Not long after Dirac discovered that to maintain the Schrodinger form ofquantum mechanics, in a way that would be consistent with the symmetryrequirements of special relativity theory, it is necessary to extend from thecomplex scalar wave function formulation to a spinor formulation, Einstein andMayer [4J asked the following question :"Is it possible that the appearance of the'spinor variable' in Dirac's electron equation was in fact necessitated by theimposition of relativistic covariance, and does not depend on the quantummechanical nature of the description per se?' To answer this question, theseauthors investigated the minimum-dimensional representations of the symmetrygroup of special relativity theory - the Poincare group.
It was found in the investigation of Einstein and Mayer that so long as oneremained with the continuous group, though removing the discrete reflectiontransformations, then the real four-dimensional representations {(X}, that relateto the transformations of a four-vector, decompose into the direct sum of twocomplex and Hermitian representations. The latter obey the algebra of quaternions whose basis functions, in turn, are the two-component spinor variables.The implication of their analysis was then that the most primitive expression of arelativistically covariant theory must be in terms of spinor field variables. Thisfollows because the theory of relativity compares the laws of nature in referenceframes that are distinguished from each other in terms of their relative motion - acontinuous set of transformations. The discrete symmetry elements corresponding to the reflections in space and time then play no role in the theory of relativity.Of course, one may always postulate the existence of scalar, vector and tensorcovariant field equations. However, Einstein and Mayer discovered that suchequations have solutions that can always be constructed from a composite ofspinor variables, but the spinor variable cannot be constructed from a compositeof any other covariant field. It is in this sense that the spinor is the most primitivetype of variable in relativity theory.
The 'spinor' was given its name when it was discovered (by G. Uhlenbeck andS. Goudsmit) that the extra degrees offreedom associated with this variable may
Spinor-Quaternion Analysis in Relativity Theory 41
be identified with the part of the angular momentum of an electron that does notdepend on its distance from a center of rotation. The need of such an 'intrinsic'spin angular momentum was required to explain the coupling of an atomicelectron to an externally applied magnetic field (the anomalous Zeeman effect).Here, one observes that to explain the data on the spectrum of atoms with oddnumbers of electrons (say hydrogen) it is necessary to postulate the existence of anintrinsic magnetic moment of the electron, in addition to its magnetic momentdue to its orbital motion about the nucleus . But the model seemed to presentlogical difficulties in the nonrelativistic view of the atom. For if the electron is apoint particle, then as far as its intrinsic angular momentum is concerned, what isit that is rotating about what?
Dirac's discovery answered this question. For according to his result , theelectron 'spin' relates to nothing more than the fact that the necessary variables todescribe the electron in a relativistically covariant way are multicomponententities - the two-component complex variable (minimally), called 'spinor'rather than the single component complex 'scalar' variable of the Schrodingertheory. Thus, one needn't think of the electron as a hard little object that isspinning on its own axis, as the earth spins on its axis. It is rather that the 'spinorvariable' has a particular algebra and calculus that, in the final analysis, predictthe existence of an extra contribution to the angular momentum in the field, thatis not dependent on its distance from some center of rotation.
When these fundamental field variables that relate to matter are examined interms of their algebra, it is found that they are the basis functions of differentialoperators that obey the algebra of quaternions. It is interesting to note that in aletter to P. G. Tait, in 1871, Maxwell made the following comment about the useof quaternions in physics [5] :
" . . . the virtue of the 4nions lies not so much as yet in solving hard quest ions as in enabling us tosee the mean ing of the question and its soluti on . . . "
Instead of reviewing other authors' analyses of spinor and quaternionvariables, I will derive them in this chapter by starting with W. R. Hamilton'sstudy, which led to the discovery of the quaternion algebra. It will then be seenhow the relationship between the quaternion, as a second-rank spinor of a specialtype, and the behavior of this type of variable under the transformations of thePoincare group, leads to the geometrical properties of the spinor variable itself.The analysis will then be extended globally, to describe the mapping of spinor andquanternion variables in a Riemannian space-time. The calculus of these fields,leading to the 'spin-affine connection', will then be derived . The development ofthese ideas.and their analysis must, however, begin with a review of the algebra ofcomplex number fields.
3.2. The Algebra of Complex Numbers
Consider the correspondence between the number field of two-dimensionalvectors and the complex number field. The correspondence follows from the
42 Chapter 3
invariants of the respective number systems. An invariant in the 2-dimensionalEuclidean space is the Pythagorean number that is the sum of the (squared)coordinate separations
(LlX)2 + (Lly)2 = inv, (3.1 )
where zlx = (x2 - xl)and Lly= (Y2 - Yl)' According to our assumedgeometry, theinvariant in (3.1)maintains its value under any continuous transformation in thetwo-dimensional coordinate system of pairs (x, y). Thus one might be led to theinvariant from the scalar product of two-dimensional vectors
(3.2)
(3.3)
(3.4)
(3.5)
To satisfy this invariance, the basis elements, ex and ey , are chosen so as to satisfythe orthogonality rules :
A A A A 1ex·ex=ey.ey=A A A A 0ex .ey=ey ·ex =
Further, since the basis elements eX'cy are independent, the equality of anytwo two-dimensional vectors
r l =cxx l +CyYl =r2 =cxx2+CyY2
corresponds to the equality of the separate components
Xl =x2 and Yl =Y2 '
While one might have a 'feeling' for the vector concept because of our 'human'reaction to perceptions of directed magnitudes, the field of numbers r(x, y) can bebrought into one-to-one correspondence with a different sort of two-dimensionalnumber field, z(x, Y), that satisfies a different type of algebra. While the 'complexnumbers', z, are not quite as 'perceivable', they do relate to the same set ofinvariants, and eventually to the same set of described observables when thismathematical language is used to represent the physical world. In this case, then,there is no special reason to prefer one mode of expression over the other.
The complex number
Z =X + iy
is defined in terms of the basis elements (1, i) such that the equality of the twocomplex numbers (analogous to the vector case) implies the equality of the 'real'and 'imaginary' components. That is,
Zl = Xl + iYl = Z2 = x2 + iY2
implies that
Xl =X2 and Yl =Y2'
Since the invariant (x 2 + yZ) is now to be constructed from complex numbers, the
Spinor-Quaternion Analysis in Relativity Theory 43
properties of the basis elements (1, i) may be determined from this invariance. Todo this, define the conjugate complex number as
Z = x + iy
so that its product with Z yields the required invariant :
zz = (x + iy)(x + ly) = XZ + y2.
Equating the separate parts of this equation, we have
if= 1 and (i + ~ =O.
Thus it follows that 1= - i so that
ii = iZ = - 1. (3.6)
To sum up, if to each complex number, Z = x + iy, there corresponds a complexconjugate Z= x - iy, then their product
zz=xz + y2
is defined throughout the two-dimensional plane. In this case, the set of allcomplex numbers, [z], is in one-to-one correspondence with the set of all twodimensional vectors {r}. In the former case, the invariant quantity (X Z + yZ
) iscalled the 'magnitude' of the vector r, in the other case it is called the 'modulus' ofthe complex number z.
3.3. Group Properties of the Set of Complex Numbers
The set of complex numbers {z- zo} obeys the rules of a group with respect tomultiplication, where Zo = 0 + iO is the null complex number.
(a) Combination rule. It follows from the fact that li = i1 = (- l)t that theproduct of any two complex numbers, Zl and Zz, is also a complex number, i.e.
(b) Associative rule. Using the definition of the basis elements (1, i), of a complexnumber, it is readily verified that
(ZI ZZ)Z3 = ZI(ZZZ3)'
where ZI' zz, Z3 are any three complex numbers.
