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# Edit detail for SandBoxHermitianIsomorphisms revision 7 of 7

 1 2 3 4 5 6 7 Editor: Bill Page Time: 2011/06/27 18:30:02 GMT-7 Note:

added:

From BillPage Mon Jun 27 18:30:02 -0700 2011
From: Bill Page
Date: Mon, 27 Jun 2011 18:30:02 -0700
Subject:
Message-ID: <20110627183002-0700@axiom-wiki.newsynthesis.org>

SandBoxHermitianIsomorphisms3


A complex vector ℂ-space possesses many different hermitian isomorphisms . In quantum mechanics a given operator may be said to be -hermitian if

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ℂ:=Complex Fraction Polynomial Integer
 (1)
Type: Type
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-- dagger
htranspose(h)==map(x+->conjugate(x),transpose h)
Type: Void
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)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame
initial

## Theorem

The necessary conditions for an operator to possess hermitean isomorphism is that and .

Two-Dimensions

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p:ℂ:=complex(ℜp,𝔍p)
 (2)
Type: Complex(Fraction(Polynomial(Integer)))
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q:ℂ:=complex(ℜq,𝔍q)
 (3)
Type: Complex(Fraction(Polynomial(Integer)))
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r:ℂ:=complex(ℜr,𝔍r)
 (4)
Type: Complex(Fraction(Polynomial(Integer)))
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t:ℂ:=complex(ℜt,0)
 (5)
Type: Complex(Fraction(Polynomial(Integer)))
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ρ:Matrix ℂ := matrix [[t/2+p,q],[r,t/2-p]]
 (6)
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
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trace ρ
 (7)
Type: Complex(Fraction(Polynomial(Integer)))
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d:=determinant ρ
 (8)
Type: Complex(Fraction(Polynomial(Integer)))
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test(p^2+r*q=(1/4)*t^2-d)
 (9)
Type: Boolean
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s0:=solve(imag d,ℜr)
 (10)
Type: List(Equation(Fraction(Polynomial(Integer))))
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eval(trace(ρ*ρ),s0)
 (11)
Type: Fraction(Polynomial(Complex(Integer)))

Given an operator , one must find the tensor for unknown manifold of hermitian isomorphisms .

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h:Matrix ℂ:=matrix [[a,complex(b,c)],[complex(b,-c),e]]
 (12)
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
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test(h = htranspose h)
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Compiling function htranspose with type Matrix(Complex(Fraction(
Polynomial(Integer)))) -> Matrix(Complex(Fraction(Polynomial(
Integer))))
 (13)
Type: Boolean
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H:=htranspose(ρ)*h-h*ρ
 (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 .

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J:=jacobian(concat( map(x+->[real x, imag x], concat(H::List List ?)) ),
[a,b,c,e]::List Symbol)
 (15)
Type: Matrix(Fraction(Polynomial(Integer)))

The null space (kernel) of the Jacobian

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N:=nullSpace(map(x+->eval(x,s0),J))
 (16)
Type: List(Vector(Fraction(Polynomial(Integer))))

gives the general solution to the problem.

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s1:=map((x,y)+->x=y,[a,b,c,e],c*N.1+e*N.2)
 (17)
Type: List(Equation(Fraction(Polynomial(Integer))))
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map(x+->eval(x,concat(s0,s1)),H)
 (18)
Type: Matrix(Fraction(Polynomial(Complex(Integer))))

SandBoxHermitianIsomorphisms3