Geometric Algebras in Physics: Eigenspinors and Dirac Theory
University of Windsor Scholarship at UWindsor
Electronic Theses and Dissertations Theses, Dissertations, and Major Papers
2008
Geometric algebras in physics: Eigenspinors and Dirac theory
J. David Keselica University of Windsor
Follow this and additional works at: https://scholar.uwindsor.ca/etd
Recommended Citation Keselica, J. David, "Geometric algebras in physics: Eigenspinors and Dirac theory" (2008). Electronic Theses and Dissertations. 8083. https://scholar.uwindsor.ca/etd/8083
This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208. Geometric Algebras in Physics: Eigenspinors and Dirac Theory
by
J. David Keselica
A Dissertation Submitted to the Faculty of Graduate Studies through the Department of Physics in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy at the University of Windsor
Windsor, Ontario, Canada 2008 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition
395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada
Your file Votre reference ISBN: 978-0-494-47108-1 Our file Notre reference ISBN: 978-0-494-47108-1
NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats.
The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission.
In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these.
While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. •*• Canada ©J. David Keselica, 2008 Windsor, Ontario, Canada Author's Declaration of Originality
I hereby certify that I am the sole author of this thesis and that no part of this thesis has been published or submitted for publication.
I certify that, to the best of my knowledge, my thesis does not infringe upon anyone's copyright nor violate any proprietary rights and that any ideas, techniques, quotations, or any other material from the work of other people included in my thesis, published or otherwise, are fully acknowledged in accordance with the standard referencing practices. Furthermore, to the extent that I have included copyrighted material that surpasses the bounds of fair dealing within the meaning of the Canada Copyright Act, I certify that I have obtained a written permission from the copyright owner(s) to include such material(s) in my thesis and have included copies of such copyright clearances to my appendix.
I declare that this is a true copy of my thesis, including any final revisions, as approved by my thesis committee and the Graduate Studies office, and that this thesis has not been submitted for a higher degree to any other University or Institution.
IV Abstract
The foundations of quantum theory are closely tied to a formulation of clas sical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formula tions of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be rep resented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation
4 C -> Ch - Clfi3 cz C/Jj ~ H ® C. indicates that the 8-dimensional even subalgebra C7/~3 of the Space-time algebra, STA is isomorphic to APS CI3, which is isomorphic to complex quaternions HI © C. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra H (B H. The algebras Cl\$ and C/3,1 are not isomorphic but their even sub-algebras are[l].
The Klein paradox is resolved in APS by considering Feynman's picture
v of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern- Gerlach magnet acts on the eigenspinor A field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved.
VI Acknowledgments
I would like to thank my supervisor, Dr. W. E. Baylis, for his complete support and encouragement through all my PhD program. I would also like to thank Dr. J. Huschilt and my colleagues for their valuable suggestions and encouragement over the years. I also would like to give special thanks to my wife Annette, my son Alexander and my family for their love and support. Finally, I would like to thank the Physics Department for its support in providing a rich environment in which to work.
VII Contents
Author's Declaration of Originality iv
Abstract v
Acknowledgments vii
1 Introduction 1
2 Algebraic Background and the Dirac Equation 4
2.1 Clifford Algebras 5 2.2 Comparison of Different Formulations of the Dirac Equation . 9 2.3 Complex Quaternions 11 2.3.1 Quaternions 11 2.3.2 Complex Quaternions 15 2.3.3 Dirac Equation 17 2.4 Space-time Algebra 20 2.4.1 The Space/time Split 21 2.4.2 Dirac equation in STA 21 2.4.3 The Opposite Metric 25 2.5 The Algebra of Physical Space (APS) 30 2.5.1 Generating CI.3 31
viii 2.5.2 Complex Structure and Paravectors 35 2.5.3 Involutions of CI3 37 2.5.4 Dirac equation 40 2.5.5 Dirac Equation Relation to Standard Form 42 2.5.6 Equivalence of Dirac Theory to APS Version 44
3 Dirac Solutions in APS 45 3.1 Momentum Eigenstates 45 3.2 Standing Waves 47 3.3 Zitterbewegung 47 3.4 Klein Paradox 48 3.5 Basic Symmetry Transformations 54
4 De Broglie-Bohm Formulation 58 4.1 Pauli-Dirac equation 59 4.2 Stern-Gerlach Measurement 62
5 Complex Algebra of Physical Space 71 5.1 Conjugations 74 5.1.1 Metric 76 5.1.2 Matrix Representation of CAPS 76 5.1.3 Comparison of Space-Time Unit Vectors 77 5.2 CAPS Form of Dirac Equation 79 5.2.1 Relation to the Standard Form 81 5.3 Fundamental Symmetry Transformations 82 5.3.1 Momentum Eigenstates 85 5.4 Canonical Quantization •. . . . 87 5.4.1 Annihilation and Creation Operators 90
IX 5.4.2 Number Operators 92
6 Conclusions 93
Bibliography 99
Vita Auctoris 103
x Chapter 1
Introduction
Clifford (geometric) algebra is a powerful mathematical language for ex pressing physical ideas. It can unify diverse mathematical formalisms and provides geometric insights as well as physical ones. Clifford algebras art? associative algebras of vectors, where the vector product is generally non- commutative. Clifford algebras are extensions of vector spaces, complex numbers, quaternions and Grassmann algebras. William Kingdon Clifford (1845-1879) introduced them in part as an effort to unite the discovery of quaternions by William Rowan Hamilton (1805-1865) and the anticommut- ing variables of Herman Grassmann (1809-1879). Clifford algebras are well suited to modeling special relativity and quantum mechanics. The approach of recasting fundamental equations into Clifford algebras often reveals new insights that were previously obscured by the choice of the mathematical formalism used. In the case of the Dirac equation, Clifford algebras have been used to add to our understanding of the foundations of relativistie quantum theory. In particular, as we show below, it provides a useful tool for investigations of the quantum/classical (Q/C) interface[2][3].
Classical relativistie physics in Clifford's geometric algebra has a for-
1 mulation that is closely related to the standard quantum formalism. Tin1 present dissertation looks at how geometric approaches have been used to represent the Dirac equation. In chapter 2, we look at three different formu lations, first those based on complex quaternions, next the real even space- time algebra Cl^3, and then that based on the paravectors on the algebra of physical space(APS) CI3. Since Clifford algebras bring some powerful tools to the table, we make use of projectors (hermitian idempotents) to facilitate the comparison. Although these formulations are isomorphic to each other, there are differences in how they are naturally interpreted and extended.
In chapter 3 we examine the algebra of physical space. The focus here is on how APS can make the quantum/classical interface more transparent. Some of the advantages of a geometric approach from the classical side are an intuitive grasp that this provides. As an example of this intuition, the Klein paradox is resolved, where plane wave solutions can exist in the region under the potential step. These represent positron solutions that can travel one way and are equivalent to negative-energy solutions propagating the other way.
In chapter 4, we see how naturally the algebra of physical space can treat the De Broglie-Bohm approach to quantum theory in an approach extended to include relativity and spin. In Bohmian mechanics a system of particles is described in part by its wave function, in this case the Dirac equation and the motion of the particle determined by the eigenspinor A which gives its proper velocity u — AA*. Fermionic spin can similarly be represented as a Lorentz transformation of a fixed rest frame direction. To illustrate its use we will look at the Stern-Gerlach problem and give a relativistic causal account of a spin measurement in APS. Chapter 5 introduces a covariant interpretation of the complex algebra
2 of physical space (CAPS), the complex extension of APS. CAPS is intro duced to solve a problem in that the PT transformation of inversion in space and time (P is parity inversion and T is time reversal), although a proper transformation, is anti-orthochronous (that the sign of the time component always changes) and thus aphysical. But in the standard theory it has the form of a physical rotation. Another problem is that the explicit form of the T and C (charge conjugation) transformations, depends on the matrix representations chosen. A representation that is independent of the matrix form would be more fundamental. As we will see CAPS provides a resolu tion to these problems. Finally, in chapter 6 we discuss the conclusions of this dissertation.
Some of the presentation is pedagogical with many explicit examples, as much of the algebraic formalism may be unfamiliar to physicists. Compar isons with conventional Dirac theory are made throughout the work. The emphasis is on the advantages of the Clifford algebra formulation over that of the standard approaches. In accord with common practice in the field, the natural units c = h = 1 are used throughout this work.
3 Chapter 2
Algebraic Background and the Dirac Equation
The Dirac equation is one of the most fundamental equations in quantum mechanics. Why do we care about different formulations of it? Some prob lems that are difficult in one formulation are easy in another. Also, different formulations provide different insights so that when extending the equation to new situations, each has different difficulties[4]. The geometric structure of the Dirac equation can best be seen using a number of different Clifford algebras. One of the advantages of the Clifford algebra formulation over the matrix version is that the particular matrix representations used is irrele vant to the physics. In this chapter we compare three different formulations, namely those based on paravectors in the algebra of physical space (APS)
Ci.i, the real, even space-time algebra Cl~t3, and complex quaternions. APS can also give insight to the other two algebras. Then the one based on paravectors in the APS, CI3, is discussed in more detail.
4 2.1 Clifford Algebras
Before defining what a Clifford Algebra is we need to define a few things first. Clifford algebras are generated by inner-product spaces, and an inner- product space is a pair (V,g), where V is a vector space and the inner product or scalar product is a symmetric bilinear map g : V x V —» K. If V is finite dimensional, then let {ei,e2, • • • ,('n} be an orthonormal basis for V with a scalar product of vectors
e* -ej = gij, (2.1) where the metric tensor g^ is diagonal with values ** = {*£/;• <2-2) The inner product has a signature (p, q) where p elements of the orthonormal basis have scalar products of +1 and q elements have scalar products — 1. In more general bases, which may not be orthogonal, the signature gives the eigenvalues of the metric tensor and is independent of the basis. A real associative algebra A is a vector space over K together with an algebraic product is a bilinear map
Ax A -> A (2.3) (a,b) i—• ab that satisfies the following two conditions. The algebraic product is distrib utive over addition, (aa + 3b) c = aac + 8be (2.4) a (3b + -,c) = 3ab + -fac, and second, the product is associative,
a(bc) = (ab)c, (2.5) for a.b.c € A and ay/3,^ & R. It should be noted that multiplication is not necessarily commutative. In a complex associative algebra R is replaced by C. If there exists 1 £ i such that la = a\ — a for every a G A, then 1 is called a unit or the identity for A. We are now at a point where we can say what a real Clifford algebra is [5]: a real Clifford algebra for an inner-product space (V,g), denoted Cl(V.g), is an associative algebra with unit 1 that contains copies of V and K = K • 1 as distinct subspaces such that
1. v2 = g{v,v) , V v e V
2. V generates CI (V,g) as an algebra over R
CI (V,g) is not generated by any proper subspace of V. These conditions uniquely define a Clifford algebra only when the bilinear form g is nondegenerate. Often Cl(V,g) is written Cl(p,q) when (V.g) is an inner-product space of signature (p,q), also written as Cllhq. If q = 0, then CI (p, 0) is usually abbreviated to CI (p); also written as Clp. The first condition tells us what the product of v with itself gives, and the right hand side g(v,v) is a real number that lies in the vector subspace of the algebra spanned by the identity. We now look at g(w,w) where for the case where w — u + v. g (w, w) = w (2-6)
g (u, u) + 1g (u, v) + g (v, v) — v2 + uv + vu 4- v2 (2.7)
9{u,v) = - (uv + TO). (2.8)
Then for the elements of the orthcmormal basis we have,
f±l :i=j c,e,^\ \. (2.9)
6 Now the second condition means that every element of CI (V,g) can be written as a linear combination of products of elements of V. Lastly the third condition, which is called the universal property of the Clifford algebras, is needed to guarantee that the algebra is the largest possible one that satisfies 1) and 2). If we do not have this condition then it is possible to generate a lower-dimensional nonuniversal Clifford algebra.
Since every element of V can be written as a linear combination of basis elements, this means that every element of CI (V, g) can be written as a linear combination of products of the orthonormal basis elements B = {e.\, .. . , e„} for V. To find a basis for CI (V,g) we need a maximal linearly independent set of products of the basis elements for V. If a product of elements of B contains an element of B more than once, then (2.9) can be used to reduce to product that contains this element. Therefore, a basis for Cl(V,g) that consists of products that contain elements of B either once or no times can always be chosen. Hence, if V is n. dimensional, the number of elements in each product must be less than or equal to n. We can then reorder the elements in the products such that the indices increase from left to right. For example,
r e se2e%e\ = -e^e^e^ei — ~e.2e\ = exe2, (2.10)
l assuming e§ = 1. There are (*fe) products of k elements that satisfy these conditions. Thus, &-vectors, also known as vectors of grade k, can be defined in the algebra as real linear combinations of a (^-dimensional basis formed from products of k distinct orthogonal basis vectors. The algebra C7;j,(/ operates in this linear space formed from the direct sum of the A:-vector spaces \\.:
n
(&Vk. (2.11) fe=0
7 A general element of the algebra is a linear combination of a scalar, a vector, a bivector (a 2-veetor), a trivector (a 3-vector), and higher-grade A;-vectors.
Therefore, a basis for CI (V.g) has fl &) = 2" elements, with k = 0 corre- fc=o sponding to the scalars. For example, the product of two linearly-independent bivectors may con tain a 4-vector. With bivectors A — eie2 and B = aeje^ + /?eae4, where a,3 are real scalars, the product is
AB = —a + 3eie2ese4 . (2-12)
It is convenient to use a bracket notation to extract parts of a general element. Then (x)k is the A;-vectors part of x. It is also possible to have larger subspaces that combine A-vector subspaces with different k. For example, a general element p of CI3 is a real sum of scalar S, vector V, bivector B, and trivector T parts or grades:
p = S + V + B + T. (2.13)
Subspaces of CI3 include the real numbers (scalars) (CI3)RS = K, the com plex numbers (scalars plus trivectors) (Cls)s = C, and the real quaternions
Q which comprise the even elements, Q — S+ B of C/3, denoted (Cii}+ = H. (Some of the notation used here is defined formally later in this chapter). To see this relationship better we use of the volume element and the Hodge dual [29]. The Hodge dual can be defined as
l *V=VeT (2.14) where, e-;- = eje2e3...e„ is the volume element of the algebra. The inverse
1 of the volume element is e^, = e„en_]...ei. In C/3, e-/- = e^e.i, and the Hodge dual of a bivector is a vector. For example,
* (eie2) = (e]e2) (eae^ei) = e:i. (2.15)
8 2.2 Comparison of Different Formulations of the Dirac Equation
The Dirac. equation is one of the most important equations in relativistic quantum theory. When Dirac developed his now famous equation in 1927, he sought an operator linear in the time derivative whose square gives the Klein-Gordon operator. He found the need for an algebra that we now recognize as equivalent to the Clifford algebra of space-time and recognized that he could realize the algebra with 4x4 matrices. The usual text-book approach to the Dirac equation uses a specific matrix representation in which the Hamiltonian is expanded in four 4x4 matrices, say -y^, and the wave function is a 4-component column vector ij.>. The Dirac equation for a free fermion of mass m can be written in the very simple form
pip ~ imp, (2.16) where p is the space-time momentum and ip is a bispinor, a four-component column matrix with complex elements. The momentum p contains both an energy component and the components for the spatial kinetic momentum. The momentum is usually represented by the 4x4 matrix
P = P"> (2.17)
= P°70 + pSl + P2l2 + P3l3 , where the four matrices 7^, p. = 0,1,2,3 represent orthonormal basis vec tors in Minkowski space-time. An obvious disadvantage to this approach is that the components of ?/; depend on the matrix representation chosen, and they do not lend themselves to direct physical interpretations. Furthermore,
9 although the matrices -^ obey the Clifford relation
))ilv + TvT/i ~ ^-^/ii-' • (2.18)
.'hero / 1 0 0 0 ^ 0 -1 0 0 (Vt*v) = rl (2.19) 0 0-10 \0 0 0 -1 J tliey are not usually interpreted as unit vectors in space-time. The space- time momentum p is the physical momentum. It is distinct from the conju gate momentum for charge e in an external field:
id = p + e,A, (2.20) where A is the space-time vector potential. In terms of coordinates xM, equation (2.20) is equivalent to four component equations
• d (2.21)
The indices are raised and lowered with the metric tensor r/^,, :
Pn = Vt*vP (2.22)
The Dirac equation can now be written as
^{iO^ - eA^t)v> = mii>. (2.23)
A more geometrical/physical approach is to use Clifford algebra directly without assuming any particular matrix representation. There are several related formulations in Clifford algebra. For the equiv alence of Daviau's, Hestenes', and Parra's formulations of Dirac theorv see
10 [7]. Hero, we compare three of them, namely those based on paravectors in APS {Cl;i), the real, even space-time algebra Clf3, and biquaternions. Since Clifford algebras bring some powerful tools to the table we make use of projectors (hermitian idempotents) to facilitate the comparison. While these formulations are formally isomorphic, there are differences in how they are naturally interpreted and extended.
