THE ALGEBRA OF SPINORS AND ITS APPLICATIONS 
TO QUANTUM MECHANICS 
by 
WILLIAM DICKEY THACKER 
Submitted in ~artial Fulfillment 
of the Requirements 1or the 
Degree of Bachelor of Sci~nce 
at the 
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
June 1978 
Signature of 
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Certified 
Accepted 
~ D~par1;In9nt of Physics, June , 1978 
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by.ea a e O e e e e ea a e e e'o e • e ea• e .-GI ea a a e e e ea e e • • .. • e e • • • • • • • 
/) , . Thesis Supervisor 
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MASSACHUSETTS lNSTITUTE 
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-2-
THE ALGEBRA OF SPINORS AND ITS APPLICATIONS
TO QUANTUM MECHANICS
by
WILLIAM D. THACKER
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Bachelor of Science
ABSTRACT
It is well known that the discussion of spin phenomena
and two state systems provides an approach to Quantum Mechanics
that is simpler than the historical path which leads at once to
the Schr6dinger equation and the continuous spectrum. This
thesis shows that these advantages are enhanced if the spinor
theory is introduced in a geometric-algebraic context before
the physical interpretation is undertaken. Moreover, among the
existing versions of spinor theory we choose one in which non-
relativistic and relativistic Quantum Mechanics are treated in
close harmony with each other.
Thesis Supervisor: Laszlo Tisza
Title: Professor of Physics
-3-
DEDICATED TO THE MEMORY OF MY MOTHER
-4-
ACKNOWLEDGEMENTS
I would like to thank Professor Laszlo Tisza for
instructing me in his approach to the topics discussed
here and for countless valuable discussions.
I would also like to thank my father for making all
this possible.
-5-
TABLE OF CONTENTS
ABSTRACT..................................................2
ACKNOWLEDGEMENTS..........................................4
TABLE OF CONTENTS.........................................5
INTRODUCTION..............................................6
I. MATHEMATICAL DEVELOPMENT OF THE PAULI ALGEBRA
AND SPINOR FORMALISM.................................8
1.1 Heuristic Introduction...........................8
1.2 The Pauli Algebra...............................12
1.3 Matrix Representations for Rotations and
Lorentz Boosts..................................13
1.4 Introduction to the Spinor Concept...............17
1.5 Unitary Spinors and the Orientable Object.......18
II. APPLICATIONS OF THE SPINOR FORMALISM TO NON-
RELATIVISTIC QUANTUM MECHANICS.......................28
2.1 The Quantum Mechanics of the Spin 1/2 State.....28
2.2 Time Dependent Spin States.......................34
2.3 The Transition from Quantum Kinematics
to Dynamics.....................................38
2.4 Extension of the Formalism to States of
Higher Angular Momentum.........................40
III. RELATIVISTIC SPINORS AND THEIR APPLICATIONS TO
RELATIVISTIC QUANTUM MECHANICS.......................50
3.1 Introduction....................................50
3.2 The Properties of Relativistic Spinors..........51
3.3 The Motion of the Spin 1/2 Null Particles.......55
3.4 Massive Spin 1/2 Particles- The Dirac
Equation.........................................62
SUMMARY AND CONCLUSIONS..................................65
REFERENCES......................................... 67
-6-
INTRODUCTION
The purpose of this paper is to present a group theoretic
introduction to Quantum Mechanics, using an approach recently
advanced by Laszlo Tisza.1, This method is based on the
concept of the object group, the formal expression of the
identity concept implicit in atomic physics. Elementary par-
ticles and even composite structures in pure quantum states
are recognized as identical representatives of distinct classes
of objects. This "principle of generic identity" is fundamen-
tal to quantum theory and has no analog in classical mechanics
where the identity of an object is based on the continuity of
its trajectory through space.
The condept of object group arises because in quantum
mechanics an object of a given identity can exist in a variety
of states. We assume that the set of transformations connect-
ing these states is closed, contains the identity element, and
has a unique inverse for each of its elements: the set of
transformations associated with a quantum mechanical object
forms a group.
The object group can be interpreted in a passive or active
sense. In the passive interpretation one considers how an
object transforms as one varies the reference frame from which
it is viewed. The active interpretation considers how an object
transforms when viewed from a single frame of reference. These
transformations are used to represent kinematic and dynamic
-7-
processes; we shall refer to them as the active kinematic
and dynamic groups, respectively. This thesis is concerned
with the Lorentz group and its subgroup of pure rotations in
three dimensional space which shall be interpreted, throughout,
in the active sense.
The standard applications of group theory to quantum
mechanics call for considerable mathematical knowledge of re-
presentation theory and Lie groups, and the physical inter-
pretation of the formalism is not always obvious. The object
group approach offers an intuitive interpretation of the
connection between group theory and quantum mechanics, which
renders it more appropriate for our purposes.
The algebra of Pauli spin matrices and spinors has been
introduced by Pauli, Dirac and many others in connection with
physical situations, such as spin and antiparticles, which
were, and to some extent still are, poorly understood. In
this thesis we follow Tisza by developing this formalism in
an elementary constructive fashion as a means of describing
simple models as they transform under the active kinematic
group. The formalism is first built up on a purely mathematical
basis. This thesis then focuses on the applications of spinorial
techniques to quantum mechanics.
-8-
CHAPTER I
MATHEMATICAL DEVELOPMENT OF THE PAULI ALGEBRA
AND SPINOR FORMALISM
1.1 Heuristic Introduction
Throughout this thesis, x0 = c t shall denote the time
component of a position four vector and p0 = E/C the zeroth
component of the energy-momentum four vector. Our task is to
develop a convenient representation for transformations which
2 2 2-leave the quantity x 0 -1 2 3 invariant, i.e. Lorentz
transformations. It is widely recognized in the advanced
literature that the algebra of two by two complex matrices is
well suited to this purpose. Recent articles in the American
Journal of Physics recognize the usefulness of this method for
3elementary purposes, also. Unfortunately, the method has been
advanced in a number of variants, and a measure uniformization
would be desirable. Accordingly, we start with elementary
heuristic considerations in which the motivation is explained
for the conventions chosen.
It is known that every Lorentz transformation can be
decomposed into the product of a pure Euclidean rotation and a
Lorentz boost. Lorentz boosts can be thought of as hyperbolic
rotations. As a first step we consider two dimensional Euclidean
rotations in the x1 , x2 plane and hyperbolic rotations in the
x0 , x3 plane.
-9-
The rotation of the point (x 1, x2 ) into the point
(x{, x) is defined by the invariance relation: x{2 + X, 2
2 2X2 + x2.-Factoring and separating this into two equations
we obtain
x - x 2 = A (x - ix2
x + i x = X*(x + i x2
The above invariance requires that A*A = 1 or A = Z , A* -
We can rewrite the above relations as a matrix equation
0 x - i x
xi+ iX 2 0
-i J 0 1 x 2
10 zl x1 + i x 2 0
The point (x1 , x2 ) is now represented by a two by two hermitian
matrix and the rotation is represented by a unitary matrix
The hyperbolic rotation of the point (x0 ' x3 ) into (x6, xi)
is defined by the equivalence, x 2 - ,2 = 2 -X2. Factoring
and separating we obtain:
x + x' =a (x0 + x 3)
3 g 
- 0 3
x x3  =a -1(x -x0 3 0 3
*Throughout this paper Z should be read as e.
-10-
By letting a = Z we exclude it from having negative values.
Negative values would represent Lorentz transformations
which reverse the order of past and future. We require
Lorentz transformations to be orthochronic so that the time
ordering is preserved. The real paramter, v, is related
with velocity as follows: tarh P = = v/c. We map the point
(x 0 , X,) onto the hermitian matrix K Xand
0 X 0 x3
represent the hyperbolic rotation as follows:
X'+X' 0 03 0 x + X 0
0 3 0 3
0 x' - x' 0 C 0 x0 -x 3
The operator effecting this hyperbolic rotation is a hermitian
rather than a unitary matrix.
