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The Pauli Algebra Cl(3) Is Frobenius In Many Ways

Linear operators over a 8-dimensional vector space representing Pauli algebra

Ref:

We need the Axiom LinearOperator library.

fricas
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial
CartesianTensor will be automatically loaded when needed from
/var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN
Arity is now explicitly exposed in frame initial
Arity will be automatically loaded when needed from
/var/aw/var/LatexWiki/ARITY.NRLIB/ARITY
ClosedMonoidal is now explicitly exposed in frame initial
ClosedMonoidal will be automatically loaded when needed from
/var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL
ClosedProp is now explicitly exposed in frame initial
ClosedProp will be automatically loaded when needed from
/var/aw/var/LatexWiki/CPROP.NRLIB/CPROP
ClosedLinearOperator is now explicitly exposed in frame initial
ClosedLinearOperator will be automatically loaded when needed from
/var/aw/var/LatexWiki/CLOP.NRLIB/CLOP
CaleyDickson is now explicitly exposed in frame initial
CaleyDickson will be automatically loaded when needed from
/var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

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-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
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-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
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-- subscript and superscripts
macro sb == subscript
Type: Void
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macro sp == superscript
Type: Void

𝐋 is the domain of 8-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

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dim:=8 (1)
Type: PositiveInteger?
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macro ℒ == List
Type: Void
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macro ℂ == CaleyDickson
Type: Void
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macro ℚ == Expression Integer
Type: Void
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𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ) (2)
Type: Type
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𝐞:ℒ 𝐋      := basisOut() (3)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
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𝐝:ℒ 𝐋      := basisIn() (4)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
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I:𝐋:=   -- identity for composition (5)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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X:𝐋:=[2,1] -- twist (6)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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V:𝐋:=ev(1) -- evaluation (7)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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Λ:𝐋:=co(1) -- co-evaluation (8)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void

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

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q0:=sp('i,) (9)
Type: Symbol
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q1:=sp('j,) (10)
Type: Symbol
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q2:=sp('k,) (11)
Type: Symbol
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QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]]) (12)
Type: Type
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B:ℒ QQ := [monomial(1,[]),monomial(1,),monomial(1,),monomial(1,),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])] (13)
Type: List(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
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M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim) (14)
Type: Matrix(CliffordAlgebra?(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
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S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
Type: Void
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ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
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Compiling function S with type CliffordAlgebra(3,Expression(Integer)
,[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]) -> Matrix(Expression(
Integer)) (15)
Type: List(List(List(Expression(Integer))))
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-- structure constants form a tensor operator
Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim) (16)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim) (17)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
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XY := X/Y;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Units

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e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication of arbitrary quaternions and fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim) (18)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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b:=Σ(sb('b,[i])*𝐞.i, i,1..dim) (19)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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(a*b)/Y (20)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication is Associative

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test(
( I Y ) / _
(  Y  ) = _
( Y I ) / _
(  Y  ) ) (21)
Type: Boolean

A scalar product is denoted by the (2,0)-tensor fricas
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim) (22)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

## Definition 1

We say that the scalar product is associative if the tensor equation holds:

    Y   =   Y
U     U


In other words, if the (3,0)-tensor:  (23)
(three-point function) is zero.

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.

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ω:𝐋 :=                 _
(    Y I    )  /  _
U        -  _
(    I Y    )  /  _
U;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

## Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

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Ú:=
(  Y Λ  ) / _
(   Y I ) / _
V (24)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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Ù:=
(  Λ Y  ) / _
( I Y   ) / _
V (25)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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test(Ù=Ú) (26)
Type: Boolean

forms a non-degenerate associative scalar product for Y

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Ũ := Ù (27)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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test
(    Y I    )  /
Ũ        =
(    I Y    )  /
Ũ (28)
Type: Boolean
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determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim) (29)
Type: Expression(Integer)

General Solution

Frobenius Form (co-unit)

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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 (30)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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Ų:= Y/d (31)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

In general the pairing is not symmetric!

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u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim) (32)
Type: Matrix(Expression(Integer))

The scalar product must be non-degenerate:

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--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1 (33)
Type: PositiveInteger?

Cartan-Killing is a special case

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ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
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Compiling function equate with type Equation(ClosedLinearOperator(
OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
-> List(Equation(Expression(Integer))) (34)
Type: List(Equation(Expression(Integer)))

Frobenius scalar product of "vector" quaternions and fricas
a:=sb('a,)*i+sb('a,)*j+sb('a,)*k (35)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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b:=sb('b,)*i+sb('b,)*j+sb('b,)*k (36)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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(a,a)/Ų (37)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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(b,b)/Ų (38)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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(a,b)/Ų (39)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

## Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

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mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
Type: Union(Matrix(Expression(Integer)),...)
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Ω:=Σ(Σ(mU(i,j)*(𝐞.i*𝐞.j), i,1..dim), j,1..dim);
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

The common demoninator is fricas
--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim) (40)
Type: Matrix(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

Check "dimension" and the snake relations.

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O:𝐋:= Ω / Ų (41)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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test
(    I ΩX     )  /
(     Ų I     )  =  I (42)
Type: Boolean
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test
(     ΩX I    )  /
(    I Ų      )  =  I (43)
Type: Boolean

Cartan-Killing co-scalar

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eval(Ω,ck) (44)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

## Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

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W:= (Y I) / Ų;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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λ:=                      _
(    I ΩX     )  /  _
(     Y I     );
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
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test
(     ΩX I    )  /
(    I  Y     )  =  λ (45)
Type: Boolean

Cartan-Killing co-multiplication

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eval(λ,ck) (46)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

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test
e     /
λ     =    ΩX (47)
Type: Boolean

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