2.3 Complex Quaternions
Before looking at the complex quaternions we should first look at quater nions. Quaternions were invented by Sir William Rowan Hamilton. He did this by extending the complex numbers. Hamilton wanted to rotate vectors in 3-D in analogy with rotations in 2-D by complex numbers.
2.3.1 Quaternions
Hamilton referred to quaternions as hypercomplex numbers. Quaternions have four components[8], one real unit and three generalized imaginary units. i, j, k that have the following relations:
i2=j2 = k2 = -l. (2.24)
ij = k = -ji, jk = i = -kj, ki = j = -ik. (2.25)
The multiplication is by definition non-commutative. Nevertheless, the quaternion multiplication is associative, with
ijk = -1. (2.2G)
A Hamilton quaternion Qa has the form
1 2 :! Qa = a° + u i + a j + a k , (2.27)
11 where the coefficients a1' are all real scalars. The quaternions form a four- dimensional real vector space. The set H of all quaternions form an associa tive algebra when distributivity is assumed, and it is also a division algebra. The only thing that prevents the quaternions from forming a field is that multiplications are not commutative. The generalized imaginary units can be denoted by i, j, k or by i, j, k, and the choice may depend on whether you wish to emphasize their action as generators of rotations (hypercomplex units associated with bivectors or rotation planes) or as "vectors" (direc tions in space). Both interpretations are possible in 3-dimensional space since bivectors are dual to vectors and generate rotations[5].
If o° is zero then Q is a "pure quaternion", or a "vector". Then it can be associated with a bivector in CI3. Since the Hodge dual of a bivector is a vector, you can associate i, j and k either with three mutually perpendicular directions ej, e2, e3 in space or with their dual bivectors *ei = e3e2 and cyclic permutations. Since there is a one-to-one relationship between the vectors and bivectors or pure quaternions in CI3 (APS) you can do this. I think that the best way to think of these objects is as bivectors in CI3 (since
(Cl3)+ EE H). Explicitly, we put
i = *ej = e3e2
j = *e2 = eie3 (2.28)
k = *e3 = e2ei.
Of course other associations, differing by a spatial rotation, are possible, but this one maintains the connection of i,j,k to directions on the x.y, z axes. It also suggests a standard 2x2 matrix representation for quaternions in
12 which the scalar part is the scalar times the unit matrix (TQ and
i -> -inu j -> -icr2, k -* -i 1 0 <7() = , O"! = , CT2 = , °"3 = 0 -1 (2.29) Vectors a and b ! 2 3 a = a ei + a e2 + a e3 (2.30) 1 2 3 b = 6 ei+6 e2 + 6 e3 (2.31) thus have duals that correspond to Hamilton's 'Vectors'': 1 2 3 1 2 3 *a=a *ei + a' *e2 + a *e3 = a ! + a j + a k l 2 3 l 2 3 *b = b *ej + b *e2 + 6 *e3 = b i + 6 j + 6 k . These have the quaternion product *a*b = -a b + * (a x b) , (2.32) where the scalar product is a • b = aj6j 4- «2&2 + a.363 and the cross product — is * (a x b) = i (a2^3 — 03^2) + j (03^1 ^63) + k (ai>2 - «2^i)- It follows that two non-zero "vectors" commute if and only if they are parallel, and anti-commute if and only if they are perpendicular, just as vectors do in Ch. 0 0 The conjugate Qu of a quaternion Qa ••= a + *a is Qa = a — *a, where wo have changed the sign of the pure quaternion part. More specifically, Qu is called the quaternion conjugate of Qa. The conjugation is an anti- automorphism of II: QuQb = QbQa for all Qa and Qi, in H. The square 13 norm of Qa is given by 3 QJTa = (a0)2 - (*a)2 = (a0)2 + a2 = ]T K)* • C2-33) The norm \Qa\ of Qa is given by \Qa\ = \/QaQa, and one can verify that 1 IQaQil = I Qui iQfcl- The inverse Q^" of a non-zero quaternion Qa is given by Qa1 = 'Q'al \Qu\2 or explicitly 0 * Q a Q-i = (2.34) Wa (a°)2+a2 l 7 This implies that if QaQb — 0 then Qa = 0 or Qi, = 0, so that the quaternion algebra is indeed a division algebra. Rotations Quaternions are very useful for rotating vectors (and their duals) without using matrices. Take the pure quaternion *r — xi + y'} + zk (2.35) of length |r| = \/x2 + y2 + z2. Then for a non-zero quaternion Q G H, the expression Q*rQ~1 is also a pure quaternion with the same length iQ'rQ-^Hrl. (2.36) Hence, the mapping •r-+Q*rQ-x (2.37) is a rotation in the quadratic space of pure quaternions. The rotation axis is found by taking Q to be a unit quaternion, \Q\ = 1, and writing it in the form Q ----- exp (*6/2) where *9 is a pure quaternion. With the help of the Eider relation, this can also be written as Q = exp (*0/2) = cos (0/2) + *0sin (0/2) (2.38) 14 where 0 = \9\ and "6 = "Qj \B\. The rotation *r -• Q*rQ~l turns r about the axis 6 by the angle 9. As we will see in section 2.5 below, analogous expressions can be written using paravectors in C/;j. The underlying metric of real quaternion space is seen to be Euclidean. 2.3.2 Complex Quaternions Since the quaternions are four-dimensional, it might be thought that they are useful in relativity, but this is not the case[9]. There is no way of treat ing the boosts with real quaternions. This is easy to see if you think of quaternions as the even subalgebra of C7,j. However, if we employ complex quaternions, a formulation of relativity is possible. The quaternions form a four-dimensional vector space over the scalar field K. The complex quater nions are obtained when the scalar field R is replaced by the scalar field C of complex numbers. The complex quaternions, like the real quaternions possess a vector-space structure. Hence, the complex quaternions are not just of the form aq with Q 6 C and { 6 1, but are taken to be of the form {q + iq' \q, q' G H } . which forms a four-dimensional vector space over C. As a real vector space, the space of complex quaternions forms a real eight-dimensional vector space C ® H. The complex quaternions are also closed under multiplication. Thus, the set of complex quaternions forms an algebra. However unlike the real quaternions, the complex quaternions do not form a division algebra. Relativity with Complex Quaternions Complex quaternions can represent space-time vectors and be used to define the Lorentz inner product to get the Minkowski space-time metric. The 15 space-time vector v — t'^e,, has the form v = v° 4- i * v = v° + i (i-1 i + v2j + r3k) (2.39) where i is tlie usual imaginary scalar, not the quaternion unit i. Therefore, in the complex quaternions, a space-time vector splits into a part that is real scalar and a part that is an imaginary pure quaternion. The complex quaternions used to represent space-time vectors employ two conjugations. The first is the bar conjugation for quaternions, which is extended naturally to the complex quaternions by 'h + iq2=W+ m, (2.40) where qi, q-i are elements of H. It is the antiautomorphism that changes the sign of the vector part of any quaternion. It thus changes the signs of i,j,k but not of i. The other conjugation, called the dagger conjugation, is the antiautomorphism that changes not only the signs of i,j,k but also of i (qi + iq-2^ = q\ - m (2.41) where vv = (v° + i*v) (v°-i*y) (2.42) = (t.°)2-vv The Lorentz transformations in complex quaternions have the form v-rv' == LvfJ (2.43) 16 where L € 51/(2, C) is a complex quaternion that satisfies LL = LL = 1. Then, i/f = (LvL]\ = LvL] = v. (2.44) Hence v' represents a real space-time vector. Also note that, v'v7 = LvlJVvL = Lt- (LL)] vl = Lvvl = vv . (2.45) Thus (2.43) is a Lorentz transformation with the properties that we need. To get the Minkowski space-time metric with the opposite signature, we could define the Lorentz inner product as -vv = -(v° + i *v) (v° - i *v) (2.46) 2 = vv-(V°) . This can be done in a similar way using the paravectors of APS. 2.3.3 Dirac Equation The Dirac equation was developed to replace Schrodinger's equation with one that included relativity. It was derived as a type of square root of the Klein-Gordon equation but differed from it as it describes a spin 1/2 particle, whereas as the Klein-Gordon equation is for spinless particles. The Dirac equation is important because it led to the prediction of antiparticles. to a fundamental understanding of the electron's magnetic moment (e.g., it predicted the correct ^-factor of 2), and to more precise predictions of atomic spectra, specifically for hydrogen. The Dirac equation (p — rn) U> = 0 for a free fennion may be viewed as a pair of coupled first-order differential equations for two-component spinors. 11 The operator p — rn is represented by ~fMp — ml, where 1 is the 4x4 unit 17 matrix and 7M are the gamma matrices. In the Weyl representation these have the form 0 &h a» 0 where conjugates of the Pauli spin matrices are an, u, — 0 cfu=< • (2-47) -o-fe, fi = k > 0 Thus, the operator p — rn has the matrix form -m j/V,, (p - rn) =| | . (2.48) v p Tv —in Writing out the equation (p — rn) i/> = 0 for the bispinor (2.49) where r/ and £ are two-component Weyl spinors, we get v p (7vT) = m£ (2.50) jfa^ = m.i). (2.51) When the momentum components are expressed as differential operators dxt, these equations become two coupled two-component differential equations. When the two-component spinors are rewritten as complex quaternions .4 ----- £ and B = n. and the quaternion differential operator is denoted id^an — V = idt + idx + jdy + kdz. 18 this becomes Lanczos's fundamental equation [10]: VA = mB (2.52) X7B = mA. Here V is the quaternion 4-gradient operator, which transforms as V = LVLf where Le SL(2,C)- However, the Weyl spinors rj and £ have only two complex components whereas complex quaternions have four. In fact, the Weyl spinors can be written as minimal left ideals of more general complex quaternions. Ele ments of the algebra can be projected onto minimal left ideals by projectors (real idempotents) of the form Pk = \(i+ ik), where k can be any unit bivector. As will become clearer when we derive the Dime equation in APS (Section 2.5), we can identify £ = A = APk. v = B = BPk and when we combine these into a single complex quaternion * = APk + &Pk f and note that V = -V, Pk = Pkik and Pk = ~Pkik. we find 1 V* = VAPk + VB Pk = mBPk - mi*Pk This is known as the Dirac-Lanczos equation and is fully equivalent to the Dime equation in the Weyl representation. It was first discovered by Lanczos 19 [l()j and then later rediscovered by Conway and Giirsey [12] and, in the STA form (see below) by Hestenes [14, 15, 16]. Characteristics of the complex quaternion model include a Minkowski space-time metric which is internally generated but seems contrived. It can also represent the Minkowski space-time metric of the opposite signature. The treatment reflects the dual nature of some quantities such as mass ( = energy component in rest frame or Lorentz scalar) since both are scalars in the quaternion algebra. The space/time split is immediate, given by the quaternion split. The complex quaternion model has a 2 x 2 matrix repre sentation, where the spatial basis vectors appear as imaginary bivectors; all are relative to the observer. However, this model has indirect geometrical interpretations and is not easily made covariant. It also cannot be readily extended to higher dimensions. 2.4 Space-time Algebra The space-time algebra, or STA, is based on the Minkowski space-time met ric of signature (1,3) [14] [17] [18]. Here we use a set of four orthonormal basis vectors {*)[,}, [J. — ()...., 3, satisfying l7,y + = v dl !l H 2 53 •V ' V = 2 ^' '),J1^ ^ ^ " ( )' ( ' ) The vectors {-y^} satisfy the same algebraic relations as Dirac's '^-matrices, but they now form a set of four independent basis vectors for space-time and not four components of a single vector as they are frequently viewed in stan dard Dirac theory. The STA has the basis set. l-/li,,rrk.I(Tk.I"n,.I (2.54) 20 In order they are 1 scalar, 4 vectors, 6 bivectors, 4 trivectors and one pseudoscalar / = 7o7i7'273 which anti-commutes with the vectors. 2.4.1 The Space/time Split Since STA works with absolute frames, the space/time split is needed to relate the results to observations[20]. The space-time split is not immediate ,bs(:,ver and is given by multiplying the reference frame by ^) . The three bivectors crk , where ak = 7t7o- satisfy the following relation, - {cTj(Tk + ak(Tj) = -- (7j-/fe + 7fc77) = Sjk (2.55) this generates the geometric algebra of C7;s, which is the 8-dimensional real even subalgebra of the STA, Clf3. The even subalgebra Clf3 is isomorphic to the algebra of physical space, APS — C/3. The identification of the algebra of relative physical space with the even subalgebra of Cl\$ gives a transition from relativistic quantities to observables in a given frame. The process of moving between a vector v in CI1.3 and its representation t>o + v in the relative spacetime is given by, v —» ?.'7o = v • 7o + v A 70 = VQ + v, (2.56) where r;() = v • 70 and v = v A 70. This is known as the space/time split. 2.4.2 Dirac equation in STA The traditional Dirac equation for a free fernhon can be expressed directly in algebraic form in the complex Dirac algebra iY7»P = ?n#, * e C7i.3 over C . (2.57) 21 where V -• rf^i^d/j. and 7,,, v = 0. 1,2.3 are tho basis vectors of Minkowski space-time[5]. One can include an interaction with the electromagnetic field F'IV via the space-time potential A by the replacement of id1* —• id^ — eAfl. Then the Dirac equation takes the form iV* - eA$ = rn*. (2.58) We want to relate solutions * to those, *//, of the Dirac equation in STA [14] -V$//7i72 - eA^H = m^ula, *// G C7^3. (2.59) In the usual equation, the Dirac spinor is a column spinor * € C4, but this can be regarded as a 4 x 4 matrix with only the first column being non zero. This can be done with the simple projectors in C/1,3 over C, [21 j each designed to project out half of the elements ^±12 = 9 ^ ±iviyi) (2.60) P±0 = -(l±7o), (2.01) where we could use the Weyl representation (Section 2.3.3) but the effects of the projectors are easiest to visualize in the Pauli-Dirac representation: / 1 0 0 0 \ 0 1 0 0 >o = as ii> ^0 = (2.G2) 0 0 -1 0 \ 0 0 0 -1 J 0 -ak Ik = -iv-2 ® fit (2.63) ak 0 22 It is easily verified that 2 _ ,2 j'o - 7- n - i-i - fa - y 2.G4) >> = -7^ for It ¥"- '<•' • We note that because of the pacwoman property[22] P±V2 = ±hil2P±Vi (2.65 P±o = ±7c)P±o • (2.66) Explicitly, P+ P-Q = k (era ± Note that P±i2 and P±o are all linearly independent and commute. The products P+oP-o — 0 = P+12P-12 vanish and the products P±oP±i2 form four primitive projectors that project elements onto minimal left ideals. Thus ^ £ A/«i(4,C)/, where / is the primitive projector[5j I \ 0 (J 0 \ 5 (^0 + ^3) 0 0 0 0 0 / = P)Pl2 (2.67' 0 0 0 0 0 0 0 0 0 0 has only four independent complex components. Explicitly, the; Dirac spinor can be either a four- component column vector or a minimal left ideal of a 23 square 4x4 matrix: V-'2 * G C4 or (2.68) V ^ J ( ipi 0 0 0 \ i/»2 0 0 0 * = G Mat {A, C)f (2.69) V3 0 0 0 \ 04 0 0 0 / The Dirac equation takes the Hestenes form for QPQPM- Thus, iV*P0Pi2 - eAtyP0P12 = -V*POA27J72 - eA*P0P12 = mtfPoi'ri'X) (2.70) which is the Dirac equation in STA. Other projector combinations give min imal left ideals that satisfy the Dirac equation in STA under charge conju gation or time reversal or both. However, the solutions $Fo-Pi2 are generally neither real nor even, as proper solutions of the Hestenes equation should be. Nevertheless, we can get the proper solutions using the following method. First apply the pac- woinan property to make the factors multiplying the projectors both real and even. For this purpose, we need to separate $ into real, imaginary and even, odd parts: (2.71) (2.72) where *± = i ( * ± $ ), ;R* = ± (# + **), and iW = i (* - **). with $ representing the grade involution of xi and $* obtained from fy by complex 24 conjugating all coefficients in the expansion in terms of the basis elements of C/1.3. Then tfP0Pi2 = (K*+ + »*_ + z3tf+ + iS*_) P0A2 = (m+ + »*_->0 - ^*+or/2 - 3#_->0"}n2) fl)^J2 Since the Dirac equation in STA is invariant under grade involution and complex conjugation, the real even part of any solution is also a solution. The real even part of P±QP±V2 is just |, so that the proper solution is just the coefficients: $// = 5ft*+ + 3?