(c) The identity. Since the basis element 1 satisfies the multiplication rule
lz=zl=z
(3.8)
44 Chapter 3
it follows that the complex number
Ze = 1 + iO
is the identity element of the multiplication group. Since this number is containedin the set of numbers of the two-dimensional plane, the group requirement thatthe set {z} must contain the identity is satisfied.
(d) The inverse. The inverse Z-l is defined such that ZZ-l = ze= 1. CallingZ -l = a + ib, we must establish that a and b exist in order to say that the set ofcomplex numbers is such that for each z there is an inverse element, z- 1.
According to the definition of the latter,
ZZ- 1=(x + iy)(a + ib) = 1.
Equating the real and imaginary parts of this equation, we find that
a = xl(x2+ y2) = xlzz
b = - YI(x2+ y2) = - ylzz.
Thus the inverse of a complex number z has the form
Z-1 = (x - iy)/zz = zlzz.
(3.9)
(3.10)
To every pair of numbers (x, y) in the infinite 2-dimensional plane, except forthe pair (0,0), there must then exist an inverse complex number. Since a group ofelements must contain an inverse element for each element contained in theset, the set of complex numbers, {z - zo}, forms a multiplication group, whereZo =0+ iO.
lt is readily verified that the set of complex numbers, including zo' forms agroup with respect to addition. In this case, the identity element is Ze = Zo and theinverse elements are z - 1 = - Z.
Finally, it follows that the complex number field is commutative with respect toboth multiplication and addition,
and distributive
zm(zn + zp)= ZmZn + zmzp'
The set {z- zo}, that obeys all of the properties described above, is called an'algebraic field'.
3.4. The Algebra of Quaternions
In view of the usefulness of the complex number field in two-dimensionalproblems in physics, it was natural to attempt to extend this algebra to the threedimensional case - the real world! This was done by W. R. Hamilton in the 19th
Spinor-Quaternion Analysis in Relativity Theory 45
century. His initial aim was then to find the generalization of the basis (1, i) of thecomplex number, in terms of the new basis, (1, i, j) so as to accommodate thenumber field of triples (x, y, z).
Following the same procedure as in the complex number field, the properties ofthe generalized basis elements would then be expected to follow from theinvariance of the metric, x 2 + y2 + Z2 of a 3-dimensional Euclidean space.
A remarkable conclusion followed from Hamilton's investigation. He foundthat the generalization of the complex number field could only be accomplishedin this way by starting with a 4-dimensional space. The three-dimensional casecould then be accommodated by taking the special case in which one of the fourvariable coordinates is held constant. [In retrospect this seems fascinatingbecause it implies that at that stage (in the early 1800's) the existence of the seeds'of a relativistically covariant formalism. The key to this intellectual jump wouldhave been Hamilton's guess that the fourth variable parameter is the fourthcoordinate that he was indeed using in his analyses in mechanics - the timecoordinate.] Hamilton's generalization then led to a very great discovery in thehistory of mathematics - quaternion analysis - though it wasn't taken tooseriously until the present time. The special case, whereby one of the coordinatesis held fixed, yielded vector analysis, which was taken seriously and used to greatadvantage in the expression of the laws of nature in three-dimensional space.Unfortunately, Hamilton did not make the small jump that would have led himto the concept of relativistic covariance. If he would have, he would havediscovered, for example, that Laplace's equation for the Newtonian potential infree space must be generalized to the wave equation in free space, in order topreserve 4-space covariance, and he would have then seen that the gravitationalforce propagates at a finite speed rather than 'acting at a distance'. Thus, ifrelativity theory would have already been developed to some extent by the timeEinstein appeared on the scene, one can only speculate how far we would haveprogressed to this day!
Hamilton discovered that the generalization from the basis elements (1, i) of acomplex number field to the basis (1, i, j, k) of a quaternion number field impliedthe very surprising result: that quaternions do not commute under multiplication,This result will be derived below, where it will be found that the basis elements(i,j, k) do not commute with each other, while they do commute with 1 - andthat these are in one-to-one correspondence with the three Pauli matrices and theunit two-dimensional matrix. (Cayley discovered matrix algebra several decadesafter the discovery ofquaternion algebra.) Thus, it will be shown that, in effect, theproper generalization of the basis element i of a complex number are the threePauli matrices (which, of course, were not identified by Pauli until about acentury later!)
The generalization from the 2-dimensional field of complex numbers to thefour-dimensional field of quaternions
(3.11)
46 Chapter 3
and the corresponding generalization of the conjugate element
z= x - iy-+Q = Ix4 -ixl - jx 2 - kx 3 (3.12)
leads to the generalization of the invariants of the respective spaces :
This generalization will now be demonstrated explicitly.Analogous to the way in which i was determined in the complex number
system, the properties of (I, i, j, k) of a quaternion are found by first substituting(3.11) and the conjugate '
Q= Ix4 + Ix 1 + jx2 + kx3
as the factors that make
QQ = (X4)2 + (X I)2 + (X2)2 + (X 3)2
invariant to rotations in the four-dimensional space. As in the two-dimensionalcase, this gives the result that if= 1, (i +I) = 0, etc. so that
i2=j2=k2= -I, 12=1
as before, but with the additional requirement that
ij + ji = ik + ki = jk + kj = 0.
(3.14)
(3.15)
(3.16)
Thus, with respect to multiplication, we see that the basis elements (i, j, k)anticommute, i.e,ij = - ji, etc., if the invariant (3.13) is indeed to be represented bythe product ofa quaternion and its conjugate. It also follows from the same analysisthat the basis elements (i,j, k) must all simultaneously commute with the basiselement I. Since real or complex numbers, in retrospect as one-dimensionalmatrices, commute under multiplication, it follows that the simplest structure forthe basis elements (I, i, j, k) must be 2-dimensional matrices.
To define the quaternion number field with an algebra, we must impose thegroup property of the combination rule - the product of any two quaternions isanother quaternion. It follows from this rule that the proper generalization of thecomplex numbers (of pairs) is a number system of quadruples (rather thantriplets) as a minimum extension . This can be seen as follows. Using (3.14), theproduct of any two quaternions has the form:
QQ' = (Lx" + ix! + jx 2+ kx 3)(lx41+ ix l l + jx 21+ kx")
= I(X4X41_ XIXII _ X 2X2 1 _ X 3X3 1) +
+ i(x4xl l + XlX4 1) + j(X4X21+ X
2X4 1) + k(X4x 3 1 + X
3X4 1) ++ ij(X IX21_ X2XII)+ik(XIX31_ X3XII)+ jk(X2X31_ X
3X 2 1).
In order that the right-hand side of this equation shall have the structure of aquaternion, the dyadic terms above, ij, ik,jk must be reducible to quaternion basiselements, I , i, j or k.
Spinor-Quaternion Analysis in Relativity Theory
Consider the dyad ij. There are three possibilities for its reduction
47
(a) ij = 1
If both sides of this equation should be multiplied by the basis element i, thenaccording to (3.14) it would follow th at - j = i.This is clearly false since (i, j , k) areindependent basis elements.
(b) ij =i (or = j).
Again using (3.14), the multiplication of both sides of this equation by i gives- j = - 1 (or = ij), which are both false because of the independence of the basiselements. Thus, we are left with the third and only possibility
(c) ij = k
Since the product of any two of the basis elements (i,j , k) must relate to the third(and different) basis element, and not to the commuting basis element, 1, we seeat this stage why the generalization of the complex number entails four, ratherthan three parameters. Three of these parameters must be associated with thenoncommuting elements (i, j, k) and the fourth with the commuting element 1.
