space time algebra second edition pdf

David Hestenes Space-Time Algebra Second Edition www.TechnicalBooksPDF.com David Hestenes Space-Time Algebra Second Edition Foreword by Anthony Lasenby www.TechnicalBooksPDF.com David Hestenes Department of Physics Arizona State University Tempe, AZ USA Originally published by Gordon and Breach Science Publishers, New York, 1966 ISBN 978-3-319-18412-8 DOI 10.1007/978-3-319-18413-5 ISBN 978-3-319-18413-5 (eBook) Library of Congress Control Number: 2015937947 Mathematics Subject Classification (2010): 53-01, 83-01, 53C27, 81R25, 53B30, 83C60 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Cover design: deblik, Berlin Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) www.TechnicalBooksPDF.com Foreword It is a pleasure and honour to write a Foreword for this new edition of David Hestenes’ Space-Time Algebra This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas Moreover, these same techniques, in the form of the ’Geometric Algebra’, can be applied in many areas of engineering, robotics and computer science, with no changes necessary – it is the same underlying mathematics, and enables physicists to understand topics in engineering, and engineers to understand topics in physics (including aspects in frontier areas), in a way which no other single mathematical system could hope to make possible As well as this, however, there is another aspect to Geometric Algebra which is less tangible, and goes beyond questions of mathematical power and range This is the remarkable insight it gives to physical problems, and the way it constantly suggests new features of the physics itself, not just the mathematics Examples of this are peppered throughout Space-Time Algebra, despite its short length, and some of them are effectively still research topics for the future As an example, what is the real role of the unit imaginary in quantum v www.TechnicalBooksPDF.com vi Foreword mechanics? In Space-Time Algebra two possibilities were looked at, and in intervening years David has settled on right multiplication by the bivector iσ3 = σ1 σ2 as the correct answer between these two, thus encoding rotation in the ‘internal’ xy plane as key to understanding the role of complex numbers here This has stimulated much of his subsequent work in quantum mechanics, since it bears directly on the zwitterbewegung interpretation of electron physics [1] Even if we not follow all the way down this road, there are still profound questions to be understood about the role of the imaginary, such as the generalization to multiparticle systems, and how a single correlated ‘i’ is picked out when there is more than one particle [2] As another related example, in Space-Time Algebra David had already picked out generalizations of these internal transformations, but still just using geometric entities in spacetime, as candidates for describing the then-known particle interactions Over the years since, it has become clear that this does seem to work for electroweak theory, and provides a good representation for it [3,4,5] However, it is not clear that we can yet claim a comparable spacetime ‘underpinning’ for QCD or the Standard Model itself, and it may well be that some new feature, not yet understood, but perhaps still living within the algebra of spacetime, needs to be bought in to accomplish this The ideas in Space-Time Algebra have not met universal acclaim I well remember discussing with a particle physicist at Manchester University many years ago the idea that the Dirac matrices really represented vectors in 4d spacetime He thought this was just ‘mad’, and was vehemently against any contemplation of such heresy For someone such as myself, however, coming from a background of cosmology and astrophysics, this realization, which I gathered from this short book and David’s subsequent papers, was