Help please concerning Dirac Hamiltonian in context of the Lippmann-Schwinger Proceedure

[Mystery111]

Due to work I have been conducting recently in regards to tachyonic neutrino's, I was investigating a mathematical proceedure found in the Lippmann-Schwinger Equation. I wanted to apply the process to the Dirac Hamiltonian. I wanted to know if I have done this right...

 

Suppose I want to describe the Dirac Hamiltonian for a free particle can be given as:

 

[math]\mathcal{H}_D = (\alpha \cdot \hat{p})c + \beta Mc^2[/math]

 

I could have chosen natural units here, but let's express it in it's full beauty. Now, effectively, there is no interaction potential term in here.

 

Because of this, I can say

 

[math]\mathcal{H}_D \phi = ((\alpha \cdot \hat{p})c + \beta Mc^2)\phi[/math]

 

The interaction could be given a number of ways, by choosing different representatives for your potential. In this case I choose a very simple electric potential [math]e\phi'[/math]

 

[math](\mathcal{H}_D + e\phi')\psi = ((\alpha \cdot \hat{p})c + \beta Mc^2)\psi[/math]

 

Due to continuity, as [math]\psi \rightarrow \phi[/math] and [math]e\phi' \rightarrow 0[/math] then it should be that a solution is

 

[math]\psi^{\pm} = \phi + \frac{1}{E- \mathcal{H} \pm \epsilon} e\phi'\psi^{\pm}[/math]

 

It automatically becomes possible solutions for retarded and advanced waves which is the idea I reached today.

Edited October 14, 2011 by Mystery111

[ajb]

[math]\psi[/math] is a Dirac spinor right? [math]\phi[/math] is a scalar potential?

 

If so I am not sure what you mean by [math]\psi \rightarrow \phi[/math].

Edited October 14, 2011 by ajb

[Mystery111]

[math]\psi[/math] is a Dirac spinor right? [math]\phi[/math] is a scalar potential?

 

If so I am not sure what you mean by [math]\psi \rightarrow \phi[/math].

 

Well, I assume the description of the Lippmann-schwinger equation is

 

[math]|\psi> = |\phi> + \frac{1}{E- H_0}V|\psi>[/math]

 

Thus I also assume the Dirac Hamiltonian can be replaced with the notation above. Because of this, I was able to freely choose which potential I wished to have. Distinguisging the term [math]e\phi'[/math] as the interaction, playing the role of [math] V[/math]?

 

I might have done it wrong, but I'm just applying the Lippmann-Schwinger proceedure to the Dirac Hamiltonian and replacing V with a new potential.

 

So in the famous Lippmann Schwinger equation, the energy eigenstates are continious in the theory which invokes the transformations you are conerned with.

[ajb]

I think one can use [math]H_{0}[/math] to be the Dirac Hamiltonian. Then you can consider [math]H_{0} + V[/math] etc.

 

[math]| \phi >[/math] is an eigenvector of the free Hamiltonian.

 

Maybe I am just misunderstanding you. [math]|\phi>[/math] has nothing to do with [math]\phi'[/math]?

[Mystery111]

Indeed! Exactly right.

 

That is why I destinguished [math]\phi[/math] from the phi in electric potential term as a prime. But I take it then I have done this right?

 

 

Thanks Ajb!

[ajb]

Formally this look ok, I have no idea of this works well for the Dirac equation.

[Mystery111]

It's completely formal in mathematics, I don't have a definition yet. It is interesting though I thought that it can be given this representation. It certainly has applications in that if you choose that the neutrino has an imaginary mass that you may be able to tranform the invariant mass term under a wick rotation which may help escape the negative mass squared term. It is a mathematical trick at best, but this is what motivated the idea.

 

It acts like an equation which will satisfy a negative and a postive solution. It is paramount to understanding the dirac equation if we are going to believe that the Hamiltonian has two different energy state solutions.

 

I might even go as far as saying it can be used as a shorthand to express these possible energy states without inferring all the complicated spinor equations which can derive the postive and negative energy solutions of the dirac equation by inferring an imaginary rotation on the Hamiltonian (energy of the system).

[Mystery111]

(Oh there might have been some confusion) why I choose an electric potential when the nuetron is electrically nuetral by theory. Well the nuetron hasa mass and a non-zero magnetic moment so it has been suggested it could possess a small charge as well.

[Mystery111]

Ok, is this ok...

 

let us take the Schrodinger equation as

 

[math]\hbar \frac{\partial}{\partial t}|\psi> = -i\hat{\mathcal{H}}|\psi>[/math]

 

Then [math]\hat{\mathcal{H}}= \mathcal{H} + i \chi[/math]

 

It can be said that the imaginary part [math]i \chi[/math] of [math]\hat{\mathcal{H}}[/math] can be divided into two parts, which are traditionally called the relaxational part and the diffusive part.

 

[math]\chi = \chi_R + \chi_D[/math]

 

A time evolution operator is given as [math]\frac{\partial}{\partial t} \hat{V}(t) = -i\hat{\mathcal{H}}\hat{V}(t)[/math] - this can satisfy the Heisenberg equation

 

[math]\frac{\partial}{\partial t}A(t) = i[\hat{\mathcal{H}}, A(t)][/math]

 

for dissipative systems. Imaginary Hamiltonian's means a non-unitary dissipation.

 

Is this right?