We have found it convenient to represent the points (xi, x2 )
and (x0, 'X 3 ) as two-by-two hermitian matrices. Putting the above
two mappings together we represent the space-time point,
(x0; x xl'2' x3), with the hermitian matrix, [0 + X3  x 1I-
X 1ix 2 x0 -X3
% -i 1 1
However if we multiply this matrix on the left byJwe do
L0
not produce a rotation in the x, x2 plane:
-11-
S0 X0  3  x 1  Ix 2
l0 Pij cx1 +ix 2  x 0 x3
([0x 3 ) Z (X'1(x 1  ix2 )1
(x1 + ix 2 ) (x 0 + x3 )
Moreover this operation does not preserve the hermiticity of
fx0 - 3  1 - 2
.0These difficulties can be removed by
x +ix2 x + x
using bilateral matrix multiplications:
L kit/2 0 x 0 +x 3  X- 2ixj (z+i'2 0
0 zi/ 2 J x + x x 0 - x 3 J lo i/ 21 2 0 3
x0 +x3 x i1
x' + ix' X 0- X3
The same device works for the Lorentz boost along the x3-axis:
P/2 0 I.0 .3 x i 2 1/2 -
0 91 / x + xi x - 0x11/2
(oZ /2J 
-
X t :2 kP/2
S2 0 3
X' + x' x -ix
0 3 1 2
x 1+ ix 2 X';-x'
-12-
In fact it can be generalized for arbitrary rotations and
Lorentz boosts, however to show this we must develop the
formalism further.
1.2 The Pauli Algebra
We introduce the three Pauli spin matrices along with
the unit matrix as the basis for our algebra .
G = 0 = ( = ( 2 1= (2 (1.1)
(1 0 ' 2 i(0 'G3 0 -1 0 1
The Pauli matrices obey the following relations:
0.0. = 6.. + iE.. 0 ; [a., a.] 2 E.. 01 j ij IJK K1 J 1]K K (1.2)
Letting a = a 1 X + a2x2 + a3X3, we can represent the
above matrix for the point, (x0 'x), as x0 1 + x-a. In general
any two by two complex matrix can be represented as A = a 1 + a-a
where a and a are now complex. The components of A can be
found by the relation: a = Tr (Aa ) where stands for trace.
The product of A = a 1 + a-a and B = b 1 + t--+ is given by:
o 0
AB = (a b + a-b)l + (a 0 + b a + La x b)-a (1.3)
Their commutator is.
[A, B] = 2 i (a x b)-' l.4)
-13-
In analogy with the operation of complex conjugation,
which acts on the complex numbers, we introduce three new
involutions which act on the set of two-by-two complex matrices.
The involutions are characterized by the fact that when they
operate twice on a matrix they leave it unchanged. The involu-
tions act on the component, a , of the matrix A rather than on
its elements, an. The Pauli matrices which make up our basis,
are left unchanged by the involutions.* Our first involution,
the hermitian adjoint, carries A = a 1 + a-a into A = a*l + a*'u.0 10
Hermitian matrices are defined by the relation, H = H. The
operation of simple reversal carries A into A = a1-a-a. This
operation is known fro- the theory of quaternions. Complex
reversal, which is derived from the composition of the above
two involutions, carries A into A = a*l - a*-y. These three0
involutions, together with the identity, form a group where
each involution is its own inverse.
1.3 Matrix Representations for Rotations and Lorentz Boosts
At this point we are ready to find matrix expressions for
arbitrary Euclidean rotations and Lorentz boosts. Tc do this
we represent the four vector to be transformed as a hermitian
matrix and operate on it with a bilateral multiplication which
preserves its hermiticity. First we try V = VXW, in this case
*This is a characteristic feature of Tisza's formalism which
contrasts with the Van der Waerden formalism. See, for instance,
Laporte and Uhlenbeck. 4
)(I+ , W+XVt. For X' to be hermitian we must have: W = V+
If we add the condition that V be unimodular, i.e. that
detiVi = 1, then we find that a transformation of the form
+ 2 -2X = VXV leaves the quantity det X = x X invariant.
Therefore X' = VXVt is a Lorentz transformation, by definition.
Our next step is to find expressions for V which produce pure
rotations and Lorentz boosts.
We have already found that a bilateral multiplication
by Ci/2and its hermitian adjoint produces a rotation
0 Ai/2
through $ about the x3 - axis. This unitary matrix can be
rewritten as:
U(x3 , $ /2) = exp[-i 4/2 x3 -t] = cos 4/2 1 -isin $/22 x3 @ca
(1.5)
Generalizing, we conjecture that a rotation through 4 about u
is produced by the following matrix:
U(u, $/2) = exp[-i $/2 ua] = cos $/2 1 - i sin $/2 u-a
The second equivalence can be demonstrated by expanding the
exponential in a power series and using the fact that (u_-)n
is 1 if n is even and u-ca if n is odd. To show that x' -a =
U(ut /2) x-a U (, ,/2) does represent a rotation, as con-
jectured, we separate x into a component parallel to u and one
perpendicular to it: x = x1 1 + x1 . x-1 1 a commutes with u-a
whereas (x.cr) (u-o) = - (u-a) (x .-Y) . Therefore x1 a and
-15-
transform as follows:
xl = 21 i sint G4') X 1 -q (cot 1I + i sin7 u'a)
(1.6a)
= (cos 1 i sin. uea .cosl(+p sin -CA +x
i (1.6b)
(cos2 sisini-51sx)
2$2
=[ (Cos - sin 7 2 i sinA cosi u-aFx -a
= O s$ x1 -y + sin $ (U X x4L ) a
x = cos1xs + sin$ u x x
xo remains invariant under this operation while xrotates
through $ about Ia. Q.E.D.
We recall that the hermitian matrix which effects a Lorentz
r22uax/2 
.boost along x3 is: 0 / We write it as H (x ,1 p/2)=
= 3(o2  .2 -2 i34
exp(p=/2 x3-a( = cosh p/2 1 + sinh p/2 X3-'. Generalizing, we
obtain for a boost along an arbitrary axis h:
H(hp/2)=exp(p/2 h-a) cosh p/2 1 + sinh p/2 hO
As we did before, we break x up into components parallel and
-16-
perpendicular to h. Now x a remains unchanged by the bilateral
multiplication:
x' 4 = (cosh 1 + sinh h-)x *e(cosh 1 + sinh 2h-)
= (csh E1 sinh h-t) (cosh L - sinhPS h-)x-
2 2 2 2 T
= x -0 (1.7a)
lx' =Gx
1 1
x and x transform together as follows:
= (cosh 1+ sinh h-t) (x++ x 1 1 a) (cosh 12 2 12
+ sinh hca)
1+ysinh 2 + +(cosh 1 + sinh -a) (x + x2 2 0 11
= [(cosh2  sinh2 +) 2 sinh1L cosh h-t]2+ +2 2
(x 0+ xll1O )
= (cosh px + sinh p x,1 ) 1 + (cosh px
+ sinh p x 0h) ea
(1.7b)
Equating the scalar and vector components we obtain:
x = cosh p x0  sinh px0 =
= cash px11 sinh px0h
(1.8a)
(1.8b)
.
-17-
These are the standard relations for how x0 and x1 1 transform
1
under an active Lorentz boost. cosh p = (1-2 ) , sinh p =6 (1-62)2
1.4 Introduction to the Spinor Concept
Thus far we have developed the Pauli algebra as a convenient
means of representing four vectors and their transformation under
Euclidean and hyperbolic rotations. A basic feature of this
formalism is the appearance of half angles, both circular and
hyperbolic, in the matrices which induce these transformations.
We have found it necessary to represent the rotation or boost of
a four vector with a bilateral.- multiplication in order that the
hermiticity of the matrix representing the four vector be
preserved. Thus, to produce a rotation through 4 we use two
multiplications, each with a matrix parameterized by q/2.
However, the half angles disappear from the equations describing
the transformation of the components of a four vector. Our next
step is to introduce mathematical entities which transform in
such a way that the half angles remain important.
