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Clifford Geometric Algebra (GA)

Clifford Geometric Algebra (GA). Vector cross product a x b. The natural algebra of 3D space. Quaternions q. Complex numbers i. Rotation matrices R. Spinors. Quaternions. j. The generalization of complex numbers to three dimensions i 2 = j 2 = k 2 = -1, i j = k,. k.

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Clifford Geometric Algebra (GA)

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  1. Clifford Geometric Algebra (GA) Vector cross product a x b The natural algebra of 3D space Quaternions q Complex numbers i Rotation matrices R Spinors

  2. Quaternions j • The generalization of complex numbers to three dimensions • i2 = j2 = k2 = -1, i j = k, k Non-commutative i j =-j i , try rotating a book i

  3. Quarternion rotations of vectors • Bilinear transformation • v’=R v R†where R is a quaternion v is a Cartesian vector

  4. Clifford’s Geometric Algebra e3 • Define algebraic elements e1, e2, e3 • With e12=e22=e32=1, and anticommuting • ei ej= - ej ei This algebraic structure unifies Cartesian coordinates, quaternions and complex numbers into a single real framework. e2 e1e2e3 e1e2 e2e3 e1e3 e1 ei~σi

  5. Geometric Algebra-Dual representation e3 e2 ι=e1e2e3 e1e2 e2e3 e1e3 e1

  6. The product of two vectors…. To multiply 2 vectors we….just expand the brackets…distributive law of multiplication over addition. A complex-type number combining the dot and cross products! Hence we now have an intuitive definition of multiplication and division of vectors, subsuming the dot and cross products which also now has an inverse.

  7. Spinor mapping How can we map from complex spinors to 3D GA? We see that spinors are rotation operators. The geometric product is equivalent to the tensor product

  8. EPR setting for Quantum games

  9. GHZ state Ø=KLM

  10. Probability distribution Where Doran C, Lasenby A (2003) Geometric algebra for physicists <ρQ>0~Tr[ρQ]

  11. References: • Analyzing three-player quantum games in an EPR type setup • Analysis of two-player quantum games in an EPR setting using Clifford's geometric algebra • N player games Copies on arxiv, Chappell A GAME THEORETIC APPROACH TO STUDY THE QUANTUM KEY DISTRIBUTION BB84 PROTOCOL, IJQI 2011, HOUSHMAND Google: Cambridge university geometric algebra

  12. The geometric product magnitudes In three dimensions we have:

  13. Negative Numbers • Interpreted financially as debts by Leonardo di Pisa,(A.D. 1170-1250) • Recognised by Cardano in 1545 as valid solutions to cubics and quartics, along with the recognition of imaginary numbers as meaningful. • Vieta, uses vowels for unknowns and use powers. Liebniz 1687 develops rules for symbolic manipulation. Modern Diophantus 200AD

  14. Precession in GASpin-1/2 Bz Z=σ3 ω θ <Sx>=Sin θ Cos ωt <Sy>=Sin θ Sin ωt <Sz>=Cos θ x= σ1 ω = γ Bz Y= σ2

  15. Quotes • “The reasonable man adapts himself to the world around him. The unreasonable man persists in his attempts to adapt the world to himself. Therefore, all progress depends on the unreasonable man.” George Bernard Shaw, • Murphy’s two laws of discovery: “All great discoveries are made by mistake.” “If you don't understand it, it's intuitively obvious.” • “It's easy to have a complicated idea. It's very hard to have a simple idea.” Carver Mead.

  16. Greek concept of the product Euclid Book VII(B.C. 325-265) “1. A unit is that by virtue of which each of the things that exist is called one.” “2. A number is a multitude composed of units.” …. “16. When two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.”

  17. e3 ι=e1e2e3 e2 e1e2 e2e3 e1e3 e1

  18. Conventional Dirac Equation “Dirac has redisovered Clifford algebra..”, Sommerfield That is for Clifford basis vectors we have: isomorphic to the Dirac agebra.

  19. Free Maxwell equation(J=0): Dirac equation in real space Same as the free Maxwell equation, except for the addition of a mass term and expansion of the field to a full multivector.

  20. Special relativity Its simpler to begin in 2D, which is sufficient to describe most phenomena. We define a 2D spacetime event as So that time is represented as the bivector of the plane and so an extra Euclidean-type dimension is not required. This also implies 3D GA is sufficient to describe 4D Minkowski spacetime. We find: the correct spacetime distance. We have the general Lorentz transformation given by: Consisting of a rotation and a boost, which applies uniformly to both coordinate and field multivectors. Compton scattering formula C

  21. Time after time • “Of all obstacles to a thoroughly penetrating account of existence, none looms up more dismayingly than time.” Wheeler 1986 • In GA time is a bivector, ie rotation. • Clock time and Entropy time

  22. The versatile multivector (a generalized number) a+ιb v ιw ιb v+ιw a+ιw a+v Complex numbers Vectors Pseudovectors Pseudoscalars Anti-symmetric EM field tensor E+iB Quaternions or Pauli matrices Four-vectors

  23. Penny Flip game Qubit Solutions

  24. Use of quaternions Used in airplane guidance systemsto avoid Gimbal lock

  25. How many space dimensions do we have? • The existence of five regular solids implies three dimensional space(6 in 4D, 3 > 4D) • Gravity and EM follow inverse square laws to very high precision. Orbits(Gravity and Atomic) not stable with more than 3 D. • Tests for extra dimensions failed, must be sub-millimetre

  26. Benefits of GA • Maxwell’s equations reduce to a single equation • 4D spacetime embeds in 3D • The Dirac equation in 4D spacetime reduces to a real equation in 3D

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