The Pauli Algebra Cl(3) Is Frobenius In Many Ways Linear operators over a 8-dimensional vector space representing Pauli algebra Ref: - http://arxiv.org/abs/1103.5113
-permuted Frobenius Algebras *Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)* - http://mat.uab.es/~kock/TQFT.html
Frobenius algebras and 2D topological quantum field theories *Joachim Kock* - http://en.wikipedia.org/wiki/Frobenius_algebra
- http://en.wikipedia.org/wiki/Pauli_matrices
- http://en.wikipedia.org/wiki/Clifford_algebra
We need the Axiom LinearOperator library. fricas )library CARTEN ARITY CMONAL CPROP CLOP CALEY Use the following macros for convenient notation fricas -- summation macro Σ(x, Type: Voidfricas -- list macro Ξ(f, Type: Voidfricas -- subscript and superscripts macro sb == subscript Type: Voidfricas macro sp == superscript Type: Void𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients. fricas dim:=8
Type: PositiveIntegerfricas macro ℒ == List Type: Voidfricas macro ℂ == CaleyDickson Type: Voidfricas macro ℚ == Expression Integer Type: Voidfricas 𝐋 := ClosedLinearOperator(OVAR ['1,
Type: Typefricas 𝐞:ℒ 𝐋 := basisOut() Now generate structure constants for Pauli Algebra The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors. The Pauli Algebra as Cl(3) Basis: Each B.i is a Clifford number fricas q0:=sp('i,
Type: Symbolfricas q1:=sp('j,
Type: Symbolfricas q2:=sp('k,
Type: Symbolfricas QQ:=CliffordAlgebra(3,
Type: Typefricas B:ℒ QQ := [monomial(1,
fricas M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j,
fricas S(y) == map(x +-> coefficient(recip(y)*x, Type: Voidfricas ѕ :=map(S, fricas Compiling function S with type CliffordAlgebra(3,
Type: List(List(List(Expression(Integer))))fricas -- structure constants form a tensor operator Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, Units fricas e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8; Multiplication of arbitrary quaternions and fricas a:=Σ(sb('a, Multiplication is Associative fricas test( ( I Y ) / _ ( Y ) = _ ( Y I ) / _ ( Y ) ) A scalar product is denoted by the (2,0)-tensor fricas U:=Σ(Σ(script('u, ## Definition 1 We say that the scalar product is Y = Y U U In other words, if the (3,0)-tensor:
Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators. fricas ω:𝐋 := _ ( Y I ) / _ U - _ ( I Y ) / _ U; ## Definition 2An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?. The Cartan-Killing Trace fricas Ú:= ( Y Λ ) / _ ( Y I ) / _ V forms a non-degenerate associative scalar product for Y fricas Ũ := Ù
Type: Variable(Ù)fricas test ( Y I ) / Ũ = ( I Y ) / Ũ General Solution Frobenius Form (co-unit) fricas d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4+εij*𝐝.5+εik*𝐝.6+εjk*𝐝.7+εijk*𝐝.8 In general the pairing is not symmetric! fricas u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), The scalar product must be non-degenerate: fricas --Ů:=determinant u1 --factor(numer Ů)/factor(denom Ů) 1
Type: PositiveIntegerCartan-Killing is a special case fricas ck:=solve(equate(Ũ=Ų), Frobenius scalar product of "vector" quaternions and fricas a:=sb('a, ## Definition 3Co-scalar product Solve the Snake Relation as a system of linear equations. fricas mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), The common demoninator is fricas --squareFreePart factor denom Ů / squareFreePart factor numer Ů matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), Check "dimension" and the snake relations. fricas O:𝐋:= Ω / Ų
fricas test ( I ΩX ) / ( Ų I ) = I Cartan-Killing co-scalar fricas eval(Ω,
Type: Expression(Integer)## Definition 4Co-algebra Compute the "three-point" function and use it to define co-multiplication. fricas W:= (Y I) / Ų; fricas λ:= _ ( I ΩX ) / _ ( Y I ); Cartan-Killing co-multiplication fricas eval(λ,
Type: Expression(Integer)fricas test e / λ = ΩX |