*_7o - 3*+0l72 - Cy*_7o7i72 A further independent solution is obtained by noting that if $// is any proper solution of the Dirac equation in STA, so is *//7i72- This shows that solutions "J/-/ to Dirac equation in STA are elements of the even part of the space-time algebra, which is isomorphic to the algebra of physical space, APS. It is also possible to relate the wave function directly with the algebra of physical space, APS, using a matrix representation and picking out the 4-component column vector see [7] [24]. 2.4.3 The Opposite Metric Since the Minkowski space-time metric of signature (1,3) is assumed and used to select the Clifford algebra C/j,;, you cannot can use the same algebra to represent the opposite metric. Instead, you need the algebra Cl\.\. which is not isomorphic with Cl]j. To go to the opposite metric (3.1) from the (1.3) space-time metric associated with the Clifford alge T bra C'li.n ~ Mat(2. il). physicists normally replace 7,, by i~u, (note that Mnt{'2.M) is the 2x2 matrix algebra over the division ring of quaternions). 25 However, such a transition to the opposite metric does not make physical souse since the unit imaginary is not part of R4. However you can perform a transition to the opposite metric and algebra C/3.1 c± Mat(4,R) {Mat(4.R) is the 4x4 matrix algebra over the reals) see [1][5]. Note that the substitu tion -yM —> ieM makes good sense in C (4) or in CL\ over C. It can explain the form of the Dime equation for the opposite metric in C (4) . But of course it can not be directly applied within the real Clifford algebras. The Clifford algebra of 67.3,1, is a geometric algebra of relativistic four- dimensional space-time. Here we use a set of four orthonormal basis vectors {e,j}, tl — 0---3 satisfying ef, • e„ = ?^„ = diag (- + + + ). (2.73) The vectors {e^} satisfy the algebraic relations as e^e,, + e,,e;, = 2r//H,, and they form a set of four independent, basis vectors for space-time. Whereas the basis {70.1\ •, 12; 73} of the space-time RJ'3 generates the Clifford algebra Cl-i^-.i. the basis {eo, ei, e-2, e$} of space-time R'5,1 generates C/3.1. with ei = -l,ei = 4 = 4 = l. (2.74) However, note that generally we want to avoid using i so that e^ 7^ ±'7;c >l Th<^ space-time vectors A "ffl £ 671,3 and A^e^ 6 Ch,\ correspond to each other but have opposite squares {A^,? = {A°f - {A'f - (A*)2 - {A*)1 (2.75) a 2 l 2 a 2 3 2 (^e//) = -(X°) +(X ) +(/l ) + (>l ) . (2.76) Here we will look at Dime equation expressed directly in algebraic form in the complex Dime algebra. For a free fermion in R,,-J the Dime equation 26 changes from (2.57) to the form -V*op = mVop. *oP € C/3,1 «ver C , (2.77) ll where V = T)" eudf,. e„, v = 0.1,2,3 are the basis vectors of Minkowski pace-time R - , and the subscript "op" labels the solutions as those of th opposite metric of signature (3,1). In a 4 x 4 matrix representation, each column of *„?, is a solution to the Dirac equation (2.77). We want to find 3,1 the Dirac equation in K and relate its solutions P±i2 = ^(l±ieie2) (2.78) P±o = ^{l±ie0) (2.79) in C/3,1 over C. We extend the equation (2.77) to include electromagnetic interactions using the traditional substitution V —*• V + icA to get (V + ieA + m) *op = 0. Now project onto the minimal left ideal of PQP\2 and use the pacwoman property to make the equation invariant under complex conjugation and grade involution: (V + ieA + m)*opP+aP+l-2 = 0 (-V#()peoeie2 - eA$>ope0 + m^op) P+0P+v2 =- 0 Next multiply by eo from the right, remembering that e|2, =- — 1. and extract the real, even part: -V*0/)//eie2 - cA^opi, = m^o/)He0 ^>opH & Cl:ij . 27 The equation is exactly the same as the Hestenes-Dirae equation (2.59) in C7|fj, only the basis vectors have been replaced. We add that Vrbik has related the Dirac equation in the opposite metric °f C'3.1 in a language that may be considered an extension of complex quaternions, namely of quaternions with quaternion elements[25j. The four generators of the Clifford algebra of C^j, namely the four orthonormal basis vectors eM, fi — 0...3, generate a 16-diinensional basis for the algebra that can be related to the 16 elements of four quaternions. In addition to the real scalar with basis element 1, Vrbik identifies the three unit pure quaternions as e0eie2e3 = i, e0 = j, e^e^ = k, allowing the 4 vectors to be written as j,kEn, 71 = 1,2,3, the 6 bivectors as ke„ = En , iEn , n — 1. 2,3, the 4 trivectors as JE„, k, and the 4-vector as k. The general multivector can be written explicitly A = (a + \b + jc + kri; f + ig + jh + kq) (2.80) where a, b, c. d: f, g, h. and q represent the 1, i, j, k; E„, iE„, jE„, and kE„ (11. ----- 1.2.3) components, resjjectively, of C/3.1. The multivector .4 can be 28 considered a collection of four quaternions a = a + lb + jc + kc/ $n = /n + IQn + jhn + kqn, U = 1,2,3. Multiplication of any two multivectors is given in terms of the quaternion components by (a; 8) (7; 5) = (07 - 0 • 6; aS + 0-y -0x S) (2.81) where the usual vector dot and cross products are meant. The conjugation of a inultivector A = (a + ib + jc -f kd; f + ig + jh + kq) is the antiauto- morphism A* ~ (a — ib — y: — kd; — f + ig + jh + kq) , which changes the sign of each imaginary unit and, at the same time, the vector part. As the notation indicates, the four quaternions can be considered a quaternion with quaternion elements, where the quaternion units at one level commute with those at the other. Vrbik shows that in this language, the Hestenes equation in the opposite metric takes the form (}0t - kV) *v + A1>vi = 771*,,, vj>t, e M •» M, (2.82) where .4 = j Summary Like the paravector model the STA model has an immediate geometrical interpretation that is one of the best features of Clifford geometric algebras. 29 Since the STA model assumes the Minkowski space-time M1,'< and uses it to select the Clifford algebra C7],3, the opposite metric is outside its scope. However, one can perform a transformation to the opposite metric shown by Lounesto [lj. The space-time vectors are also homogeneous elements of the algebra. The formulation is covariant with a 4 x 4 matrix representation of the Dirac theory. Furthermore, the basis vectors are absolute, not relative: and 70 is hermitian, whereas the 7^ are antiherinitian. The space/time split is not immediate but is given by multiplying the reference frame by 7ol,s<*rvf'r. The formalism can also be easily extended to pseudo-Euclidean spaces of higher dimensions. 2.5 The Algebra of Physical Space (APS) C7;j is W. K. Clifford's (1845-79) geometric algebra of physical (three-dimension Euclidean) space (APS). It is an associative algebra of vector products. It also allows inverses, square roots, and other functions of vectors, as with square matrices. The main differences between fields such as K and C, and APS of vectors are that the product is noncomrnutative and that divisors of zero exist, i.e., there exist elements u 7^ 0, v ^ 0, with uv = 0. This allows us to construct projectors which give APS a richer structure than fields possess. APS provides very natural ways for expressing geometrical relationships, e.g. rotations and reflections, as well as four-dimensional geometrical rela tionships in relativity. Because of this close relationship between algebra and geometry, general Clifford algebras are often called geometric algebras. Calculations in geometric algebra often avoid the explicit basis dependence of coordinate^ systems. This is true for any geometric algebra, but what is 30 interesting about the APS is that it is the smallest possible Clifford algebra that can incorporate all possible relativistic phenomena in space-time, from relativistic quantum theory to electromagnetism[26] [27]. 2.5.1 Generating C7a You can generate CI3 by the real vectors of three-dimensional space, K,! and an associative vector product, satisfying u2 = u • u (2.83) for any vector u G RJ, where the dot product is the usual symmetric scalar product[29]. Since u is any real vector, we can put u = ej+ek. The axiom then expands to 2 e] -f e,efc + eke3 + e , = e, • e,- + 2e, • ek + ek • ek (2.84) This axiom (2.83) thus determines the structure of a Clifford geometric algebra, e,-efc + ekej = 28jk. (2.85) The product of any two vectors can be expressed as the sum of symmetric and antisymmetric parts: ab = I (ab + ba) + ± (ab - ba) 2 2 , (2.86) = a • b + a A b where, from the axiom for Clifford algebras, the symmetric part is just a- b. Note that if a and b are parallel then ab = ba = a • b, (2.87) and if a and b are orthogonal then ab=-ba = aAb. (2.88) 31 The product ab is seen to be a scalar if and only if a and b commute, ami if and only if a and b are proportional and hence collinear. Also, in addition to the scalar part, the product ab generally has an antisymmetric biveetor part a A b = —b A a, which represents a part of a rotation plain1 containing a and b. Any biveetor in the algebra can be expanded as a linear combination of the three unit basis bivectors e^, exe3, e2e3 which represent three perpendicular rotation planes in K'H. This means that APS contains a three-dimensional linear vector space of bivectors that is distinct from the three-dimensional linear vector space of vectors from which the algebra that generated it. The product of all three vectors eie2ea is a pseudoscalar, representing the volume element. A full basis of Cl% has 8 elements {l,ej,ejefc,eie2e3},l < j < k < 3. (2.89) where 1 is the unit scalar, {e^}, 1 < j < 3, is an orthonormal vector basis for R'\ {eie2,e,e3, e2e3}, is the basis for the biveetor space, and {e^e;)} is the trivector representing a volume element. Hence any element of APS can be written in geometrical terms as the linear combination of a sealars, vectors (lines), bivectors (planes) and a pseudoscalar (volume). Rotations A basic but very important concept is the use of bivectors to generate ro tations in a plane. Consider the effect the biveetor e2ei has on the basis vectors ei and e^, (e2ei)ei=e2, (e^ei) e2 = -ej : (2.90) it rotates the vectors in the plane by 90°. Both basis vectors are rotated in the same sense by a right angle in the eje^ plane (see Fig. 2.1). 32 ve^ If ©2 Ci C Figure 2.1. Vector v is rotated by a right angle when multiplied by the unit bivector eie2 for the plane. Any vector in the e\e-2 plane is a linear combination of ej and e-2 and will be similarly rotated. For example, the vector v = i,,Jei 4 v'2e,2 becomes t'(eie2) = v e2 — v e\ (2.91) To rotate a vector in the plane of rotation by 6 we use, linear combinations of rotations of 0 and 90 degrees: v —•> v [cos B + (eiey) sin 0] (2.92) The rotation of any vector v — v'ej •+- r^e^ by an angle ft in the e[ev> 33 Figure 2.2. Vector v lies outside the eie-2 plane. It is rotated by angle 0 in the eie<2 plane. plane can be expressed in a few ways: v —> vexp (eie2#) = exp (e^e^) v (2.93) = exp (e2ei^/2) v exp (eie2#/2) • The last form is useful, since it also correctly describes the effects of rotating any vector perpenticular to the plane, it leaves them invariant: exp (e2e]6'/2) .rexp (ele20/2) = xexp (e2e{0/2) exp (eie26»/2) = x (2.94) for any x that commutes with e\e-2- To rotate any vector r that does not 34 necessarily lie in the rotation plane, one can use r'= exp (e2ei0/2) r exp (eie20/2) (2.95) Note that the form of the rotation is r-^RrR11, (2.96) where R is the rotor R = exp (e2eid/2) = cos - + e2el sin -. (2.97) 2.5.2 Complex Structure and Paravectors The volume element eie2e3 squares to —1 and commutes with all elements[28]. Therefore we set eie2e3 = i. (2.98) The volume element can also be identified with the unit imaginary in Cl~, Cl\\, C/4,1, Cl-2,3, C!,Q,5, CI52, and some other algebras of odd n = p + q but not in C/13. Hence any bivector of CI3 is an imaginary vector, eie2 = eie2e;je3 = ie$ (2.99) Just as we identified multivectors of vector grades 0 to 3, we can identify other elements as multiparavectors of paravector grades 0 to 4. The relation is given in table 2.1. We use here {• • •) subscripted by S, V, Iff.or Cs! or combinations thereof to denote the scalar-like, vector-like, real, or imaginary parts of the bracketed expression (as defined in the subsection below in involutions). Note that whereas paravector grades 1, 2, and 3 have contributions from two neigh boring vector grades (see Table 2.1), the spacetime scalars (paravector grade 35 Table 2.1. Relations of paravector grade (pv-grade) to vector grade (v-grade). There exists a linear space for each vector and paravector grade. The number (no.) of independent elements is the dimension of the corresponding linear space. pv-grade pv-type no. v-grades basis elements 0 scalar 1 0 1 = e0 = (eo>9,\s e fc 1 paravector 4 0+1 fi = ( 7i/i(f 2 biparavector 6 1 + 2 3 triparavector 4 2 + 3 (exe^e^cf or iep 4 pseudoscalar 1 3 (exK,i('u('P)^s or i 0) are the same as spatial scalars (vector grade 0), and that the pseudoscalar element in spacetime is i, the same as the vector pseudoscalar. This permits a simple calculation of Hodge-type duals of elements: if x is any element of APS, even a multigrade one, its Clifford-Hodge dual[5, 29] is defined to be *x - —ix. A general element of C7;; is a complex paravector, which can be expanded in the basis {en, ej, e2,e3} over C. Pai'avectors[6, 29] are scalars plus vectors of the form p = p° + p = p''elt l> with p = (p)Q, p = (p)j, and eo = 1. They get double duty out of Cl\, reflecting a 2 —» 1 mapping of Cl\a —> C7f3 ^ CI3. Starting with a 3- dimensional vector space, APS generates a 4-dimensional linear paravector space with a metric determined by the quadratic form pp = (pp)0 = pfp" (ef,e,y)f| -(P°)2-P2=//V • (2-100) where p ---- p° — p = p^e^ is the Clifford conjugate of p and r/ul, = (e„e,,)0 — i (e„e,; + e,.e„) is the metric tensor of paravector space. Working out the values of )j/u/ explicitly, we see that the Clifford algebra C7;! of Euclidean 3- 36 space RA contains a 4-diniensional paravector space with a Minkowski space- time metric where r) = diag {1, -1, -1, -1} . (2.101) To get the opposite metric in APS, we redefine the paravector space as a 4-dirnensional linear space with a metric determined by the quadratic form -PP = - (pP)o = P'V (~e^)o = -(P°)2 + P2=PV^- (2-102) Now, the Clifford algebra Cl$ of Euclidean 3-space K contains a 4-diniensional paravector space, and it has a Minkowski space-time metric with the opposite metric (~ene„)s = -- (eue„ + e„eu) = 0,tt, where g ~ diag {-1,1,1,1}. (2.103) It is natural to represent space-time vectors by paravectors; the scalar part represents the time component. 2.5.3 Involutions of CI3 There are three types of involutions for APS, and with each having natural actions on elements of the Clifford algebra. An involution is an invertible transformation mapping from an algebra to itself, whose double application gives the identity map. We will use two basic types of involutions in C7;j Clifford conjugation (bar conjugation) and reversion (dagger conjugation). Clifford conjugation is defined as: P = Pa + P ~^P - po - P (2.104) where pq = qp. This is used to split p into scalar-like and vector-like parts: P=r2(p + P) + \(P-P) (2.105) =-(p)s + (P)v (2-106) Reversion p —-> ;^, reverses order in product of vectors and changes the sign of i = eje-^es. In the standard representation of CI3 , reversion is equivalent to hennitian conjugation. This is used to split elements into "real" and '•imaginary" parts: P=\(p + P1) + \{P-Pt) (2.107) = :(P>H + (P>a- (2-108) The combination of the involutions reversion and Clifford conjugation gives another involution which is an automorphism map called the grade involu tion. This map is denoted by bar-dagger, and preserves the order of the multiplication. The grade automorphism is p —> p*. It is used to split elements into even and odd parts: P=^(p + tf) + ^(p-p]) (2-109) = -(p)+ + (p)-- (2.H0) All three involutions are very important ami are used to split the Pauli algebra into important parts and subalgebras. Lorentz Transformations Linear transformations of elements that leave their space-time length invari ant are called Lorentz transformations[29j. Physical (''restricted") Lorentz transformations include spatial rotations, boosts (velocity transformations), 38 and combinations of boosts and rotations. The physical Lorentz transfor mations of a space-time vector p are sometimes called Lorentz rotations by analogy to spatial rotations (2.96): p-+p' = Lptf. (2.111) Here L. which may be called a Lorentz rotor, gives the motion and orien tation of the initial frame with respect to the final one. It can be taken to have the form L = ±exp(W/2), (2.112) where W = w — i6 is a biparavector (spacetime bivector). For an active transformation, where w = 0, then L describes a pure rotation in the plane if) by the angle 0, whereas if 9 ~ 0, then L is a boost of the object with rapidity w. Proper Lorentz transformations may be taken to be unimodtdar: LL = 1. (2.11.3) If L is real (Z, = L*), it represents a boost, B = exp (w/2) = cosh (w/2) + wsinh (w/2) (2.114) whereas if it is unitary f I, = L 1, it represents a rotation: R = exp(-i0/2) = cosf)/2 +if) m\0/2. (2.115) Every restricted Lorentz transformation L can be written as the product L = BR (2.1LG) of a boost and a rotation. As an example, one space-time-vector transforma tion is the transformation of the proper velocity from one frame to another. 39 Ill the rest frame of the particle, its proper velocity is urefil -= 1. In the lab frame, its proper velocity is u — LurestU = LL'. Since L can be written as the product of a boost with a rotation: L = BR, then u is independent of the rotation u = LL] = B2 = exp (w). (2.117) We can relate any boost to the time-like square root of a proper velocity: B = u1/2. (2.118) Matrix Representation A faithful matrix representation of the Ch is obtained by associating the Pauli spin matrices 0 1 —i (7l = \,*2=\, 0-2 = , "3 = , (2.U9 1 0 / r\ i 0 with the elements of a right handed orthonormal basis {ei, e2,e;j}, and the identity matrix with the unit element. The matrix representation of a par- avector p is then • \ o , I' (2'120) /r — ip p — p 2.5.4 Dirac equation In this section we look at the paravector approach to the Dirac equation. This relates classical dynamics to the quantum Dirac equation and is there fore useful in studies of the quantum/classical (Q/C) interface. The classical Dirac equation [22, 30. 31J follows directly from the space-time momentum relation ;; = inu for a particle of rest mass m and proper velocity u. We 40 take r = 1 so that u is a unit paravector liu = 1. A particle at rest with respect to the observer has proper velocity eo = 1 and momentum in. More generally, the momentum p is related to the rest-frame value by a Lorentz rotation, given in the spinorial form p = AmAf =mu, (2.121) where A 6 SL(2, C) is a unimodular element of CI3 : AA = 1. Equation (2.121) can be rewritten in the form pAt = mA, (2.122) which is real linear in A, the classical eigenspmor of the particle, that is, the Lorentz rotor relating the particle rest frame to the lab. The current density associated with the rest-frame distribution pr (x) is j(x) = AprA1 = **t, (2.123) where from (2.122), since pr is a real scalar, $ — pr A also obeys the classical Dirae equation p*f = m*. (2.124) Now equation (2.121) implies that the particle has positive energy. Rel ativity also admits negative-energy solutions, for which p=-AmAt. (2.125) The proper velocity of the particle is u = p/rn = Ae'^A* = e':iB2. (2.126) The proper velocity is the tangent vector to the world line x (T) of the particle, and this makes an Yvon-Takabayasi angle (i with respect to the 41 local forward tangent. In particular, 8 = 0 for positive-energy solutions dt/dr = (AA^) > 0, whereas 8 — n for negative-energy solutions dt/dr = -(AAt) < 0. This result (2.125) is compatible with (2.124) if we put ^ = ipr A for particles of negative energy. The differential form of the standard formulation Dirac equation is obtained by replacing the conjugate momentum ;/' + e.A'1 by id1*. In the paravector formulation, (p + cA) "P is replaced by 18$ e, where e is the spin axis in the electron's rest frame, and 0 :— e^dfl is the paravector gradient operator. This results in the APS Dirac equation /;*f = idrfe - eA$ = ?n*, (2.127) and since physical space-time vectors are represented by real paravectors A we can equivalently, after taking the bar-dagger conjugation, write i<9#e - e.4* = m*f. (2.128) Under a spin transformation L, d —> L dL, $ —> L$>, A —> L AL, and e is invariant because it always represents the spin in the rest frame of the electron. Lorentz covariance is easily shown by transforming these quantities in (2.128). Note that when the momentum is replaced by a differential operator we move to the quantum side of the Q/C interface. 2.5.5 Dirac Equation Relation to Standard Form To verify that the paravector Dirac equation is equivalent to the standard Dirac equation we use projectors to eliminate the factor e = e;j in (2.127) and (2.128). Thus we project the equations onto the minimal left ideal CI.\P of the algebra, where P = ^ (1 -f e:j) . Using the pacwoman property 42 of projectors, e^P — P. we find id^P - eA^P = mMP (2.129) iMP - eAU!P = mtf1P (2.130) By combining the two 2-component ideal spinors into a single 4-component Dirac spinor ip, ( ^P \ w (2.131) q,p we can write the pair of coupled equations as the single equation tp = mil! (2.132) 0 (i&* - eA») W; ip = mv (2.133) {iir-eA")^ 0 x o i 0 -°k (i (2.134) 1 0 "k 0 which gives l Y (idfi -eAIL)iJ> = mi/> (2.135) ith 0 70 = 'U= ° "" (2.136) 10/ \ Tfc 0 which is the usual Weyl representation of the Dirac equation. If $] and $2 are two solutions of (2.127). then any real combination of ^j and $2 is also a solution. However, complex linear combinations are not generally solutions because the appearance of the dagger on only one side of (2.127) keeps it. from being complex linear. However, if * is a solution, so is 'i>e'ea, where n is a constant real scalar, and a complex 43 linear combination of Dirac spinors aie"t[il'i + a,2<-Mi2i>2 corresponds to the ftie !a2e APS spinor «,*e' + a2*e , which is a solution to (2.127). The factor exp(/ne) represents a rotation of the reference frame about the spin axis by an angle 2a. It is a global gauge transformation. Making it a local transformation is of course a reason for introducing the gauge potential A, hint that's another story. 2.5.6 Equivalence of Dirac Theory to APS Version Because of the isomorphisms between APS, the complex quaternion algebra, and the C£~['3. it is fairly obvious that treatments of the Dirac equation in these three algebras are equivalent. It may be less obvious that they are also equivalent to standard Dirac theory, whose representation in C (4) offers twice as many degrees of freedom. Note, however, that the Dirac spinors have four complex entries for a total of eight real variables, exactly the number in APS or isomorpic algebras. Although there is more freedom in choosing matrix reps of operators acting in C (4), it is not used. Tin- projected APS equations are exactly the same as those of standard Dirac theory, and from the projected APS equation, one can derive the full APS equation, complete with the spin term. 44 Chapter 3 Dirac Solutions in APS In this chapiter, the APS Dirac equation is explored in more detail. First, we look at plane-wave solutions to the Dirac equation in APS. Then the APS approach to the Klein paradox is solved, and we see that this gives a simple geometrical interpretation. In last section, we point out a problem with the geometric interpretation of the symmetry transformations in APS. It is here where some limitations of the standard approach, namely PT transformations, become apparent. 3.1 Momentum Eigenstates We now look for solutions of the Dirac equation in free space (A = 0) p^ = id^e3 = mcV (3.1) which are characterized by a given constant momentum p: *„(*) = *„(()) exp (-i{px)„e3) (3.2) The Lorentz scalar (px) s is just mr, where we have assumed that the particle and lab frames have the same coordinate origins. So what does this mean'.' 45 It describes a rapid rotation, at the frequency w — 2m, which is called the Zitterbewegung frequency, about the direction e3 in the particle frame. The spinor field is a plane wave of phase velocity E/p whose magnitude is greater than the speed of light. It. represents de Broglie waves. [32] The factor l 2 *p(0) = e ^ Pr(0)B(0)R(0) (3.3) may describe an additional constant rotation and a boost. Substitution into the Dirac equation gives p*J(0)=7n*p(0). (3.4) Multiplying from the right we get £*P(0)*P(0) = *P(0)*P(0). (3.5) 1 and using 2 ui jp = *P(0)*P(0) = prB = %r The solution may be normalized to a rest frame density of unity: pr = 1. For a positive-energy solution, j3 = 0 then *,(0) = (-)I/2i?(0) = P + m R(0), E > 0 (3.7) wliere R(0) is an arbitrary rotation together with the boost (p/in)^2 gives the direction of the spin in the lab frame. Using the standard matrix repre sentation, ^p(x) is m + E + Vz Px ip v {r) z= ( ~ y \ ^(0)fixp(-»^)ae3) 8 y px + ipy m + E -pz J \/2m(E + in) To find the negative-energy solutions let 3 = 7r but with p — E + p and E replaced by —p = \E\ — p and —E = \E\ , respectively, for \£p(0): 46 3.2 Standing Waves A linear superposition of plane waves of the same spin and energy but op posite spatial momenta gives standing waves. In the low velocity limit, * (J;) = [*p (x) cos a + *„p (x) sin a] i1 + 2^)exp('ip-xs)cosrt +- (l - 5^7) exp (-ip • xs) sin a with probability current J **t - 1 + P.cos2tt+ (cos2p-x + s x psin2p-x)sin2« (3.11) where s = R(Q)e^R(0). Here we have sheets of current lying in planes parallel to the plane of reflection and moving in the directions ±s x p. These sheets of opposite current densities are separated by twice the de Broglie wavelength. 3.3 Zitterbewegung Linear superpostions of the postive and negative energy solutions 5.58 and 5.59 of the same p give Zitterbewegung at the angular frequency 2m in the spin direction s = R(())e:iR({)): *(.r) = [#p (x) cos n + *_p (x) sin a] (l + ^) R(0) exp [—imte;i] cos a f (l — 2^7) R(0) exp [iinte^j isina with a probability current •j = vpO'1 = 1 -f — cos 2o - s sin (2mt) sin 2a (3.1 m where terms of order p2/;u2 have been ignored. When a =• 0. then the current, is in a purely positive energy state 1 + p/m and when a = 7r/2 it is of a purely negative energy state 1 — p/m. The current of a negative energy state is thus in the opposite direction to its vector momentum p. The rapid Zitterbewegung oscillations come from the interference of the postive and negative energy components. 3.4 Klein Paradox We show here how the Klein paradox can be resolved by considering Feyn- man's picture of antiparticles as negative energy solutions moving backwards in time[33]. In the APS approach we see that this gives a simple geometrical interpretation. Stationary states are energy eigenstates, that is solutions of idtVe3 = EV. (3.15) These describe normal modes of the system in which all parts are rotating with a single frequency 2E radians/s about the particle-frame direction ey. <£ (x) = * (x) exp (-ie3Et) . (3.16) The time-independent form of the Dirac equation in the Weyl eigenspinor representation is {E - V -p)# = m^ with its bar-dagger conjugate (E - V + p)*f --= m$ (3.17) 48 The differential form of the stationary-state Dirac equation with a vector potential A = 0 is f {E - V')* + zV*e3 = ?»* , (3.18) where the potential V is time independent. If V is also constant and uniform, the solutions to this equation are plane waves, *{x) = V(0)exp{i{px)se3). (3.19) In the presence of a constant paravector potential A. the exponential fac tor becomes becomes exp (i ((p + eA) x) ^ e;j). With A = 0 and the time dependence removed, the spatial solution is: * (x) = Since we know that p# = —iV\I/e3 we can verify that \f/ (x) satisfies the Dirac stationary-state equation (??). With the solutions yB = e$p?BR (3.20) f * = e-%p?BR (3.21) substituted into ??, we get (E- V -p)B = niBe~f —2 _i2 _ (E — V — p) = inB e 2 = rnu (E-V + p) xi = m so that from uu -- 1, we have {E - V)2 - p = m2 (3.22) which gives p = ±\J{E - V)2 - m2. (3.23) 49 We consider a potential step in one spatial dimension: Vb for x > 0 V = (3.24) 0 for x < 0 Now suppose that our boundary conditions are for a wave that is incident on the step from the left. v0 2 mc R T 1 X Figure 3.1. An incident plane wave I approaches a large potential step from the left, giving rise to a transmitted wave T and a reflected wave 7?. The solution is Aeik-™* + Btrik™3 for :r < 0 *(x) (3.25) Ce!/°'e'! for x > 0 50 where A' and K are given by k = \lE2 - m2, K = \J(E - Vb)2 - m2 (3.26) and A.B and C are constant spinors of the form A = auARA (3.27) B = bulB,B (3.28) C = cu^Rc . (3.29) Here a, b and c are real constants, uA) UQ and uc are proper velocities, n A = — (E + fcei) m uy = — (E — kei) = UA (3.30) in uc — — (E - VQ + A'ef) , in and R.\, RB and Re are rotations of the form, exp(—id/2). By matching the spinors at x = 0, we obtain A + B — C, and when written out. this can be manipulated to give ill 2 au\RA + bxi BRB = cu'cRc (3.31) -u\R4c + -uIRBc = ul, (3.32) c c where R\c = RARC and -Rye = RBRC- Note the rotors R are even ele ments. Furthermore, one can expand the boost factors as 1 + U B = J= , = Ji±l + —JL= (3.33) Consequently, the even and odd parts of (3.32) give - (ORAC + bRiu•) = ^7=£ (3.34) I, „ ,„ > U,1U(V1 + 7.4 U,1U(; u^iVI + Of (7.4 - 1) ^/(l f-,i)(l +"c) (3.35) 51 Adding and subtracting gives 2_a \/l + 1c u.