A Propagator In A Solution To The Dirac Equation

[QuantumTantrum]

The Dirac Hamiltonian for a free particle can be given as a Hamiltonian

 

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

 

The interaction is given by the potential. In this case I choose a very simple electric potential [math]e\phi[/math]

 

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

 

[math]V = e\Phi[/math]

 

Due to continuity, as [math]\psi \rightarrow \phi[/math] and [math]V \rightarrow 0[/math] then a solution lies in

 

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

 

Which is really just a Lippman-Schwinger equation. What makes it interesting though, is that the term

 

[math]\frac{1}{E- \mathcal{H} \pm i\epsilon}[/math]

 

Is just a form of a Feyman Propagator 

 

[math]G_F = \frac{1}{(pc - mc^2)^2 + i\epsilon}[/math]

 

Imaginary numbers are not really wanted in denominators, so some algebra can actually remove it by multiplying it with the ratio complex conjugate of the denominator with itself, which takes the imaginary number into the numerator with a real number. 

 

Another way has already been written in history, an alternative propagator without using any complex numbers is

 

[math]G_F = \frac{(pc - mc^2)^2}{(p^2c^2 - m^2c^4)^2 + \epsilon}[/math]

 

If we had done it the way I had suggested previously, instead we'd have an equation to solve

 

[math]G_F = \frac{1}{(pc - mc^2)^2 + i\epsilon}\frac{(pc + mc^2)^2 - i\epsilon}{(pc + mc^2)^2 - i\epsilon}[/math]

 

http://en.wikipedia.org/wiki/Propagator#Position_space

 

http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation

 

http://en.wikipedia.org/wiki/Dirac_equation

Edited May 8, 2015 by QuantumTantrum

[Rade]

Question.  Suppose a matter particle with 3 mass units (pnp) and an antimatter particle with 2 mass units (n^p^), where ^=antimatter.  So, what would be the predicted Feynman Propagator that would allow the two particles to form stable quantum superposition, and what would be the predicted real and virtual dimensions of the superposition ?  The important constraint is that any type of annihilation between matter and antimatter is impossible during the interaction. 

[QuantumTantrum]

Question.  Suppose a matter particle with 3 mass units (pnp) and an antimatter particle with 2 mass units (n^p^), where ^=antimatter.  So, what would be the predicted Feynman Propagator that would allow the two particles to form stable quantum superposition, and what would be the predicted real and virtual dimensions of the superposition ?  The important constraint is that any type of annihilation between matter and antimatter is impossible during the interaction. 

 

 

 

Ok.... Ok.... Be back in a min while I type something up :)

[QuantumTantrum]

You know, it's not an easy answer... there might be a few different ways all going to separate means. For instance, say you were a sadistic intelligent mathematician and work out the dubious road to the answer/solution... though they could have just some used some algebra to remove the harassment. 

 

For instance if you really wanted to keep the imaginary number in the equation, usually physicists do this for non-unitary dissipation equations, so the list is long in application, the real interpretation of the equation I derived doesnt tell me really why a propagator term is in there, only that it has some subtle notion to an equation of motion.

Edited May 14, 2015 by QuantumTantrum

[QuantumTantrum]

 

The Dirac Hamiltonian for a free particle can be given as a Hamiltonian

 

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

 

The interaction is given by the potential. In this case I choose a very simple electric potential [math]e\phi[/math]

 

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

 

[math]V = e\Phi[/math]

 

Due to continuity, as [math]\psi \rightarrow \phi[/math] and [math]V \rightarrow 0[/math] then a solution lies in

 

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

 

Which is really just a Lippman-Schwinger equation. What makes it interesting though, is that the term

 

[math]\frac{1}{E- \mathcal{H} \pm i\epsilon}[/math]

 

Is just a form of a Feyman Propagator 

 

[math]G_F = \frac{1}{(pc - mc^2)^2 + i\epsilon}[/math]

 

Imaginary numbers are not really wanted in denominators, so some algebra can actually remove it by multiplying it with the ratio complex conjugate of the denominator with itself, which takes the imaginary number into the numerator with a real number. 

 

Another way has already been written in history, an alternative propagator without using any complex numbers is

 

[math]G_F = \frac{(pc - mc^2)^2}{(p^2c^2 - m^2c^4)^2 + \epsilon}[/math]

 

If we had done it the way I had suggested previously, instead we'd have an equation to solve

 

[math]G_F = \frac{1}{(pc - mc^2)^2 + i\epsilon}\frac{(pc + mc^2)^2 - i\epsilon}{(pc + mc^2)^2 - i\epsilon}[/math]

 

http://en.wikipedia.org/wiki/Propagator#Position_space

 

http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation

 

http://en.wikipedia.org/wiki/Dirac_equation

 

 

 

Another way to write the solution,can be

 

 

[math]\psi^{\pm} = \phi + G_F V\psi[/math]

 

 

 

REF:

 

1. http://en.wikipedia.org/wiki/Transactional_interpretation

 

2. http://en.wikipedia.org/wiki/Wheeler%E2%80%93Feynman_absorber_theory

 

3. http://arxiv.org/pdf/quant-ph/0507269v1.pdf

Edited May 15, 2015 by QuantumTantrum

[QuantumTantrum]

Another way to write the solution,can be

 

 

[math]\psi^{\pm} = \phi + G_F V\psi[/math]

 

 

 

 

 

Since 

 

 

[math]\psi^{\pm} = \phi + G_F V\psi[/math]

 

 

uses a  wave function [math]\psi^{\pm}[/math] which denotes a positive and negative wavefunction [in time], we have [math]\psi^{+}[/math] the interaction time at any time in the the future and [math]\psi^{-}[/math] which measures the time after the interaction, in other words, it describes a propagator in motion. Also, it might be interesting to note that the solution chosen for the Dirac Equation [the Lippmann-Schwinger equation] results naturally in the form of a Schrodinger equation for single particle states [1]

 

 

 

[1] http://en.wikipedia.org/wiki/Lippmann%E2%80%93Schwinger_equation