login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

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

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

fricas
dim:=8

\label{eq1}8(1)
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k,'ij,'ik,'jk,'ijk], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k , ij , ik , jk , ijk ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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,[2])

\label{eq3}i^{2}(3)
Type: Symbol
fricas
q1:=sp('j,[2])

\label{eq4}j^{2}(4)
Type: Symbol
fricas
q2:=sp('k,[2])

\label{eq5}k^{2}(5)
Type: Symbol
fricas
QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])

\label{eq6}\hbox{\axiomType{CliffordAlgebra}\ } (3, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , [ [ i [ ; 2 ] , 0, 0 ] , [ 0, j [ ; 2 ] , 0 ] , [ 0, 0, k [ ; 2 ] ] ])(6)
Type: Type
fricas
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])]

\label{eq7}\left[ 1, \:{e_{1}}, \:{e_{2}}, \:{e_{3}}, \:{{e_{1}}\ {e_{2}}}, \:{{e_{1}}\ {e_{3}}}, \:{{e_{2}}\ {e_{3}}}, \:{{e_{1}}\ {e_{2}}\ {e_{3}}}\right](7)
Type: List(CliffordAlgebra(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
fricas
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq8}\left[ 
\begin{array}{cccccccc}
1 &{e_{1}}&{e_{2}}&{e_{3}}&{{e_{1}}\ {e_{2}}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}
\
{e_{1}}&{i^{2}}& -{{e_{1}}\ {e_{2}}}& -{{e_{1}}\ {e_{3}}}& -{{i^{2}}\ {e_{2}}}& -{{i^{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}
\
{e_{2}}&{{e_{1}}\ {e_{2}}}&{j^{2}}& -{{e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{1}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}
\
{e_{3}}&{{e_{1}}\ {e_{3}}}&{{e_{2}}\ {e_{3}}}&{k^{2}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}}&{{k^{2}}\ {e_{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}
\
{{e_{1}}\ {e_{2}}}&{{i^{2}}\ {e_{2}}}& -{{j^{2}}\ {e_{1}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}}
\
{{e_{1}}\ {e_{3}}}&{{i^{2}}\ {e_{3}}}& -{{e_{1}}\ {e_{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}}& -{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{i^{2}}\ {k^{2}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}}
\
{{e_{2}}\ {e_{3}}}&{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{j^{2}}\ {e_{3}}}& -{{k^{2}}\ {e_{2}}}&{{j^{2}}\ {e_{1}}\ {e_{3}}}& -{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}}
\
{{e_{1}}\ {e_{2}}\ {e_{3}}}&{{i^{2}}\ {e_{2}}\ {e_{3}}}& -{{j^{2}}\ {e_{1}}\ {e_{3}}}&{{k^{2}}\ {e_{1}}\ {e_{2}}}& -{{i^{2}}\ {j^{2}}\ {e_{3}}}&{{i^{2}}\ {k^{2}}\ {e_{2}}}& -{{j^{2}}\ {k^{2}}\ {e_{1}}}& -{{i^{2}}\ {j^{2}}\ {k^{2}}}
(8)
Type: Matrix(CliffordAlgebra(3,Expression(Integer),[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]))
fricas
S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
Type: Void
fricas
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
Compiling function S with type CliffordAlgebra(3,Expression(Integer)
      ,[[i[;2],0,0],[0,j[;2],0],[0,0,k[;2]]]) -> Matrix(Expression(
      Integer))

\label{eq9}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
\left[{\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \:{j^{2}}, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \:{k^{2}}, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}}, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {k^{2}}}, \: 0, \: 0 \right]}, \: \right.
\
\
\displaystyle
\left.{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{j^{2}}\ {k^{2}}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}\ {k^{2}}}\right]}\right] 
(9)
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, i,1..dim), j,1..dim), k,1..dim)
There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name. Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

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;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))

Multiplication of arbitrary quaternions a and b

fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
There are no exposed library operations named I but there is one 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)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