-iuc R > = (3.36) c AC ^l + 1A (7^-l)v/(1+7.4)(l+7c) 1 U — ft , = ^ + "^ -4»C (3.37) c CC v/T+^1" (7.4-l)\/(l+7.4)(l+7c)' Since the vector parts of the proper velocities are aligned, the bivector parts of these equations vanishes, and we can take R,\c = RBC = 1- Using values in (3.26) and (3.30), this leaves a = ^ { v/(# - Vb + m) (E - m) + \/(E - V0 - m) (E + m)} (3.38) 6 = |r { v^ - V0 + m) (£ - m) - \/(£ - Vb - m) (E + m)} . We then get the reflection R and transmission T coefficients, 12 k (k - K) - EV0 (A: - K) v{ R = 2 (3.39) k (k + K) - EVQ (k + Kf - V 2kK T = (3.40) k{k + K) - EVo We see that the reflection and transmission coefficients sum to unity. We can put the reflection and transmission coefficients into a more standard form by looking at the low-velocity limit. Suppose that Vb is very small compared to mc2 ~ E. Then 2 2 2 2EV0 ^ 2EVa - V0 = k - A" which gives the low-energy limits. This then gives us k - A" R (3.41) k + K 4k K T (3.12) [k + A)- 52 Now in the situation that the energy E is below the barrier, E < V'o but lb — E > m we have the Klein paradox: K is real and despite the large barrier, reflection and transmission occur. To resolve this problem we use Feynman's picture that antiparticles are particle states of negative energy (relative to the background potential VQ) moving backwards in time. This implies here that inside the barrier, the solution represents an antiparticle of momentum — A". The interpretation of the wave function with plane- wave factor exp(JA'.re3) for x > 0 as a transmitted fermion is replaced by that of an anti-fermion moving to the left! The non-vanishing probability density deep inside the barrier represents the presence of antiparticles that then annihilate some of the particles approaching tiie barrier from the left. The transmission coefficient T represents the pair annihilation rate, and the reduced reflection coefficient R is a result of total reflection of the iionan- uihilated part of the beam. Unitarity gives 1 — T = R. Of course, pair annihilation is the time reverse of pair creation, which we could calculate with different boundary conditions. The Dirac theory does describe a single fermion, and indeed the com bination of particle and antiparticle above is a superposition of states for a single fermion of given energy E. One might expect that in a sufficiently large electric field, such as needed for the barrier step above, more than one fermion/anti-fermion pair could be created. However, the creation of more than one pair at a given energy and spin is excluded by the Panli exclusion principle. The desire to be able to treat more general cases with the pos sible creation and annihilation of many fermions led to the development of quantum field theory, as discussed below. You may be asking why should we bother with the Klein paradox or the Dirac equation when we are eventually going to use field theory? The very 53 simple case of an electromagnetic step seemed to reveal a contradiction about the nature of the Dirac current. Most of the time in such circumstances we would conclude that this indicated a deficiency in the theory, and that we should go back and correct it. It seems, however, that here in quantum theory only a reinterpretation of the mathematics is needed. This is more than a little odd, but in any case, it is unreasonable to suppose that a contradiction will go away just by making the theory more complicated. There seems to be a fundamental truth in the Dirac equation. This gives the same results that are summarized by Holstein [33], although we believe our interpretation is somewhat more straightforward. 3.5 Basic Symmetry Transformations APS offers simple geometric interpretations of the symmetry transforma tions. These interpretations are hidden in the traditional matrix formula tions. In this section we point out an apparent problem with the geometric interpretation of the symmetry transformations in APS. First, note that Clifford conjugation is not the same as parity inversion P. This is the case since, for example, spatial bivectors should be invariant under inversion. Since only odd elements of APS change sign under P, we need to represent inversion by the automorphism p —> p^[29]. There are also simple forms for space-time inversion (PT) and its product with charge conjugation C, CPT: P : tf -+ ^ (3.43) PT : * -» vp exp zTrn = i*I>n (3.44) CPT : * -^ i*, (3.45) 54 where the unit vector n lies in the rest-frame spin plane ie and is therefore perpendicular to the spin axis e. These transformations can be combined to give the other transformations. T : vp -> <£' exp and C : # -* 4»n (3.47) These transformations correspond mathematically to standard transforma tions of the Dirac theory, but in APS they have simple geometrical interpre tations: PT rotates the rest frame by 180 degrees about an axis perpendic ular to the spin and thus flips the spin direction; CPT changes B —> B + TT (there are some problems with the interpretation of this). To show that these reproduce the standard transformation forms in the usual Dirac the ory, consider for example parity inversion P: and for CPT, where ',5 = — i i>(H) = -= -^ (3.50) but in APS, every element x G Cl;\ in the standard representation, ] e2x* = x e2 (3.51) 55 and with e = e;j and a the angle of rotation about e,;j that takes e2 into n, ln n? = e2e2nP = e e2P. (3.52) Then. (3.53) (3.54) a W fy(w)* = e'* >n/,( ') Time reversal is then given by, 1 (3.55) ie2^*P N/2 \ ?:e2vI/*P ia w) ta (W) = e (T0 0 icr2^ * = e ln^ * = -e?a7(i727()7i72^:i^(H }* = 7oe 7i75V- ' • l;j/2 In the rest frame of the particle, B = 1 and * = pre R = c''% . Then, for positive-energy particles, which have 3 = 0, have vp = The problem arises from the geometrical interpretation of PT (3.44). The PT transformation is seen to correspond to flipping the spin direc- 56 tion in its rest frame. It appears problematic if one can change a par ticle to an antiparticle by a physical rotation! However, the full particle to antiparticle transformation also requires charge conjugation C, which is represented by a reflection, an unphysical transformation. Furthermore, it may be difficult to flip the spin in the rest or the lab frames because of the Zitterbewegung-frequency rotation. Compared to other physical rota tions, it might require a synchronous electromagnetic wave, which would have photons with 2mc~ energy, just enough for pair creation. Therefore, it does make some sense. Now here is where the PT transformation prob lem becomes apparent. The PT transformation, although a proper Lorentz transformation, is anti-orthochronous; the sign of the time component is changed, and is not physical. The PT transformation should be distinct from a physical transformation. We see below in Section 5 on CAPS how this can be resolved. 57 Chapter 4 De Broglie-Bohm Formulation The algebra of physical space provides a natural framework for an extension of the de Broglie-Bohm formulation of causal but non-local quantum me chanics. Louis de Broglie is the one who proposed the pilot-wave approach at the Solvay Congress in 1927[35], and it was Bohm who developed these ideas independently into a practical theory [36] [37]. One of the common crit icisms of Bohmian mechanics[38] [39] is that it is mathematically inelegant, but APS suggests a simple and rather elegant form. However, the existence of a simple formalism does not imply that Bohm's causal interpretation is required. In the pilot-wave approach, quantum theory is about the behavior of particles, described by their positions (or fields) and secondarily about wave fruitions. Here we take the approach of Durr, Goldstein, and Zanghi[40] but in a fully relativistic form. In this formulation the wavefunction, which obeys the Dirac equation, does not provide the complete picture. The the ory is defined by two evolution equations: the Dirac equation for the current amplitude 41 and an equation for the space-time rotation rate of the eiqc.n- spiuor A. Recall that the eigenspinor A (r) of a particle is the active Lorentx transformation from the inertial reference frame, in which the particle is in- 58 stantanoously at rest, to the lab frame. It gives the motion and orientation of the reference frame as seen in the lab. The Lorentz transformation of the eigenspinor is A->LA. (4.1) The eigenspinors at two proper times T\,T-I are related by another Lorentz transformation: A(T2) = L(r2,T1)A(r1). (4.2) The proper time-rate of change of the eigenspinor can always be written A = ^ftA, (4.3) where the biparavector Vt = 2AA is the space-time rotation rate. If A(r) is known, then the time evolution of any properties fixed in the particle's reference frame can be found. Both particles and antiparticles satisfy the same real-linear equation that relates fy to the eigenspinor A : l 2 2 * = e ^ Py A = e'^ip^BR, (4.4) where for positive-energy particles ;.j — 0 (modulus 2TT), and for negative- energy ones 8 = n (modulus 2ir). 4.1 Pauli-Dirac equation Since the Stern-Gerlach experiment is usually performed at speeds much below the speed of light, we need the low-velocity limit of the Dirac equation in APS. It takes the form of the Pauli-Dirac equation. As usual. *(.;:) -= ph'^BR . (4.5) 59 For a slowly moving, positive-energy particle, we have ;3 = 0 and r -C c so that the boost p....l/2_ U+l ~ 1 V2 (7 + 1) and *(a:)~pr" (l + ^v^) " The even and odd parts of ^ are 2 (*)+~pr /? (4.6) In this limit, \p « ^t and (*P)+ = ± ( t) is generally much larger than (^)_ = 5 (^ — H^). However, negative-energy particles have eigenspinors with ;3 = 7T, so that energy is £" « —rnc?, and $ w — ^i. For these particles 4'+ is asinall" and \t- is much larger. The transformation*!' —> ^ represents spatial inversion, and because positive-energy particles are invariant under spatial inversion in their rest frame, they have, as mentioned previously, an intrinsic parity of + 1. Negative- energy particles, have an intrinsic parity of —1. In general, (^)± are the parts of tf> which are even and odd under a parity inversion. W± = 5(*±*t) (4-7) r)±=±<*)± (4.8) Adding and subtracting the two equivalent forms of the Dirac equation in APS p& = id&ez - cA& = m* (4.9) pvp — id^e-A - eAV = m&. (4.10) 60 we find (idt <*) + + iV (#>_) e3 - (7/i + V) (*>+ - eA <#)_ 4.11) (idt (*>_ + *V (*) ) e3 = (-7n + V) (*)_ - eA (#)_ (4.12) whore we have set V7 = e.40. There are both space and time parts of par- avectors, but these are not separately covariant. Also note that the unit, vector e.3, is the spin direction in the particle frame, and it is unchanged by a change in observer. Now consider energy eigenstates for which -idt^e-i = E®. (4.13) For such states, p (*>_ = -iV (*)_ e3 - eA {*)_ = (E- m - V) (*>. ;4.14) p (*) = _iV (*), e3 - eA <*) , = (£ + m ~ V) (*). (4.15) We can eliminate the small component by setting (,P)_ — (E + m — V)~ p ($)_ We then get a differential equation of second degree which looks like the Sehrodinger equation. V p (E + m - VY p (*>+ = [E - m - V) <#) + (4.16) In the low-velocity limit, E 4 m — V ~ 2m and the above e([uation take-; the form of the Pauli-Schrodinger equation for the ''large" component ,2 El (4.17) V m+ =(E-m)(*) + 2v 61 This litis the same form as the Sehrodinger equation when the vector poten tial A = 0. Otherwise we need to use, 2 2 2 2 p (vp)+ =-. (fi' A - V ) (*>+ + c (i (V • A) + 2iA • V - B) (*)+ e:i (4.18) and then project the result onto a minimal left ideal of the algebra with P = TJ (1 + e;j) to get the usual equation. Here B = V x A is the magnetic field. In the coulomb gauge, where V • A = 0 and we can choose a gauge condition with A =jB x r, then 2 2 2 2 p Zi (B • L + B) 2m represent the interaction of the magnetic moments associated with the or bital and spin angular momenta with the magnetic field. Also note that ^ is the Bohr magneton. The term B is commonly written as Bka^. ----- B • rr, which makes it look more like the coupling of the magnetic field with the spin. The current J is the future-directed paravector f j J = *e0* = prAA = p,.u (4.20) gives the flow lines, and the spin distribution is given by t 5 = *e:i* . (4.21) 4.2 Stern-Gerlach Measurement The Stern-Gerlach experiment is a measurement of spin, in which a beam of ground-state silver atoms is split by a magnetic-field gradient into distinct 62 beams of opposite spin polarization[41]. Here wo use a filter method of treating Stern-Gerlach measurements in APS. Consider a nonrelativistic beam of ground-state atoms that travels with velocity v = v&\ through a static magnetic field B ca Be;; that vanishes everywhere except in the vicinity of the Stern-Gerlach magnet. The net effect of the magnet on the beam is a vertical impulse proportional to the z component f.iz of the magnetic dipole moment, which we take to be the magnetic dipole moment of an electron, corresponding to an atomic ground state with an unpaired electron in an S state. The negative gradient of the magnetic-field interaction is the force, given on the average by r-, ™ &B, OB. 1.22) V (// • B) ~ fiz—f-ea = -/i0s • e3-—e3 , and the impulse is the time integral of this. We assume that OBz/dz < 0 so that spin-up atoms are deflected upward and spin-down ones, are deflected downward. The ideal state spinor for the initial beam in the rest frame is •0 = plfBRP3 . (4.23) Any rotation operator can be expressed in the Euler-angle form ' 0' R exp ?:e3 exp —i- exp — ie3- (4.24) r 2j 2j . 0 ( ° • XP exp _te3 1 cos - — in sin -/ea . 2. V 0 " . (0 + X) . e (<;> + x) cos - exp -'ea—r— rn sin - exp -ie;r 2 cos - — msin - ] exp -ie:r 1 1 where n -- exp — /'e3| e2 exp ue3^ is a unit vector in the ejC2 plane 63 Therefore, any rotor R is a real linear combination of rotors: 9 9 R = cos -i?[ -f sin -R[ (4.25 whercfff = exp . R\ = —inRf. The rotors are inutuallv or- -ie3^ H thogonal, RlR\) =(-in)s = 0. (4.26) The rotor Rf maintains the e3 component whereas R[ flips it. The key point is that the rest-frame state spinor can be expanded in spin-up and spin-down terms: w = fifBRP3 = plf (cos |flT + sin ^ ) P3 (4.27) 0 9 cos -0 + sin -i'[ (4.28) 2r'T 2 ( e3V'r = i>] (4.29) and can therefore be "filtered" by the projection operators P±3 applied from the left: 9 P:iy< = cos-t/-'T ;4.30) — 9 P3V- = Sill -?/'j. Every rotor R is a linear combination of "up" and ''down" rotors Rt and 7?i with opposite spin directions. For a state in which the spin direction s makes an angle 9 with respect to e3. R = cos TJRJ -|- sin ^7?; with R.re:iR\ e3 and 04 /?|e;j/?