It seems natural to extend our formalism by introducing
two component complex column vectors upon which the two by two
matrices operate. These two component entities are called
spinors. We use Dirac's bra-ket notation denoting the spinor,
with the ket 1>. Spinors transform under Lorentz boosts
a2
and rotations as follows: 1 '> = V|$>. Here the half angles are
important because only a single multiplication is used in
transforming the spinor. In what follows we concern ourselves
exclusively with Euclidean rotations and omit the discussion
of relativistic spinors for a later chapter.
1.5 Unitary Spinors and the Orientable Object
Our present task is to use the algebraic methods developed
so far, together with the spinor concept, to find a convenient
parameterization for the orientation of objects in ordinary
space. The mathematical model we consider is the triad, which
consists of three orthogonal unit vectors attached to a
common origin. We consider two frames: the space fixed frame,
associated with the triad .K(x1 , x2 , x3)' and the body fixed
frame, associated with the triad E(Q1 , E2' z 3).
AA A
In order to define the orientation of the body, E( 1 k, 2 3)'
with respect to fixed space, E(x1 ,x2 ,x 3 ), we suppose that the
two triads originally coincided and the body has been brought
to its present position through a sequence of rotations through
the three Euler angles. The Euler angles refer to a rotation
first through the angle a about its Z3 - axis, then through S
about its Z2- axis, and finally through y about its 3 -axis.
This sequence of rotations in the body frame corresponds to
the same sequence, performed in the opposite order, in the
space frame. The unitary matrix which describes the sequence
of Euler rotations is therefore given by:
-19-
V (ay,) =U( 3 , c/2) U (x2 1 '/2)
ia/2 0 cos 12
0 i/2 sin /2
-ia/2 cos (/2 k-iy/2
k+ia/2 sin (/2 k-iy/2
U x 3, y/2)
-sin /2 -iz /2 0
cos (/2 0 g4iy/2
9,-i*a/2 sin (/2 k iy/2
a/2 cos (/2 iy/2
V(a,r,y) describes the orientation of an object in space.
However V(a, f,y) is the same kind of mathematical entity as the
matrices which induce rotations. It is preferable to continue
to use matrices to denote rotation operators and to introduce
spinors which describe orientable objects. We therefore
introduce the two basis spinors'
1> = ( and I> = (. Operating upon them with V(a,(,y)
we01 oban
we obtain:
H> = V(cc,(3,6)|J1>
(9 -ia/2 cos (/2 -i<2
P, ic/2 sin (/2
2.9
(1 .9a)
i;> = V(a,,y)|11>Ck -ia/2 sin (/21V
._ ,iy/2iot/2/ cos (/23
(1.9b)
-20-
We have defined the spinor compoents:
K = Z-ia/2 cos 61/2 k-iy/2 and C2 =+ia/2sin V/2 A-iy/2
t > and its conjugate, i>, contain the same information as
V(a, (,y) and therefore describe orientation.
V(a,46,y) can be written as the juxtaposition of IC> with
its conjugate: V(a,ty) = (j> |A. Taking the hermitian
adjoint we obtain: * *
V+ ,y,y) _ _2
We have thus introduced the bra spinors <J = (
= (-C 2 ' 'Taking the inner products between &C>, IT>
and their hermitian adjoints we obtain:
2 2
l' 2 H1 + |ti21 (l-l0a)
2
*2 2
= 2 ' * 2|44j= + || = 1 (1.10b)
<CIE> - 2' 1  = -C 2 + t1 E 2 = 0 (1.10c)
<> = (CE2  * l2 + C 02 (l.10d)
These relations show that I> and I> are orthogonal to each
other and are each normalized to unity;j|> and its conjugate
therefore form an orthonormal basis for our two dimensional
spinor space. Because of their normalization we shall call |C>
-21--
and IF> unitary spinors.
We now consider the operation of spin conjugation, which
1
carries JF> into jt>. This operation can be represented as
follows: ia2 KI = ( =} J [2}= > where K denotes
22 1l0
complex conjugation. The hermitian adjoint <t can be conjugated
similarly: K <Cicr 2 =(l' (2- i 2'l
Performing the operation of spin conjugation twice we find
that:
ia2K 2 _ = - (1.lla)
1 2 1 (l.llb)
These relations show that spinors are two valued under the opera-
tion of spin conjugation.
It will be shown in Chapter III that when a relativistic
spinor transforms according to the matrix V, i.e. I'> = VIa>,
the conjugate spinor transforms according to the complex reflected
matrix V, |1> = VI>. However, since a unitary matrix is un-
changed by complex reflection, the transformation properties
of unitary spinors are invariant under spin conjugation.
A A A
We have started with a triad, E(, 1 1 2 'X3 ) , and associated
it with a pair of conjugate spinors. Now we work the other way
starting with |t>, IT> and finding the unit vectors of the triad.
-22-
To do this we consider the unit configuration, |l> = |> =(}, 1 = (1, 0), ij = (0,1), and find the outer products of
these basis spinors:
o10ol><lI 0
01
l><ll 00
100i><j 
= [0)
x 34(1 
- x3c7
S1 2~
The unit configuration is related with the body frame through
the sequence of Euler rotations: |(> = V((cd,6) |1>, > =
V(ccIy) IT>, <J = <1IV+(cf,y), <Il = <fTV+ (a.,,y), K-C =
V(ajy) xKU- V (a,f,y). Transforming the outer products for
the unit configuration, we obtain:
= V(a,43y)
= V(cxry)
= V(afy)
= V(czBy)
+1 1b+
Ii><I v (,,) = 4 (1 + Q3 ) (1.12a)
|i><11 V+(any) = (1 - ) (1.12b)
-2 1 ^ ^2|'><Yi v~ (a, ,y = ( + t2 ).a (1.12c)
+ 1 (%+Ii><l| v (,, = (Z- i9 2 )a (1.12d)
One can use these outer products to derive expressions for the
unit vectors 1, Z2 3. For example:
-23-
3. 1 | <l -
E l22
2 
IC1 IC2
2E 1E2
The important point here is that vectors are related to
quantities quadratic in the spinor components.
The outer products listed above can be interpreted as
operators on the space of spinors. We rewrite them as follows:
E3 = LE><E
E+= >
E3 (1.13 a,b)
(1.13 c,d)
E3 and E3 are projection operators which act on an arbitrary
spinor In> as follows:
E3 i> a |
E 3 |n> = a2 |>
where a1 =
where a2 =
SinceE 3 +E 3 -21 (1 + k3-a) +21 - k3-)= 1, |f> can be
expanded in terms of |t> and I1>: In> = [E3 + E3 ]jn>, which
neans- that In> = a0IC> + a 1 C> . E+ and E- operate
on IC> and its conjugate as follows:
E IK> = C> E_ C> = 0.
E C> = 0 E_ > = |C>
(1.15 a,b)
(1- 14a)
(1.14b)
(1. 15 c, d)
-24-
We now introduce a short hand notation for
spinors as an alternative to the Euler angle para.meterization.
The first two Euler angles, a and t, define the direction in
space of the preferred axis of the triad, Z A rotation of3.
the triad through a about 9 3 followed by one through about
Z2 leave the Z3 axis pointing in the (a,S) direction where c
is the azimuthal angle and a the polar angle. The third angle,
y, defines the orientation of the triad about the preferred
axis. We can therefore write the spinor, F , in the axis
angle form as It3, y>. It is easily seen from our expression
for IC> that: It3' i>/t3. The conjugate spinor,
fT>, is written in axis angle form as: |=t3' i=/2k3 >
Let us determine the action of the operator t3 3'a on
jZ3 ,y> and |t3,y>. For the moment we omit the third angle, y,
from the discussion. We shall re-insert it into our results.
We recall that |i( + - and It3><9 31 7(l 3
and we use the fact that =33 1Z 3 > and I'3 ><9,3It3>
It 3 >
1 + i3 ji> = 3A (1.16a)
T(1 3 3> 3
1 A +A
(1 =3 3 3> (1.16b)
3 33
-25-
Multiplying |k3> and g > by.a phase factor does riot change
our results and therefore y can be reintroduced. Thus, j 3 ,'y>
and 14 3 'y> are eigen-spinors, for the direction operator
3 'a, with eigenvalues 1 and -1, respectively.