To sum up, the basis elements of a quaternion must sat isfy the followingproperties:
i2 = j2 = k2 = _ 1, 12 = 1
ij = - ji = k, jk = - kj = i, ki = - ik = j
Ii = iI, I] =[I , lk = kl.
(3.17a)
(3.17b)
(3.17c)
With these rules for its basis elements, the quaternion set of numbers obe ys therules of an algebraic group.
Substituting (3.17) into (3.16), it follows that the product of an y twoquaternions, Q and Q', are the third quaternion Q" that has the form:
QQ' = Q" = (X4X4
' - r .r') + i (X4 X I ' + Xl x'") + j (X4
X2 ' + X2
X4
' ) +
k
+ k(X4 x 3 1 + x3 x 4' ) + Xl x 2 x 3
X li x 2 ' x 3 '
(3.18)
where r = el Xl +e2x2 +e3x3 is the three-dimensional vector and the last term onthe right side of Equation (3.18) is the determinant, equivalent to the crossproduct of vectors, r x r'. Thus we see that the product of quaternionsencompasses the scalar and cross products of vectors, of vector analysis in a threedimensional space . Indeed, the subject of vector analysis was an outgrowth ofquaternion analysis, even though the former was much more widely used inphysical applications after Hamilton's discovery of the quaternion algebra. [Thiswas primarily because the scientists did not yet ha ve a physical interpretation ofx 4
.]
48 Chapter 3
With the properties (3.17) of the quaternion basis elements, they may be chosento have the form in terms of two-dimensional matrices as follows:
1 = a0 = ( ~ ~). j = - iu2 = - {~ - ~ ).
i=-iUl=-i(~~). k=-iU3=-{~ _~} (3.19)
The two-dimensional form of the quaternion basis elements, (u l ' U2 ' o3) are(one of the representations of) the Pauli matrices. By themselves (without thefactor i) they satisfy the algebra
(3.19')
where j =F k =F I = 1,2,3. Aside from the common factor i, the Pauli matrices are inone-to-one correspondence with Hamilton's basis elements (i,j, k). Thus, as it wasasserted earlier, the proper generalization of the basis elements (1, i) of a complexnumber, in the extension from a two-dimensional Euclidean space to a fourdimensional Euclidean space, is
With this notation, the quaternion Q may be expressed in the form :
Q= - iu"x" = uox4 - iu , x ' - iU2X2 - iU3X3
=(uox 4- ia -r), (3.20)
(3.22)
where we have used the notation - iU4 = Uo .
With the definition (3.19) of the matrices a", Q may also be expressed in theform
(iX4 + x3 Xl - ix2\
Q= - i Xl + ix2 ix" _ x 3)" (3.21)
The simplest representation of the quaternion is then in terms of such twodimensional matrices. The one-to-one correspondence between the four-vector,as a set of four parameters (X4,X l, X 2,X3), and Q is clearly exhibited here. Thecorresponding metric of the four-dimensional space is obtained from the productof Q and its conjugate
Q = uox4 + ia- r
to give
QQ = QQ = [(X4)2+ r2Juo = - (det Q)uo = inv .
The multiplication of an invariant by any constant is, of course, also invariant.Thus, the multiplication of Q by i, which is the matrix in the right-hand side ofEquation (3.21) without the factor i, has basis elements (iuo;Uk) with k = 1,2,3. Itsdeterminant is the invariant, - [(X4)2 + r2].
Spinor-Quaternion Analysis in Relativity Theory 49
3.5. Group Properties of the Set of Quaternions
With the definition of the equality of two quaternions
(Q = Q') ¢>(x4 = x 4 ' ), (X I = x'"), (x 2 = X2l),(X3 = x 3 ,)
it will now be shown that the set {Q - Qo} forms an algebraic group with respectto both addition and multiplication - thus defining an 'algebraic field',just as theset of real numbers and complex numbers form algebraic fields.
(a) The Combination Rule. This rule was already shown to follow from the formalproperties of the basis elements of a quaternion.
(b) The Associative Rule. It follows from the expression (3.18)for the product ofany two quaternions that
Q(Q'Q") = (QQ')Q".
(c) The Identity. Since the matrix 0"0 is the only one that will commute with allquaternions, simultaneously, i.e.
O"oQ = QO"o = Q
it follows that the unit matrix 0"0 must be the identity with respect to themultiplication group of quaternions. Since the set of quaternions clearly containsthe element 0"0 = Q(x4 = 1, Xl = x2 = x3 = 0), the group requirement for thecontainment of the identity is obeyed.
(d) The Inverse. Analogous to the features of the complex number field, theinverse quaternion in regard to the multiplication group is related to theconjugated quaternion as follows:
Q-l = Q/QQ . (3.23)
As in the case of the complex number field, we see that there is no inverse elementfor the null quaternion, Qo= Q(O, 0,0, 0), since in this case the right-hand side of(3.23) would not exist. Thus the set of quaternions that excludes the nullquaternion, {Q - Qo}, forms an algebraic group.
In regard to the addition group, the inverse of a quaternion Q is its negative,and the identity is the null quaternion Qo' Thus, according to its definition, the setof quaternions {Q- Qo} is an algebraic field. The only essential differencebetween the field of quaternions and the field of complex numbers is that theformer is a noncommutative set under multiplication while the latter iscommutative.
3.6. The Spinor Field and Special Relativity
The invariance of the norm of a quaternion,
QQ = (X4)2+ (XI)2 + (X2)2 + (X3)2
50 Chapter 3
is independent of the initial choice of the signs of each of the squares on the righthand side. Hence, one could, in principle, define the vector components in such away that any of them are purely imaginary numbers, without altering theinvariance properties of QQ. The implication would be that the components ofthe general type of such a quaternion would be complex. But the salient point isthat whether or not Qentails complex components, Q is so constructed so as tomake QQ a sealar with respect to a predetermined invariance group. Differentmetric spaces may then be defined in terms of the norm QQ by choosing differentdistributions of plus and minus signs in the combination of four squares above.
For example, the 4-dimensional orthogonal space that is characterized by thecompact group 04' would be represented by an invariant with all signs positive.The particular metric that we are concerned with in special relativity theory,however, entails the noncompact invariance group in which one (or three) signabove is negative and the remainder are positive (or one is positive and the threeremaining are negative). This corresponds to making the choice x" = - ixo == - iet, and the remaining parameters, x", are each purely real numbers. In thiscase, the invariant metric takes the Lorentzian form
(XO)2 - r2 = inv ¢> - QQ. (3.24)
The matrix structure of Q(Equation (3.22)) implies that it is made up of thedirect product of two-component entities:
(e ) (ae be)Q- (a b) ® d == ad bd . (3.25)
We now wish to determine the nature of the two-component entities into whichthe quaternions will factorize in this manner - these are called spinor variables.
Consider first the case in which the determinant (and norm) of Q is equal tozero. This corresponds to the points on the 'light cone'
(3.24')
This equation may be re-expressed in terms of the matrix elements of Qas follows:
Xl - ix 2 Xl
XO - x 3 X2'(3.26)
where we have labelled the ratios of quaternion components as XI /X2' Thecomplex conjugate of Equation (3.26) is:
Xl + ix 2 X!XO - x 3 xi'
(3.27)
Solving Equations (3.26) and (3.27), we have:
iQll = XO + x 3 = KXIX!
iQ22 = XO - x 3 = KX2xi
Spinor-Quaternion Analysis in Relativity Theory 51
iQ12 = Xl - ix2 = KX1X!
iQ21 = Xl + ix 2 = KX2XT,
where K is an arbitrary constant. Thus, the quaternion Q has matrix elementsthat may be expressed in the form
QafJ = - iK XaX;, (ex, P= 1,2). (3.28)
Clearly, Equation (3.28) corresponds to Equation (3.26) in terms of the nullinvariant of the light cone:
- det Q = - K 2(Xl xTX2X! - XlX!X2XT) = (XO)2 - r2 = O. (3.29)
Consider now the intervals in space-time that are off of the light cone. In thiscase, define the two-component entity, (, as one with the same algebraicproperties as X in the sense that it combines with X in the quaternion structure(3.28) as follows :
( 0)2 _ 2= ' = -d tQ=d t(X1XT+(1(! X1X!+(1(!)X r mv e e * * * r r* 'X2Xl + (2(1 X2X2 + \'2\'2
Where, in this case, we have normalized the arbitrary constant K to be equal to i.This normalization corresponds to the points inside of the light cone, i.e,intervalsthat are a timelike distance apart. If the normalization had been chosen withK = 1, this would correspond to the points outside of the light cone - i.e. thespacelike intervals.