a revelation, and showed me that one could cut through pages of very unintuitive spin calculations in Dirac theory, which only experts in particle theory would be comfortable with, and replace them with a few lines of intuitively appealing calculations with rotors, of a type that for example an engineer, also using Geometric Algebra, would be able to understand and relate to immediately in the context of rigid body rotations A similar transformation and revelation also occurred for me with respect to gravitational theory Stimulated by Space-Time Algebra and www.TechnicalBooksPDF.com Foreword vii particularly further papers by David such as [6] and [7], then along with Chris Doran and Stephen Gull, we realized that general relativity, another area traditionally thought very difficult mathematically, could be replaced by a coordinate-free gauge theory on flat spacetime, of a type similar to electroweak and other Yang-Mills theories [8] The conceptual advantages of being able to deal with gauge fields in a flat space, and to be able to replace tensor calculus manipulations with the same unified mathematical language as works for electromagnetism and Dirac spinors, are great This enabled us quickly to reach interesting research topics in gravitational theory, such as the Dirac equation around black holes, where we found solutions for electron energies in a spherically symmetric gravitational potential analogous to the Balmer series for electrons around nuclei, which had not been obtained before [9] Of course there are very clever people in all fields, and in both relativistic quantum mechanics and gravity there have been those who have been able to forge an intuitive understanding of the physics despite the difficulties of e.g spin sums and Dirac matrices on the one hand, and tensor calculus index manipulations on the other However, the clarity and insight which Geometric Algebra brings to these areas and many others is such that, for the rest of us who find the alternative languages difficult, and for all those who are interested in spanning different fields using the same language, the ideas first put forward in ’Space-time Algebra’ and the subsequent work by David, have been pivotal, and we are all extremely grateful for his work and profound insight Anthony Lasenby Astrophysics Group, Cavendish Laboratory and Kavli Institute for Cosmology Cambridge, UK References [1] D Hestenes, Zitterbewegung in Quantum Mechanics, Foundations of Physics 40, (2010) [2] C Doran, A Lasenby, S Gull, S Somaroo and A Challinor, www.TechnicalBooksPDF.com viii Foreword Spacetime Algebra and Electron Physics In: P.W Hawkes (ed.), Advances in Imaging and Electron Physics, Vol 95, p 271 (Academic Press) (1996) [3] D Hestenes, Space-Time Structure of Weak and Electromagnetic Interactions, Found Phys 12, 153 (1982) [4] D Hestenes, Gauge Gravity and Electroweak Theory In: H Kleinert, R T Jantzen and R Ruffini (eds.), Proceedings of the Eleventh Marcel Grossmann Meeting on General Relativity, p 629 (World Scientific) (2008) [5] C Doran and A Lasenby, Geometric Algebra for Physicists (Cambridge University Press) (2003) [6] D Hestenes, Curvature Calculations with Spacetime Algebra, International Journal of Theoretical Physics 25, 581 (1986) [7] D Hestenes, Spinor Approach to Gravitational Motion and Precession, International Journal of Theoretical Physics 25, 589 (1986) [8] A Lasenby, C Doran, and S Gull, Gravity, gauge theories and geometric algebra, Phil Trans R Soc Lond A 356, 487 (1998) [9] A Lasenby, C Doran, J Pritchard, A Caceres and S Dolan, Bound states and decay times of fermions in a Schwarzschild black hole background, Phys Rev D 72, 105014 (2005) www.TechnicalBooksPDF.com Preface after Fifty years This book launched my career as a theoretical physicist fifty years ago I am most fortunate to have this opportunity