Problem involving imaginary numbers in the denominator

[Mystery111]

I followed a link recently by ajb http://en.wikipedia....wiki/Propagator . There was notation in the work in the position space part. In the solutions, you can see there is an imaginary part in the denominator; [math]\pm i\epsilon[/math]. Now my memory is vague, but there was a reason why equations like this would have an imaginary part in the denominator.... of course, someone figured this out.... it involved a problem with a certain equation which could be fixed by assuming part of the denominator was imaginary... but who it was who assumed this and what equations it referred to is escaping me now.

 

I wanted to look more into this, but I can't get anywhere further to understanding this without some information! Can anyone get their fingers on who I am talking about, or what equations I could be referring to?

 

Thanks

 

In fact that must sound really vague, I know the imaginary part plays roles in many parts of physics, such as the Green Function... How to go about this without being vague with little to go on is hard to do!

 

I'm also under the impression that the term is also a wick rotation... using imaginary numbers to solve problems is pretty standard in QM, so the more I read my post, the less translational I think it is!

 

(I'm going to get into trouble... this twice I've posted something and found my own answer! Upfront, I apologize:)

 

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

Edited October 14, 2011 by Mystery111

[swansont]

Why not multiply by the complex conjugate, divided by itself? The denominator becomes real and the numerator has a real and imaginary term.

[questionposter]

Why not multiply by the complex conjugate, divided by itself? The denominator becomes real and the numerator has a real and imaginary term.

 

That's correct. Having a complex or imaginary number in the denominator is sort of similar to having an irrational number in the denominator. You want to try and get rid of it if you can.

 

So you have something like

 

[math]4/2+3i[/math]

 

 

And what you do to get rid of i is

 

 

[math]((4/(2+3i))((2-3i)/(2-3i))[/math] It's the same as multiplying by "1", but it changes the look, and the result is

 

[math](8+12i)/(4+12i-12i-9i^2)[/math]

 

Which simplifies to be

 

 

[math](8+12i)/(13)[/math]

 

since i^2 is negative 1, which is a real number.

Edited October 15, 2011 by questionposter

[Mystery111]

Why not multiply by the complex conjugate, divided by itself? The denominator becomes real and the numerator has a real and imaginary term.

 

Yeah, you could. Indeed!

 

There maybe fruitful ways to keep the idea that the Hamiltonian as negative and positive solutions. According to a doctor I was speaking to recently:

 

''Imaginary Hamiltonian means non-unitary dissipation. Such models are used in nuclear physics to model decaying quantum states.''

 

So we be invoking a non-unitary solution to understand the initial equation. It is only by being able to to make the wick rotation in the denominator make the mass term real and positive. There may be a problem with simply multiplying the complex conjugate with the equation in context of the Dirac Equation.

 

Mind I have been focused on describing neutrino's which can be resprented as spin 1/2 particles in a Dirac Equation which had been modified by Tsao Chang. Normally, when you take the adjoint matrix of the Dirac equation, (the hermician transpose) you effectively neglect all the imaginary parts and not the real parts. We would loose the mass term (as it is formally by definition in this theory an imaginary object) and that would not be good.

 

Seems simple enough right? It just seemed problematic to assume a hermitian solution for equations describing particles with a negative mass sqaured straight away. That is like just like sweeping it under the carpet, or dressing up your variables so that you can deal with them easier.

 

I think Tsao was right in what he did, that we often stare at the equation with a negative mass term and get scared of its implications. He just said ''deal with it another way,'' or I assume that is what went through. The good thing about treating the Hamiltonian as either positive or negative allows you to have the freedom to make the change in the mass term to being negative or indeed back to positive at will.

Edited October 15, 2011 by Mystery111