Writing out the determinant of the right-hand side ofthe matrix above, we findthat we have the following correspondence between the components of a fourvector and those of spinors :
(xO)2 - r2 = inv = IX1(2 - X2(112 > O. (3.30)
Since the square root of an invariant is still invariant, it follows that the twocomponent entities, that we now call 'spinors', combine in the following way toform an invariant
Ixe(1 == IX1(2 - X2(11 = inv,
where
(3.31)
is the Levi-Civita symbol.In summary, it has been shown that there is a one-to-one correspondence
between a four -vector of real components, IxO Xlx2 x3>and a Hermitian, twodimensional matrix representation of the quaternion Q(X, X*). The vector isexpressed as a column of four real numbers, with a metric <xix> = (XO)2 - r2,
while the quaternion depends on these same four real numbers, but in the form of
52 Chapter 3
(3.32)
an Hermitian matrix, with determinant equal to the same invariant, (XO)2 - r2•
The four-vector transforms as the basis functions of the four-dimensional (real)representations of the Poincare group, while the two-dimensional Hermitian(complex) representations of the same group - the set {Q}, which are 'secondrank spinors' - obey the algebra of quaternions.
Since the quaternion transforms as the direct product of a spinor and itsconjugate, X® X*, it follows that the basis functions of the quaternion representations of the Poincare group must be the complex, two-component spinorfield variables X. The next step, then, is to derive the transformation properties ofthe spinor field variables , with respect to the Poincare group of special relativitytheory.
3.7. Transformation Properties of a Spinor Variable
Suppose that there exists a spinor transformation T, defined in such a way thatwhen the space-time transformations of the Poincare group are applied, thefollowing spinor transformations are induced :
x'"-+ x'"' = O(~xv =>X(x)-+ X'(x' ) = TX(x).
The first thing that we see is that the set of transformations {T} is unimodular.This is derived as follows : (from Equation (3.31))
IXl (2 - x2( 11= inv = IX~(2 - X2(~ I
= I(Ttl Xl + T12X2) (T21(l + T22(2)
- (T21Xl + T22X2) (Ttl (1 + T12(2)1
= [det Tllxl(2 - X2(t! .
Thus it follows that [det TI = 1 - i.e., the transformations {T} are unimodular,corresponding to the fact that the group {O(} of the four-dimensional representation of the Poincare group is unimodular.
It is also readily verified that if we assume that {T} is assumed at the outset tobe unimodular, then the corresponding metric in spinor space (Equation (3.31))isverified. Thus, the unimodularity of the spinor transformations {T} is a necessaryand sufficient condition to ensure the invariance of the metric (3.31).
Using the form (3.20)for the quaternion structure, and its definition in terms ofa second-rank spinor (3.28), the following relationship is obtained (choosingK = i for timelike intervals)
XlIX; = (a,"x'")IIP' (3.33)
Since x'"-+ xr' = O(~xv =>X(x)-+ TX(x), X*(x) -+ T*X*(x), it follows that the relation(3.33) looks as follows in any other inertial frame of reference :
Spinor-Quaternion Analysis in Relativity Theory 53
Using (3.33) once more (dropping the spinor index notation), this transformedequation takes the following form:
xP(Tap Tt) = xVlXealt'
Finally, equating the coefficients of the independent parameters x P in thepreceding equation, the following relationship appears between the spinortransformations {T} and the transformations {IX} of the four-vectors in spacetime :
(3.35)
It will be found below that the correspondence between {T} and {IX} is two-toone, rather than one-to-one ; that is, for every element IX of the vectorrepresentation of the Poincare group, there are two representations, T+ and T-,of the spinor representation of the same group. We will now solve Equation (3.35)for the explicit form of these representations.
3.8. The Explicit Form of the Spinor Representations of the Poincare Group
Since the four-dimensional representations of the Poincare group correspond tothe set of spinor transformations, the latter must also form an algebraic group.The rule of combination of the group must then apply - implying that any finitetransformation may be generated by the successive application of an infinitelylarge number of infinitesimal transformations of the same group. Thus, to derivethe finite transformation elements of the group we need only calculate theinfinitesimal transformations.
An infinitesimal coordinate transformation, to order Be is:
x lt' = lXex · = (be + Be)Xv• (3.36)
To this same order, the spinor transformation may be expressed as :
T(Be) = ± (ao + BeA(ltv)), (3.37)
where A(ltv) is a two-dimensional matrix in the expansion of T, which multipliesthe infinitesimal parameter Be, characterizing the transformation. (Note thatthere is no summation over 'J1.V' implied here.) The plus and minus sign appearbecause of the quadratic dependence on Tin (3.35). Substituting Equation (3.36)into the invariant relationship, (XIt')2 = (XIt)2, it follows that Be = - B;. Finally,with (3.36) and (3.37) in (3.35), it follows that to order B;, the latter equation is:
(Bp)* apA:~/l) + (Bp)A(~/l)aP = B~ . (3.38)
Since ap are Hermitian matrices (i.e. self-adjoint) , the left-hand side of thisequation is the sum of a matrix and its Hermitian conjugate, thus it is Hermitian.Therefore, the right-hand side ofthis equation must also be Hermitian. Equatingthe right-hand side of this equation to its own Hermitian conjugate it then follows
54 Chapter 3
that the parameters e~ are real numbers. The factor e~ can then be expressed as acommon multiplier on the left-hand side of (3.38), yielding the relation:
(3.39)
Note once again that 'rx{3' are not summed over in this equation.If we should now equate the coefficients of the independent parameters eli on
both sides of Equation (3.39), we obtain the following equation
(3.40)
In view of the anticommutation properties of the Pauli matrices, it follows thatthe (only) solution of Equation (3.40) is:
A.(IlP) = to"Il0"o :
Thus, the infinitesimal transformations that are saught are:
T(e~)= ±(O"o+te~O"IlO"v)' (3.41)
representing an infinitesimal Poincare group element.The finite transformations of the group may now be constructed from a
superposition of such infinitesimal elements, in accordance with the rule ofcombination :
T(e~) = T(e~(l»T(e~(2»'" T(e~(n», (3.42)
where e~ is the sum of infinitesimal transformation parameters, e~(I) + e~(2) + ...+ e~(n), where n is arbitrarily large. The function that uniquely satisfies theproperty (3.42)is the exponential function. Thus, the infinitesimal transformation(3.41)represents the first two terms of a power series expansion (in e~) for the finitetransformation
T(e~) = exp <to"1l00ve~) . (3.43)
The set of two-dimensional transformations {T(e~) } of the Poincare group thuscorrespond to its four-dimensional representations { rx(e~ )} .
3.9. The Three-Dimensional Rotation Group and Spinor Transformations
To exemplify the preceding result, consider the subgroup of the Poincare groupthat corresponds to the linear orthogonal transformations in three-dimensionalspace (the 'rotation group'°3 ), The spinor transformations (3.43) that correspondto spatial rotations are the subset of elements with indices iu;v) running over1,2,3 only.
