Functions of in 3D Euclidean geometric 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: of , idempotent, nilpotent, spectral decomposition, unipodal , , Clifford conjugation, multivector amplitude, bilinear transformations

1. Numbers

Geometric algebra is a promising platform for 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 ([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 . Non-commutative 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 of elements of 23 – dimensional (Clifford basis ) e e e ee ee ee eee {1, 1 , 2 , 3 , 12 , 31 , 23 , 123 }, where we have a , three vectors, three and . According to the 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 , 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 .

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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, or . 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 are:   1) grade involution: Mtˆ =−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 . An application of involutions is easy now.

 2   Defining p= t + x we have ppt=+x =+()() t x t − x =− tx2 2 and we have a usual metric of .  From MM=ˆ ⇒ Mtj =+ n , the even part of the algebra (spinors ).  † From MM=† ⇒ Mt =+ x , paravector; reverse is an anti- (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

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F= F F2 F 2 = FFˆ= −= F2 FF =− FF ˆ 2 =− we have ()1/ , () 1 1 , and / /j ()1 , 1 (complex = u=± uuuu2 = = ). 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 ), 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 (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 ) ).

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− 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 -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 =1 + , =+ /1 + , γγγ12( ϑϑ 12) ϑ( ϑϑ 12) ( 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 ([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 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 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 . Otherwise, from the series expansion

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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 . We also have a special cases = ± 1 fu( ±) f( ) u ± , = −=1 +− 1 f( f ) fuu( +−) fuf( ) +( ) u − , Fˆ =−+ =− + f( ) f()()() ju+− ju f ju + fju − .

Obviously, for an odd function we have f()()f= f1 fF , f( ˆ ) = − fj() f and for even functions f()()f= f1 , f( Fˆ ) = fj() f .

For an inverse function we have

−1 =⇒ =⇒ =⇒= − 1 fyxfxyfx( ) ( ) ( ±±±) y x fy( ± ) .

For a light-like multivectors ( MM = 0 ) we have

M=+ z 2 2 == Ff=z+zF f , z- F 0( z-z F)( z+z F ) , and two possibilities:

=⇒= =⇒ = 1) zzFF M+2 zM , − 0 fMfzu( ) ( 2 F ) + =−⇒ = =− ⇒ =− 2) zzF M+0, M − 2 zF fMfzu( ) ( 2 F ) −

Once a spectral decomposition of a function is analyzed there remains just to use the well known properties of functions of a complex variables.

5. Examples

2 Here we (again) define Mz=+=+F z Ff =+ zz F f and

M+=+ zzF, M − =− zz F .

For the inverse of M we have ( M ≠ 0 )

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1 MuMu+ MuMu + u u M −1 ==+− −+ ==+ +− −+ + − , MuMu+ MuMuMuMu + + MM M M ++ −−()() ++ −− +− −+ +− + − as expected. Now it is obvious that

1 u u M −n = =+ + − . MuMu+ n M n M n ()++ −−() +() −

We can find square root using

MSSuSu==+⇒= MMuMu + = Su2 + Su 2 ⇒=± S M ++−− ++ −−()() ++ −− ± ± . M±1/ n =⇒= SSM±1/ n n ∈ ℕ So, generally we have ±() ± , . As a simple example (an e= ± + e / 2 interested reader could compare with [ 3]) i( j i ) j .

M M+ M − The exponential function is easy one, e= eu+ + eu − and now we have exponentials of complex numbers (just use ordinary i = − 1 and replace i→ j at the end). Logarithm is the inverse function to exponential, so we have

MXeMMuMu=⇒==+=X Xu + Xu ⇒= X M log++−− exp( ++) exp( −−) ± log ± . In [ 3] is derived formula logM= log M + ϕFˆ , ϕ = arctan( F / z) , but those are equivalent:

2 ˆ zF =F =− j FF, =−⇒ j f logM+ log M log M − log M Mu+= Mu ( +) ( −) +( +) ( − ) = log()()++ log −− f 2 2 FFˆ F FFˆ F ˆ logMj− log(  1 − jzjz() /  / 1 +() /  ) =+ log M arctan() / zM =+ logϕ .

