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SandBoxHermitianIsomorphisms

Edit detail for SandBoxHermitianIsomorphisms revision 6 of 7

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Editor: Bill Page Time: 2011/06/26 16:09:30 GMT-7
Note: documentation

added:
A complex vector ℂ-space $V$ possesses many different hermitian isomorphisms
$h^\dagger=h \in iso(V,V^\dagger)$. In quantum mechanics a given operator
$\rho \in End(V)$ may be said to be $h$-hermitian if
$$
\rho^\dagger \circ h = h \circ \rho
$$

changed:
-p:Complex Fraction Polynomial Integer:=complex(ℜp,𝔍p)
-q:Complex Fraction Polynomial Integer:=complex(ℜq,𝔍q)
-r:Complex Fraction Polynomial Integer:=complex(ℜr,𝔍r)
-t:Complex Fraction Polynomial Integer:=complex(ℜt,0)
-ρ:=matrix [[t/2+p,q],[r,t/2-p]]
ℂ:=Complex Fraction Polynomial Integer
-- dagger
htranspose(h)==map(x+->conjugate(x),transpose h)
)expose MCALCFN
\end{axiom}

Theorem

  The necessary conditions for an operator $ρ$ to possess hermitean isomorphism
$h$ is that $trace ρ \in ℝ$ and $det ρ \in ℝ$.

Two-Dimensions

\begin{axiom}
p:ℂ:=complex(ℜp,𝔍p)
q:ℂ:=complex(ℜq,𝔍q)
r:ℂ:=complex(ℜr,𝔍r)
t:ℂ:=complex(ℜt,0)
ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]]

added:
test(p^2+r*q=(1/4)*t^2-d)

removed:
-test(p^2+r*q=(1/4)*t^2-d)

added:

Given an operator $ρ \in End V$, one must find the tensor $H=0$
for unknown manifold of hermitian isomorphisms $h$.

changed:
-h:Matrix Complex Polynomial Integer:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
-htranspose(h)==map(x+->conjugate(x),transpose h)
h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

added:
H:=htranspose(ρ)*h-h*ρ

added:

We wish to find expressions for $h$ in terms of the components of
$ρ$. To do this we will determine how the components of $H$ depend
on the components of $h$.

changed:
-H:=htranspose(ρ)*h-h*ρ
-Hlist:=concat(H::List List ?)
-Hreal:=removeDuplicates(select(x+->(x~=0),concat(map(x+->real x, Hlist),map(x+->imag x, Hlist))))
-)expose MCALCFN
-H1:=jacobian(Hreal,[a,b,c,e]::List Symbol)
J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
     [a,b,c,e]::List Symbol)

added:
The null space (kernel) of the Jacobian

changed:
-s1:=solve(determinant subMatrix(H1,2,5,1,4),ℜr)
-H2:=map(x+->eval(x,s1),H1)
-N:=nullSpace(H2)
-H2*(c*N(1)+e*N(2))
-s2:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)
-h1:=map(x+->eval(x,s2),h)
-map(x+->eval(x,s1),htranspose(ρ)*h1-h1*ρ)
N:=nullSpace(map(x+->eval(x,s0),J))

added:
gives the general solution to the problem.
\begin{axiom}
s1:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)
map(x+->eval(x,concat(s0,s1)),H)
\end{axiom}

A complex vector ℂ-space possesses many different hermitian isomorphisms $h^\dagger=h \in iso(V,V^\dagger)$ . In quantum mechanics a given operator $\rho \in End(V)$ may be said to be -hermitian if

$\rho^\dagger \circ h = h \circ \rho$

axiom

ℂ:=Complex Fraction Polynomial Integer

$\label{eq1}\hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Type

axiom

-- dagger
htranspose(h)==map(x+->conjugate(x),transpose h)

Type: Void

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial

Theorem

The necessary conditions for an operator to possess hermitean isomorphism is that $trace ρ \in ℝ$ and $det ρ \in ℝ$ .