Since
(3.44)
it follows from Equation (3.43) that the corresponding spinor transformations
Spinor-Quaternion Analysis in Relativity Theory
are :
55
(3.45)
where the exponential function has been expanded in a power series of itsargument and the commutation properties (3.19') applied.
Consider, for example, a rotation 0 about the x3-axis- i.e, a spatial rotation inthe Xl - Xl plane. This corresponds to the spinor transformation (3.45) asfollows:
T(O) = exp(ti00'3) = (costo + i0'3 sin to)
(exp t O 0)
= 0 exp( _ to) . (3.46)
Since this function has the property that T(O) = - T(O + 2n), the spinortransformations are 'double-valued'; thus this is a double-valued representationof the Poincare group, even when the full set of transformations is includedbesides the spatial rotations.
An important property of the spatial rotations that follows from thehermiticity of the Pauli matrices is the feature that this subgroup of the Poincaregroup is unitary, i.e.
(3.47)
On the other hand, the space-time transformations that connect the temporalmeasure XO with the spatial coordinate measures, xk, are not unitary. (This followsfrom the Hermiticity of the product 0'00'k compared with the antihermiticity of theproduct O'Pk in the rotation group representations.)
To exemplify the latter case of space-time transformations, consider twoinertial frames moving relative to each other with the constant speed v alongthe colinear Xl and Xl' axes. In this case the transformation parameter isO? = vic = - O~ . Thus, the spinor transformation (3.43) takes the form:
T(O?) = exp(tO? 0'1) = cosh to~ + 0'1 sin to?
= (cosh to? sinh to~ ) (3.48)sinh to~ cosh to~ .
In this case, it is clear that while the transformation is unimodular (itsdeterminant is unity) it is not unitary (Tt =1= T- l ) .
3.10. Lack of Reflection Symmetry in the Spinor Formulation
It is readily verified that there is no solution T ofEquation (3.35) that correspondsto reflection transformations in space or time. Consider the inversion of thespatial coordinates,
(XO, x', Xl, x 3 ) -+ (X O, _ x", _ Xl , _ x 3 ).
56 Chapter 3
This corresponds to the four-dimensional transformation matrix (in the vectorrepresentation)
o-1
-1(3.49)
Calling the corresponding spinor transformation the 'parity' f!i'.. Equation (3.35)(with the index J1 = 0) yields the equation
f!i'a0 f!i' t == f!i'f!i't = ao- (3.50)
Thus it follows that (if f!i' exists) it must be unitary, i.e.
f!i' t = f!i'- 1 • (3.51)
With J1 = k = 1,2,3, substitution of (3.49) and (3.51) into Equation (3.35) gives thefollowing relation:
f!i'ak+ akf!i' = O. (3.52)
However, according to the algebra of quaternions discussed previously, there areno two-dimensional matrices that anticommute with all three Pauli matricessimultaneously. Thus, the space-reflection (parity) operator, f!i', does not existwithin the spinor-quaternion formalism.
It is readily verified in the same way that the time-reflection operator .r alsodoes not exist in the spinor transformation group T.
The preceding results are consistent with the fact that the f!i' and ff operatorscorrespond to transformations that ha ve det cx = - 1. As such, they are notincluded in the specific group that underlies the symmetry of relativity theory the group of continuou s transformations in space and time. It is also interesting tonote, however, that another discrete transformation that is the combination of r!Jand ff corresponds to det a = 1, but also does not exist within the covariancegroup. This may be seen as follows: The vector transformation for the combinedspace and time reflections is
(
- 1
" .¥ Y ~ ~
o-1
-1(3.53)
Substituting this into Equation (3.35), we have
(f!i'Y )all(f!i'y)t = (cxgo.r);av •
With J1 = 0, this corresponds to the equation
(f!i'.r)(r!Jy)t = - ao.
(3.54)
Spinor-Quaternion Analysis in Relativity Theory
Thus,
(.?J,r)t= _(f}l,r) -I.
Now with jJ.= k = 1,2,3 the following relations are obtained:
(.?J,r)CTk - CTk(.?J,r) = O.
57
(3.55)
(3.56)
The only two-dimensional matrix that simultaneously commutes with all threePauli matrices, according to the algebra of quaternions, is a multiple of the unitmatrix, that is,
(3.57)
where b is an arbitrary (complex) constant. With this result, Equation (3.55)requires that b* = - l ib, or
b*b = -1.
However, since b*b is a positive-definite number, there is no solution bandEquation (3.57) has no solution. That is, the combined transformation (.?J,r)cannot be contained in the covariance group.
This result demonstrates that while det (X = 1 may be a necessary condition forthe underlying group representations, it is not sufficient. The reason for theexclusion ofthe transformation ess" in fact goes back to the requirement that theirreducible representations of the Poincare group of special relativity theory mustbe generated by the successive application of infinitesimal transformations,starting from the identity. This was the procedure that was used in deriving thespinor transformations (3.35).
3.11. Spinors and Quaternions in a Riemannian Space-Time
We have seen that the analysis of Einstein and Mayer indicated that the mostprimitive (irreducible) representations of the symmetry group that underliesrelativity theory obeys the algebra of quaternions. They discovered this byremoving the reflection elements from the full Lorentz group, yielding thePoincare group of special relativity theory. Further, it was found that the Einsteingroup - the invariance group of general relativity theory - is a 16 parametercontinuous group. That is, there are 16 essential parameters here that characterize the transformations from any observer's reference frame to any other referenceframe in which he compares the laws of nature, so as to preserve their form where the frames move arbitrarily with respect to each other.
The algebraic properties of the representations of the Poincare group mustpersist when this is globally extended to the nonlinear transformations of theEinstein group. Thus it is concluded that the most primitive representations ofthe Einstein group entail 16 parameters and these representations must obey thealgebra of a quaternion number field.
58 Chapter 3
In the preceding chapter, the Einstein group was defined as the set oftransformations that leave invariant the squared differential increment of aRiemannian space-time
ds;, = gllV{x)dxll dx, = gllv{x)dxll dx",
With e" = gVIl there are only ten independent metric field amplitudes todetermine at each space-time point x. On the other hand, a 16-parameterinvariance group implies that there must be sixteen relations to solve at eachspace-time point rather than ten. That is, the full exploitation of the symmetrygroup of general relativity theory implies that the metric field entails 16independent components rather than ten. It then follows that the invariance ofthe differential increment
ds = + (gllV dx dx )tg' - Il v
is too restrictive to fully represent the Riemannian space in question. That is tosay, it should be possible to find a different way of 'factorizing' ds;" rather thanmerely taking its square root and throwing away the negative value, that wouldindeed reveal the full set of sixteen parameters that underlie the Einstein group.
It is, of course , a weakness of the invariant metric determined from the squareroot of a number that it is double-valued - it has a plus and a minus value at eachspace-time point. For some purposes it may be sufficient to take only one of thesevalues (say the positive one), ignoring the other. However , there are otherapplications ofthe theory of general relativity that would not allow one to do thisarb itrarily.
How then should we proceed to determine the 16-component metric field thatis single-valued from the outset, yet underlying the Riemannian space-time. Thehint is in the principle of relativity itself. Aswe have seen in regard to the algebraicfeatures of the Poincare group of special relativity theory, this is most primitivelyrepresented by a quaternion number field. Then in proceeding to generalrelativity the geometry changes from that of a Euclidean axiomatic system to aRiemannian axiomatic system - corresponding to a change in a part of thelogical structure of space-time in the sense of geometry. But another part of thelogical structure ofspace-time is in the sense of algebra - and this does not changein globally extending from special to general relativity.
It will be shown in Chapter 4 that the d'Alembertian operator, 0 = o~ - \72, isa particular representation of an operator of the Poincare group that is, in fact,Jactorizable into a product of first order conjugated quaternion differentialoperators, allOil and allOil. The basis functions of the quaternion operators, inturn, are the two-component spinor field variables that Dirac discovered for theelectron. Thus, Dirac's factorization of the d'Alembertian operator led to amore general representation since it is no longer covariant with respect toreflections in space or time (as shown earlier) while the d'Alembertian operatoris. Dirac's result, which was later studied regarding the group stucture of specialrelativity by Einstein and Mayer , as discussed above, then led to the prediction
Spinor-Quaternion Analysis in Relativity Theory 59
that the electron field has more degrees of freedom than were revealed by theKlein-Gordon (or Schrodinger) theories. These extra degrees of freedom werethen referred to as the 'spin' of the electron, since they had previously beenintroduced only in an empirical sense by Pauli in the Schrodinger equation inorder to correctly describe some of the electron properties. But, as mentionedearlier, their explanation did not come until Dirac derived them from arelativistically covariant form of the Schrodinger equation.
Similarly, in proceeding to the irreducible representations of the Einsteingroup of general relativity theory , and keeping the quaternion algebra for itsform, the invariant line element of the space-time may be expressed in thefollowing quaternionform:
d.9" = q"(x) dx", (3.58)
where q"(x) are a set of four quaternion field variables that transform in spacetime as a four-vector under the Einstein group . Thus, d.9" is a geometricalinvariant, though it is algebraically a quaternion. The latter feature is equivalentto that of a second-rank spinor of the type X(8)X*. That is, at each space-timepoint, d.9" has spin degrees offreedom in addition to its dependence on the spacetime coordinates x.
To construct a function that is invariant with respect to changes in 'spincoordinates' as well as space-time configuration, one must find the 'real number 'field that corresponds to the quaternion field. This is done with the conjugatedquaternion
The symmetrization occurs here, as in the discussion of the vector-tensor calculusin a Riemannian space-time (in the preceding chapter) because of the commutability of the product of real numbers dx, and dx, in the summation over all(J1., v). The factor ( - t) is introduced in anticipation of the normalization of thequaternion field variables, as will be defined below. Thus we see that thefactorization of the metric tensor of a Riemannian space-time is as follows:
g"V <=> _ t(q"qV + qVq"). (3.59 ')
Let us now proceed to a generalization of the field of quaternions, Q = (1"x"(Equation (3.20)) to the quaternion field
(3.60)
This is, then, the general form of a two-dimensional, Hermitian matrix in whicheach of the matrix elements is a four-vector. The field vv" is then a set offour unit
60 Chapter 3
four-vectors - called a 'tetrad' or a 'vierbein' field-with oVI' having a positive normand kVI' having negative norms. That is, with the summation over 11, we definethese vectors with the norms
(k = 1,2,3). (3.61)
(3.62)
Thus we see that ql' is a set of four quaternions (one for each four-vectorcomponent) that transform as (1) a contravariant four-vector in space-time and(2) a second rank spinor of the type X®X* in 'spinor space'.
The conjugate quaternion Q, which was defined earlier in terms ofthe invariantnorm QQ, was set up by changing signs ofthe basis elements Uk' while leaving thebasis element U 0 unchanged. Recall, however, that a change in the sign of U 0 underconjugation, while leaving Uk all unchanged would stiIllead to an invariant - thenegative of the previous one. To distinguish these different types of quaternionconjugation, we will denote the latter with the 'tilda',
Since ql' transforms in space-time as a contravariant four-vector, it follows thatql'ijv must transform as a second-rank tensor, contravariant in both indices - asrequired by the invariant structure of the left-hand side of Equation (3.59). Toobtain the proper local limit of the metric field, we require that the 'matrixelements' of ql' have the following local limit :
vVI'(x)~ ± be where bg = boo = 1,b~ = - bkk = - 1. (3.63)
In this limit, then,
1'( ) Joclim I'q X -+U , (3.64)
It then follows from (3.59') that the local limit of the metric tensor is
~·(x) ~.(i ~ ~ - I J (365)
which is the required form in special relativity.With this quaternion formalism, it is readily verified that
ql'ijl' =inv = - 4uo. (3.66)
It might be remarked at this point ofthe discussion that the factorization (3.59)can be considered either in terms of the 16-eomponent tetrad field vVI'(x), or in
Spinor-Quatern ion Analysis in Relativity Theory 61
terms of the quaternion fields.The latter, for each four-vector component, entailsfour real fields, since they are Hermitian matrices. Since there are four quaternionsdefined at each space-time point, the quaternion metric variable ql'(x) also entails16 real fields. Thus there seems to be a one-to-one correspondence between usingthe quaternion field as the metric variable or using the tetrad field as the metricvariable. However , they are not really equivalent since the quaternion representation is restricted because of its uniquely prescribed matrix properties (e.g.Hermiticity) and its relationship to the algebra and calculus of first-rank spinorfields, as basis functions. Because of the attempt in this study to couple thequaternion metric fields to the spinor matter and spinor electromagnetic fieldvariables, in a unification of the inertial and force manifestations of elementarymatter, within the theoretical framework of general relativity (tQ be followed insucceeding chapters) we will utilize the quaternion fields to represent the variablerelationship between the space-time points of a Riemannian manifold .
The correspondence between the fundamental quaternion and tensor fields of ametric space (Equation (3.59» will be exploited further in Chapter 6, where theEinstein field equations will be re-expressed in a factorized form, where space andtime reflection symmetry are no longer included in the covariance group. Thisgeneralization, together with the re-interpretation of the field equations that jointhe gravitational, electromagnetic and inertial features of interacting matter, willthen be studied.
3.12. Conjugation and,Time Reversal of Spinor and Quaternion Fields
Because the time reversal transformation is not defined within the covariancegroup of the spinor-quaternion formalism, one may still utilize the relationshipbetween distinguishable time-reversed spinor variables . Indeed, it will be shownin the next chapter, dealing with the inertial properties of interacting matter, thatthe relationship between the time-reversed spinor fields, mapped in a Riemannianspace-time, leads directly to the generally covariant form of the Dirac equation,which in turn, approaches the Schrodinger form of wave mechanics in thenonrelativistic limit.
Since the conjugated quaternion is obtained from the quaternion by reversingthe sign of the basis element (10' the same result could have been obtained bychanging the sign of the time components of the quaternion fields, leaving (10
unchanged. Thus, the conjugated quaternion is equivalent to a time-reversedquaternion. It is also readily verified by direct substitution that the time-reversedquaternion relates to ql' as follows:
(3.67)
where e is the Levi-Civita symbol (see (3.31».Since ql' behaves algebraically as a second-rank spinor,of the type i @X*, it
follows from (3.67)that the time reversal of the first-rank spinor field variable is as
62
follows:
f/x =ex*·
3.13. Quaternion Calculus
Chapter 3
(3.67')
It was emphasized earlier that the invariant differential interval of a Riemannianspace-time, in the quaternion calculus, is itself a quaternion, d5l' = q"dx", ratherthan the real number, ds = ± (g'" dx" dx.)t. The metric field q" entails 16independent real lield variables (whereas g'" entails 10 independent real fieldvariables). Thus to completely prescribe the space-time according to thequaternion representation, we need to determine the amplitudes of 16 differentfield variables at each space-time point x. Indeed, this is the expected result froman analysis of the irreducible form of the underlying symmetry group - a 16parameter Lie group - which requires that there should be 16 field relations ateach space-time point in order to unambiguously specify the space-time.
