Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics (Mathematics and Its Applications, 290)

detailed contents is added through MasterPDFI don't know who Gigerenzer is, but he wrote something very clever that I saw quoted in a popular glossy magazine: "Evolution has tuned the way we think to frequencies of co-occurances, as with the hunter who remembers the area where he has had the most success killing game." This sanguine thought explains my obsession with the division algebras. Every effort I have ever made to connect them to physics - to the design of reality - has succeeded, with my expectations often surpassed. Doubtless this strong statement is colored by a selective memory, but the kind of game I sought, and still seek, seems to frowst about this particular watering hole in droves. I settled down there some years ago and have never feIt like Ieaving. This book is about the beasts I selected for attention (if you will, to ren­ der this metaphor politically correct, let's say I was a nature photographer), and the kind of tools I had to develop to get the kind of shots Iwanted (the tools that I found there were for my taste overly abstract and theoretical). Half of thisbook is about these tools, and some applications thereof that should demonstrate their power. The rest is devoted to a demonstration of the intimate connection between the mathematics of the division algebras and the Standard Model of quarks and leptons with U(l) x SU(2) x SU(3) gauge fields, and the connection of this model to lO-dimensional spacetime implied by the mathematics. Contents Preface 1. Underpinnings 1.1. The Argument. SCREED I SCREED II SCREEDIII 1.2. Clifford Aigebras. 1.3. Conjugations and Spinors NILPOTENT CLIFFORD ALGEBRAS SYMPLECTIC NILPOTENT CLIFFORD ALGEBRA 1.4. Aigebraic Fundamentals ofthe Standard Model 2. Division Algebras Alone. 2.1. Mostly Octonions. 2.2. Adjoint algebras. 2.3. Clifford Algebras, Spinors 2.4. Resolving the Identity of O_L 2.5. Lie Algebras, Lie Groups, from O_L 2.6. From Galois Fields to Division Aigebras: An Insight. 3. Tensor Algebras 3.1. Tensoring Two: Clifford Aigebras and Spinors. 3.2. Tensoring Two: Spinor Inner Product. 3.3. Tensoring Three: Clifford Aigebras and Spinors 3.4. Tensoring Three: Spinor Inner Product RESOLVING THE IDENTITY OF T THE TRACE OF X 3.5. Derivation of the Standard Symmetry. 3.6. SU(2) X SU(3) Multiplets, and U(1) 4. Connecting to Physics. 4.1. Connecting to Geometry DIMENSIONAL REDUCTION 4.2. Connecting to Particles 4.3. Parity N onconservation LEFTHANDED DIRAC OPERATOR 4.4. Gauge Fields. 4.5. Weak Mixing. 4.6. Gauging SU(3). 5. Spontaneous Symmetry Breaking. 5.1. Scalar Fields. 5.2. Scalar Lagrangians. 5.3. Fermions and Scalars 6. 10 Dimensions 6.1. Fermion Lagrangian. MATTER/ ANTIMATTER MIXING 6.2. More SU(3). 6.3. Freedom from Matter-Antimatter Mixing. 6.4. (1,9)-Scalar Lagrangian 6.5. Charge Conjugation on T_L (2) 6.6. Charge Conjugation on T^2 THE MEANING OF MAJORANA 6.7. 10 Other Dimensions. THE CLIFFORD ALGEBRA 7. Doorways. 7.1. Moufang and other Identities TWO IDENTITIES THE MOUFANG IDENTITIES 7.2. Spheres and Lie Algebras. SPHERE FIBRATIONS 7.3. Triality. TRIALITY REPRESENTATIONS OF so(8) THE Tri IN TRIALITY FREUDENTHAL'S PRINCIPLE OF TRIALITY 7.4. LG2 and Tri. LG2 TRIALITY TRIPLET 7.5. LG2 Triplets and the X-Product LG^X_2 GENERAL SOLUTION LG^X_2 AND THE X-ADJOINT ALGEBRA O_{LX} 8. Corridors. 8.1. Magie Square. 8.2. The Ten MS_{KK'}. R®R R®C R®Q R®O C®C C®Q C®O Q®Q Q®O O®O 8.3. Spinor_{KK'} Outer Products C OUTER PRODUCTS Q OUTER PRODUCTS O OUTER PRODUCTS 8.4. LF_4 ~ MS_{RO}. 8.5. J^O_3 and F_4 8.6. More Magie Square Appendix i. O_L Actions: Product Rule e_a * e_{a+1} = e_{a+5}. Appendix ii. O_R Actions: Product Rule e_a * e_{a+1} = e_{a+5}. Appendix iii. O_L Actions: Product Rule e_a * e_{a+1} = e_{a+3}. Appendix iv. O_R Actions: Product Rule e_a * e_{a+1} = e_{a+3} Bibliography Index

the Four Real Division Algebras (reals, Complexes, Quaternions And Octonions) Are The Most Obvious Signposts To A Rich And Intricate Realm Of Select And Beautiful Mathematical Structures. Using The New Tool Of Adjoint Division Algebras, With Respect To Which The Division Algebras Themselves Appear In The Role Of Spinor Spaces, Some Of These Structures Are Developed, Including Parallelizable Spheres, Exceptional Lie Groups, And Triality. In The Case Of Triality The Use Of Adjoint Octonions Greatly Simplifies Its Investigation. Motivating This Work, However, Is A Strong Conviction That The Design Of Our Physical Reality Arises From This Select Mathematical Realm. A Compelling Case For That Conviction Is Presented, A Derivation Of The Standard Model Of Leptons And Quarks.
The Book Will Be Of Particular Interest To Particle And High Energy Theorists, And To Applied Mathematicians.

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a Monograph That Demonstrates And Effects A Connection Of The Division Algebras To Physics. Half Of The Volume Is Devoted To The Mathematical Underpinnings And Some Applications That Demonstrate Their Power. The Rest Is Devoted To A Demonstration Of The Intimate Connection Between The Mathematics Of The Division Algebras And The Standard Model Of Quarks And Leptons With U(1)xsu(2)xsu(3) Gauge Fields, And The Connection Of This Model To 10-dimensional Spacetime Implied By The Mathematics. Annotation C. Book News, Inc., Portland, Or (booknews.com)

I don't know who Gigerenzer is, but he wrote something very clever that I saw quoted in a popular glossy magazine: "Evolution has tuned the way we think to frequencies of co-occurances, as with the hunter who remembers the area where he has had the most success killing game." This sanguine thought explains my obsession with the division algebras. Every effort I have ever made to connect them to physics - to the design of reality - has succeeded, with my expectations often surpassed. Doubtless this strong statement is colored by a selective memory, but the kind of game I sought, and still seek, seems to frowst about this particular watering hole in droves. I settled down there some years ago and have never feIt like Ieaving. This book is about the beasts I selected for attention (if you will, to renƯ der this metaphor politically correct, let's say I was a nature photographer), and the kind of tools I had to develop to get the kind of shots Iwanted (the tools that I found there were for my taste overly abstract and theoretical). Half of thisbook is about these tools, and some applications thereof that should demonstrate their power. The rest is devoted to a demonstration of the intimate connection between the mathematics of the division algebras and the Standard Model of quarks and leptons with U(l) x SU(2) x SU(3) gauge fields, and the connection of this model to lO-dimensional spacetime implied by the mathematics