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.
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(1) -> )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
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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)
Type: Type
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𝐞:ℒ 𝐋 := basisOut()
There are 1 exposed and 0 unexposed library operations named
basisOut having 0 argument(s) but none was determined to be
applicable. Use HyperDoc Browse, or issue
)display op basisOut
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 no-argument definition or library operation named
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
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q0:=sp('i,[2])
Type: Symbol
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q1:=sp('j,[2])
Type: Symbol
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q2:=sp('k,[2])
Type: Symbol
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QQ:=CliffordAlgebra(3,ℚ,matrix [[q0,0,0],[0,q1,0],[0,0,q2]])
Type: Type
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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])]
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)
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))
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)
>> System error:
#<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.fasl" {1002F94153}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.
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;
𝐞 is declared as being in List(ClosedLinearOperator(
OrderedVariableList([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)))
but has not been given a value.
Multiplication of arbitrary quaternions and
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a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
>> System error:
#<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.fasl" {1003111923}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.
Multiplication is Associative
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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 "$" to specify which version of the function you need.
A scalar product is denoted by the (2,0)-tensor
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U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)
There are no library operations named 𝐝
Use HyperDoc Browse or issue
)what op 𝐝
to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" 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 "$" to specify which version of the function you need.
Definition 1
We say that the scalar product is associative if the tensor
equation holds:
Y = Y
U U
In other words, if the (3,0)-tensor:
(three-point function) is zero.
Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.
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ω:𝐋 := _
( Y I ) / _
U - _
( I Y ) / _
U;
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
Perhaps you should use "@" to indicate the required return type,
or "$" 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
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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 "$" to specify which version of the function you need.
forms a non-degenerate associative scalar product for Y
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Ũ := Ù
Type: Variable(Ù)
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test
( Y I ) /
Ũ =
( I Y ) /
Ũ
There are no exposed library operations named Y but there are 2
unexposed operations with that name. Use HyperDoc Browse or issue
)display op Y
to learn more about the available operations.
Cannot find a definition or applicable library operation named Y
with argument type(s)
Variable(I)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
General Solution
Frobenius Form (co-unit)
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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
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 "$" to specify which version of the function you need.
In general the pairing is not symmetric!
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u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
>> System error:
#<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.fasl" {10031E5D23}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.
The scalar product must be non-degenerate:
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--Ů:=determinant u1
--factor(numer Ů)/factor(denom Ů)
1
Cartan-Killing is a special case
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ck:=solve(equate(Ũ=Ų),[ε1,εi,εj,εk,εij,εik,εjk,εijk]).1
There are no library operations named equate
Use HyperDoc Browse or issue
)what op equate
to learn if there is any operation containing " equate " in its
name.
Cannot find a definition or applicable library operation named
equate with argument type(s)
Equation(Symbol)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Frobenius scalar product of "vector" quaternions and
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a:=sb('a,[1])*i+sb('a,[2])*j+sb('a,[3])*k
Type: Polynomial(Integer)
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b:=sb('b,[1])*i+sb('b,[2])*j+sb('b,[3])*k
Type: Polynomial(Integer)
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(a,a)/Ų
There are 15 exposed and 15 unexposed library operations named /
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)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)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Definition 3
Co-scalar product
Solve the Snake Relation as a system of linear equations.
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mU:=inverse matrix Ξ(Ξ(retract((𝐞.i*𝐞.j)/Ų), i,1..dim), j,1..dim);
>> System error:
#<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP.fasl" {1003457603}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.
The common demoninator is
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--squareFreePart factor denom Ů / squareFreePart factor numer Ů
matrix Ξ(Ξ(numer retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)
There are no library operations named 𝐝
Use HyperDoc Browse or issue
)what op 𝐝
to learn if there is any operation containing " 𝐝 " in its name.
Cannot find a definition or applicable library operation named 𝐝
with argument type(s)
PositiveInteger
Perhaps you should use "@" to indicate the required return type,
or "$" 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 "$" to specify which version of the function you need.
Check "dimension" and the snake relations.
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O:𝐋:= Ω / Ų
Cannot convert right-hand side of assignment
Ω
-
Ų
to an object of the type ClosedLinearOperator(OrderedVariableList
([1,i,j,k,ij,ik,jk,ijk]),Expression(Integer)) of the left-hand
side.
Cartan-Killing co-scalar
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eval(Ω,ck)
There are 10 exposed and 6 unexposed library operations named eval
having 2 argument(s) but none was determined to be applicable.
Use HyperDoc Browse, or issue
)display op eval
to learn more about the available operations. Perhaps
package-calling the operation or using coercions on the arguments
will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval
with argument type(s)
Variable(Ω)
Variable(ck)
Perhaps you should use "@" to indicate the required return type,
or "$" 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.
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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)
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or "$" to specify which version of the function you need.
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λ:= _
( I ΩX ) / _
( Y I );
There are no exposed library operations named I but there is one
unexposed operation with that name. Use HyperDoc Browse or issue
)display op I
to learn more about the available operation.
Cannot find a definition or applicable library operation named I
with argument type(s)
Variable(ΩX)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
Cartan-Killing co-multiplication
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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)
Perhaps you should use "@" to indicate the required return type,
or "$" to specify which version of the function you need.
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test
e /
λ = ΩX
Type: Boolean