These relations will be determined in Chapter 6, specifying the explicit way inwhich the matter of a closed system and the geometry of the space-time mustcorrespond, in accord with general relativity theory.
At this stage of development of the quaternion calculus, it becomes necessaryto define the 'quaternion total derivative', did51', and the quaternion pathintegral, <t d5l'. The total quaternion derivative is defined as follows:
di d51' = q,,-l d/dxll
,
where q"- 1 = ij"Iq"ij" is the inverse quaternion, evaluated at the space-time pointx where the coordinate derivative is determined. Thus, by definition, if F(.'f') is anyfunctional form in terms of 51',
This is analogous to the definition of the derivative of a complex variable,df(z)/dz, in the complex plane. Here, iff(z) is analytic in the domain where dfldzis evaluated, its value is independent of any particular path in z-space along whichthe increment dz may be defined. Similarly, in the quaternion calculus, dF Id5l' isdefined in the domain of quaternion space where F is nonsingular and thederivatives are path-independent.
Since the derivative did51' is algebraically a quaternion, its minimal representation must be in terms of an Hermitian, two-dimensional operator,
(IX, P= 1,2).
Thus, by definition, the quaternion derivative stands for four independent realnumber derivatives, though related according to their Hermitian symmetry, LS
indicated above. Once again, this is analogous to the complex derivative, d /dz, as
Spinor-Quaternion Analysis in Relativity Theory 63
a combination of two derivatives - one along the real axis and the other along theimaginary axis.
The quaternion line integral, ~ d9', is similarly equivalent to four real lineintegrals, such that
( f Y'b ) (f Y'b )*d9' = d9',Y' a <1.(1 Y' a (1<1.
where (9'a,9'b) are the end points in 'quaternion space', for the path integral.Note that, according to this definition, the path ~ d9' is parameterized by a fourelement set, rather than a single element set {s}, as we have for the path of thestandard formulation in Riemannian geometry. That is, in the quaternionformulation, at each point along a path in space-time one needs four numbers(rather than one) to specify a further infinitesimal translation along this path.These four parameters then play the role of the (single time) parameterization ofthe Newtonian trajectory in a Euclidean space, or ofthe Einstein trajectory in thestandard formulation in general relativity that utilizes the tensor calculus.
In this context, it is interesting to recall W. R. Hamilton's comment, more thana century ago, on a possible interpretation of the quaternion algebra in terms of'time' [6] :
"It early appeared to me . . . to regard ALGEBRA as being no mere Art, nor Language, norprimarily as Science of Quantity ; but rather as the Science of Order in Progression. It was,however, a part of this conception, that the progression here spoken of was understood to becontinuous and unidimensional : extending indefinitely forward and backward, but not in anylateral direction . And although the successive states of such progression might (no doubt) berepresented by point upon a line, yet I thought that their simple successiveness was betterconceived by comparing them with moments oftime, divested, however, of all reference to causeand effect ;so that the 'time' here considered might be said to be abstract, ideal, or pure,like that"space" which is the object of geometry. In this manner I was led, . .. , to regard Algebra as theSCIENCE OF PURE TIME.
. . .And with respect to anything unusual in the interpretations thus proposed, .. . it is mywish to be understood as not at all insisting on them as necessary, but merely proposing themas consistent among themselves, and preparatory to the study of quaternions, in at least oneaspect of the latter."
This generalization of the trajectories in a Riemannian space, in terms of a 4parameter set characterizing a generalized 'pro per time ' in terms of a quaternionalgebra, was not introduced in an ad hoc fashion. It is a result of fully exploitingthe symmetry group that underlies the theory of general relativity - and itappears in conformity with Hamilton's expectations when he invented thequaternion algebra, more than a century ago! Also in conformity withHamilton's interpretation, the generalized trajectory is an abstract feature ofspace-time - it is not an observable quantity, as it is in classical approaches suchas that of Newton. Neither, however, is the abstract path in a Riemannian spacetime in the tensor formulation supposed to be an observable.
These results will be applied in Chapter 7 to the astronomical planetary motionproblem and to the problem of cosmology.
64
3.14. Spin-Affine Connection
Chapter 3
Recall from our discussion in Chapter 2 that it is because the vector field has morethan one component that it is non-integrable in a non-linear space-time. Thisnecessitates the introduction of the affineconnection {r~v} in the definition of the'covariant derivative' of a vector field in general relativity.
Similarly, the spinor field, having more than one component, requires theintroduction of the spin-affine connection in order to define its derivativescovariantly in a nonlinear space-time. The spin-affine connection {Q~Xl} is thusdefined as follows in terms of the covariant derivatives of the two-componentspinor variable X in a nonlinear space-time:
x., = 0/lX + Q:flX . (3.68)
Just as is the case ofthe four-vector field, the spin-affine connection is determinedexplicitly from the requirement that X;/l transforms covariantly as a four-vector ina Riemannian spacetime as well as a first-rank spinor under the same group.
It follows (from the invariance of q/lq/l) that the covariant derivatives of thequaterion fields must vanish - analogous to the vanishing of the covariantderivatives of a" in the tensor formalism . This feature will be used to determinethe explicit form of the spin-affine connection. Note first that q/l is a second-rankspinor on the type X® X*, and that their transformations with respect to theEinstein group must be in terms of both the spinor indices and the coordinateindices. The coordinate transformations of the quaternion, q/l as a four-vector,leads to the usual formulation that involves the ordinary affineconnection. Thus,the covariant derivatives of a (conjugated) quaternion field, ijll , with respect toboth the spinor and the coordinate indices, is as follows:
(3.69)
where iX, [3 = 1, 2 are the spinor indices and p, u, r denote the space-timecoordinate indices. The term on the right-hand side of (3.69)that depends only onthe spinor changes is:
where the 'dagger' denotes the Hermitian adjoint. Dropping the (e, [3) indices, thecombination of the preceding equation with (3.69) yields the following matrixequation :
q/l;p = 0 = 0pq/l + Q~Xlq/l + q/lQ~Xlt + r~pqt .
Contracting this equation from the left with the quaternion q/l then leads to the
Spinor-Quaternion Analysis in Relativity Theory 65
It is readily verified with Equations (3.60) and (3.62) that if A is any twodimensional matrix, then
ql'Aijl' = - 2Tr(A),
where 'Tr' denotes the trace of the matrix. It also follows that
Ae+ eA = Tr(A),
where the 'tilda' is the transposed matrix and s is the Levi-Civita symbol.Substituting Q~X) into the preceding equations, it follows that
Q~X)e + d2~X) = (TrQ~»e . (3.72)
From the vanishing of the covariant derivatives of the fundamental quaternionfields and Equation (3.67),it follows that e;p= O. Further, since xex is an invariantof the spinor space (Equation (3.31» , it follows that s is a second-rank spinor ofthe type X®X . Finally, since e is independent of the space-time coordinates, itfollows from an analysis that is analogous to the one that led to Equation (3.70)that
(3.73)
(3.74)
(3.75)
With Equations (3.73) in (3.72), the following general result is obtained:
TrQ~X) =0.Combining Equation (3.74) with (3.71) we obtain the result:
qI'Q~X )ijl' = O.
Using this with Equation (3.70), and the value of the invariant, ql'ijl' = - 4, itfollows finally that
Q~xJt =tql'(8pij l' + r~pijt).
Taking the Hermitian adjoint of this equation and using the fact that thequaternion fields are Hermitian, the following result is obtained for the spinaffine connection field:
Q{X) = 1.(8 q-I' + F': q-t)qp 4 P tp I'"
An equivalent form to this is obtained by using Equation (3.73) and theproperty that e2 = - (Jo. Equation (3.73) may then be expressed as follows:
Q(x) = e!'2 s =>Qw t = eQ* s (3.76)p p p p .
