20) it is readily verified that the matrices rr (n) thus defined satisfy _m II + i e II = 0 ( 1 < m, n < 4; n = +, -) • fJl Bibliography [1] ATIYAH, M. F., R. BOTT and A. SHAPIRO. "Clifford M:>.dules." Topology Vol. 3. Suppl. 1 pp. 3-38. [2] BRILL, D.R., and J.A. WHEELER. "Interaction of Neutrinos and Gravitational Fields." Rev. Mod. Phys. Vol 29, 3, 1957, pp. 465-479. [3] BUCHDAHL, H.A. "On the calculus of four-spinors." Proc.Roy. Soc.A, 303, 1968, PP 355-379. [4] CHEVALLEY, c.c. The A.lgebraic Theory of Spinors. New York. Columbia University Press. 1948. [5] EDDINGTON, A.S. Fundamental Theory. 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