for reflecting on its influence and status today Let me begin with the title Space-Time Algebra John Wheeler’s first comment on the manuscript about to go to press was “Why don’t you call it Spacetime Algebra?” I have followed his advice, and Spacetime Algebra (STA) is now the standard term for the mathematical system that the book introduces I am pleased to report that STA is as relevant today as it was when first published I regard nothing in the book as out of date or in need of revision Indeed, it may still be the best quick introduction to the subject It retains that first blush of compact explanation from someone who does not know too much From many years of teaching I know it is accessible to graduate physics students, but, because it challenges standard mathematical formalisms, it can present difficulties even to experienced physicists One lesson I learned in my career is to be bold and explicit in making claims for innovations in science or mathematics Otherwise, they will be too easily overlooked Modestly presenting evidence and arguing a case is seldom sufficient Accordingly, with confidence that comes from decades of hindsight, I make the following Claims for STA as formulated in this book: (1) STA enables a unified, coordinate-free formulation for all of relativistic physics, including the Dirac equation, Maxwell’s equation and General Relativity (2) Pauli and Dirac matrices are represented in STA as basis vectors in space and spacetime respectively, with no necessary connection to spin ix www.TechnicalBooksPDF.com x Preface after Fifty years (3) STA reveals that the unit imaginary in quantum mechanics has its origin in spacetime geometry (4) STA reduces the mathematical divide between classical, quantum and relativistic physics, especially in the use of rotors for rotational dynamics and gauge transformations Comments on these claims and their implications necessarily refer to contents of the book and ensuing publications, so the reader may wish to reserve them for later reference when questions arise Claim (2) expresses the crucial secret to the power of STA It implies that the physical significance of Dirac and Pauli algebras stems entirely from the fact that they provide algebraic representation of geometric structure Their representations as matrices are irrelevant to physics—perhaps inimical, because they introduce spurious complex numbers without physical significance The fact that they are spurious is established by claim (3) The crucial geometric relation between Dirac and Pauli algebras is specified by equations (7.9) to (7.11) in the text It is so important that I dubbed it space-time split in later publications Note that the symbol i is used to denote the pseudoscalar in (6.3) and (7.9) That symbol is appropriate because i2 = −1, but it should not to be confused with the imaginary unit in the matrix algebra Readers familiar with Dirac matrices will note that if the gammas on the right side of (7.9) are interpreted as Dirac’s γ-matrices, then the objects on the left side must be Dirac’s α-matrices, which, as is seldom recognized, are × matrix representations of the × Pauli matrices That is a distinction without physical or geometric significance, which only causes unnecessary complications and obscurity in quantum mechanics STA eliminates it completely It may be helpful to refer to STA as the “Real Dirac Algebra”, because it is isomorphic to the algebra generated by Dirac γ-matrices over the field of real numbers instead of the complex numbers in standard Dirac theory Claim (3) declares that only the real numbers are needed, so standard Dirac theory has a superfluous degree of freedom That claim is backed up in Section 13 of the text where Dirac’s equation is given two different but equivalent formulations within STA In equation (13.1) the role of unit imaginary is played