{ ••= — e^. More generally, the spin unit vector for the unboosted system is s = Refill (4.31) The components of the spin unit vector s along e,-j are / + \ f +1- R = R\ s-e:i = (Re3We3) ={ (4.32) X s { _1> R= Rl and are seen to be opposite for the rotors R^ and R[. The Stern-Gerlach magnet forces the state spinor components V-'t arKl V-'| upwards and down wards, respectively. This splits the incident beam into two isolated branches, analogous to the way a birefringent crystal splits a beam of light into two branches of orthogonal polarization. The fraction of the initial beam in the upper branch is found from the amplitude (VT |t/>) = 2 / d3x (v'V'l) ., = cos - (4.33) 2 to be 2 2 (T/>ThO| =COS ^ = i(l+COS0) (4>34) whereas in the lower branch the amplitude is A 0 (! \i>) = 2 d x (Wj)s = sin - (4.35) giving a fraction |(^|w)|2 = sm2- = i(l-coH(9). (4.30) The two-valued property of the measurement is a direct result of the de composition of any rotor R into rotors for rotations about any two opposite; directions, rotations that correspond to "spin-up" and ''spin-down" compo nents. 65 As an aside, note that a nonrelativistic boost of the system perpendicular to 63 mixes the spin states by a small amount: VJ = PifBRP3 ~ p\ef (l + M RP3 (4.37) i 2 i1 + -vV 1 | COcoSs -RtKr ++ S sim n -7?. ! i ) P3 ; Pre} I + 2 ) I 2 2 / , l W 0 . 0 . 2vy i™8^ + sm2VJ The spin-up part of ip is 0 1 0 P3V; = cos -ipl + -v sm - ^ (4.38) whereas the spin-down part is — 10 0 P-i^P = gvsm 9 V;r + sin oV'J1 • (4.39) The generic; form of the Stern-Gerlach experiment can be put into a form analogous to the action of a birefringent crystal on a beam of polarized light: the ideal state spinor is split into 2 parts V = {Pi + Pi) 4' (4.40) which are separated spatially. In the Stern-Gerlach case, each part is asso ciated with a distinct boost, so that the full state spinor, becomes *=p*efBR = 2B(4>)+ (4.41) = 2B+(P3ti,)++2B-(ft«)+, where the boosts combine the velocity v = vei of the beam before the magnetic-field gradient with increments ±Ave;j given by the gradient: B± ----- exp (±Att-e3/2) exp (iuei/2) (4.42) 1 , . . 1, 1 + - (i'ei ± Are ) = 1 + -V. 2 3 H 66 an< n e 2 ex With 0 = 0, x = (px)s' l = 2, w« have u'-'r = P, ef l- -ie-j^Pi PA and (/)j = —n0f. The "up" and "down" spinors can therefore be selected by the filters ( projection operators) P±3 applied from the left: i5f/2 P3^ = co8|vT=/'i/e- cos|p3 (4.43) l (9 2 (P«H>)_ pi e x/'->cos rref ixl 2 P3t/> = sin -V-'i = Pre.fe~ ' sin -P3 (4.44) 5 —i\ 12 • 6 sm e e 2 The full state spinor is *=2B+ If the initial beam has finite profile pref(x), the action of the Stern-Gerlach magnet will eventually split the beam into two distinct beams moving with velocities V± and with opposite spin directions and distinct profiles f>± ~ prrf (.r — ~V±t). The proper-velocity is given by the current density ,/ = ^Pi)¥ (4,46) e e \ ( e e = ( P>+ ™S X + B~ Sm 9eie3 J Pref [ B+ C0S o + SU1 Q6^1-0- (4.47) ,0 - + B_p- sin -e + Prej sinfl {B e- e B _) (4.48) =-- p + p+. cos" + A { K since the cross terms cancel. At some distance down the beam past the magnet, the 2 sub-beams become nonoveilapping and there are two distinct 67 beams. The corresponding spin-density profile is S = Ve3& (4.49) o o \ foe B+ cos - + B- sin -eje3 1 e3pref I B+ cos - + sin -e3ej B. ft ft 2 2 p+B+e3B+ cos - + p^B-e^B- sin - + pref sin 9 (B+e\ B-)-% Algebraically p± = pref but the argument is appropriately transformed 1/2 from the rest-frame coordinates. For consistency, we can take? Bpr'Z = 1 2 {BprejB) ' so that the pref factors in the sin ft terms in the above expres ! sions for J and ^e^* are to be calculated as the geometric mean (p+p_) . Using B± = 1 + |V± and discarding terms quadratic in V± we get B% ~ 1 + V± B+e3B+ = e3 + 2V+ • e3 = e3 + A?; Z?_e35_ = e3 - A?; {B+e3eiB-)n = - (V+ - V^)e3e{ = Avei {B+e{B-)n = ei + v. This gives ./ = \p+ (1 + V+) (1 + cos B) + !/,_ (1 + V_) (1 - cos 0) + y/p+p^Avei sin 9 (4.50) 5 -•= -/9+ (e3- A r) (1 -f cos 0) - -/>_ (e3 - Av) (1 - cosfl) + yjp+p- (ei +'i;)sin(9. (4.51) In the case 0 = TT/2 (same fraction of each spin direction in the initial beam). 68 this reduces to J = -p+ (1 + V+) + -p. (1 + V_) + v/pTpTAue, 5 = -/J+ (e3 - Ac) - -p_ (e3 - Av) + JpTi^ (ei + i') which makes good sense. The last term in each expression is evidently an interference contribution. It will die away as the distributions separate and cease to overlap. The scalar terms in the spin distribution arise because of relativity: it is really a distribution of the Pauli-Lubanski spin, and the boost of the rest-frame spin has a scalar contribution, but this is small because the velocities involved are much less than 1 (the speed of light). At different times t for a given finite distribution prej of the incident wave packet gives the current J as shown below (see Fig. 4.1): Current Density 0.7" -G- Figure 4.1. Ferniion beam splits into two beams in the eoe;i plane. 69 and the spin distribution as shown here(see Fig. 4.2): 3- 1 ll / / - 1 J" 1> s-S D b.1^ 0,4 0 6 0 3 1 12 14 IB 18 2 f | 3- 1 1 Figure 4.2. The fermion beam is split by the magnetic-field into spin-up and spin-down parts. The approach taken is different then that of Dcwdney[42] and Challinor[43] who look at wave packets that receive a time-dependent impulse from a field gradient F = i (dB/dz) ze^S (t) . Although our approach gives similar re sults to those of Dewdney and Challinor, it is a time-independent plane-wave approach that has the added benefit of illuminating the quantum/classical interface. 70 Chapter 5 Complex Algebra of Physical Space The complex algebra of physical space, CAPS, is the complex extension of APS. It is also the hyperbolic extension of APS. Here we will introduce a covariant interpretation of CAPS for use in relativistic problems. We look at CAPS to solve a problem, namely that the PT transformation of inversion in space and time, although a proper transformation, is anti-orthochronous and thus aphysieal; yet, it has the form of a physical rotation in the tradi tional Dirac theory as seen geometrically in APS. A further problem is that the explicit form of the T and C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. CAPS, the complex extension of APS, offers a resolution to these problems. Another important function that the hyperbolic numbers provide is the separation of the paravector grades. The basis of the ground vector space is {ei,e2,e:{}. In APS. the volume element eje263 is assumed equal to the unit imaginary i because it shares its basic properties: it squares to —1 and commutes with the vectors {ej, e-j. e.j} and their products. There are times in physics, however, when it is useful to introduce a unit imaginary free from any geometrical association with the 71 volume element, and there is no fundamental reason that eje^e.'j has to he i. Thus, in CAPS, which is the complex extension of APS, we assume that independent of the volume element, which we denote j = e^e^, there is a unit imaginary scalar i that squares to — 1. In CAPS, we can therefore use complex numbers that are unrelated to the volume element. Since i' ----- — 1 — j2, we can write j = ih where h = —ij is the unipotent element (hyperbolic unit) with h2 = 1[44][45, 46]. We can use the hyperbolic unit to distinguish a space-time vector p — p'e,, from the paravector pa + p. We put v p = h(p° + p)=p ev. (5.1) The space-time basis is therefore {e,,,} = {h,hei,he2,he^}. Note that the volume element in space-time is the same as the volume element in physical space: e0eie2e3 = j = eie2e3 , (5.2) but that the spatial volume in space-time differs. It is just the unit imagi nary: eiC2e3 = -eiC2f;3 =-' hj = i. (5.3) The geometric significance of h is that it is the time axis, that is, the ob server's own time axis with respect to the observer. To recover the original APS we use the minimal two-sided ideal of CAPS formed by the abelian projector P+ — T> (1 + h). A fully equivalent subalgebra, called anti-APS, is the ideal of CAPS formed by the projector P_ = i (1 — h) . APS does quite well for most physics so that a distinct unit i is not needed in most applications. The use of h is mainly a convenience for identifying in covarianf usage the paravector (space-time) grades of nmltivector elements. 72 For example, a spatial vector v with no h factor must be part of a space-time ll bivector whereas /tv is part of a space-time vector v = v eft. This coincides k with the view that v = v eiie$ is a persistent vector that sweeps out a time like plane (a space-time bivector) in space-time. Where i is introduced, it needs to be consistent with the algebra and with i = hj. which identifies its space-time (Clifford-Hodge) dual as the time axis of the observer. Thus. i =- t'iC2fi3 — ~*eie-2R3 is the volume element of the spatial hypersurface of the observer, formed from the product of space-time unit vectors rather than from the persistent unit vectors e^. The element i can also be called the space-time pseudovector along the time axis of the observer. In CAPS, i and j both have geometric significance as volume elements but they are distinct. In APS, they are equal, whereas in anti-APS they have the opposite sign. To make sense of the physics described by anti-APS, recall that the unit vectors in APS are relative vectors, that is, relative to the observer. In anti- APS, h = —1 so that the time flow is opposite to that in APS relative to the observer. The time components of particle momenta, that is, the energies, also are negative relative to the APS observer so that such particles are identified as anti-matter. Our approach differs from that used by Ulrych[47]. Ulrych has defined a hyperbolic number with the form ; e H z = x + iy + h (v + iw) , x. y, v. w £ M (5-J) (Ulrych uses j in place of h here). His volume element is given by / = hi. His conjugation is an anti-involution that changes the signs of both i and h z = x + iy + h (v + iw) —* 3 — ,r — iy — h (r — iw) . (J>-^>) 73 Ulrych's paravectors are written in the form, X = i° + /ix. (5.6) The conjugation X is equivalent to Clifford conjugation. The 'hyperbolic coniplexification' of CI3 is the same as the usual coinplexification: C73®H~C73®C (5.7) where a general element of H (K) is z = x + hv, x, v G M. 5.1 Conjugations The involution of Clifford conjugation reverses the sign on the vector parts: p = h (p° + p)->p=h (p° - p) , (5.8) in other words h — h. The Clifford conjugate should not change the sign of j = eie2e3, j = j and therefore it should be an antiautomorphism, it reverses the order of multiplication, pq = qp. It follows that 1 = i. The scalar-like and vector-like parts of p are the time and space components: (p)s = T2(P + P) = hp° (5.9) (l>)v = l(p-p) = hp. (5.10) Reversion (f conjugation) reverses the order of multiplication and is thus an antiautoinorphism. Evidently ] t j = (e1e2e:i) = e3e2ei =- ^eje2e:, = -j (5.11) i* --= {ci('2'--s) — ('-,AC-2('\ = -f-'if'2^.'t = -?' (5.12) 74 X X .rt a-t X* X'1 = X xt* = 1 1 1 1 1 1 1 1 I ?' — i —i — i i i 3 3 -3 -i 3 ~j ~3 h h h /i -h -h -h e/c -e/t efc -efe efe e/c -efc yt~t XJZ ?.V *V J/*i* ~y p Table 5.1. The effect of conjugations in CAPS on basis elements. Various conju gations and their combinations are listed across the top row . h) = h. (5.13) We use the dagger conjugation to isolate "real" and "imaginary" parts of an element x : (x)x=\(x + x1) (5-14) <*>* = £ (*-**) (5-15) Complex conjugation (*) is an automorphism that changes the sign of i but not of j. For consistency, it also changes the sign of h. It is the space-time vector grade involution. The combination of bar and dagger conjugations is the vector-grade automorphism for space-time even (relative- vector) expressions. To summarize in table form: Iu CAPS, unlike APS, reversion x = x'* (the tilde conjugation) is dis tinguished from Hermitian conjugation x^. Note that Lorentz rotors are space-time-even and therefore are even in h. Since Lorentz rotors can al ways be expressed as products of rotations in space-time L ----- RB and are generated by biparaveetors (space-time planes), they are real: L = L* and 5.1.1 Metric Vectors of the ground space are "persistent, vectors'' and represent time-like planes in space-time: ek=ekeQ = hek. (5.1G) These correspond to Hestenes' relative vectors. The metric is determined by the '"square length" pp = l/lpuV^ (5.17) where r)fW is just the Minkowski space-time metric tensor. It is given explic itly by {e^e„)s = - (eMe„ + evetl) = r/^ (5.18) CAPS is APS over the complex field C , but it can be considered the 8- dimensional algebra with basis {l.e*..,^} over C , or as the 4-D algebra {l,e^} over the commuting ring {l,i,j,h} , or as a 16-dimensional algebra with commuting centre {l,i,j, h} over the real number field R. 5.1.2 Matrix Representation of CAPS The matrix representation of CAPS may be formed with 2x2 matrices over the abelian ring {l,i,j.h} : e;i / h 0 \ / 0 h \ 0 -i \ 0 h ) \ h 0 J \i 0 76 Note however that {l,i,j%h} is not a division ring because 1 ± h is not. invertible: they are divisors of zero: (l + /i)(l-/i)=0. (5.20) One could represent the ring by 2 x 2 complex diagonal matrices: 1 = (TQ,i = i<70, h — The result of inserting these into the 2x2 representation above is a set of 4x4 matrices. Then for CAPS we have, / 1 0 0 0 \ I % o o o \ 0 1 0 0 0 i 0 0 1 = . I = .22) 0 0 1 0 0 0 i 0 \ 0 0 0 1 / \ 0 0 0 i J / 1 0 0 0 \ ( i 0 0 0 ^ 0 -1 0 0 o -?: o o J (5.23) 0 0 1 0 0 0 i 0 \Q 0 0 -1 j \ 0 0 0 -i / 5.1.3 Comparison of Space-Time Unit Vectors The space-time unit vectors of the different algebras used here are compared in the table below: As noted above, H ® H ~ C&s.i which is not isomorphic to STA {Cf\.^). On the other hand, complex quaternions are isomorphic to APS: M (<) C ~ CT3 - APS. In PIYDC, a space-time vector, called a minquat, is, as discussed above, / / en APS M®C CAPS H fio 1 1 /i j 70 «i a i ii h(T\ kl 71 e2 CT2 (j /ifJ2 kJ 72 c-'A V'A ik /;cr;j kK 73 f'0123 i i j = /" i 75 Table 5.2. The space-time basis vectors used in APS compared to those in other algebras used here. a complex quaternion with a real scalar part and an imaginary vector part: x = x° + i (x1! + x2j + x3k) (5.24) 2 :i xc = x = x° - i (Vi + x j + x k) Complex conjugation x —> x* is an automorphism that, only changes the sign of i. A minquat is defined by xc = x*. The scalar product is (x,x) = xx (5.25) (.T, y) = - {xy + yx) = x • y = y • x The wedge product is x A y = —y A x = - {xy — yx). (5.26) In !H3 C) H, a space-time vector has the form x = j.x° + k (Vl f x2J + x3K) (5.27) Xr = — X . where conjugation a changes the sign of space-time vectors and bivectors (space-time grades 1 and 2) while leaving scalars. pseudoscalars, and trivcc- 78 tors (space-time grades 0,3,4) unchanged. The quaternion units i.j.k com mute with the units I, J,K. The asterisk now means the dual, defined by A* = \A (5.28) for any element A. Since the volume element i anticoinmutes with space- time-odd elements, space-time vectors x obey :r* = \x = -xi (5.29) 5.2 CAPS Form of Dirac Equation The classical Dirac equation arises from the transformation of the momen tum from the rest frame to the lab: p = AhmA, (5.30) using the invertibility of the eigenspinors, we find the equivalent form pA = hmA. (5.31) Then multiplying by p2, where p is the rest-frame density and defining ^ = f>2 A for positive-energy states, we get vq> = hm$>. (5.32) Note that the positive-energy spinor $ contains no i (is space-time even) and does not depend on the sign of h, so that "I1 = ty*. A negative-energy \&, however is space-time odd (odd in i) and does depend on the sign of h, and * = -v]/t. The first-order CAPS form of the Dirac equation is derived from the classical Dirac equation. This is done by replacing the momentum operator 79 bv its differential form pty = jd^e-i - e.A'i = /mi* (5.33) or its bar-tilde conjugate, which is fully equivalent, |j* = j#*e3 - eA'H — hm^. (5.34) If we were to express the equation in terms of the bar-dagger conjugate rather than the bar-tilde one, we would get different signs depending on whether the spinor was orthochronous or anti-orthochronous. In the orthochronous case, f p* = jd^e3 - e.A^ = -/im*. (5.35) Recall that h is the time axis of the observer. For regular particles, the momentum p (and the proper velocity -^) is directed with its principal component in the forward (future) lightcone: p = A/trnAt. (5.36) However, if the sign of h is reversed, the momentum and proper velocity lie in the backward (past) lightcone. Particles traveling backward in time are equivalent to antiparticles traveling forward. Changing the sign of h also changes the vector parts of space-time vectors, but not the persistent vectors that are formed from a single tetrad. However, space-time vectors in CAPS are relative to the observer. Consequently, a change in the sign of h changes the How of time relative to the observer. 