Taking the complex conjugate of Equation (3.75), applying s from the left and theright to this equation, using Equation (3.76) and taking the Hermitian adjoint ofboth sides of the resulting matrix equation, the following matrix equation isobtained :
Q(X) = _1.q- (8 ql' + rl' qt)p 4 I' P tp ' (3.77)
66 Chapter 3
Equations (3.75) and (3.77) are equivalent expressions for the spin-affineconnection field Q~X).
The transformations of ql', as a contravariant vector, implies that the spinaffine connection field must transform as a covariant vector in space-time,according to its expression in Equation (3.75) or (3.77). This is to be comparedwith the ordinary affine connection r;v' whose elements do not transform as anysort of relativistically covariant entity (as shown in the preceding chapter).
In the local (Euclidean) limit , the quaternion fields become the constant Paulimatrices and the unit matrix, and the ordinary affine connection terms vanish.Thus, in this limit, the spin-affine connection field must approach a null matrix :
n(x)( ) loc lim>~ p X ----+ O. (3.78)
One other important property of Qp follows from the time-reversalproperty (3.67) of the quaternion fields. Combining (3.67) and (3.75) and utilizingthe Hermiticity of the ql' fields, it follows that
:TQ(X) = _ n(x)tp >~p'
3.15. Spinor Transformations in General Relativity
(3.79)
(3.80)
The generalization of the relation (3.35),which defines the transformations {T} asa 2-dimensional representation of the Poincare group, follows in the same wayfrom the corresponding generally covariant equations. The nonlinear transformations of the four -vector components, in this case, imply that {T} is a set ofcoordinate-dependent matrices. Thus,
Xl' -> x'" = (X ~xv = X(x)-> X'(x ') = T(x)X(x),
where T(x) is a solution of the equation
T(x)ql'(x)T(x)t = (X~q Vf(X ')
Consider now the form
'1tql''1, (3.81)
where '1(x) is a two-component spin or variable, and require it to be invariantunder the transformations in spinor space that are induced by the transformations of the space-time coordinates in general relativity. The spinor '1transforms.as follows:
xl' -> x" ='1(x) -> '1 '(x') = S(x)'1(x).
The form (3.81), with the use of (3.80), then transforms in the following way:
'1 t ql''1 -> '1 t st T ql'Tt S'1, (3.82)
Spinor-Quaternion Analysis in Relativity Theory 67
where we have taken oce = be in order to represent the spinor transformationsalone at this stage.
The right-hand side of Equation (3.82) is equal to the left side (representing theimposed invariance) only if
(3.83)
Hence, the invariance of the form (3.81) in spinor space implies the existence of twokinds of spinor variables: one transforming according to the rule
X(x) -+ X'(x') = (st) -1 X(x)
and the other according to the rule
(3.84)
(3.85)
We will find that 1] and Xare in fact related to each other in terms of the timereversal operation
X= 81]*
when, in the next chapter, we examine the structure of the two-component spinorform of the Dirac equation, both in special relativity and in general relativity.Since the sets of spinor transformations {S} and {T} relate to each other inaccordance with (3.83), one need only refer, from now on, to one of theserepresentations. We will refer to S in the ensuing discussions. Thus, in accordancewith (3.80), the matrix fields, S(x), for the transformations of a spinor in aRiemannian space-time solve the following matrix equation :
st(x ')q"'(x')oceS(x') = qll(X). (3.86)
According to the limit (3.64) of the fundamental quaternion fields, it followsthat Equation (3.86) reduces in the local limit to
st (J' S = ocll(J' (3.86')P P IJ
which is the form for the linear transformations in a Euclidean space-time .It should be noted at the present stage that the generally covariant spinor
transformation solutions S(x) of(3.86) are not uniquely determined - because thefundamental quaternion fields, qll(X) are yet unspecified. To remove thisredundancy, we need four more field equations whose solutions are thecomponents of the quaternion variable s".This is indeed the role that will beplayed by the metrical field equations, in quaternion form. It will be seen inChapter 6 that the usual tensor form of Einstein's fieldequations (whose solutionsare the metric tensor,gllV) factorize into a set oftwo time-reversed quaternion fieldequations. The latter restriction on the quaternion field variables, that theyshould be the solutions of the factorized equations that relate the metrical field toa matter field,together with (3.86), then yields an unambiguous set of relations forthe group of generally covariant spinor transformations {S(x)}.
68 Chapter 3
Finally, the transformation properties of the spin-affine connection field maybe determined from the requirement that the form
(3.87)
must be invariant in both spinor space the space-time. In this expression,
Q = ,rQ(x) = 1(0 qP + rp qt)q- = _1.q (0 q-P + rp q-t)I' I' 4 I' tl' P 4 P I' tl'
follows from Equations (3.75), (3.77), and (3.79).It then follows that the fully transformed form (3.87) is
YJ'tqW'(o~ + Q~)YJ ' = YJ t s t ((St)-1 qPS-1)iX~a;(o. + Q~) SYJ
= YJ tqPb;S -1(0. + Q~)SYJ
= YJ t q'S -1(0 . + Q~)SYJ .
(3.88)
(3.89)
The double-prime symbol is used above to denote transformed variables withrespect to both the spinor and the vector indices.
The vector transformations of QI' in space-time have been taken care of in theequation above. The remaining prime symbol refers to a transformed field inregard to spinor indices alone. (This corresponds to a particular 'reshuffiing' ofthe matrix elements of QI" leaving the coordinates unchanged). The invariance ofthe left-hand side of Equation (3.89) then implies the following:
Multiplying (3.90) on the left with S and on the right with S-1 , the following resultis obtained for the (spin-) transformed spin-affine connection field:
Q~ = SQ.S-1 - (0.S)S-1.
Finally, combining this result with the transformation property of Q. as acovariant vector in coordinate space, the following general transformation rulefollows for the spin-affine connection:
Q~(x')=a;(SQ.S-1_(0.S)S-1). (3.91)
It is important to note that while Qp. transforms covariantly in space-time (as acovariant four-vector) it is not covariant in spinor space. That is, QI' could be anull-matrix in one spin frame (defined within the Riemannian system) but itwould be equal to - (0I'S)5- 1 =1= 0 in other spin frames. Thus, the lack of
Spinor-Quaternion Analysis in Relativity Theory 69
covariance that resides in the ordinary affineconnection field of the vector-tensorformalism in general relativity is relegated to an 'internal space' (i.e. spinor space)in the spinor-quaternion formalism, thereby leaving covariance with respect tocoordinate transformations unmarred! This lack of spinor-space covariance inthe affine connection does not affect the invariance properties of physicalobservables, according to their definition in this theory, since the latter alwaysfollow from energy density (and other field density) terms of the types shown in(3.87). In fact, it is the lack of spinor space covariance that is displayed inEquation (3.91)that is required in order to set up the density fields as invariant, asdemonstrated above.
With the mathematical apparatus that has been developed in this chapter and inthe preceding chapter, we are now in a position to derive a theory of interactingmatter that fully exploits the ideas of general relativity theory - as a theory offundamental matter, in which the inertial, electromagnetic and gravitationalmanifestations of matter are unified in terms of a common set of spinor andquaternion fields variables, solving a set of coupled, self-consistent fieldequations - that are to be taken as the basic laws of nature.
In the next chapter, we begin with a derivation of the generally covariantmatter field equations and study their mathematical and physical implications,including the derivation of the inertial mass field and demonstrating theasymptotic approach of these field equations toward the form of quantummechanics, as a low energy, linear approximation.