Now we can find, for a ∈ℝ , MXa =⇒log XaMXe = log ⇒= alog M , but the same appears to be correct for a= z + F and one can find, for example,

   MXv =⇒log X =v log MXe ⇒= v log M ,  although here some caution is needed because of possibility logX=( log M )v ⇒  X= e (log M )v . Also, relation M= e log M is generally not valid and needs some care due to the multivalued nature of the logarithm . Nevertheless, expressions like e e3 j1 =expe log jj = expπ e / 2 = j e , or e e = j are quite possible in Cl 3. Simple ( 1) ( 1) 1 ()1 2 u =1 ± e / 2 examples ( ± ( 1 ) ):

e1 1. e1=X ⇒ e 1log e 1 = log X , e=−⇒ e = + −=π ⇒ 1uu+−log 1 u +− log1 u log( 1 ) ju − e e =−π ⇒= − π = − π =− 1log 1 juX− exp( ju −) exp ( juu) −−

(solution e1 is not valid because of e1log e 1=−jπ u − =− log e 1 ).

e2 2. e1 =⇒Xlog X = eee2 log 1 = juπ 2 − , euu e= ee uu = 0 =πe = + π e but 2−− 2 22 +− , so Xexp( ju2−) 1 ju 2 − ,

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=ee =π e ⇒=+ π e or logX( log1 ) 2 ju− 2 Xju 1 2 + , n and finally X=1 + jπ e2 u ± and X=1 + jnπ e2 u± , n ∈ ℤ

(solution 1 is not valid because of e2 log e 1 ≠ 0 , it is a nilpotent).

Trigonometric and hyperbolic trigonometric functions are straightforward and ctg one could obtain as inverse of tan. For example

sinFF= sin2u +− sin F 2 u = sin F 2 uu −= FF sin 2 ( ) +( ) −( )() +− 1 ( ) , cosFF= cos2u +−= cos F 22 u cos F uu += cos F 2 ( ) +( ) −( )() +− ( ) . For F= N there is no spectral decomposition, but using exp( zN) = 1+ zN we have

sinNN= , cos N = 1, sinh NN = , cosh N = 1 . There is no N-1, so ctg N is not defined.

Series in powers of argument are crucial for all analytic functions (in geometric algebra too) and we can use presented spectral decomposition to obtain components of such functions in spectral basis.

Conclusion

Geometric algebra of Euclidean 3D space (Cl3) is really rich in structure and gives the possibility to analyze functions defined on multivectors, extending thus theory of functions of real and complex variables, providing intuitive geometrical interpretation also. From simple fact that for a complex vector ( F 2 ≠ 0 ) we can write FF/2= ff , 2 = 1, F 2 ∈ ℂ follows u =1 ± /2 nice possibility to explore idempotent structure ± ( f ) and spectral decomposition of multivectors. Using the of the spectral basis vectors (idempotents) u± it is shown that all multivectors ([1]) can be treated as the unipodal numbers (i.e. hypercomplex numbers over a complex field). A definition of functions is then quite simple and natural and strongly counts on the theory of functions of complex variable. Complex numbers and vectors (bivectors, trivectors) are thus united in one promising system.

Appendix

A1 . Bilinear transformations

Regarding that bilinear transformations of multivectors do not change some property of multivectors one could ask yourself: what property? In [ 2] was shown that the multivector amplitude, defined using the Clifford conjugation, is the unique involution that is commutative ( MM= MM ) and belongs to the center of the algebra. From text we see that multivector amplitude is defined as M+ M − ∈ ℂ using just natural conjugation ab+f → ab − f , where f is our “hypercomplex unit”. But this is just Clifford conjugation and we see now new meaning for it: it is just a “hypercomplex conjugation”. This is a strong argument for regarding Lorentz transformations to be a group of bilinear transformations that preserve multivector amplitude. It is verified on giving known special relativity, but now we should extend it on a whole multivector including in this way all multivector of Euclidean 3D space. For multivector X (transformation) now we have

X+ X − =1 and

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Xe=⇒=M Mlog X = log X +=+==ϕFˆ log1 ϕ FFF ˆ ϕ ˆ ,

  giving thus a general bilinear (twelve parameters) transformation M′ = epq+j Me rs + j .

We started from a new definition of a vector multiplication (Clifford or geometric product) and we see that 3D has a rich complex and hyper-complex structure. The Clifford conjugation is the natural choice to define a multivector amplitude. A bilinear transformations that preserve multivector amplitude form a group containing the ordinary Lorentz transformations (restricted), but it is really natural to assume that a physical reality should be richer. Some consequences of that assumption are discussed in [1] and [2].