The spinor |1> = '>= |,4,y> , and its conjugate,
transform in a particularly simple way under rotations about
the body axis, Z3 , and the space axis, x 3 :
3 2 I3,y> = [cos 7 3 - i sin 3c 3
= [cos - i sin!]J9,y> (1.17a)
-i4/2 -iy/2
L 3,y + q>
U(Z 3  3 ,y> =[cos - i sint 3*Ji
3' )IZ'y> 1co1 12 3. 3 '
[cos + i sin!-]JJQy>(l
2 3 (1.17b)
= +i$/2 9tiy/2
3 ,y + 4
-I0/2 1 cos/2 A_/2 iy/2
U(x3 , 7 , ty = gip/2 sin/2  ia/2
(1.18a)
[cos,/2 A-i(c /2) iy/2
~sinS/2zki(a $)/2
=[zi/2 0 }f-sin /2 t-i/2 iy/2
31 2o k /2cosV2 ki%/
(1.18b)
sin /2 Ci(ct+ p)/21iy/2
Icos3/2 Zi(a + $)/2
=Ia + $,3,y>
Unitary spinrs therefore have a biaxial charactor where both',.
z3 and x3  play a privileged role. A rotation about 23 changes
the angle y and one about x3 changes the angle a.
Finally, it should be mentioned that unitary spinors have
the peculiar property that a rotation through 27T reverses their
sign. This property is rooted in the appearance of the half
angle in the matrices which induce spinor transformations. The
matrix for a rotation through 2R about the arbitrary axis, u,
is given by: U(u, -T ) = cosTrl - i sinT u-a =-1. Therefore:
A 27
I '> = U(u, -. )|E> = | and t'> = -| > However, since the
-27-
unit vectors of the triad, associated with | > and |(>, are
related with quantities quadratic in the spinor components,
the factor of ,1 cancels out and these vectors are left
unchanged by a rotation through 2 r.
-28-
CHAPTER II
APPLICATIONS OF THE SPINOR FORMALISM TO
NON-RELATIVISTIC QUANTUM MECHANICS
2.1 The Quantum Mechanics of the Spin 1/2 State
Up to this point we have developed the formalism on a
purely mathematical basis -Unitary spinors have been introduced
within the context of a geometric problem, the parameterization
of the orientation of a triad. We now make the transition to
quantum mechanics by introducing a new prototype for the orien-
table object, the spin 1/2 state. This program shall be carried
out in a series of steps. We first consider the description of
stationary spin 1/2 states and measurements made on them. Then
we discuss the kinematics of time dependent spin states and
derive an analogue to the Schrtdinger equation. A transition
to dynamics follows in which spin is related to angular momen-
tum. Finally, we extend the spinor formalism to account for
states of higher angular momentum.
The transition from the triad, taken as a prototype for
the orientable object, to the spin 1/2 state necessitates a new
conceptual interpretation of the formalism. We recall that the
conjugate spinors, |4> and |IC> were used for a complete descrip-
tion of the triad, but these ingredients always appeared jointly,
without an independent meaning. By contrast the conjugate spinors
take on a separate meaning in quantum mechanics: t > refers to
-29-
a spin up state in the (ca, ) direction and IF> refers to a
state with spin down. The third angle is irrelevant to the
direction of the spin axis, and we omit it from the present
discussion.
Spin in quantum mechanics differs from the triad in that
one can only consider its orientation within the context of a
measurement process. The component of spin in a particular
direction is measured by means of the Stern-Gerlach experiment.
A beam of spin 1/2 magnetic dipoles is passed through an
apparatus which contains an inhomogeneous magnetic field. This
apparatus deflects the particles in one of two possible directions,
thus splitting the beam in two. We do not go into the dynamical
details of the Stern-Gerlach effect; this phenomenon is rooted
in the fact that spin may be associated with a magnetic moment
which is coupled to an external magnetic field. Rather, we
treat the measurement of spin in a purely formal manner.5 The
important point is that the inhomogeneous magnetic field sets
up a preferred direction in space with respect to which a spin
can take on two values, spin up and spin down, and the particles
with these spin states are deflected in different ways.
A Stern-Gerlach apparatus can be used to produce a beam
of particles in identical spin states. Now suppose that a .
beam in the state jg> = tY'11> + C21~> is sent through a Stern-
Gerlach apparatus which splits it into two beams, one in the
-30-
state 1l> and the other in the state 11>. Each particle in
the incident beam is in a state which can be thought of as a
mixture, or more properly a superposition, of the two outgoing
states. However, it is basic to quantum mechanics that an
individual particle cannot be split into its constituent states;
each particle must be considered as a unit when its spin is
measured. For this reason the spinor components, which still
describe orientation, must now be interpreted as probability
amplitudes; l and E2 are related to the probabilities that a
particle come out in the 11> or |T> state.
The fact that the spin 1/2 state can take on two orienta-
tions, spin up and spin down, when it is sent through a measur-
ing device renders it particularly suitable for treatment with
the algebra of two-by-two matrices and two component spinors.
Let us now translate the formalism developed in the first
chapter into the language of quantum mechanics. We associate
the Pauli matrices with the three components of the quantum
mechanical operator for spin:
1 1 1(2. )
s= rl ,s2 = r2' 63 732(2.1
These operators obey the standard commutation relations for
angular momentum: [s., s.] = is.. s . Spinors are now inter-
1 J 1JK K
preted as state vectors associated with spin. The operator
for spin :an be defined in terms of the generator for an
infinitesimal rotation of the spin 1/2 state. An infinitesmial
rotation about the k-axis is generated by the unitary operator:
-31-
U(k, = exp[-i k-c ]2 2 (2.2)
= 1 - i k-c
= i - i S s-k
The Pauli matrices are chosen so that our representation
+ 1 1 01
is diagonal in the third component of s , s3 = j -( .We there-
fore choose as our basis spinors the two eigenstates of s3
11> = I> = 3. These states are associated with the
eigenvalues 1/2 and -1/2, respectively: s3 11> = 1/2 1>,
s3 I'> = -1/2 |>. We call them spin up and spin down states
with respect to the x3 - direction. The spin up arid spin down
states are connected by the raising and lowering operators,
s+= l><I = sl + is2 and s_ = IT><l = s - is2 :
S+I >= fI1> SJY> = o
S+|1> = 0 s_Il> = |1>
(2.3 a,b) (2.3 c,d)
+*2
It should also be mentioned that the operator s is diagonal:
+2 -1 2 2 2 3
s =-(a + a2 + a34P1 + 2 +3) =
The results we have found for spin along the x3 direction
can be generalized for the arbitrary direction k. The operator
sk = }k-a has two eigenstates |k> and 1k> with eigenvalues
1/2 and -1/2, respectively. In addition, raising and lowering
A W n|
operators can be defined: S+ k>k and s-= Ik><kI.
-32-
Let us now go back to the problem of how spin is measured.
A Stern-Gerlach apparatus with its preferred axis in the k-direc-
tion can be described by the projection operators:
P = k><k| and = Ik><kl. A spin in the state Jn> will
emerge from this apparatus in the state |k> or Jk> with the
respective probabilities given by: <lP^Jr> = I<kr>1 and
2
<nPI^n> = |<kjn>j . When an ensemble of such spins is sent
through this apparatus all the spins projected into the state
|k> will contribute a value of + 1/2 to the expectation value
<s-k> and all those projected into the 1k> state will contribute
a value of -1/2. The average value of spin in the k direction
for this ensemble is therefore:
+1 
- 1 (2.4)< = <alPk^H> - <nlj|n>(4
-<n1 I [Pj?-^ ]I-n>
<n j4[ (l + k -t) - ( - k -a)]>
= <rJskJn>
It is instructive to consider as an example the measure-
ment of the average value of s3 for an ensemble of spins in the
-33-
state I 3>. The measurement of this would be carried out
with a Stern-Gerlach apparatus oriented along the x3-axis.
