A Mathematica Program for Geometric Algebra Gordon Erlebacher Garret Sobczyk

Mi vida a Udla Learn Spanish Learn to dance Learn about life at Udla Work with Garret Not necessarily in the above order!!

Hard at work

Learning spanish a la Fiesta y a la Iglesia

Learning spanish while dancing

My influence on Garret

Gordon repents for his fun

Gordon starts serious work

Objectives Develop a new algebra of 2x2 matrices whose elements are in the geometric algebra G3 Develop a Mathematica program to perform symbolic calculations in this algebra Reduce calculation errors Check calculations done manually Try out new ideas Peforms complex computations

Contents Algebras Mathematica Program Application

What is an Algebra? Combination of a Field of scalars , a vector space (+) and a ring (x) Commutativity of elements of the Field and elements of the algebra

Examples of Algebras All non-invertible matrices Unitary matrices Integers Real numbers Set of polynomials with integer coefficients

Concepts from Geometric Algebra Let (vector space) Graded Algebra: Non-commutative Geometric interpretations are vectors are scalars are bivectors (tangent plane) are trivectrors (volume elements) Scalar product Outer product

Geometric interpretation 2D vector space Scalar  point (grade 0) Directed line  vector (grade 1) Directed plane  bivector (grade 2) Directed volume  trivector (grade 3)

2D: G2 Basis : 4=22 elements one scalar: 1 two vectors: one pseudoscalar, bivector:

3D: G3 Basis : 8=23 elements one scalar: 1 three orthonormal vectors: three bivectors: one pseudoscalar, bivector:

Generalizations Geometric algebra generalizes nicely to higher dimensions n-D

Element of G3 An element of takes the form . has properties of imaginary number, i.e., for all Rewrite element g as 4 complex numbers encode the same information as the 8 elements of G3 1 2,3,4

Paravector Element g of G3 is written as called a paravector

Operations on G3 Geometric product Rewriting in terms of complex coefficients, we find that /\ is associative, is not is symmetric part of is the antisymmetric part of

Algebra: G41 algebra The algebra is generated by the vector space V41 Signature of V41 is (++++-), or Pseudoscalar: 5-D volume element 32 basis functions: 1 scalar, 5 vectors, 10 bivectors (2-vector), 10 trivectors, 5 4-vectors, 1 pseudoscalar (5-vector)

Isomorphism between Gn and matrix algebra It can be shown that every G2n (in ) is isomorphic to the algebra Mm of mxm matrices of reals (in ) . Thus, if then (homomorphism)

Isomorphism G41 M2(G3) Element g of G41 is isomorphic to Element a of G3 is isomorphic to

Degrees of freedom G4,1 has 25=32 degrees of freedom A 2x2 matrix has 4 degrees of freedom G3 has 8 degrees of freedom Therefore, M2(G3) has 32 degrees of freedom, consistent with G41

Why M2(G3)? Combine the advantages of matrix algebra with that of Geometric (Clifford) algebras 2x2 matrices are small and simple to manipulate Extensive literature on matrices G3 is closely related to the standard vector algebra of Gibbs

Why G41? 5-dimensional vector space is a superset of the following useful spaces: Euclidean space Quaternions Affine space Projective space Horosphere Space of special relativity

Mathematica Parent company: Wolfram Powerful software package for symbolic manipulation Exists for more than 15 years Main competitor: Maple (with similar capabilities, but a different programming style)

Paravector

ParaMatrix

Operations in G41 The display is independent of internal representation elements

Expansions Display depends on internal representation Choose the representation that is most convenient pV[a] operates the fastest but it is not possible to work with scalar and vector components

More complex operations Expansion Simplification scalarPart, vectorPart Determinant Characteristic polynomial Conversion routines between G41 and M2(G3) more …

Expansion expandAll[x_] := FixedPoint[expandAllOnce,x] expandAllOnce[x_] := Module[{y=x}, (* Conjugation should probably be done near the y = y //. flattenGeomRules; y = y //. geomRules; y = y //. expandGeomRules; y = y //. expandpVRules; y = y //. expandDotRules; y = y //. expandWedgeRules; y = y //. conjugationRules; y = y //. inversionRules; y = y //. reversionRules; y= y //. tripleRules; (* Need a display for y= y //. orderRules; y = y // ExpandAll; y ]

Matrix multiplication

Expansion

Futher expansion

Techniques for simplification Identify scalar components in products and extract them Isoloate scalar components using scal[] to avoid problems in complex expressions Flatten out geometric products to take associativity into account

Checking identities Checking through use of random numbers (not demonstrated here)

We expect that o1 = o2

= 0 ??? Yes!!! Convert to 4-component form

Determinant of g in M2(G3) Consider a 2x2 matrix Element of G3: g = g0+gi ei has the matrix representation The determinant of M is simply the determinant of the 4x4 matrix

Goal Compute determinant using only matrix elements and their conjugates Use Gauss elimination, taking non-conjugation into account Reminder, conjugation of is

Determinant of . The determinant is simply (in )

Objective Express Det[m44] using only and their conjugates How to use Mathematica to do this automatically?.

Determinant Properties Multiplication by a vector of G3 Linear combinations of row (or columns):

Gauss elimination Remember: elements of G3 are not commutative

Gauss Algorithm This form led to a Gauss-like algorithm with the result which can be rewritten as

General Inverse

How about . Try Gauss elimination Leads to very complex formulas Pivot[1,1] Leads to very complex formulas

Using Mathematica

Simplify further

Final answer But …

How about So far, we are not able to find a formula for the determinant (by hand or with Mathematica!) This is our next goal Try to find a recursive formula to compute as a function of

First step Understand better the properties of as a function of lower order traces. Mathematica is required for this.

Examples Tr (a,b) = <a,b>0=scalarPart[oGeom[ab]] Tr (a,b,c) Tr (a,b,c,d) Tr(a,b,c,d,e) Tr (a,b,c,d,e,f) Higher order …

Possible Approach Define Find formulas for and as a function of and where m < n So far we have been unsuccessful. Formulas get complicated very fast

First few traces <ab>0 <abc>0 <abcd>0 <abcde>0

First few vector parts <ab>1 <abc>1 <abcd>1 <abcde>1

Conclusions Potentially powerful new algebra Subalgebras include: Euclidean space, affine space, projective space, horosphere, relativity, twistors, quaternions, etc. Powerful Mathematica program available that operates similarly to pen and paper This work has many potential extensions Search for more general formulas is underway

Muchas Gracias por todo!!! Preguntas???