Two-Dimensions

axiom

p:ℂ:=complex(ℜp,𝔍p)

$\label{eq2}� � p +{�� p \ i}$

(2)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

q:ℂ:=complex(ℜq,𝔍q)

$\label{eq3}� � q +{�� q \ i}$

(3)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

r:ℂ:=complex(ℜr,𝔍r)

$\label{eq4}� � r +{�� r \ i}$

(4)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

t:ℂ:=complex(ℜt,0)

$\label{eq5}� � t$

(5)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]]

$\label{eq6}\left[ \begin{array}{cc} {{{� � t +{2 \ � � p}}\over 2}+{�� p \ i}}&{� � q +{�� q \ i}} \ {� � r +{�� r \ i}}&{{{� � t -{2 \ � � p}}\over 2}-{�� p \ i}}$

(6)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

trace ρ

$\label{eq7}� � t$

(7)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

d:=determinant ρ

$\label{eq8}{{{4 \ �� q \ �� r}+{4 \ {�� p^2}}+{� � t^2}-{4 \ � � q \ � � r}-{4 \ {� � p^2}}}\over 4}+{{\left(-{� � q \ �� r}-{� � r \ �� q}-{2 \ � � p \ �� p}\right)}\ i}$

(8)

Type: Complex(Fraction(Polynomial(Integer)))

axiom

test(p^2+r*q=(1/4)*t^2-d)

$\label{eq9} \mbox{\rm true}$

(9)

Type: Boolean

axiom

s0:=solve(imag d,ℜr)

$\label{eq10}\left[{� � r ={{-{� � q \ �� r}-{2 \ � � p \ �� p}}\over �� q}}\right]$

(10)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

eval(trace(ρ*ρ),s0)

$\label{eq11}{{{\left(-{4 \ {�� q^2}}-{4 \ {� � q^2}}\right)}\ �� r}+{{\left(-{4 \ {�� p^2}}+{� � t^2}+{4 \ {� � p^2}}\right)}\ �� q}-{8 \ � � p \ � � q \ �� p}}\over{2 \ �� q}$

(11)

Type: Fraction(Polynomial(Complex(Integer)))

Given an operator $ρ \in End V$ , one must find the tensor for unknown manifold of hermitian isomorphisms .

axiom

h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]

$\label{eq12}\left[ \begin{array}{cc} a &{b +{c \ i}} \ {b -{c \ i}}& e$

(12)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

axiom

test(h = htranspose h)

axiom

Compiling function htranspose with type Matrix(Complex(Fraction(
      Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
      Integer))))

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

axiom

H:=htranspose(ρ)*h-h*ρ

$\label{eq14}\left[ \begin{array}{cc} {{\left(-{2 \ b \ �� r}-{2 \ a \ �� p}-{2 \ c \ � � r}\right)}\ i}&{{e \ � � r}-{a \ � � q}+{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}} \ {-{e \ � � r}+{a \ � � q}-{2 \ b \ � � p}+{{\left(-{e \ �� r}-{a \ �� q}+{2 \ c \ � � p}\right)}\ i}}&{{\left(-{2 \ b \ �� q}+{2 \ e \ �� p}+{2 \ c \ � � q}\right)}\ i}$

(14)

Type: Matrix(Complex(Fraction(Polynomial(Integer))))

We wish to find expressions for in terms of the components of . To do this we will determine how the components of depend on the components of .

axiom

J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
     [a,b,c,e]::List Symbol)

$\label{eq15}\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ -{2 \ �� p}& -{2 \ �� r}& -{2 \ � � r}& 0 \ - � � q &{2 \ � � p}& 0 & � � r \ - �� q & 0 &{2 \ � � p}& - �� r \ � � q & -{2 \ � � p}& 0 & - � � r \ - �� q & 0 &{2 \ � � p}& - �� r \ 0 & 0 & 0 & 0 \ 0 & -{2 \ �� q}&{2 \ � � q}&{2 \ �� p}$

(15)

Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

axiom

N:=nullSpace(map(x+->eval(x,s0),J))

$\label{eq16}\left[{\left[{{2 \ � � p}\over �� q}, \:{� � q \over �� q}, \: 1, \: 0 \right]}, \:{\left[ -{�� r \over �� q}, \:{�� p \over �� q}, \: 0, \: 1 \right]}\right]$

(16)

Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

axiom

s1:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)

$\label{eq17}\left[{a ={{-{e \ �� r}+{2 \ c \ � � p}}\over �� q}}, \:{b ={{{e \ �� p}+{c \ � � q}}\over �� q}}, \:{c = c}, \:{e = e}\right]$

(17)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

map(x+->eval(x,concat(s0,s1)),H)

$\label{eq18}\left[ \begin{array}{cc} 0 & 0 \ 0 & 0$

(18)

Type: Matrix(Fraction(Polynomial(Complex(Integer))))