The respective probabilities for a particle to be projected
into the state, 1l> or |1>, are given by:
prob (spin up) = <9 1 3 l><1It3
ia/2 - ia/2 k_ iot2cos /2
= (k cosl/2, sin/2) i/2
1 ia/2sin '/2
= cos21/2
prob (spin down) = <k 311>c lZ3
= sin 2 /2
1 2 1 2 1
Therefore: <s3> > 9cos /2 + (mT)sin /2 = osa.
Notice that the half angle, which plays an integral role in the
spinor formalism, appears in the expressions for the probabili-
ties that 1t 3 > be projected into the spin up or down state.
However, only the full angle, 1, appears in the expression for
the expectation value of spin. 2cos is just equal to the x3 -
component of a length 1 vector in the direction of the spin
axis, 3 This example clarifies the role that spinors and
1
vectors play in describing spin T states. A spinor is associ-
ated with the quantum mechanical state which gives the probabi-
lities for the various possible outcomes of a measurement
process. When the expectation value of a component of spin is
-34-
taken, a vector can be associated with the spin axis.
2.2 Time Dependent Spin States
The phenomena we have considered involve measurements
taken on stationary spin states. Only two angles are required
to specify the orientation of the spin axis. In order to
consider the time evolution of a spin state we must reintroduce
the third angle and express it as a frequency, w, multiplied
by time: y = ot. We thus describe a spin state with the time
dependent spinor:
| Ft)>= ia/2coss/2 -iwt/2 1
cia/2 sin/2(t) (2.5)
This spinor can be interpreted as a two dimensional isotropic,
harmonic oscillator. The two spinor components, 5(t) and
C2 (t) ,oscillate with the common frequency, o. These modes
of oscillation have a phase difference of a, and their respective
amplitudes are cosf/2 and sinf/2. Therefore, the angles a and
0, which describe the orientation in space of the spin axis,
also determine the configuration space of an oscillator. The
isotropic, harmonic oscillator is only a physical model which
arises from our mathematical description of the time dependent
spinor. However, it shall become apparent in what follows that
-35-
this mathematical description is useful in dealing with physical
phenomena such as the spin procession. This writer suspects,
therefore, that spin may be associated with a process of
oscillation. Much more work needs to be done to understand the
underlying dynamics of this oscillation, but that carries us
beyond the scope of this thesis.
In dealing with time dependent spinors it is convenient
to work in our short hand notation. We choose as our basis
set the spinors: Ik,Q> and Jk,Q>. Then, as we derived in the
first chapter:
Ik,wt> = U(k, 2)|k,0>
=-iwt/2 k,"
-Z"" 1"jO>
Wt^
and |k,Lot> = U(k, S)Ik,o>
= +iwtk,
An arbitrary time dependent spinor, Irn(t)>, can be written as
the linear combination:
In)(t)> = a1 C-it/2 k,o> + a2 iWt/ 2 jk,o>,
where (2.6)
a1 = <k,Ojn (0)> and a 2 = <k,|n(O)>
-36-
We now seek a differential equation for the time depend-
ence of |n(t)>. Infinitesimal time displacements are generated
by the unitary operator:
odt .wdt^+U(k, 2) expf-i 2 k-a]
. wdt^- +1 -6 2 k-a
The choice of the axis. k depends on the physical situation.
We proceed by operating on Iqi(t) with the infinitesimal time
displacement operator:
odt
Jr(t + dt)> = U(k, yt)In(t)>
. dt ^ _+
= 1 - 2 k-]In(t)>
Jn(t + dt)> - |,(t)> = - i k-aTn(t)>dt
. W ^ (2.7)
i jr(t)> 2k-4 Jrt)>(2.
This is the analogue to Schrddinger's equation where the
Haniltonian is: H = 2 k-a. A similar equation can be derived
for the bra spinor, <n(t)f
-37-
<n(t + dt) =
cn(t + dt)I -r(t)| =
-1 <n(t)J =
+ A wdt
<n(t)IU (k, 2
<q(t)Ii k-a dt
4
The bra and ket spinors, |n(t)> and <n(t)I, are related
with the spin axis as follows:
1 Ajn(t) ><r, (t) | = s(t) = 2(l + s(t) -t) (2.8)
The Schrodinger-like equations can be used to derive a
differential equation for the time dependence of the spin axis:
S(t) = - s ( ) I=T|n(t)><n(t)I + (t) ><2
WA +
= -i k- iin><nl + iln><nIS k-a
= i[|r><n|,~ k-tI
'2
Since n> < J = 1 + s(t)ha) and the commutator [1, 2ka
we have:
s(t) -= i[s(t) ea, k*al.
dt "'0+
= <n t |[ 2 k-cy
(2.9)
-38-
2.3 The Transition from Quantum Kinematics to Dynamics
We have obtained a kinematic analogue to the Schrbdinger
equation by considering infinitesimal time displacements of the
state In(t)>. In our Schrodinger-like equation the Hamiltonian
has the dimensions of frequency. We can translate the kinematic
Schrodinger-like equation into a dynamic one by multiplying it
by Planck's constant Ti.
K w in > (2.10)
in"|> = K! k - a I n>
In addition, we associate spin with angular momentum through
Planck's constant: S = = a. Then the Hamiltonian
associated with spin is: H = wJ-k.
In classical mechanics angular momentum is related to
angular velocity about a spin axis through the moment of inertia.
This physical interpretation does not apply in quantum mechanics
because one cannot define an angular velocity associated with
spin. However the angular momentum of a spin state does have
observable consequences when it is linked with a magnetic moment.
It is an experimental fact that a charge distribution with
angular momentum has a magnetic moment given by: v = yJ, where
y is a phenomenological constant. This relationship also
holds in quantum mechanics where p and S are interpreted as
operators. The Hamiltonian associated with a magnetic moment
is: H = - p'B. If we let the magnetic field be directed in
the k-direction the Hamiltonian is
-39-
H - ByJ-k Bk = B
2
We recognize that this Hamiltonian has the form H = fk-a
where the frequency is defined as: w = -yB. This Hamiltonian
has the eigenstates |k, > and 1k, '2> with eigenvalues
1f and - iwrespectively. Therefore, while we cannot ascribe
an intrinsic frequency to spin, we can define a "beat frequency"
between spin up and spin down states in a magnetic field.
Let us now use these results to derive a relation for
the time evolution of spin axis 9(t). Recalling our equation
for the time rate of change of the spin axis.
s(t)-t = i[s(t)-4, k-d]
= i - [s-a, k-a]2
= i (2i(s x k)-a)
= - w(s x k)-o
An A A (2.11)
s = - ws x k
Therefore the spin axis precesses about the magnetic field with
a frequency Iwi = yB.
-40-
2.4 Extension of the Formalism to States of Higher Angular
Momentum
We have found the two component spinor formalism to be
well-suited for a description of the spin 1/2 state. The
effectiveness of this formalism is rooted in the fact that
the spin 1/2 particle is a two state system when coupled with
*
an external field. An arbitrary spin 1/2 state can be expressed
as the superposition of two conjugate basis states representing
spin up and spin down with respect to a given direction in space.
The question now arises as to how one can use our formalism
to deal with states of higher angular momentum. How can our
two component formalism be extended to account for states
which take on more than two configurations when coupled with an
external field? In what follows, a procedure shall be outlined
for using the algebra of spin 1/2 to build up states of higher
angular momentum. This procedure is well known within the
context of the theory of group representations. It is a standard
method for building up unitary representations of the rotation
group. We shall attach an intuitive interpretation to this
procedure.
We recall that our spin 1/2 representation is spanned by
two basis states representing spin up and spin down. We now
interpret these basis states as the two components of a wave
*This has been extensively demonstrated by Richard Feynman.6 ,7
-41-
function describing the motion of a spin:
IT-(x,t)> = u = u(x,t)1> + v(x,t)I1>.
v(xt)
(2.12)
The quantities
-+ 2 + 2
ju(x,t)j and Iv(x,t)I
give the probability densities for a spin to be found in the up
or down state, respectively, at the point (x,t). The total
probability density for a spin to be found at (x,t) is therefore:
2 2
W(x,t)I f(x,t)> = ju(xt)I + Iv(xt)I . This quantity, which
we write as uu* + vv*, must be invariant under spacial rotations.