80 5.2.1 Relation to the Standard Form To relate the CAPS form of the Dirac equation to the standard form, we split the Dirac equation into two complementary minimal left ideas, (CAPS) P:i '5.37 (CAPS)Wi where p$ — ~,(l + e^). Also note that e^P-j = P3. We now apply P^ and P-i from the right to the CAPS form of the Dirac equation and take the bar-tilde conjugate of the latter, jd^lPz - eA^/P-i = hmVP3 (5.38) jd^Pi - eA^Pj, = hm$>PA. (5.39) Note that the bar-tilde of P3 is P3 and vice versa. We now stack these together to get a 4 x 4 matrix operator in block form: 0 jd-eA \ I *P3 \ I *P3 5.40) jl) - e~A 0 ) \$>Pi / \ $P3 Expanding the space-time vectors, so that d = /i (#° + e,tdfc), and d h (0() — ekdk) we get the form, *Pi (jd* - eAk) 10/ V efc 0 v|p. This can be expressed in the usual quantum form (5.41) where 1;: the four-row bispinor is given in the Weyl representation, 1 (5,12) 7! ^Ps 81 Also. e;3 is given by the 2x2 matrix 03, and the gamma matrices are given the usual Weyl representation[34]: 70 = ax ® (T0 = I I - V> (5.43) (To 0 (5.44) (5.45) In Clifford algebra it is natural to define 75 in terms for the volume element, so the extra factor of ±j common in many representations is omitted. However, the matrix representation of ijAw> is a 4 x 2 matrix, where the second column is zero, and since it is only operated on the left, it can be represented by a single column. Now do we have a problem if APS is equivalent to the usual Dirac equation and CAPS is as well? In this case CAPS doubles the degrees of freedom, allowing all the basic symrnetre operations within the algebra since the elements are bicomplex and can be expressed as i/>i +/i't/>2 or as ip'\+H'2 > where 'i, '(/>2, V-'i, and ''''2 are J-complex functions. 5.3 Fundamental Symmetry Transformations There are a number of possibilities of representing the fundamental sym metry transformations in CAPS. In CAPS we can use complex conjugation independent of reversion, which is not available in APS. There are several ways to avoid the problems encountered in APS of a nonphysical transfor mation having a physical form. To choose among these, we are constrained 82 by the rule that the CAPS transformations should reduce to the APS ones in the ideal with h = 1. We can represent parity inversion P of multivectors in CAPS by bar-tilde conjugation, and space-time inversion PT by complex conjugation. The Dirac equation p$ = jd^e3 - eAV = hm^ (5.46) is invariant under complex conjugation (all terms are space-time-odd and thus change sign), although the APS (particle) and anti-APS (antiparticle) components are interchanged: P±tf — (P±tf)* = PTvp* (5.47) where P± = ^ (1 ± h). Consequently, if \1/ is a solution to the Dirac equation so are ty* and i$>. More generally, Vl/e1^ is also a solution. The bar-tilde transformation ^ —> $ transforms the Dirac equation into p* = j<9#e3 - eA^> = km® (5.48) and the bar-tilde of this equation gives us the original CAPS Dirac equation with 0 and A replaced by d and A, respectively, which is equivalent of parity inversion P. -pip = -jd^e^ - e^A~qj = -hm^ (5.49) The bar-tilde combined with complex conjugation, which can be written xi —+ *i' —- ^ , effects the same change in the Dirac equation. The duality transformation $ —* — jfy effectively changes the sign on the mass term, as does its combinations with complex conjugation and rotation. * -^ -j**. * —> /i* and * —> /i**. Any of these might be considered as CPT. If we combine CPT and P. we find d —> —d and e.A —* — R.4 which is 83 indeed effectively CT. (Recall that A behaves like the current density under T: A —» A, the same change as under P.) It remains to find separate C and T transformations of ^. The transfor mation 4< —-> ^n with n any unit vector perpendicular to the spin direction e.-j in the rest frame, has the effect of changing the charge and thus might he associated with C, as might variants such as $ —* j^n or 1i/*n. There are thus several ways to avoid the problem encountered in APS of nonphysical transformations having a physical form. To choose among these or others forms, we would like these transformations to reduce to the APS forms when h = 1. We are also constrained by the classical transformations such as those of the momentum p = hArnA and of the Pauli-Lubahski spin w = ZiAegA . Thus P: ty —-> ^f works since for p since it gives p —> — p = fJ. For w it gives w —> —Hi, which agrees with the fact that the spin is a pseudovector whose vector part does not change sign under P. Under T, using T : *P —» jvP n, we get -t -f -t -f _t p —> hjA n'mnA j = -hA rnA = -;jT (5.50) w —+ hjA neanA j = hA e^A = w^ We summarize our choice in the following Table: The CAPS transformations do correspond to those for APS in the ideal with It ---- 1. since there / -= j. 84 CAPS APS P: * -* 1 * _• $t C: * -» #*n vp —> *n X T: * -+ j^n PT : vj; _^ jty*n * —> i*n CT : \J/ -» j\j> * —• i^t PC : * -> -*fn * —> -^n PC'T : tt—j* * —• i* Table 5.3. Basic symmetry transformations used in APS compared to those in CAPS. 5.3.1 Momentum Eigenstates Consider solutions of p* = jd^e-i — eA$ = h'tnty in free space (.4 = 0) characterized by a given constant momentum p: *PW = *P(0)exp(-je3(pi)J (5.51) Since {px)s — rnr, where we have assumed coincident coordinate origins in the particle and lab frames, typ (x) describes a rapid rotation at the Zit- terbewegung frequency u = 2m about the direction e3 in the particle frame. The spinor field is a plane wave of phase velocity E/p, whose magnitude is greater than the speed of light. The factor *;) (()) = e*p? (()) B (0) R (0) may describe an additional constant rotation and boost. Substitution into the Dime equation gives p*p (0) - hm*p (0). (5.52) ji Multiply from the left by ^~ and use the normalization ** = prc to 85 obtain the constant current 2 J = q> (0) /i* (0) = prB = ?-pTe-rt. (5.53) m The solution may be normalized to the rest-frame density of unity: pT = 1. For a positive-energy solution, 3 = 0 and we can write 5 *P (*) = (£) flexp (-je3 (px>s) (5.54) i where /? is an arbitrary rotation which, together with the boost (p/m)'2-, determines the direction of the spin in the lab frame. The bar-tilde conjugate is 2 lp (x) = (j£\ Rexp (-je3 (px)s) (5.55) In the limit p —> hm, \& —* i?exp (jmiea), which has positive parity. The momentum eigenstate with momentum — p is *~p (x)=(~^02 ^exp (je3 ^s)=?:*p (_-';) • (5-56) Wliereas ^ is space-time even, the anti-APS $!-p (x) is space-time odd. The standard matrix representation of ^lp (x) is , m + £+P, Px-ipy \ Rexp(-je3(px)s) Px+iPy m + E-pz) \J2m(E + m) Negative-energy solutions are similar, but with p = E + p and E replaced by —p — \E\ — p and — £ = |£|. In the limit of low velocities, 1 *», + ,, (a ) = f 1 + ^-) R (0) exp (-je:l (mt - p • x)), £ X) (5.58) and *»,-P (J") - j f 1 - ^-) i? (0) exp (je;s (m< + p • x)) . E < 0. (5.59) 86 Note that p is the momentum of both the positive and negative energy electrons, and consequently, the corresponding positron has momentum —p. Linear superposition of the positive and negative energy solutions 5.58 and 5.59 with the same p give Zitterbewegung at the angular frequency 2m = 2mc2/ft in the spin direction s = R (0) e^R (0): *p (x) = [#m+p (x) costv + *_m+p (x) sin ft] (5.60) This is the same solution found in APS. 5.4 Canonical Quantization The Dirac equation has had considerable success when treated as a single particle equation. It gets the correct magnetic moment of the electron and the energy levels of the hydrogen atom, but one of the initial difficulties with the Dirac equation was the existence of negative energy levels. From these, Dirac predicted antiparticles. He did this by redefining the vacuum as having all negative energy levels filled, the exclusion principle being crucial here. The problem is the prediction of pair production amounts to the abandonment of the Dirac equation as a single particle equation. One way to overcome this is to think of the Dirac equation as a field equation, and then to quantize the field. The field is then an operator. APS and CAPS seem well suited for such an approach since the eigenspinor is already an operator, one proportional to a particular Lorentz rotor that describes the particle motion and orientation. Furthermore, there appears to be no inherent restriction to a single particle as seen from the classical origin of the equation. To treat multiple-particle systems we can introduce annihilation and creation operators. Here we show how this is done within the CAPS approach. 87 Following the established method[48], we need to look at the energy of the Dirac field. In order for the energy to be positive, the Dirac field must obey Fermi-Dirac statistics. We also need to find a Lagrangian density. First, it is easy to show that the Dirac equation jd^e3 - eA* = hmfy (5.01) and its bar-dagger version j5*e3 - eA^ — hmty (5.62) follow from the Euler-Lagrange equation d*L-d(da*L)=ti. (5.63) Taking the Lagrangian density as L = - /j*e3*# - jd^e^ + m (*¥ + ** J + 2eA**\ , (5.64) we find the required derivatives to be — = - (je^d - jdVea) + -in f* + #) + eAV . (5.66) The Hamiltonian operator in CAPS Dirac theory is then defined by //* = jdt^e3 = {V + p)# + hmV (5.67) This has the expected form, but it can be shown that the negative energy and plane wave solutions of the Dirac equation 5.33 give a negative contribution to //, which is therefore not positive definite. This problem can only be removed by quantization. 88 The general solution to the Dirac: equation as plane-wave superposition of annihilation and creation operators is: 1 * 4 + Batc-je3{kx)» (5.(38) v ""v a=£ 1. 2 I' -"*"' 1 3 m ^-> (5.69) a=1.2 where ^ is now an operator and (kx) s = ut — k • x. Also note that /r = UJ (1 f k) is a null paravector such that kk = 0. The Hamiltoiiian is //^ = jhOt'i/ej,. The total energy is given by 2 E (fxH = / d x*j5f*e3 (5.70) 3 3 (a+ (fc) A (fc) e^ + aQ (A;) B (k) e~^ 1 x (a,y (k) A (k) {jJ) eJ''°* + a+ (k) B' (k) {-jj) c ->'"=• ^5Z{ - (a+ (k) A (k) (JLJ) e>«* + an (k) B (k) (-JLU) e~J*°* x (aa, (k) A' (k) e~J*<* + a+ (k) B' (k) ej™) where we have set s = (kx)s. For the energy to be positive definite need to introduce the anticommutators {a, b] = ab + ba (5.71) with the anticommutation relation, first proposed by Jordan and Wigner: {<**.«£} =/>k.k> (5.72) {a. a} == {a+,a+\ = 0. Now the total energy E is positive definite and the anticommutation rela tions imply fermi statistics. Finally, we need to show that the equal-time 89 antkomnmtator is {* (x, t) . * (x', t)\ — 6 (x - x'). Since * is a four com- ponent spinor, more precisely we need to calculate *i(x 2 1 , m aa (k) A? (k) e-J«» + a+ (k) B? (k) e^. E ^ / / <^* JUJ a I o+(fc')^ {k')eJ °* + aa,{k')Bj' (k') The only surviving anticomnmtators are Af(k)Aj (k'){aa(k),a+(ki)}c->^+^ = Zv* [[****£+B« (k) B? (A;') {a+ (k), a„, (*')} e^~^ (5.75) 1 where {«„ (fc), a+ {k')} = %5(k- k ) Sa.a,. This gives r= £ _L /" ^ ^«^"e-j(—')e3 + ^B/V(*-"'>•*) (5.7G) d3;r fc l j(s 8 )e3 = ^2 / rj (>' - + " )ij e- - ' + (w + * - m))J eJ'(»-"')^) . Considering that the integral over the volume contains ~ I d3!^'-8')"* = I / d^e^-"'''11 = 5 (:r - x') , (5.77) this then gives = _L J ^xS'-^Hij = 6 (,: - ./) S„. (5.78) 5.4.1 Annihilation and Creation Operators As an addition, we note that even though the annihilation and creation op erators are outside the CAPS algebra, they can he represented by a Clifford algebra in the following way. For a system of N fermion states, we define a 90 : symmetric 2N dimensional anti-abelian basis (C/iv,,v) {61,6_1!62,«'_2,...,6A-,6-.V} (-3.79) 2 with b'l = 1 = ~b _k, % {bjbk + bkbj} ~ sgn (k) Sjk. The basis elements of positive index are assumed to be hermitian, whereas those of negative index- are antihermitian (imaginary): b+ = bk (5.80) btk - -b.k. The combinations give the annihilation and creation operators «A- = \(h~ b-h) = \h (1 - hb-k) = r2 U + bkb-k)bk (5.81) 4 = ^ {bk + 6_fe) = ^fc (1 + fefc6_fc) = ^ (1 - bfcft-fc) h which are nilpotent: 4 = 0=(a+)2. (5.82) Note that the annihilation and creation operators for different states anti- commute a,jak = - (bjbfc - 6J6_A; - b-jbk + b-jb_k) (5.83) "fc«j = 4 (-&A- + 6-A + ftj*M- " &-J&-A-) (5-84) so that ajak+aka, =0 (5.85) and the factors P± = ^(l±bkb.k) (5.80) are idempotent hecausr {bkb-kf = bkb-kbkb_k = -bkbkb..kb-k = 1 (5.87) 91 The P± are also real: Pi = \ (1 ± btkbt) = \{\T b-kh) = P± . (5.88) If we start with the vacuum state |0) . application of a^ creates one feiniion in the kth state and ak annihilates it. 5.4.2 Number Operators A single-particle operator of particular interest is: nk = a{ak (5.89) The coninmtation rules for creation and annihilation operators give the fol lowing commutation relations, a\8kk, (5.90) -ak8kk, (5.91) ' ,t m nkAak, m(a{) (5.92) These hold for both bosons and fermion operators. From the last equation, m m »ifc(4) |())=m(4) |0) . (5.93) The state describing in particles in the quantum state A: is an eigenfunction of the operator nk with the eigenvalue m. The eigenvalue is the number of particles in the state k. 92 Chapter 6 Conclusions In chapter 2, we looked at three different Clifford (geometric) algebra for mulations to the Dirac equation. We showed that these formulations are isomorphic to each other. The main characteristics of the paravector model include an immediate geometrical interpretation, which is a strength of most Clifford geometric algebras. The classical Dirac equation, the correspon dence of A and >J/, and its relation to quantum formalism illuminate the Q/C interface. The Minkowski space-time metric is also internally gener ated and appears naturally. Since standard physics does not distinguish between the space-time metric signatures (1,3) and (3, 1) the ability to have both in Clifford geometric algebra Ch is satisfying. Relativity is essential part of this approach to the Q/C interface. The formulation is also eovari- ant and has a 2x2 matrix representation. The basis vectors are also all