by the pseudoscalar i, while, in equation (13.13) it is played by a spacelike bivector www.TechnicalBooksPDF.com 88 Conclusion ultimate justification of our space-time calculus as a “theory” awaits a physical prediction which depends on our identification of i as a pseudoscalar www.TechnicalBooksPDF.com Appendixes A Bases and Pseudoscalars In this appendix we give some miscellaneous conventions and formulas which are useful for work with arbitrary bases in Vn , together with some special formulas in the Dirac and Pauli algebras A set of linearly independent vectors {ei : i = 1, 2, , n} will be called a frame in Vn The metric tensor gij of Vn is defined by gij = ei · ej = 12 (ei ej + ej ei ) (A.1) A vector a can be written as a linear combination of the ei : a = ei The inner product of vectors a and b is a · b = bj (ei · ej ) = bi = bi (A.2) = aj gij (A.3) where Consider the simple r-vector ei1 ∧ .∧eir The number of transposition necessary to obtain eir ∧ .∧ei1 is 1+2+3 +(r−1) = 21 r(r−1) It follows that the reverse (defined in section 5) of any r-vector Ar is A†r = (−1) r(r−1) Ar (A.4) The pseudoscalar of a frame {ei } is an n-vector e defined by e = e1 ∧ e2 ∧ ∧ en (A.5) The determinant g of the metric tensor is related to the pseudoscalar e by (3.17): e† e = e† · e = (en ∧ ∧ e1 ) · (e1 ∧ ∧ en ) = det gij = g Ó Springer International Publishing Switzerland 2015 D Hestenes, Space-Time Algebra, DOI 10.1007/978-3-319-18413-5 www.TechnicalBooksPDF.com (A.6) 89 90 Appendixes By (A.4), e2 = (−1) n(n−1)+s |g|, (A.7) where the signature s is the number of vectors in Vn with negative square In order to simplify algebraic manipulation with base vectors, it is convenient to define another set of base vectors ej related to the ei by the conditions ei · ej = δij , (A.8) where δij is a Kronecker delta The raised index on ej is strictly a mnemonic device to help us remember the relations (A.8) between the two bases, and is not meant to indicate that the ej are in any sense vectors in a “space” different from that containing the ei Each ej can be written as a linear combination of ei : ej = g ij ej (A.9) from which it follows immediately that g ij gjk = δki (A.10) g ij = ei · ej = 12 (ei ej + ej ei ) (A.11) and We can construct the ej explicitly in terms of the ei in the following way: Define the pseudovector E j by E j = (−1)j+1 ei ∧ ∧ ˆej ∧ ∧ en , (A.12) where the circumflex means omit ej Observe that ei ∧ E j = δij e (A.13) Also define the reciprocal pseudoscalar e−1 by e−1 = (−1)s e e2 (A.14) Now we can define ej by ej ≡ E j e−1 www.TechnicalBooksPDF.com (A.15) A Bases and Pseudoscalars 91 The frame {ej } is often called dual to the frame {ei } because of the duality operation used in constructing it With (A.15) we can verify (A.8): ei · ej = (ei ∧ E j )e−1 = δij ee−1 = δij We can also write e−1 = (−1)s en ∧ ∧ e2 ∧ e1 (A.16) We now restrict ourselves to the Dirac algebra The pseudoscalar for the frame {γ µ : µ = 0, 1, 2, 3} will be denoted by γ , γ ≡ γ 0123 = γ ∧ γ ∧ γ ∧ γ (A.17) Similarly, the pseudoscalar for the dual frame is γ5 ≡ γ0123 (A.18) All of the following formulas can be proved with the aid of formulas in section 3, γ γ5 = 1, γ5 = γ5 , (γ5 )2 γ µ γµ = 4, γ µ γ5 γµ = −4γ5 (A.19) (A.20) (A.21) For any vectors a, b, c, γ µ aγµ = −2a, γ µ abγµ = 4a · b, γ µ abcγµ = −2cba (A.22) (A.23) (A.24) The last two formulas admit the following special cases: γ µ a ∧ bγµ = 0, γ µ a ∧ b ∧ cγµ = 2a ∧ b ∧ c (A.25) (A.26) The completely antisymmetric unit εµναβ can be defined by εµναβ ≡ γ γµναβ www.TechnicalBooksPDF.com (A.27) 92 Appendixes Similarly, εµναβ = γ5 γ µναβ (A.28) It follows that ε0123 = 1, γµναβ = γ5 εµναβ , γµναβ γ β = γµνα , γµναβ γ αβ = 2γµν , γµναβ γ ναβ = 6γµ (A.29) (A.30) (A.31) (A.32) (A.33) The magnitude of γ5 can be expressed in terms of the metric tensor, g ≡ (γ5 )2 = g0µ g1ν g2α g3β γ µναβ γ5 = g0µ g1ν g2α g3β εµναβ = det gµν Thus γ5 = ±i √ − g (A.34) (A.35) The reader is invited to find formulas analogous to those above in the Pauli algebra Bases in D and P can be related by σ µ ≡ γµ γ0 (A.36) The pseudoscalar in space for the frame {σ k } is σijk = γ5 εijk , (A.37) εijk ≡ εijk0 (A.38) where Of course, for a righthanded orthonormal frame, σijk = iεijk B (A.39) Some Theorems Theorem If R = R and RR = 1, then R = ±eB , (B.1) where B is a bivector B can be chosen so that the sign in (B.1) is positive except in the special case when B = and R = −eB www.TechnicalBooksPDF.com B Some Theorems 93 Proof Since R = R, R is the sum of a scalar S, a bivector T and a pseudo-scalar P , R = S − T + P R = S + T + P, If T = 0, we can express T as a linear combination of orthogonal simple bivectors T1 and T2 , T = S T1 + S T2 , T1 · T2 = 0, T12 0, T22 The condition RR = gives us two equations: S = S12 T12 − S22 T22 + P = 1, S1 S2 T1 T2 = SP We can adjust the magnitude of T1 T2 so T1 T2 = P , then S1 S2 = S This enables us to factor R, R = (S1 + T1 )(S2 + T2 ) So the condition RR = becomes (S12 − T22 )(S22 − T12 ) = We can take S12 − T22 = 1, so, since T22 < 0, S1 + T2 = eB2 = cos B2 + sin B2 Similarly, since T12 > 0, S2 + T1 = eB1 = cosh B1 + sinh B1 Since T1 · T2 = 0, T1 commutes with T2 , and B1 commutes with B2 Therefore R = eB1 eB2 = eB2 eB1 = eB1 +B2 = eB (B.2) This equation shows slightly more than (B.1), because B is expressed as the sum of orthogonal simple bivectors To complete our proof, we must treat the special case T = The condition RR = gives S + P = 1, SP = www.TechnicalBooksPDF.com 94 Appendixes Since P 0, we must have P = and S = Consider the case S = +1 Then, since T = 0, ∞ R=1+T = n=0 n T = eT n! (B.3) This is in the form (B.1) If S = −1, then R = eT Theorem If U U = and U = U = U ∗ , then U = eib , (B.4) where b is a simple timelike bivector Proof From theorem we have U = ±eB , so ∗ U ∗ = γ0 U γ0 = ±eγ0 Bγ0 = ±eB = U implies B = B ∗ = γ0 Bγ0 Therefore γ0 · B = 12 (γ0 B − Bγ0 ) = 0, which means that B is orthogonal to γ0 and so is simple and spacelike We can write B as the dual of a simple timelike bivector b: B = ib ˆ ˆ ˆ and B ˆ = −1, Since −eB = eBπ eB = eBπ+B = eB , where B = |B|B B we can always choose B so U = e Theorem If H H = and H = H = H † , then H = ±ea , (B.5) where a is a simple timelike bivector Proof From theorem we have H = ±eB , so H † = γ0 Hγ0 = ±e−γ0 Bγ0 = H implies B = −B ∗ = −γ0 Bγ0 Therefore γ0 ∧ B = 12 (γ0 B + Bγ0 ) = 0, which means that B is proportional to γ0 and so is simple and timelike Hence we can write B = a www.TechnicalBooksPDF.com B Some Theorems 95 Theorem If RR = and R = R, then a timelike unit vector γ0 uniquely determines the decomposition R = HU, (B.6a) U = U ∗ = eib (B.6b) H = H † = ±ea (B.6c) where and Proof Note that A ≡ RR† satisfies AA = and A = A = A† , so by theorem we can write RR† = e2a Define H by H = (RR† ) = ±ea Note that R† R∗ = 1, so R = RR† R∗ = H(HR∗ ) Therefore define u by U = HR∗ We verify that U U = HR∗ R† H = H H = and U ∗ = H ∗ R = H ∗ HHR∗ = HR∗ = U So, by theorem 2, we can write U in the form (B.6b) Theorem If SS † = 1, then S = eB (B.7) S = veB , (B.8) or where v is a unit vector and B is a bivector for the five-dimensional Euclidean space E5 spanned by vectors em , defined by ei = σi = γi γ0 , e4 = γ0 , e5 = e1234 = iγ0 , www.TechnicalBooksPDF.com (B.9) 96 Appendixes where {γµ : µ = 0, 1, 2, 3} is a righthanded orthonormal basis of vectors in the Dirac algebra Relative to the basis (B.9), B = B mn 12 [em , en ], v = v m em (B.10) (m, n = 1, 2, 3, 4, 5) The Clifford algebra of a 4-dimensional Euclidean space E4 is isomorphic to the Clifford algebra of a 4-dimensional Minkowski space (we call the latter the Dirac algebra) The two algebras differ only in geometric interpretation, i.e in what elements are called