To find the basis functions for states of higher angular
momentum j, we raise this invariant to the 2j th power:
2j
(uu* + vv*)2j _=Y 2j u2j-n vn U*2j-n v*n
n=l
Dividing through by (2j)! and relabeling the terms of the sum:
j
1 (u*+ 2j - 1 (j+m) (j-m) Uj+m j-m
(2j)! (uu* + vv, = + m C! -j M! u v u v
M=-j
(2.13)
This sum contains 2j + 1 terms each of which is associated with
a probability density. Again the sum of these probability
densities must be invariant under rotations. Thus, we write
the 2j + 1 basis functions for angular momentum j in the form:
--42-
M (j + m)!j 
- M) m)
$3 is interpreted, in our approach
m
spin 1/2 excitations, j + m of whi
and j - m of which are in the stat
the suffix m shall become apparent
The angular momentum operator
2j 2j
S +(n)- 1 z +(n)
n=l sl Thec
n=1 n=1
. The state described by
., as a collection of 2j
ch are in the state u
e v. The importance of
shortly.
is now given by:
ormutation relations for
the components of J can be readily found from the commutation
relations for the components of S:
2j
i ] = In=l
2j
(n) (n) .=(n)= E
[s ,Iskikksp = iikk Z
n=1
Recalling that s3U = -1U and s3V = . V, we now consider
the action of the operator j3 on the basis function 4,j
m
2 j
.j(n) (0 M) -M)
33M n2 + 1 - u (jm (Pm
(2.15)
= [j+ m) -10 - im)]
= in$
*We are now working in units where -i = 1.
(2.14)
-43-
The quantum number m therefore denotes the eigenvalue of J3
associated with the eigenfunction $ .M
The eigenfunctions cFL can be connected with their
adjacent states $Z .and r- through the step operators
m+l M-1
J+ and J_. These operators are defined in terms of the spin
raising and lowering operators, s+ = S1+ i s2 ands_ = s 1-is2
2j
J = s+ = J + 2J(2.16a)
n=l
2j
~ (n) .~-i
J_=ls_ = - 1 J2 (2.16b)
n=12
Each term of these two sums operates on a separate excitation.
We recall the action of s+ and s_ on the states u and v:
s+ v = u sv = 0
s + U = 0 s-U = v
Now consider how J acts on $ :
+ M
J $= s(n)1u(j+m)v(j-m)
+ M I +
n=l VQj + M!(j 
- M)!
Since * contains j-m excitations in the state v, the above
m
mum will have j-m non-zero terms, each with a v-state raised
to a u-state:
-44-
-jj-n(jM)1
.(irn)fu ! -n-l) 0
4
- ( j - )
(j+m) ! (j-m)!
(j+M+1) v (j-n-1)
(2.17a)
(j~m+l) (j-n-l)
(j~i+l+) (j-m-l)!
j +m+l
j -m
(j+m+l) (j -M)
Similarly:
J_
It should
2j (j+m) (j-m)
= (n) u v
n=l (+m) (j-m)!
j +m u ( j+M-1) v ( j-m+ 1)
(-j+m-) ! (j-m+l) j
(j+m) ! (j-m) ! M-
= (j-m+I) (j+m) I 1
be noted thatJ+0$ =QandJ_$ =0,
+ J
restricted to take on integral or half integral values between
(2.17b)
mis
M+,l
JL
-45-
j and --j.
-*2 +-+
The $4 s are eigenfunctions of the operator J = J-J
with eigenvalues j(jtl). To show this we find an expression
for 42 in terms of operators whose action on $ has already
been shown. The operator JJ operates on P as follows:
J J+ - (j+M+1)(j-m)$+
= (j+m+l) (j-m) qi
Note that: JJ+ = ( 1 - iiY2)(J 1 + iJY2)
= J +J JJ +i[J ,2
2 2
= J2 + J 2 
_ 3
Therefore:
~2 2 29
1 2 3 -+ 3 +J 3 S (2.18)
This operator acts of $3 as follows:m
J2=j JJ- J + J2 + J]j (2.19)
m- + 3 3 m
= j(j+l) $
Let us now consider the behavior of the basis functions
under rotations. We recall that a rotation of the spin 1/2
m
state is generated by the operator:
-46-
U(u, 9 = expt-i ~u-]. We rewrite this operator as:
U( 1/2)(u,fl = expl-i pS-ul. The rotation of an angular
momentum j state should be generated by the operator:
U (u,c) = expl-i J-u] (2.20a)
2j
. "'^-+ (n)
= exp[-i -f n]
n=l
This operator can be written ap the product:
2j
U)(u,$) = I exp[-i u-cx (2.20b)
n=l
Each of the above. factors operates on an independent excitation.
Therefore, to induce a rotation of the state
1 (j+m) (j-m) (2.21)
S= u____ U V
V(j+m) (j-m)!
one operates on each excitation sending u into u' = u a11 + va2 1
and v into v' = ua1 2 + va2 2 . Then the rotated state is given by:
j' 1 , (j+m) , (j-m)
fm =uv(2.22)
(j~m)jI (j(m- )
= _(u a1 1 + v a21) (j+m)u a1 2 + v a 2 2 )
(j+m)!a(j-m)
-47-
The ak9 are matrix elements for exp[-i u-g].
To demonstrate this method for a concrete case we consider
the problem of parameterizing the orientation of a spin 1 object
with the 3 Euler angles. The eigenfunctions for angular
momentum 1 are :
1 = 1 U2 1 - u, = 1v2 (2.23)
1 -20 
- P2
A rotation through the three Euler angles sends u into
* *
U'=V u+ E 2v and v into v' = - 2 u + 'Iv, where
K = Z-a/2 cos /2  -ia/2 and = = +i/2sina/2 -io/2 The
rotated angular momentum 1 eigenfunctions are therefore:
I' 1 2
= - (Cu + 2)
1 2 2 2 2 (2.24a)
(C1 U + 2C 1C 2 uv + C2
1 4+ 2 + E2-1
1'
= ilu + C2v) (-Cu + C*v)
(2.24b)
* 2 2 * 2
=/C1 C2  + ( 2 )1 uv + /2
1 2 JL+ (| - 2 21 0 + y C2* 1
V' 1 * * 2 (2.24c)
1 -2 + % v)
1 *2 2 ** *2 2
- ( 2 u - 2%f 2 uv + 1 V
= *2 1 42* + *2 1
-"2 v2 CC2  t1  c.1
We can represent the angular momentum basis states as three
component column vectors:
1 1 -- 0 0
$110 , 0 1- 10
0 (0t1"
A rotation of these basis states through the Euler angles is
effected by the matrix:
V(cS,y) = E E 2
2
2
1 2
1 2 2 12
1 2
ia1+coss 
-iy
2
X-ly sin
ia 1-cos3 -iy
2
_ ia sin
/x2
cos g
ia sinS
22
ia 1-cos3iy
2
iy sin
ia 1+cosSkiy
2
22
*2
(2.25)
-48-
-49.-
We can write this matrix as the juxtaposition of three column
vectors, each of which describe the orientation of an object with
angular momentum 1. Notice that only the full angles a, 3, and
y appear in this representation. In general, the half angles
will appear only in representations for half-integral angular
momentum.
-50-
CHAPTER III
RELATIVISTIC SPINORS AND THEIR APPLICATIONS
TO RELATIVISTIC QUANTUM MECHANICS
3.1 Introduction
In the past several sections we have restricted ourselves
to a consideration of unitary spinors and their applications to
non-relativistic quantum mechanics. Unitary spinors are defined
by the normalization condition: <tjE> = 1. They can only trans-
form meaningfully under pure rotations, which are represented by
unitary matrices, in order that their normalization be preseved.