hermitian and relative to the observer. The space/time split is immediate and is given by the vector grades. The paravector model also reflects on the dual nature of some quantities such as mass, which is equal both to the energy component in the rest frame and to the Lorentz scalar representing the "length" of the space-time momentum. The formalism is easily extended 93 to Euclidean spaces of higher dimension. Like the paravector model in APS, STA has an immediate geometrical interpretation. Since the STA model assumes the Minkowski space-time metric and uses it to select the Clifford algebra C/1,3. the opposite metric is outside its scope. While you can perforin a transition to the opposite metric, as shown by Lounesto, the process is hardly straightforward. The space-time vectors are also homogeneous elements of the algebra. The formulation is covariant with a 4 x 4 matrix representation of the Dirac theory. But the basis vectors are absolute, not relative; and -70 is hennitian, whereas 74. is antihermitian. Furthermore, in contrast to APS, the volume element in STA anticommutes with vectors and thus acts more like an additional dimension than as a true pseudoscalar. The space/time split is not immediate and is given by multiplying the reference frame by ^ol)Server. The formalism can also be easily extended to pseudo-Euclidean spaces of higher dimensions. The characteristics of the complex quaternion model includes a Minkowski space-time metric that is internally generated but seems contrived. How ever, it can be easily modified to represent the Minkowski space-time metric of the opposite signature. The treatment also reflects the dual nature of some quantities such as mass (= energy component in rest frame or Lorentz scalar). The space/time split is immediate and is given by the quaternion split. The complex quaternion model also has a 2 x 2 matrix representation, but here the spatial basis vectors are imaginary and the time-like one is real: all are relative to the observer. However, this model has indirect geometri cal interpretations and is not easily made covariant. It also cannot be easily extended to higher dimensions. Dirac spinors are four-component complex entities, and must therefore 94 he represented by objects containing 8 real degrees of freedom. The relation 4 C -» C/3 ~ C7+3 ^ Cl+A ~ffsC. (6.1) indicates that the 8-dimensional even subalgebra of STA (C/+3) is isomor phic to the algebra of physical space C/3, which is isomorphic to complex quaternions H ® C Both C/3 and Clf3 offer simple and direct geometrical interpretations. The Minkowski space-time metric arises naturally in C/3 but is imposed in the ground space of Cl~l-A . Two conjugations are required in C/3 whereas in C/j~3 one (reversion) is sufficient. Both provide eovari- ant treatments of relativistic phenomena, but C/3 is closer to observational results whereas Cl^3 has a tighter integration of space and time. To re late results of C/^3 to observation, one usually creates the inhomogeneous elements of C/3 by space-time splits. Geometrical algebras are ideally suited for providing insight into the Dirac equation. When using geometrical algebras, no matrix representation is needed. In fact, it is usually best to avoid such a representation. The algebra of physical space contains the complex numbers and quaternions as subalgebras, and is itself isomorphic to the complex quaternions. Thus, it is no surprise that the geometrical concepts for space, and applications of complex numbers and quaternions, are all easily expressed using it. The geometric algebra for space-time has twice the dimension of the Pauli al gebra, and would seem to be a natural tool for relativity. Therefore, it is a pleasant surprise that all aspects of relativity can be formulated easily within the smaller and simpler Pauli algebra C/3. In multiparticle systems, multiple copies of C/1.3 are needed with each particle given a separate time coordinate, whereas in C/3 we can include many particles with a single time coordinate, so that C/3 may be more appropriate for a Hamiltonian formu- 95 lation of a many-particle theories. APS also provides a natural extension of the de Broglie-Bohm formula tion. The theory is defined by two evolution equations. The wave equation is given by the Dirac equation and determines *I< as its solution, and the closely related eigenspinor A gives the proper velocity u — AA^. The eigenspinor A(T) of a particle is the active Lorentz transformation from the reference frame to the lab frame. It gives the motion and orientation of the refer ence frame in the lab. and equivalently, the current is given by J = ^CQ^ and the spin distribution by 5 = ^'e^'. A relativistic causal account of a spin measurement in APS was discussed. The Stern-Gerlaoh magnet in duces a space-time rotation rate on the spin, acting on the eigenspinor field in analog}' to the way that a birefringent medium acts on a beam of light. In the last chapter, we introduced a covariant interpretation of CAPS, the complex extension of APS. We used the complex algebra of physical space, CAPS, to solve a problem of the standard theory, namely that the PT transformation, although proper, is not physical. Yet, it has the form of a physical rotation in the traditional Dirac theory inn PT : * ->i$n = *exp ~2~ The CAPS form of the PT transformation, on the other hand, lias the nonphysical form, PT : * — * -» jvf*n. A further problem addressed was that in APS and the standard Dirac theory, the explicit forms of the T and C transformations depend on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS, these symmetry 96 operation take the following forms T : * -» jtf fn C : * -» **n , which are independent of matrix representation. Finally, it was shown that the CAPS form of Dirac equation can be treated as a field equation, which can then be quantized to complete the model. A path integral formulation would be the next step. One possible reason for the introduction a unit imaginary free from any geometrical as sociation with the volume element is its application within a larger algebra. For example, in Cl-j which can be used as a basis for the standard-model [26], li corresponds to e4esefie7 and the projectors 5 (1 ± h) are the clii- ral projectors. These project onto particle and antiparticle spaces. The spatial volume element j — e]e2e3 anticommutes with any vector in the span {e,j, e5,e6.e7} , and the volume element of CI7 can be taken to be i = eie2e;je4e5e6e7. Indeed, the algebra CI7 is the next Clifford algebra in which the volume element can be associated with the imaginary unit (this occurs in every algebra Cl^+An with n a non-negative integer). CAPS also resolves a potential problem in APS, since APS has nonhomogeneous space- time vectors and makes double use of vector grades, associating each vec tor grade with two possible paravector (space-time vector) grades, whereas CAPS allows the different space-time grades of elements to be distinguished. More generally, as we have seen, the foundations of quantum theory are closely tied to a formulation of classical relativistie physics. The algebra of physical space, APS and CAPS, the complex extension of APS, gives some clear insight into the quantum/classical interface. The new ideas presented in this dissertation included the use of prujec- 97 tors to facilitate the comparison between the even subalgebra Cl{ :j of STA. APS C/3, and the complex quaternions H o-i C on the one hand and stan dard Dirac theory on the other. The solutions of the Dirac equation in the APS formulation were also new for Zitterbewegung, the resolution of the Klein paradox, and the geometric interpretation of the symmetry transfor mations in APS. It is here where some limitations of the standard approach, namely PT transformations, became apparent. Also newly shown was that APS can naturally describe an extended version of the de Broglie-Bohm approach to quantum theory in a mathematically elegant form. Here, the fermionic spin was represented as a Lorentz transformation of a fixed rest frame direction. This was illustrated in the solution of the Stern-Gerlach problem using time-independent plane-wave approach. The key point was that the rest-frame state spinor can always be expanded in spin-up ami spin-down terms. Finally, the covariant interpretation of CAPS was intro duced to solve a problem in the traditional Dirac theory, namely that the PT transformation, although a proper transformation, is anti-orthochronous and thus aphysical; yet, it has the form of a physical rotation in the standard theory as seen geometrically in APS. 98 Bibliography [1] P. Lounesto, "Clifford Algebras and Hestenes Spinors", Found. Phys. 23, 1203 1237 (1993). [2] W. E. Baylis, "Quantum/Classical Interface: A Geometric Approach from the Classical Side", in Computational Noncommutative Al gebra and Applications, Proceedings of the NATO Advanced Study Institute, NATO Science Series, Sub-series II, Vol. 136, ed. J. Byrnes, Kluwer Academic, Dordrecht, the Netherlands, 2004, pp. 127 154. [3] W. E. Baylis and J. D. Keselica, "A Classical Spinor Approach to the Quantum/Classical Interface", Can. J. Phys. 86, 629-634 (2008). [4] D. F. Styer, M. S. Balkin, K. M. Becker, M. R. Burns, C. E. Dudley, S. T. Forth, J. S. Gaurner, M. A. Kramer, D. C. Oertel, L. H. Park, M. T. Rinkoski, C. T. Smith, T. D. Wotherspoon, "Nine Formulations of Quantum Mechanics", Am. J. Phys. 70, 288-297 (2002). [5] P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 2001. [6] W. E. Baylis, "Introduction," in Clifford (Geometric) Algebras with applications in physics, mathematics, and engineering, edited by W. E. Baylis, Birkhauser. Boston 1996. [7] B. Fauser, "Equivalence of Daviau's, Hestenes', and Parra's Formula tions of Dirac Theory", Int. J. Theor. Phys.. 40, 399 411 (2001). [8; M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, Oxford. 1972. |9j J. Lambek, "'If Hamilton had Prevailed: Quaternions in Physics.'- Math ematical Intelligencer, 17(6), 7 15 1995. 99 [10] C. Lanczos, Zeits. f. Phys. 57. 447 473, 474 483, 484 493 (1929) Reprinted and translated in W.R. Davis et al., OKIS.. Cornltus Lanc zos Collected Published Papers With Commentaries (North Carolina State University. Raleigh, 1998), pp. 2-1132 2-1225. [11] A. W. Conway, "Quaternion Treatment of the Relativistie Wave Equa tion", Proc. Roy. Soc. A162, 145-154 (1937). [12] F. Giirsey, "On Two-Component Wave Equations", Phys. Rev. 77, 844 ' 845 (1950). [13] F. Giirsey, "On a Conform-Invariant Spinor Wave Equation", // Nuovo Cimento 3, 988 1006 (1956). [14] D. Hestenes, Space-Time Algebra, Gordon and Breach, New York, 1966. [15] D. Hestenes, "Real Spinor Fields", J. Math, Phys. 8, 798 808 (1967); "Spin and Isospin", J. Math. Phys. 8, 809 812 (1967) [16] A. Gsponer and J.-P. Hurni, "Comment on Formulating and General izing Dirac's, Proca's, and Maxwell's Equations with Biquaternions or Clifford Numbers", Found. Phys. Lett. 14, 77-85 (2001). [17] C. Doran, A. Lasenby, S. Gull, "Linear Algebra", in Clifford (Geo metric) Algebras with applications in physics, mathematics, and engineering, ed. by W. E. Baylis, Birkhauser, Boston 1996. [18] C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cam bridge Univ. Press, Cambridge, UK, 2003. [19] I. M. Benn R. W. Tucker, An Introduction to Spinors and Geomety with Applications in Physics. Adam Hilger, Bristol and New York 1987. [20] W. E. Baylis. G. Sobczyk, "Relativity in Clifford's Geometric Algebras of Space and Spacetime", Int. J. Theor. Phys., 43, 2061 2079 (2004). [21] W. E. Baylis •'Comments on "Dirac theory in spacetime algebra' ". ,/. Phys. A 35. 4791 4796 (2002).; quant-ph/0202060. \Y. E. Baylis. "Classical Eigenspinois and the Dirac Equation". Phys. Pee. A45. 4293 4302 (1992). 100 [23] W. E. Baylis, ''Geometry of Paravector Space with Applications to Relativistic Physics", in Computational Noncommutative Alge bra and Applications, Proceedings of the NATO Advanced Study Institute, NATO Science Series, Sub-series II, Vol. 136, ed. J. Byrnes, Kluwer Academic, Dordrecht, the Netherlands, 2004. [24] C. Daviau, "Dirac Equation in the Space Clifford algebra", in Clifford Algebras and their Application in Mathematical Physics, ed. by V. Dietrich, K. Habetha and G. Jank, Kluwer, Dordrecht. 1998. pp. 67 87. [25] J. Vrbik, "Dirac equation and Clifford algebra". ,/. Math. Phys., 35, 2309 2314 (1994). [26] G. Trayling and W. E. Baylis, "A Geometric Basis for the Standard- Model Gauge Group", J. Phys. A 34, 3309 3324 (2001). [27] W. E. Baylis, Yuan Yao, "Relativistic Dynamics of Charges in Electro magnetic Fields: An Eigenspinor Approach", Phys. Rev. A 60, 785 795 (1999). [28] W.E. Baylis, J. Huschilt, Jiansu Wei, "Why i?", Am. ./. Phys. 60, 788 797 (1992). [29] W. E. Baylis, Electrodynamics, a Modern Geometrical Approach, Birkhauser, Boston, 1998. [30] W. E. Baylis, "Eigenspinors in Quantum Theory," in Clifford (Geomet ric) Algebras, edited by W. E. Baylis, Birkhauser, Boston, 1996. [3Lj W.E. Baylis, "A Relativisitc Algebraic Approach to the Q/C Interface-: Implications for 'Quantum Reality'", Adv. Appl. Clifford Alt). 18, 395 4L5 (2008). [32j W. E. Baylis, "De Broglie Waves as an Effect of Clock Desynchroniza- tion". Can. J. Phys. 85, 1317-1323 (2007). [33] B.R. Holstein, "Klein's Paradox", Am. J. Phys. 66, 507 512 (1998). [34] A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press. Princeton, 2003. [35] L. De Broglie, in Solvay, 1928. 101 [36j D. Bohm, B.J. Hiloy, The Undivided Universe, Routlcdge. London, L993. [37] D. Bohm, "A Suggested Interpretation of the Quantum Theory in Terms of 'Hidden' Variables, I and II", Phys. Rev 85, 166 193 (1952). [38] P.R. Holland, The Quantum Theory of Motion, Cambridge University Press, Cambridge, 1993. [39] S. Goldstein, "Quantum Theory without Observers, Part One", Phys. Today 51 (3), 42-46 (1998); "Quantum Theory without Observers, Part Two", Phys. Today 51 (4), 38-42 (1998). [40] D. Durr, S. Goldstein, N. Zanghi, "Quantum equilibrium and the origin of absolute uncertainty", ,/. Stat. Phys. 67, 843-907 (1992). [41 j E. Merzbacher, Quantum Mechanics, J. Wiley and Sons, New York, 1961, 1970. [42] C. Dewney, P. R. Holland, A. Kyprianidis, "What Happens in a Spin Measurement?" Phys. Lett. A 119, 259 267 (1986). [43] A. Challinor, A. Lasenby, S. Gull, C. Doran, "A Relativistic, Causal Account of a Spin Measurement", Phys. Lett. A 218, 128 138 (1996). [44] G. Sobczyk, "The Hyperbolic Number Plane", The College Math. Jour nal. 26, 268-280 (1995). [45] D. R.ochon, S. Trernblay, "Bicomplex Quantum Mechanics: II. The Hilbert Space", Adv. App. Clifford Alg.16, 137-157 (2006). [46] D. Rochon, S. Trernblay, "Bicomplex Quantum Mechanics: I. The Gen eralized Schrodinger Equation", Adv. App. Clifford Alg., 14, 231 248 (2004). [47] S. Ulrych, "Relativistic quantum physic-s with hyperbolic numbers", Phys. Lett. B 25, 313 323 (2005). [48] L.H. Ryder, Quantum Field Theory 2nd eel, Cambridge University Press, Cambridge. 1985, 1996. 102 Vita Auctoris NAME: J. David Keselica PLACE OF BIRTH: Toronto, Canada YEAR OF BIRTH: November 16, 1968 EDUCATION: BSc honors from the University of Windsor, 1997 103