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.

(1) -> )library CARTEN ARITY CMONAL CPROP CLOP CALEY

   >> System error:
   The value
  15684
is not of type
  LIST

Use the following macros for convenient notation

-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])

-- list
macro Ξ(f,i,n)==[f for i in n]

-- subscript and superscripts
macro sb == subscript

macro sp == superscript

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

dim:=8

(1)

macro ℒ == List

macro ℂ == CaleyDickson

macro ℚ == Expression Integer

𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)

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                        )what op ClosedLinearOperator
      to learn if there is any operation containing " 
      ClosedLinearOperator " in its name.

   Cannot find a definition or applicable library operation named 
      ClosedLinearOperator with argument type(s) 
                                    Type
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

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

q0:=sp('i,[2])

(2)

q1:=sp('j,[2])

(3)

q2:=sp('k,[2])

(4)

QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])

(5)

B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[3]),monomial(1,[1,2]),monomial(1,[1,3]),monomial(1,[2,3]),monomial(1,[1,2,3])]

(6)

M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

(7)

S(y) == map(x +-> coefficient(recip(y)*x,[]),M)

ѕ :=map(S,B)::ℒ ℒ ℒ ℚ

Compiling function S with type CliffordAlgebra(3,Expression(Integer)
      ,[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]) -> Matrix(Expression(
      Integer))

(8)

-- 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)

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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
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      with argument type(s) 
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
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      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

   Cannot find a definition or applicable library operation named 𝐞 
      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

Units

e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4; ij:=𝐞.5; ik:=𝐞.6; jk:=𝐞.7; ijk:=𝐞.8;

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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

Multiplication of arbitrary quaternions and

a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
   There are no library operations named 𝐞 
      Use HyperDoc Browse or issue
                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

Multiplication is Associative

test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

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      unexposed operation with that name. Use HyperDoc Browse or issue
                                )display op I
      to learn more about the available operation.

   Cannot find a definition or applicable library operation named I 
      with argument type(s) 
                                 Variable(Y)

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      or "$" to specify which version of the function you need.

A scalar product is denoted by the (2,0)-tensor

U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

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                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.
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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.
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      Use HyperDoc Browse or issue
                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

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:

(9)

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.

ω:𝐋 :=                 _
     (    Y I    )  /  _
           U        -  _
     (    I Y    )  /  _
           U;

   𝐋 is not a valid type.

Definition 2

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

The Cartan-Killing Trace

Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

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      unexposed operations with that name. Use HyperDoc Browse or issue
                                )display op Y
      to learn more about the available operations.

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      with argument type(s) 
                                 Variable(Λ)

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      or "$" to specify which version of the function you need.

forms a non-degenerate associative scalar product for Y

Ũ := Ù

(10)

test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

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      unexposed operations with that name. Use HyperDoc Browse or issue
                                )display op Y
      to learn more about the available operations.

   Cannot find a definition or applicable library operation named Y 
      with argument type(s) 
                                 Variable(I)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

General Solution

Frobenius Form (co-unit)

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

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                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

In general the pairing is not symmetric!

u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)

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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.
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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

The scalar product must be non-degenerate:

--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1

(11)

Cartan-Killing is a special case

ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1

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      to learn if there is any operation containing " equate " in its 
      name.

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      equate with argument type(s) 
                              Equation(Symbol)

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      or "$" to specify which version of the function you need.

Frobenius scalar product of "vector" quaternions and

a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k

(12)

b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k

(13)

(a,a)/Ų

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      having 2 argument(s) but none was determined to be applicable. 
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                                )display op /
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named / 
      with argument type(s) 
                         Tuple(Polynomial(Integer))
                                 Variable(Ų)

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      or "$" to specify which version of the function you need.

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);

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                                 )what op 𝐞
      to learn if there is any operation containing " 𝐞 " in its name.
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      with argument type(s) 
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      to learn if there is any operation containing " 𝐞 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

The common demoninator is

--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)

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      to learn if there is any operation containing " 𝐝 " in its name.
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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.
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                                 )what op 𝐝
      to learn if there is any operation containing " 𝐝 " in its name.

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      with argument type(s) 
                               PositiveInteger

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      or "$" to specify which version of the function you need.

Check "dimension" and the snake relations.

O:𝐋:= Ω / Ų

   𝐋 is not a valid type.

Cartan-Killing co-scalar

eval(Ω,ck)

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      having 2 argument(s) but none was determined to be applicable. 
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                              )display op eval
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named eval 
      with argument type(s) 
                                 Variable(Ω)
                                Variable(ck)

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      or "$" to specify which version of the function you need.

Definition 4

Co-algebra

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

W:= (Y I) / Ų;

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      unexposed operations with that name. Use HyperDoc Browse or issue
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      to learn more about the available operations.

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      with argument type(s) 
                                 Variable(I)

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      or "$" to specify which version of the function you need.

λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );

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      unexposed operation with that name. Use HyperDoc Browse or issue
                                )display op I
      to learn more about the available operation.

   Cannot find a definition or applicable library operation named I 
      with argument type(s) 
                                Variable(ΩX)

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      or "$" to specify which version of the function you need.

Cartan-Killing co-multiplication

eval(λ,ck)

   There are 10 exposed and 6 unexposed library operations named eval 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op eval
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named eval 
      with argument type(s) 
                                 Variable(λ)
                                Variable(ck)

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      or "$" to specify which version of the function you need.

test
         e     /
         λ     =    ΩX

(14)