Geometric Algebra Techniques in Mathematics, Physics and Engineering S. Xamb´o UPC IMUVA 16-19 Nov 2015 · · S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 1 / 47 Index Main points Background. Lorentz transformations. Introduction. Definitions, notations and conventions. The Dirac representation of . D + Lorentzian GA. = 1;3 = ΛηE. Involutions and . Complex structure. G G G The nature of the even subalgebra. Inner products in and . Relative P D involutions. Space-time kinematics. Space-time paths. Timelike paths and proper time. Instantaneous rest frame. Relative velocity. Lorentz transformations. Lorentz spinors. Rotations. Lorentz boosts. Relativistic addition of velocities. General boosts. Lorentz group. Appendix A. Appendix B. EM field. Maxwell's equations. Gradient operator. Riesz form of Maxwell's equations. GA form of Dirac's equation. References. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 2 / 47 Background Lorentz transformations The Lorentz boost for speed v is given by t0 = γ(t βx); x 0 = γ(x βt) − p − where β = v=c and γ = 1= 1 β2. − This transformation leaves invariant the Lorentz quadratic form (ct)2 x 2 y 2 z 2. − − − S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 3 / 47 Background Lorentz transformations Let γx ; γy ; γz ; γt be the unit vectors on the axes of the reference S 0 0 0 0 0 and γx ; γy ; γz ; γt be the unit vectors on the axes of the reference S . Then the above Lorentz boost is equivalent to the linear transformation 0 0 0 0 γt = γ(γt + βγx ); γx = γ(γx + βγt ); γy = γy ; γz = γz ; which is a Lorentz isometry. One basic results of elementary relativity theory is that any proper Lorentz isometry that preserves the temporal orientation is, up to a roation of the x; y; z axis and a rotation of the x 0; y 0; z 0 axis, a Lorentz boost. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 4 / 47 Introduction Definitions, notations and conventions Historically, the spacetime algebra was the first modern implementation of geometric algebra. This is because it provides a sythetic framework for studying spacetime physics. Doran-Lasenby-2003, Ch. 5. A Lorentzian spacetime is a real quadratic space (E; η) of signature (1; 3). We express this by writing E1;3 instead of E. We also write a b to mean η(a; b). The elements a E are called events. We will use· the customary terms time-like, space-like2 and light-light to refer to vectors such that η(a; a) is positive, negative or null, respectively. An (inertial) frame of E1;3 is an orthonormal basis γ0; γ1; γ2; γ3: γ0 γ0 = 1; γ0 γj = 0; γj γk = δj;k (j; k 1; 2; 3 : · · · − 2 f g Or, in the familiar relativistic notations, γµ γν = ηµν (µ, ν 0; 1; 2; 3 ): · 2 f g S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 5 / 47 Introduction Definitions, notations and conventions Instead of the indices 0; 1; 2; 3 , we will also use the labels t; x; y; z . The componentsf ofg an event a in the frame γ are f gµ µ denoted a , and so a = a γµ (Einstein's summation convention). Instead of a0 = at we often write t (actually ct, but we can choose k units for which c = 1), so that a = tγ0 + a γk . Remark. In Dirac's theory, the γµ are certain 4 4 matrices (Dirac matrices), but here they are vectors. The Dirac× matrices yield (see next slide) a representation of the space-time algebra = 1;3 = ΛηE1;3, and so is Dirac's algebra without matrices. The beautyD G and usefulness of Dwill be apparent along the way, much as it happened with the treatmentD of quaternions by geometric algebra. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 6 / 47 Introduction The Dirac representation of D Proposition. Define σ0 σk Γ0 = and Γk = − , k = 1; 2; 3. σ0 σk − Then there exists an algebra isomorphism R(4) such that G! γµ Γµ. 7! Proof. The Γµ satisfy the Clifford relations ΓµΓν + ΓνΓµ = 2ηµνI4. This follows from the Clifford relations σj σk + σk σj = 2δjk satisfied by the Pauli matrices σ1; σ2; σ3 and straightfoward matrix computations. So there is an algebra homomorphism R(4) such G! that γµ Γµ. Finally, this homomorphism is an isomorphism: the 7! images ΓI of the γI (I running over the multiindeces of 0; 1; 2; 3) turn out to be lineraly independent. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 7 / 47 Lorentzian GA D = G1;3 = Λη E In terms of the frame γ0; γ1; γ2; γ3, the basic rule in is D γµγν + γνγµ = 2ηµν: Let i be the pseudo-scalar unit associated to that frame: i = γ0γ1γ2γ3: By GAT-03.24, Theorem 5, i anticommutes with vectors and trivectors and commutes with scalars, bivectors and pseudo-scalars. In a compact form, xi = ( 1)r ix for x r . Moreover, we have (see also the Remark after its proof):− 2 G Proposition 1) i2 = Q(i) = det(diag(+; ; ; )) = 1: 2) The Hodge duality map −r − − 4−r , x− x ∗ = xi = ( 1)r ix is an antiisometry for r = 0D; 1;!D2; 3. This implies7! that the− signatures of 2, 3 and 4 are ( 3; 3), (3; 1) and (0; 1) = 1.¯ D D D − S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 8 / 47 Lorentzian GA D = G1;3 = Λη E Let σ1 = γ1γ0 = γ10, σ2 = γ2γ0 = γ20, σ3 = γ3γ0 = γ30. Then a ∗ short computation shows that σj = σj i = γk γl = γkl (j; k; l a cyclic permutation of (1; 2; 3)).1 Explicitely,− − ∗ ∗ ∗ σ = γ23; σ = γ31; σ = γ12: 1 − 2 − 3 − ∗ The σj and σj have signatures 1 and +1, respectively, and together form a basis of 2. − D ∗ 3 The γµ = γµi form a basis of and they have signatures 1 for µ = 0 and +1 otherwise. NoteD that − ∗ ∗ ∗ ∗ γ0 = γ123; γ1 = γ023; γ2 = γ031; γ3 = γ012: With these notations, we finally have i = γ0123: 1 ∗ This is often condensed as σj = jkl γk γl , where jkl denotes the sign of the permutation jkl of 123 (Levi-Civita symbol).− S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 9 / 47 Lorentzian GA D = G1;3 = Λη E The preceding facts are summarized in the following table: 1 γ0 γ1 γ2 γ3 x∗ = x i σ1 σ2 σ3 σ1∗ σ2∗ σ3∗ Q(x)=+1 γ0∗ γ1∗ γ2∗ γ3∗ Q(x)= 1 − 1∗ S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 10 / 47 Lorentzian GA D = G1;3 = Λη E ∗ ∗ ∗ ∗ ∗ ∗ ∗ G γ0 γ1 γ2 γ3 σ1 σ2 σ3 σ1 σ2 σ3 γ0 γ1 γ2 γ3 i ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ0 1σ ¯1 σ¯2 σ¯3 γ¯1 γ¯2 γ¯3 γ¯1 γ¯2 γ¯3 i σ¯1 σ¯2 σ¯3 γ0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ1 σ1 -1σ ¯3 σ2 γ¯0 γ3 γ¯2 γ¯0 γ¯3 γ2 σ1 -i σ3 σ¯2 γ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ2 σ2 σ3 -1σ ¯1 γ¯3 γ¯0 γ1 γ3 γ¯0 γ¯1 σ2 σ¯3 -i σ1 γ2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ3 σ3 σ¯2 σ1 -1 γ2 γ¯1 γ¯0 γ¯2 γ1 γ¯0 σ3 σ2 σ¯1 -i γ3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ1 γ1 γ0 γ¯3 γ2 1 σ3 σ¯2 i σ¯3 σ2 γ1 γ0 γ3 γ¯2 σ1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ2 γ2 γ3 γ0 γ¯1 σ¯3 1 σ1 σ3 i σ¯1 γ2 γ¯3 γ0 γ1 σ2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ3 γ3 γ¯2 γ1 γ0 σ2 σ¯1 1σ ¯2 σ1 i γ3 γ2 γ¯1 γ0 σ3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ1 γ¯1 γ¯0 γ¯3 γ2 i σ¯3 σ2 -1σ ¯3 σ2 γ1 γ0 γ¯3 γ2 σ¯1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ2 γ¯2 γ3 γ¯0 γ¯1 σ3 i σ¯1 σ3 -1σ ¯1 γ2 γ3 γ0 γ¯1 σ¯2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ σ3 γ¯3 γ¯2 γ1 γ¯0 σ¯2 σ1 i σ¯2 σ1 -1 γ3 γ¯2 γ1 γ0 σ¯3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ0 -i σ1 σ2 σ3 γ¯1 γ¯2 γ¯3 γ1 γ2 γ3 1σ ¯1 σ¯2 σ¯3 γ¯0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ1 σ¯1 i σ¯3 σ2 γ¯0 γ¯3 γ2 γ0 γ¯3 γ2 σ1 -1σ ¯3 σ2 γ¯1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ2 σ¯2 σ3 i σ¯1 γ3 γ¯0 γ¯1 γ3 γ0 γ¯1 σ2 σ3 -1σ ¯1 γ¯2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ γ3 σ¯3 σ¯2 σ1 i γ¯2 γ1 γ¯0 γ¯2 γ1 γ0 σ3 σ¯2 σ1 -1 γ¯3 ∗ ∗ ∗ ∗ ∗ ∗ ∗ i γ¯0 γ¯1 γ¯2 γ¯3 σ1 σ2 σ3 σ¯1 σ¯2 σ¯3 γ0 γ1 γ2 γ3 -1 This is the multiplication table of in terms of the described basis. The bar over D symbols indicates minus sign, not Clifford conjugation. S. Xamb´o (UPC) GAT 04 Space-time GA IMUVA · 16-19 Nov · 2015 11 / 47 Lorentzian GA D = G1;3 = Λη E A product γI γJ is determined by the rule GAT-03.25, Corollary 1. In this case it yields γK , K = I M J (symmetric difference) and 1 = ( 1)ν, where± ν is the sum of the number of elements in ±1; 2; 3− I J and the number of inversions in the sequence I ; J. f g \ \ By GAT-03.25, Corollary 2, the table is symmetric up to sign, c rs because γJ γI = ( 1) ( 1) γI γJ , where r = I , s = J and − − j j j j c = I J .