Unitary spinors are therefore quite useful in dealing with non-
relativistic phenomena in quantum mechanics but are not suffi-
cient for use in relativistic quantum mechanics, which considers
objects that transform under the Lorertz group.
In this chapter we introduce relativistic spinors and pre-
sent some of their applications to relativistic quantum mechanics.
The theory of relativistic spinors was first introduced by Van
der Waerden in connection with the Dirac equation. The problem
was that the four component Dirac wave functions did not fit into
the tensor formalism based on Dinkowski's space-time. Van der
Waerden remedied the situation by introducing a complex spinor
space along with a use of indices setup to resemble those in
the traditional tensor calculus of relativity theory. 'Tisza has
recently introduced an alternative approach in which the theory
-51-
of relativistic spinors is developed in a manner harmonious
with our treatment of unitary spinors. In Tisza's formalism,
relativistic spinors are written in the same bra-ket notation
which we have been using, and Lorentz transformations are
represented by the matices derived in section 1.3.* In
addition this method makes use of the involution group together
with the operation of spin conjugation. This method seems to
obtain equivalent results to the standard method within the
domain of special relativity and relativistic quantum mechanics,
even while maintaining the basic unity of the formalism of
quantum mechanics. The Van der Waerden formalism has been applied
also to general relativity, while no attempt has been made as
yet to extend the present formalism in this direction. In this
chapter we shall apply Tisza's method to the description of
the motion of neutrinos and anti-neutrinos and give an intuitive
motivation for the Dirac equation.
3.2 The Properties of Relativistic Spinors
Relativistic spinors can be defined formally by their rela-
tionship with null four vectors. We recall that the outer product
between a unitary spinor and its hermitian adjoint defines a unit
*Remember that an arbitrary Lorentz transformation can be
expressed as the composition of a pure rotation and a boost:
V = H U. h'sis a result of the polar decomposition theorem.
See Tisza pp. 18-20.
-52-
vector: + = ). Similarly, we equate the outer
product of a relativistic spinor and its adjoint with one half
times the matrix for a real four vector:
1 1 + +|T><nI = A = (a01 a-) (3.1)
Notice that the four vector (a0, a)is null because the determinate
of In><nI is zero:
( 2 *Jn><qJn= il 2|n1n2 a0 +a 13 a - ia2
* 
1
l1n2  n21 a1 +ia2 a -a3
2 2 2 2
therefore 2 det (fn><n) = a - a - a2 - a3 =0.
We call it a null vector. In analogy with our relation
|T><Tl = (1 -k^3-a) for unitary spinors, we can define the
conjugate spinor In> by the relation:
| ><I|= 1A
(3.2)
1 1- +
=- ~(a0  a-s )2 0
Expressions for the transformation of relativistic spinors
and their conjugates can be derived from the transformation
properties of four vectors:
-53-
1 1
-A' = V A V22
TI> <ni= V In><uI Vt
This is satisfied if
|n'> = Vln> and <ni'! = <IV+ (3.3)
Therefore the Lorentz transformation of a null four vector can
be dissociated into two spinor transformations and, again, the
half angles, both circular and hyperbolic, come into play. If
A transforms according to the matrix V then A transforms accord-
ing to the complex reflected matrix V:
-A' =-V AV
Therefore
n'> = <q> and c<i'I = <lVF- (34)
The transformation properties of the conjugate spinor In>
can also be derived by considering the operation of spin conjuga-
tion: -iac 2 K, Suppose, again, that In> transforms according to
the matrix V: |n'> = Vln>. Then, taking the spin conjugate of
both sides of this equation:
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2-iO
I' = -ia 2 kV) (KIn>)
= J-ia 2 2KV) icr2 l-iG2 ln>
[-ia2 (K2) icr2 ]ij>
To evaluate -ia 2 (KV) iG 2  we ncte that if V = v 0 1 + v-a,
* * * * * * *
then: KV = v1 + vca + va2* + v*a3o 1ii 22 3 3
* * * * *
=v0 + vcr 
- v22 + v3r3
-icr2 (KV)iG 2 = a 2 2(KV)c2
- a2 [ 0 1
2 *
=cr 2 [v 0 1
*
= 0
* **
+ 2 
a 22 + v 3 Y3 ] a2
+v 1Cr-v cr 2+v 3a 2
* *1*
- va 
- va2 
-3c3]
* * *
vlcr - v2 2 v 3 r 3
-V
Therefore: ji'> = flij>, the conjugate spinor transforms accord-
ing to the complex reflected matrix.
-55-
It is of interest in relativity to consider quantities
which are invariant under Lorentz transformations. One such
invariant is the inner product of a conjugate spinor and a spinor
<x|n> = X1r12  x 2 n1
<x In'> = Kijv' 1VI p>
= <XIl>. (3.5)
In the Van der Waerden formalism this invariant is written as:
2 2
X2 2+ X2n
3.3 The Motion of the Spin 1/2 Null Particles
Let us apply the relativistic spinor formalism to the
task of describing the motion of spin 1/2 null particles. Null
particles are characterized by the fact that they travel at the
speed of light in all inertial reference frames, they cannot
be transformed to rest. (See also Tisza 111.2). The wave
four vector (k0 ,iZ) is introduced to describe the motion of a
o .+2 7r
null particle through space-time; k0 = - and k= 2 where
A and o are the wave length and angular frequency of a wave.
For a null particle the wave four vector must satisfy the
2 2 2 2
conditions: k2M"k20 1 k2 -k3 = 0, k0 > 0. The wave function
describing the motion of a null particle will be some function
times the phase factor ! exp[-i(k0x 0 - k-x)].
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Null particles with spin have an additional degree of
freedoxm known as helicity. Relicity is defined by the
relative orientation of the spin axis and wave vector
helicity =S( r~IutI(3.6)
where s is a vector rather than an operator in this equation.
It can take on two values for spin 1/2 null particles:
+1 when s is parallel to t and -l when s is anti-parallel to k..
Null particles with spin 1/2 are realized in natures as neutrinos
and anti-neutrinos. (Actually there are two varieties of neut-
rino, anti-neutrino pairs but this has not been accounted for.)
It is known that the neutrino has negative helicity and the
anti-neutrino has positive helicity.
Since the neutrino and anti-neutrino have spin 1/2, we
would expect each to be describable by a two component spinor.
FOr the moment we denote the spinors associated with the
neutrino and anti-neutrino by |y> and |a>, respectively, and
do not distinguish between conjugate spinors. Since ki =k
we have from the definition of helicity:
tekk -r, for the anti-neutrino
0 S
Isk
and k1 = - ---,for the neutrino.
s
-57-
If we interpret s as an operator, again, this means that:
k |a> = k'ja>, which implies that [k 1 - kcYlja> = 0 (3.7a)
kly> = k-ajy>, which implies that [k 1 + k-u]jy> = 0 (3.7b)o Y>0 yI>=0(.b
These equations are very suggestive of the formalism we have
developed. Notice that k 1 - k-a and k 1 + k-S can be inter-
0 0
preted as projection operators. If we redefine Jy> as jK> and
la> as JK> and letJIK><KJ = (k1 + i-t) and JH><HJ = ( kl - k-t),
then
IK><CKK> = 0, for the anti-neutrino (3.7a)
and |K><KjK> = 0, for the neutrino (3.7b)
follow as identities.
Let us now find explicit expressions for |K> and I>, the
spin parts of the wave functions for the anti-neutrino and
neutrino. We first solve for |K> using the outer product
relation:
(k 1+ k+
JK><l = 4 (k 1 + k8t)
K |2K1 2 k + k3 k - ik2
K2 2 Q1 3.
k + k k0 +k 3  i$P2o 3 .___ 3_K1 2 implies that K 1 = 2
K2 k0 - k3 - k0 - k3  i$2
K 2 2 implies that c2  2
2 2
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S Kk1 +ik iU2P) k . + k2
and K1K2 2 implies thatk 1 -
22 k2 k2
0 k3
The phases Z 1 and t 2 are arbitrary and only the phase
difference is determined. If we set $= 0 then we obtain:
_ _ __ I
Jk +k
|K> = 2
k1+ ik2
2(k +ko +k3)
Note that if another phase convention had been chosen then
our expression for JK> in terms of the components of (k0, i
would have appeared quite different.