vectors, bivectors, etc An isomorphism between the two algebras is given by (B.9) If we take e1 , e2 , e3 , e4 as an orthonormal basis for E4 , then e5 can be identified as the unit pseudoscalar We can identify the transformation u → u = SuS † (B.11) as a rotation of a vector u in E5 if S is given by (B.7) and as a reflection if S is given by (B.8) The reflections can be distinguished from rotations in E5 by the method used in section We can prove that for rotations S has the form (B.7) in the same way that we proved theorem of this appendix In fact, the proof here is somewhat simpler because the scalar part of B is negative definite We also find, just as in theorem 1, that B can be written as the sum of two orthogonal simple bivectors This represents the fact that the rotation group in five dimensions is a group of rank Theorem If a and b are timelike vectors and a · b > 0, then, for any d-number A, (aAbA)S 0, (B.12) and equality holds only if A vanishes Proof We lose no generality by taking a and b to be unit vectors Write a = γ0 Since, as we saw in section 18, b can be obtained from γ0 by a time-like rotation, we can write b = Rγ0 R Now define a new d-number B = AR, so (aAbA)S = (γ0 Bγ0 B) = (BB † )S www.TechnicalBooksPDF.com (B.13) C Composition of Spatial Rotations 97 By continuing our discussion following theorem 5, we identify (B.13) as a Euclidean norm, and we recall that we proved this was positive definite in section Theorem If S S = 1, then S = ±eB+A (B.14) S = (V + P )eB+A , (B.15) or where B is a bivector, A is a pseudovector, V is a vector, P is a pseudoscalar and (V + P )2 = If u is an element of the 5-dimensional linear space V5 consisting of the vectors and pseudoscalars, we can identify the transformation u → u = SuS (B.16) as a rotation if S is given by (B.14) and as a reflection if S is given by (B.15) Theorem can be proved in the same way as theorem 5, with due account taken of the metric (+ − − − −) on V5 C Composition of Spatial Rotations Recall from section 16 that a spatial rotation of a vector p can be written 1 p → p = e− ib pe ib (C.1) This is a righthanded rotation through an angle b = |b| about the vector b The exponential can be written as a sum of scalar and bivector parts, e ib = cos 12 b + i sin 12 b (C.2) The series expansion for the exponential clearly shows cos 12 b = cos 12 b, ˆ sin b, sin b = b 2 www.TechnicalBooksPDF.com (C.3a) (C.3b) 98 Appendixes ˆ is the unit vector in the direction of b where b The composition of rotations in Euclidean 3-space is reduced by equation (C.1) to a problem in composition of exponentials, which is fairly simple Suppose the rotation (C.1) is followed by the rotation 1 p → p = e− ia p e ia (C.4) Then the rotation of p into p is given by 1 1 1 p → p = e− ic pe ic = e− ia e− ib pe ib e ia (C.5) From 1 e ic = e ib e ia (C.6) we can find c in terms of b and a The scalar part of (C.6) is cos 12 c = cos 12 a cos 12 b − (sin12 a) · (sin 12 b) (C.7) From the bivector part of (C.6) we get sin 12 c = cos 12 a sin 12 b + cos 12 b sin 12 a + (sin 12 a) × (sin 12 b) (C.8) Or, in more explicit form, (C.7) and (C.8) become ˆ sin a sin b, ˆ·b cos 12 c = cos 12 a cos 12 b − a (C.7 ) 2 1 1 1 ˆ cos a sin b + a ˆ sin a sin b ˆ cos b sin a + a ˆ×b cˆ sin c = b 2 2 (C.8 ) The ratio of (C.8) to (C.7) gives the law of tangents tan c tan 12 a + tan 12 b + (tan 12 a) × (tan 12 b) = − (tan 12 a) · (tan 12 b) (C.9) When a and b are collinear these formulas reduce to the familiar trigonometric formulas for the addition of angles in a plane From these the full angle formulas can be obtained, but the half-angle formulas are simpler and adequate in most problems www.TechnicalBooksPDF.com 99 D Matrix Representation of the Pauli Algebra D Matrix Representation of the Pauli Algebra In this appendix we give the correspondence between operations in the Pauli algebra