Edited October 17, 2011 by Mystery111

[Mystery111]

The dynamics which have had me spending great deal of effort over the last couple of days is a way to treat the squared

negative mass of a nuetrino under the dirac formalism where we are going to treat the part of [math]-i\hat{\mathcal{H}}[/math] in the schrodinger equation as being the same as the denominator description of [math]\hat{\mathcal{H}_D} + i\epsilon[/math] in the equation

 

[math]\psi^{\pm} = \phi + \frac{1}{E- \hat{\mathcal{H}_D} \pm i\epsilon} e\phi'\psi^{\pm}[/math]

 

since [math]\hat{\mathcal{H}}_D = \mathcal{H} + i\chi[/math] from breaking the complex part into two descriptions as shown

before, will allow us to treat the Lippmann-Schwinger as having two solutions as well for the imaginary part of the hat-Hamiltonian.

 

Knowing that [math]\chi = \chi_R + \chi_D[/math]

 

Then

 

[math]\hat{\mathcal{H}_D} = \mathcal{H} + i(\chi_R + \chi_D)[/math]

 

[math]\psi^{\pm} = \phi + \frac{1}{E- \hat{\mathcal{H}_D}+i(\chi_R + \chi_D)} e\phi'\psi^{\pm}[/math]

 

Since in the most mathematical formal definition of quantum waves desribed under a time-symmetric analysis of the wave function

in the Transactional Interpretation, the deformed wave function [math]\psi^{\pm}[/math] when under integration has the solution

of a wave function for the in-phase and out-phase both extending to the infinite past and infinite future. These are analgous

to understanding wave functions where one half retarded and one half advanced wave propogate into the past and the other into

the future. This means that the equation

 

[math]\psi^{\pm} = \phi + \frac{1}{E- \hat{\mathcal{H}}_D + i(\chi_R + \chi_D)} e\phi'\psi^{\pm}[/math]

 

can be decoupled to make to solution for the advanced and retarded wave solutions of

 

[math]F_1 = e^{[i(kr - \omega t)]}[/math]

 

and

 

[math]G_1 = e^{[-i(kr - \omega t)]}[/math]

 

If I have done this right so far, which is what I am hoping on of the revered scientists around here could help clarify for

me, then I will begin to attempt to formulate this properly in terms of the Transactional interpretation.

 

 

(the two solutions should not be F_1 = but should be F_1 ~ but I am unsure how to write the latter notation in latex)

Edited October 17, 2011 by Mystery111

[Mystery111]

I am now going to take us back to the Tsao-Dirac Equation for fermions with a negative-mass sqaured which has been modified for the proper mass [1][2].

 

[math]-i(\alpha k)c\psi + \beta_s M_s c^2 \psi = i\hbar \partial_t \psi[/math]

 

Where [math]M_s[/math] is the proper mass. Eq 1 can be re-written as a pair two-component equations:

 

[math]i\hbar \frac{\partial \phi}{\partial t} = -ic\hbar \vec{\sigma} \cdot \nabla_{\chi'} + M_s c^2 \chi'[/math]

 

and

 

[math]i\hbar \frac{\partial \phi}{\partial t} = -ic\hbar \vec{\sigma} \cdot \nabla_{\chi'} - M_s c^2 \chi'[/math]

 

Slight differences will be seen when this equation is under a Weyl representation

 

[math]i\hbar \frac{\partial \eta}{\partial t} = -ic\hbar \vec{\xi} \cdot \nabla \eta + M_s c^2 \xi[/math]

 

and

 

[math]i\hbar \frac{\partial \xi}{\partial t} = -ic\hbar \vec{\eta} \cdot \nabla \xi - M_s c^2 \eta[/math]

 

[math]\eta[/math] and [math]\xi[/math] are in fact coupled to a limit where [math]M_s[/math] is zero. Under mathematcial strutiny, the fact that the Nuetrino has such a ridiculously small mass incorporates the similar contention that the nuetrino could act more or less like a particle with no mass.

 

With the limit where [math]M_s=0[/math] reduces to the Weyl equation

 

[math]\frac{\partial \xi v'}{\partial} = -c\vec{\sigma} \cdot \nabla \xi v'[/math]

 

This is because [math]\phi = \chi'[/math] when [math]M_s = 0[/math]. It turns out that the Weyl representation of his equation only satisfies antineutrino description [math]v'[/math]. This places some understanding in how in treating the Hamiltonian of the Dirac Equation Hamilonian in the Lippmann-Scwinger proceedure and whether it satisfies a neurtino or an antinuetrino. The way to do this, is by changing the sign of momentum part of the Tsao-Dirac equation. A negative representation of the momentum part will refer to antinuetrino's, while a positive representation of the momentum plays the role of neutrinos.

 

 

[1] http://www.mendeley....article/#page-1

[2]http://arxiv.org/PS_...1/0011087v4.pdf

 

Now, this would suggest that the full equation taking into the Tsao-Dirac Equation in the Lippman-Schwinger proceedure is then:

 

[math]\psi^{\pm} = \phi + \frac{1}{E- ((\alpha \cdot \hat{p})c + \beta M_sc^2) + i(\chi_R + \chi_D)} e\phi'\psi^{\pm}[/math]

 

And this would then, under the same principles of the Weyl equation would only satisfy an antiparticle solution since the hat-Hamilton has a negative description, making the momentum part as negative.

Edited October 18, 2011 by Mystery111

[Mystery111]

.. sorry... accidently posted something when I copied and pasted

Edited October 18, 2011 by Mystery111