Functions of multivectors in 3D Euclidean geometric algebra via spectral decomposition (for physicists and engineers) Miroslav Josipović [email protected] December, 2015 Geometric algebra is a powerful mathematical tool for description of physical phenomena. The article [ 3] gives a thorough analysis of functions of multivectors in Cl 3 relaying on involutions, especially Clifford conjugation and complex structure of algebra. Here is discussed another elegant way to do that, relaying on complex structure and idempotents of algebra. Implementation of Cl 3 using ordinary complex algebra is briefly discussed. Keywords: function of multivector, idempotent, nilpotent, spectral decomposition, unipodal numbers, geometric algebra, Clifford conjugation, multivector amplitude, bilinear transformations 1. Numbers Geometric algebra is a promising platform for mathematical analysis of physical phenomena. The simplicity and naturalness of the initial assumptions and the possibility of formulation of (all?) main physical theories with the same mathematical language imposes the need for a serious study of this beautiful mathematical structure. Many authors have made significant contributions and there is some surprising conclusions. Important one is certainly the possibility of natural defining Minkowski metrics within Euclidean 3D space without introduction of negative signature, that is, without defining time as the fourth dimension ([1, 6]). In Euclidean 3D space we define orthogonal unit vectors e1, e 2, e 3 with the property e2 = 1 ee+ ee = 0 i , ij ji , so one could recognize the rule for multiplication of Pauli matrices. Non-commutative product of two vectors is ab= a ⋅ b + a ∧ b , sum of symmetric (inner product) and anti- symmetric part (wedge product). Each element of the algebra (Cl 3) can be expressed as linear combination of elements of 23 – dimensional basis (Clifford basis ) e e e ee ee ee eee {1, 1, 2, 3, 12, 31, 23, 123 }, where we have a scalar, three vectors, three bivectors and pseudoscalar. According to the number of unit vectors in the product we are talking about odd or even elements. If we define = e e e j 1 2 3 it is easy to show that pseudoscalar j has two interesting properties in Cl 3: 1) j2 = − 1, 2) jX= Xj , for any element X of algebra, and behaves like an ordinary imaginary unit , which enables as to study a rich complex structure of Cl 3. This property we have for n = 3, 7, … [3]. Bivectors can be expressed as product of unit pseudoscalar and vectors, jv . 1 We define a general element of the algebra (multivector ) Mt=+++=+xn jjbz F, ztjb =+, Fxn =+ j where z is a complex scalar and an element of center of algebra , while F , by analogy, is a ∗ ∗ complex vector . Complex conjugation ( j ∈ ℂ ) is z= z† = t − jb , F= F† = x − j n , where dagger means reversion , to be defined later in the text. The complex structure allows a different ways of expressing multivectors, one is Mt=++xn j + jbtj =+ n + jbj( − x ) , where multivector of the form a+ vj vˆ belongs to the even part of the algebra and can be associated with rotations, spinors or quaternions. Also we could treat multivector as ([ 12]) 3 M =α + αe α ∈ ℂ 0 ∑ i i, k and implement it relying on ordinary complex numbers. i=1 Main involutions in Clifford algebra are: ˆ 1) grade involution: Mt=−x + j n − jb ∗ ∗ 2) reverse (adjoint ): M† =+− tx j n − jb = z + F 3) Clifford conjugation: Mt=−−xn j + jbz =+ F =− z F , where an asterisk stands for a complex conjugate. Grade involution is the transformation ˆ x= − x (space inversion ), while reverse in Cl 3 is like a complex conjugation, x†= x , j † = − j . Clifford conjugation is combination of two involutions MM=ˆ † , x =− x , jj = . Bivectors given as a wedge product could be expressed as x∧ y =j xy × , where x× y is a cross product . An application of involutions is easy now. 2 Defining paravector p= t + x we have ppt=+x =+()() t x t − x =− tx2 2 and we have a usual metric of special relativity. From MM=ˆ ⇒ Mtj =+ n , the even part of the algebra (spinors ). † From MM=† ⇒ Mt =+ x , paravector; reverse is an anti-automorphism (MM†) = MM † , so MM † (square of multivector magnitude , [ 2]) is a paravector. From M= M ⇒ M =+ t jbz = , a complex scalar. Clifford conjugation is anti-automorphism, MM= MM , so MM (square of multivector amplitude , [ 2]) is a complex scalar and there is no other “amplitude” with such a property ([ 1]). We define a multivector amplitude M (hereinafter MA) 2 2 2 2 2 MMM= =−+−+ txnb2 jtb() −⋅x n , M ∈ ℂ (3) which we could express as MMMz==+()()FF z −=− z2 FF 2, 2 =−+⋅∈ xnj 22 2 xn ℂ . 