and operations in its matrix representation As we saw in section 12, a p-number φ can be represented by a two by two matrix Φ over the field of the complex numbers, φ = φ0 + φi σ i = φ11 u1 + φ12 v1 + φ21 u2 + φ22 v2 , φ11 φ12 φ21 φ22 Φ= (D.1) (D.2) The matrix elements of Φ can be looked upon as matrix elements of φ, φab = (u†b φua )I = (vb† φva )I , (D.3) where the subscript I means invariant part, i.e scalar plus pseudoscalar part Using (12.7) and (12.5) we can find the matrix representations of the base vectors σ i : σ1 = 1 0 i , σ2 = −i , σ3 = −1 (D.4) Let ΦT be the transpose of Φ, let Φ× be the complex conjugate of Φ, and let Φ† be the hermitian conjugate Φ The following correspondences are easily established: φ∗ = φ∗0 − φ∗i σ i ∼σ Φx σ , (D.5) φ† = φ∗0 + φ∗i σ i ∼Φ† , (D.6) φ = φ0 − φi σ i T ∼σ Φ σ , (D.7) φφ = φφ ∼ det Φ, (D.8) φI = φ0 ∼ 12 T rΦ (D.9) For the two-component spinors we have the correspondences: u1 or v1 ∼ , (D.10) u2 or v2 ∼ (D.11) www.TechnicalBooksPDF.com 100 Appendixes The operations of complex conjugation and transpose while of interest to matrix algebra are only of minor interest to Clifford algebra, for these operations are defined and meaningful only relative to a particular basis in the algebra On the other hand, the operations of inversion and hermitian conjugation are independent of basis; for this reason they may be considered to be more fundamental than the operations of complex conjugation and transpose www.TechnicalBooksPDF.com Bibliography [1] J Gibbs, The Scientific Papers of J Willard Gibbs, Dover, New York (1961) [2] E B Wilson, Vector Analysis, Dover, New York (1961) [3] A Wills, Vector Analysis, Dover, New York (1958) [4] H Grassmann, Die Ausdehnungslehre von 1862 [5] W K Clifford, Amer Jour of Math 1, 350 (1878) [6] E R Caianiello, Nuovo Cimento 14 (Supp.), 177 (1959) [7] H Jauch and F Rohrlich, The Theory of Photons and Electrons, Addison-Wesley, Cambridge (1955) [8] M Riesz, Dixi´eme Congr´es des Mathematicians Scandinaves, Copenhague, 1946, p 123 [9] M Riesz, Clifford Numbers and Spinors, Lecture series No 38, The Institute for Fluid Dynamics and Applied Mathematics, University of Maryland (1958) [10] F Gă ursey, Rev Fac Sci., Instanbul A20, 149 (1956) [11] O Heaviside, Electromagnetic Theory, Dover, New York (1950) [12] L Silberstein, Theory of Relativity, Macmillan, London (1914) [13] C Misner and J Wheeler, Ann Phys 2, 525 (1957) [14] L M Brown, Lectures in Theoretical Physics, Interscience, N.Y., IV, 324 (1962) [15] F Gă ursey, Rev Fac Sci Univ Istanbul A21, 33 (1956) [16] F Gă ursey, Nuovo Cimento 7, 411 (1958) Ó Springer International Publishing Switzerland 2015 D Hestenes, Space-Time Algebra, DOI 10.1007/978-3-319-18413-5 www.TechnicalBooksPDF.com 101 102 Bibliography [17] T D Lee and C N Yang, Nuovo Cimento 3, 749 (1956) [18] H Green, Nuclear Phys 7, 373 (1958) [19] V Fock, Space, Time and Gravitation, Pergamon Press, New York (1958) [20] H Flanders, Differential Forms, New York, Academic Press (1963) [21] A Einstein and others, The Special Principle of Relativity, Dover, New York (1923), p 115 [22] C Yang and R Mills, Phys Rev 96, 191 (1956) [23] R Utiyama, Phys Rev 101, 1597 (1956) [24] J J Sakurai, Annals of Physics 11, (1960) www.TechnicalBooksPDF.com ... the title Space- Time Algebra John Wheeler’s first comment on the manuscript about to go to press was “Why don’t you call it Spacetime Algebra? ” I have followed his advice, and Spacetime Algebra. .. “Minkowski” signifies the flat space- time of special relativity 12 A www.TechnicalBooksPDF.com The Algebra of Space- Time 21 algebra We begin our study of the Dirac algebra D by selecting a set... Pauli algebra appears as a subalgebra of the Dirac algebra and is simply the vector algebra for the 3 -space of some inertial frame The vector algebra of Gibbs is seen not as a separate algebraic