A2 . Hyperbolic inner and outer products

Given two multivectors M1= z 1 + z 1 F f and M2= z 2 + z 2 F f we define a square of multivector (conjugate products , [ 13 ]) as

MM− =− zz zz + =− zzzz + zz − zz =+ hiho 12( 11Fff)( 22 F) 1212 FFFF( 12 21 ) ff ,

where hi and ho stands for hyperbolic inner and hyperbolic outer products. If M1= M 2 = M MMz− =−+22 z zz − zz =− z 22 z we have FFF( ) f F , just square of multivector amplitude. This suggests that hi and ho have to do something about being “” or “orthogonal” besides being “near” and “close”. For complex and hypercomplex plane (with real coordinates) meaning is obvious (fig A2).

With ho = 0 multivectors are said to be “h-parallel”, while for hi = 0 they are “h-orthogonal”. For hi = ho = 0 multivector distance is null and we said it to be “h-light-like”, where h- stands for hyperbolic. In “boost” formalism we have

hi = − ρργγ1212(1 ϑϑ 12 ) , ho = −/ 1 − . h-orthogonal h-parallel ρργγ12122( ϑ ϑ 1) ( ϑϑ 12 )

fig A2: Yellow represents h-numbers parallel So, “h-parallel” multivectors have equal hi = or orthogonal to given h-number (red dot). “velocities” and ρ1 ρ 2 , while for “h- orthogonal” multivectors “velocities” are reciprocal and ho becomes infinite (orthogonal multivectors belong to different hyper- quadrants delimited by light-like hyper-planes).

− Lema : Let M1 M 2 = 0 for M1 ≠ 0 and M 2 ≠ 0 .Then M1 M 2 ≠ 0 , and vice versa .

MM=+ Mu Mu Mu += Mu MMu + MMu , 121( ++ 1 −−)( 2 ++ 2 −−) 12 +++ 12 −−− MM− =+ Mu Mu Mu += Mu MMu + MMu 121( +− 1 −+)( 2 ++ 2 −−) 12 −++ 12 +−− , and we have MM12−+=0 or MM 12 +− = 0 , which means M1−=0 and M 2 − = 0 or

M1+=0 and M 2 + = 0 , but either case gives M1 M 2 ≠ 0 . Converse proof is similar.

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A3.

Suppose we have simple equation M 2 +1 = 0 , then objects that squares to -1 are solutions. Using spectral decomposition we could explore it further, so,

M2 +=1 MuMu +2 ++= uuM2 +1 uM + 2 + 1 u =⇒ 0 ()++ −− +−( +) +( −) − 2 2 M++=1 0, M − += 1 0 , so we have two equations over complex numbers. Obvious solutions ˆ are −1, j, je i , F , but it is possible to investigate further. There is an infinite number of solutions, obviously, due to algebraically expanded paradigm of number.

2 2 2 Another simple equation is M= Mu++ + Mu −− = 0 . One obvious solution is M = N (nilpotent) which we cannot obtain using spectral decomposition (because there is no one for a multivector N ), so, some caution is necessary.

In we are using a lot of special polynomials and their roots and we probably should reconsider those in geometric algebra.

References:

[1] Josipović, Some remarks on Cl3 and Lorentz transformations , vixra: 1507.0045v1 [2] Chappell et al, Exploring the origin of Minkowski , arXiv: 1501.04857v3 [3] Chappell, Iqbal, Gunn, Abbott, Functions of multivector variables , arXiv:1409.6252v1 [4] Chappell, Iqbal, Iannella, Abbott, Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime , PLoS ONE 7(12) [5] Dorst, Fontijne, Mann, Geometric Algebra for Computer Science (Revised Edition), Morgan Kaufmann Publishers, 2007 [6] Hestenes, New Foundations for , Kluwer Academic Publishers, 1999 [7] Hestenes, Sobczyk, Clifford Algebra to : A Unified Language for Mathematics and Physics , Kluwer Academic Publishers, 1993 [8] Hitzer, Helmstetter , Ablamowicz, Square Roots of -1 in Real Clifford , arXiv:1204.4576v2 [9] Mornev, Idempotents and nilpotents of Clifford algebra (russian), Гиперкомплексные числавгеометрии и физике, 2(12), том 6, 2009 [10] Sobczyk, New Foundations in Mathematics : The Geometric Concept of Number , Birkhäuser, 2013 [11] Sobczyk, Special relativity in complex , arXiv:0710.0084v1 [12] Sobczyk, Geometric algebra , and its Applications 429 (2008) 1163–1173 [13] Sobczyk, The Hyperbolic Number Plane , http://www.garretstar.com [14] Sobczyk, The Missing Spectral Basis in Algebra and Number Theory , JSTOR [15] Sobczyk, A Complex Gibbs-Heaviside Vector Algebra for Space-time , Acta Physica Polonica, B12, 1981

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