2 2 2 2 2 For F=0, or F = N = 0 fallows M= z ( N=x − n , x ⋅= n 0 is a nilpotent in the F 2 =c ∈ ℝ x⋅ n = 0 F F 2 ∈ ℂ algebra). For ( , whirl [1]) here is used a designation ()c . From 2 F= F F2 F 2 = FFˆ= −= F2 FF =− FF ˆ 2 =− we have ()1/ , () 1 1 , and / /j ()1 , 1 (complex = u=± uuuu2 = = unit vector ). With f F ()1 we also define ±(1f ) / 2, ±±+−, 0 , idempotents 2 of the algebra ([ 1, 9]). Every idempotent in Cl 3 can be expressed as u±= p ± +N ±±, N = 0 , =1 ± e / 2. where p± are simple idempotents, like p± ( 1 ) For example, u=1 +e + e + j e / 2, N= e + j e (figure below). + ( 1 2 3 ) 2 3 2. Implementation 3 M=+α ααe =+ A α ∈ ℂ From 0∑ i i 0 , k , it is i=1 easy to implement the algebra on computer using ordinary complex numbers only. In [15 ] are defined products: 3 A B = ∑αi β i (generalized inner product ), i=1 A⊗ B = e det[ i , α i, β i ], (generalized outer product ), and AB= AB + A ⊗ B (generalized geometric product ). α+A α += B ααα +++ ABAB α α Now we have ( A)( B) ABB A . We can find i for multivector Mt=+x + j n + jb using linear independency. 3. Spectral decomposition F 2 = 1 Starting from multivector (1) 2 2 "whirl" Mz=+=+F z Ff =+ xy fF, ≠ 0 we see the form of unipodal-like numbers. Defining M± = x ± y and recalling a relation (easy to proof) fu±= ± u ± follows =± =± = Mu±( xyuf ) ±( xyu) ±±± Mu , so we have a projection. Spectral basis u± is very useful because the binomial expansion of a multivector is very simple M2 = Mu + Mu2 =+⇒=+ Mu2 Mu 2 Mn Mu n Mun n ∈ ℤ ()++ −− ++ −− ++ −− , , where n < 0 is possible for M ≠ 0 ( M−1 = M/ ( MM ) ). 3 − Defining conjugation ()ab+f = ab − f (obviously the Clifford conjugation) we have − ()()ab+f ab +=− f ab2 2 , where a2− b 2 = MM− = MM is a multivector amplitude. In u− = u MM− =+ Mu Mu Mu += Mu MM spectral basis using ± ∓ we have ( ++ −−)( +− −+) +− . Starting from M= z + F we have 2 x y 2 2 M=+ zFff =+=+= x y ρ f ρϕ()cosh + f sinh ϕρ , =−x y , ρ ρ obtaining the polar form of a multivector. If a multivector amplitude is zero we have light-like multivector and there is no a polar form. Now defining tanh ϕ= ϑ (“velocity”) we have M =ρ()()cosh ϕ +f sinh ϕργϑγ =+ 1 f , −1 =−1 ϑ 2 and in the spectral basis M=1 + = kukuk + ⇒= 1 ±= K ±1 ργϑ( f ) ++ −− ± ργϑρ( ) , where K =()()1 +ϑ / 1 − ϑ is a generalized Bondi factor (ϕ = log K ). Now we have Ku+ Ku−1 Ku += Ku − 1 KKu + KKu −− 11 =+⇒= Ku Ku − 1 K KK ( 1+−+− 1)( 1 1) 1 2 + 1 2 −+− 1 2 , or 1+ 1 += 1 ++ += γγ12( ϑ 1f)( ϑ 2 f) γγ 12( ( ϑϑ 12) f ϑϑ 12 ) 1+ 1 ++ /1 + ⇒ γγ12()()() ϑϑ 12() ϑϑ 12 ϑϑ 12 f = + =+ + γγγ12(1 ϑϑ 12) , ϑ( ϑϑ 12) /1( ϑϑ 12 ) , and we have “velocity addition rule”. So, every multivector could be mathematically treated like an ordinary boost in special relativity. For ρ = 1 we have a “boost” 1+ =Ku + K−1 u γ( ϑ f ) + − as transformation that preserves multivector amplitude and should be considered as a part of the Lorentz group ([1, 2]). As a simple example of a unit complex vector we already mentioned fee=++1 2j e 3 =++ ee 1 2 ee 12 , completely in Cl 2, suggesting F that one could analyze problem in basis (1, (1) ) or related spectral basis in e1 e 2 plane and rotate all elements to obtain relations for an arbitrary orientation of a plane, using powerful apparatus of geometric algebra for rotations. Mapping basis (1, f ) to (eφf , f e φ f ) we obtain new orthogonal basis and new components of multivector ab+→f ae′φf + be ′ f φ f =+( abe ′′ f ) φ f , aae′=−φf, bbe ′ = − φ f , ab +=+f ab ′′ f . 4. Functions of multivectors Using series expansion it is straight forward to find a closed formulae for (analytic at least) functions. If F 2 = 0 we have fM( ) = fz( ) and it is easy to find closed form using theory of functions on the complex field. Otherwise, from the series expansion 4 n f( ) (0) x n f()() x= f 0 + ∑ n n! 2 n n using M= Mu++ + Mu −− we have = + fM( ) fMu( ++) fMu( −−) 2 and, again, it is “easy” to find a closed form because of M ± ∈ ℂ . For M =F = F f we have =±⇒F2 = F 2 +− F 2 . If function is even we have M± fMf() ( ) uf+( ) u − F= F2 + = F 2 and similarly for odd functions f() f( )() uuf+ − ( ) =2 − = 2 . For M=+ z F, F2 = N 2 = 0 there is no spectral f()FF f( )() uuf+ − ( Ff) nn n n −1 decomposition ( f is not defined), but we have M=+() zN = znz + N , giving fz( +N) = fz( ) + fz′( ) N .