We can express the components of k in terms of the spherical
angles (a,H):
k = k sine cosa
k2 = k0 sinS sina
k3 = k0 cos5
-59-
Then:
/Vk l+cosS A cosf/2
o 2
K2=kzia 1sin ia 2 sing/2 cosf/2
22(l+coss) 2 coso/2
/RT sinS/2 zkia
|K> = cos/2 (3.8a)
sinS/2 zia
Notice the similarity between this expression and the expression
we found for Z 3 >, they differ only in their normalization and
the phase convention used. Taking the spin conjugate of 1K>
we obtain:
-sin$/2 Z
0
cosS/2 (3.8b)
Our expressions for 1K> and |1> underscore the point that the
anti-neutrino and neutrino are orientable objects with their
orientation given by the direction of their wave vector and
their helicity about that direction.
Reintroducing the space-time part of the wave functions
for the anti-neutrino and neutrino, we obtain:
-60-
Ta > = Aexp[-iCk x -k.xjK> ( 3 .9a)
a 00o
-A expi-i (k x -ib-XK> , (3.9b)y 00o
respectively, where A is a normalization constant. To obtain
differential equations for the space-time dependence of
T a> and |T>, similar to the Schrtidinger-like equation
derived in section 2.2, we introduce the differential cperators:
D = - 1 - -( . a9*0 (3.10a)
and
~3X0 1 + -a .r (3.10b)
When i D and i D operate on |Wa > or ITY> they bring out a factor
a y
of
K = k01 + k- and K = k01a-
respectively:
i D exp[-i(k0 x0 -k-x)]I> = tk 01+k-x]expl-i(k 0 x0 -kx)]I>
expj-i(.k0 x0 -k x)1 = Ik 1-kc]exp[-i(k 0x 0 -k-x)]
where It can be any spinor. This means that D transforms like
K and D transforms like K under the Lorentz group. Referring to
equations (3.7 a and b), we obtain the differential equations:
-61-
i D Ta> = k 0 1-k3-)Ta> = 0 (3.lla)
i D Ty> = (koltkS)t > = 0 (3.11b)
In order to translate these kinematic equations of motion
into dynamic ones we note that the wave four vector is related
to the four momentum through Planck's constant:
(p0 , ') = -fi(k0, it) (3.12)
The operators associated with the four momentum are:
P = i i D = i i( 3 -4-4) (3.13a)
x0
and P = i -h D = i ti(ax0 + Voo) (3.13b)
Using these operators, we rewrite the equations of motion for
the anti-neutrino and neutrino as follows:
PTTa>= a> = 0 (3.14a)
=PIT ~~ )~> = 0+_ 0--
P1%,> = Cp 0 l3pIa)T > = 0 (3.14b)
Notice that mass does not appear at all in these equations.
Operating on (3.14a), we obtain:
-62-
PP IT> = (P0l+p)(P01Ta> = 0
(p 2 P 2) >T=a 0
Therefore:
P0 -P 2 = 0
It is apparent that a single two component spinor can account
only for massless particles. It will be shown in the next
section that a pair of conjugate spinors is needed in order to
introduce mass in a relativistically covariant fashion.
3.4 Massive Spin 1/2 Particles- The Dirac Equation*
The equations of motion for the neutrino and anti-neutrino
describe waves of well defined helicity which travel at the
speed of light. The neutrino and anti-neutrino have a Lorentz
invariant helicity because their spin and wave vector have the
same relative orientation in all inertial frames; these particles
cannot be transformed to rest.
Mass appears in our approach because there exist objects
which can be transformed to rest. Massive objects with spin do
not have a Lorentz invariant helicity because their wave vector
*The argument presented here is similar to that which appears
in Kaempfer. 5 Section 19. HOwever sgme aspects of the inter-
pretation are different. (see Tisza ).
-63-
can be transformed away altogether while the absolute value of
the spin is invariant. This is another reason why a single
relativistic two-component spinor cannot describe a massive
spin 1/2 object.
The question now arises as to how we can account for mass
within our scheme of massless waves which travel at the speed
of light. In answer to this, we hypothesize that a state with
rest mass can be thought of as a standing wave pattern
achieved through the superposition of two waves traveling in
opposite directions. We therefore expect that a massive spin
1/2 particle will be describable by a pair of spinors obeying
two wave equations of a form similar to (3.14 a and b). These
equations must be relativistically covariant and must also be
consistent with the fact that
p 0 -|pI =2-m2c2, (3.15)
for a particle with rest mass m.
Let us investigate the transformation properties of the
two uncoupled equations (3.14 a and b):
P I>V = VIT > = V FIT >.a a a
Therefore 'FITa> transforms like a conjugate spinor.
Similarly:
P'IT> =V p V+VF'7 > =V PIT>Y ~YY
-64-
Therefore PI> transforms like a spinor. The spinors IT a
and |TY> describe two different massless spin 1/2 particles.
We expect a massive spin 1/2 particle to be described by the
two conjugate spinors Iu> and jV> coupled by equations
with the same transformation properties as (3.14 a and b):
T~|u> = Alv>
PIV> = Alu>
where A is a coupling constant. Iterating each of these
equations we obtain:
PP Iu> = (p2 - Ip1 2)|u> = A U>
PPIV>= (4 - 2 )2 > = A2 1K>
2 2 2The condition (3.15) is satisfied if A = m c . We omit negative
energy solutions by selecting A = mc. We have therefore found
that a spin 1/2 particle of mass m should be describable by the
two coupled equations!
Flu> = mcI-v> (3.16a)
PjV> = mcju> (3.16b)
Equations (3.16 a and b) are normally written as a single four
component equation known as the Dirac Equation.
-65-
SUMMARY AND CONCLUSIONS
In keeping with our stated purpose we have built up the
mathematical formalism sufficiently to serve as the basis for
subsequent physical applications. The Pauli algebra was
developed as a convenient means of representing Lorentz trans-
formations. Spinors were then introduced and used to represent
objects which transform first under pure rotations (Chapter I)
and then under Lorentz transformations (Chapter III). Only
after we had developed the mathematics as a coherent structure
did we undertake its physical interpretation. This allowed us
to take full advantage of the inner unity of our formalism
supplied by the involution group, in particular. An example
of the unifying power of the involutions is provided by the
operation of spin conjugation which was found to be associated
with complex reflection. This operation was first introduced
in reference to the triad and then found to convey the trans-
formation between spin up and spin down states, in non-relati-
vistic quantum mechanics, and finally the transformation
between neutrino and anti-neutrino states in relativistic
quantum mechanics
A further unification was made possible by our use of the
object group concept. Throughout this thesis we have focused
on the problem of how to characterize objects which transform
under the active kinematic group. The transition between the
geometric-algebraic treatment of Chapter I and the applications
-66-
to quantum mechanics involved the introduction of a new kind
of "object", the quantum mechanical state, and a new inter-
pretation of the formalism in terms of the measurement process
and probability amplitudes.
This thesis deals only with a fraction of the problems
involved in the foundation of quantum mechanics. Many more
problems can be treated within the existing framework and the
framework itself is susceptible to extension, in particular,
to deal with the continuum aspects of quantum theory.
REFERENCES
1. L. Tisza, Lecture Notes.
2. L. Tisza, "Toward a Reconstruction of the Mathematical
Principles of Physics", (to be published).
3. W. G. Harter and N. Santos, Am. J. Phys., 46 (1978) pp.251,
264.
4. 0. Laporte and G. E. Uhlenbeck, Phys. Rev., 37 (1931) D. 1380.
5. F. A. Kaempfer, Concepts in Quantum Mechanics, Academic Press,
N.Y., 1965.
6. R. P. Feynman and F. L. Vernon, J. Appl. Phys., 28 (1957) p. 4 9.
7. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman
Lectures on Physics, Vol. 3, Addison-Wesley, Reading, Mass.,
1965.