A scalar product is denoted by the (2,0)-tensor U = \{ u_{ij} \}

fricas
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name. Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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:


\scalebox{1} % Change this value to rescale the drawing.
{
\begin{pspicture}(0,-0.92)(4.82,0.92)
\psbezier[linewidth=0.04](2.2,0.9)(2.2,0.1)(2.6,0.1)(2.6,0.9)
\psline[linewidth=0.04cm](2.4,0.3)(2.4,-0.1)
\psbezier[linewidth=0.04](2.4,-0.1)(2.4,-0.9)(3.0,-0.9)(3.0,-0.1)
\psline[linewidth=0.04cm](3.0,-0.1)(3.0,0.9)
\psbezier[linewidth=0.04](4.8,0.9)(4.8,0.1)(4.4,0.1)(4.4,0.9)
\psline[linewidth=0.04cm](4.6,0.3)(4.6,-0.1)
\psbezier[linewidth=0.04](4.6,-0.1)(4.6,-0.9)(4.0,-0.9)(4.0,-0.1)
\psline[linewidth=0.04cm](4.0,-0.1)(4.0,0.9)
\usefont{T1}{ptm}{m}{n}
\rput(3.4948437,0.205){-}
\psline[linewidth=0.04cm](0.6,-0.7)(0.6,0.9)
\psbezier[linewidth=0.04](0.0,-0.1)(0.0,-0.9)(1.2,-0.9)(1.2,-0.1)
\psline[linewidth=0.04cm](0.0,-0.1)(0.0,0.9)
\psline[linewidth=0.04cm](1.2,-0.1)(1.2,0.9)
\usefont{T1}{ptm}{m}{n}
\rput(1.6948438,0.205){=}
\end{pspicture} 
}
 


\label{eq10}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (10)
(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.

fricas
ω:𝐋 :=                 _
     (    Y I    )  /  _
           U        -  _
     (    I Y    )  /  _
           U;
There are no exposed library operations named Y but there are 2 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.

Definition 2

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

The Cartan-Killing Trace

fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V
There are no exposed library operations named Y but there are 2 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(Λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq11}��(11)
Type: Variable(Ù)
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
There are no exposed library operations named Y but there are 2 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)

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
There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝 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.

In general the pairing is not symmetric!

fricas
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

The scalar product must be non-degenerate:

fricas
--Å®:=determinant u1
--factor(numer Å®)/factor(denom Å®)
1

\label{eq12}1(12)

Cartan-Killing is a special case

fricas
ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
There are no library operations named equate Use HyperDoc Browse or issue )what op equate to learn if there is any operation containing " equate " in its name.
Cannot find a definition or applicable library operation named equate with argument type(s) Equation(Symbol)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Frobenius scalar product of "vector" quaternions a and b

fricas
a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas
mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

The common demoninator is 1/\sqrt{\mathring{U}}

fricas
--squareFreePart factor denom Å® / squareFreePart factor numer Å®
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name. Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. There are no library operations named 𝐝 Use HyperDoc Browse or issue )what op 𝐝 to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝 with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Check "dimension" and the snake relations.

fricas
O:𝐋:= Ω / Ų

\label{eq13}�� \  ��(13)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ų I     )  =  I
There are no exposed library operations named I but there is one 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)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing co-scalar

fricas
eval(Ω,ck)

\label{eq14}��(14)
Type: Expression(Integer)

Definition 4

Co-algebra

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

fricas
W:= (Y I) / Ų;
There are no exposed library operations named Y but there are 2 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.

fricas
λ:=                      _
     (    I ΩX     )  /  _
     (     Y I     );
There are no exposed library operations named I but there is one 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)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing co-multiplication

fricas
eval(λ,ck)

\label{eq15}��(15)
Type: Expression(Integer)

fricas
test
         e     /
         λ     =    ΩX
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)




  Subject:   Be Bold !!
  ( 13 subscribers )  
Please rate this page: