What is the role of the minus sign in particle and antiparticle propagation?

[malawi_glenn]

Hello

I am trying to understand how one sees that particles are propagated forward in time and antiparticles propagated backward in time.

http://www.phys.uAlberta.ca/~gingrich/phys512/latex2html/node77.html

Eq 6.75 and 6.76 are the sources for my confusion. The text says "We see that $S_F(x^\prime-x)$ carries the positive-energy solutions $\psi^{(+)}$ forward in time and the negative-energy ones $\psi^{(-)}$ backward in time "

If t' > t, then positive freq. are prop. from t to t' and then the step function for the neg. freq. sol.'s is zero and hence no propagation. The negative freq. are propagated only if t' < t

So let's say we start at t = 5. Then pos. freq can be propagated to t > 5 and then neg. freq to t < 5.

Is this the correct observation one should make? Or is it the minus sign? Since the text says "The minus sign in the second equation results from the difference of the direction of propagation in time between (6.75) and (6.76). "

This is why I get lost I think, the text stresses that minus sign in some way I can't really appreciate

 

[RedX]

i couldn't derive anything in your link. relativistic quantum mechanics can be really hard.

in quantum field theory, it's easy to see that a antiparticle is a negative energy particle going backwards in time. you do two things: in your feynman diagrams draw all the arrows backwards, and have the 4-momenta of every external particle be it's negative. That Feynman diagrams have all the arrows going backwards: this means that the final state becomes the initial state, and the initial state becomes the final state - this is in keeping with going backwards in time. Also mutliplying the 4-momentum by negative one changes the negative energy solution (corresponding to antiparticles) to positive energy: this is in keeping that a negative energy antiparticle is like a positive energy particle when it goes backwards in time. Finally, when multiplying the 4-momentum by negative one, the 3-momentum also reverses direction, and this is in keeping with the notion that 3-momentum reverses when going backwards in time.

 

[malawi_glenn]

But this IS feynman diagrams and propagator. Feynman diagram lines ARE these propagators.

Now I know fore sure that the stressing on that minus sign that this author does is just confusing.

 

[RedX]

I looked at the introduction to your link and it has this:

"The field theoretical approach to quantum mechanics is not investigated in this course but rather a heuristic approach using the propagator formalism developed by Feynman and Stückelburg is used. The Feynman rules for quantum electrodynamics are developed by example. This is an intuitive and practical approach that gets the student doing calculations quickly."

So it is Feynman diagrams, but it is a different way than the field theoretical approach.

You're trying to see the "backwards in time thing" at this juncture, before the Feynman diagram rules are derived. I'm just saying that once the Feynam rules are derived, it becomes easy to see. So although it may or may not be clear to you now, once you derive the Feynman rules using your alternative approach, you will get another chance to see how the "backwards in time thing" plays out, so don't worry too much over it now.

 

[malawi_glenn]

Well the propagators here are derived in the same way as one does in e.g. Mandl's textbook. Also we had the same propagators in my quantum field theory in statistical mechanics class.

But maybe you are correct that one will see this clearer when doing quantum field theory, on a more modern approach.

But at the moment, I am satisfied, it was just that stress that the author seemed to have on that minus sign which really confused me.

 

[RedX]

Well the propagators here are derived in the same way as one does in e.g. Mandl's textbook. Also we had the same propagators in my quantum field theory in statistical mechanics class.

But maybe you are correct that one will see this clearer when doing quantum field theory, on a more modern approach.

But at the moment, I am satisfied, it was just that stress that the author seemed to have on that minus sign which really confused me.

I read Mandl's book before, and I don't remember him doing it that way, but it was awhile ago, so maybe I forgot. There are two editions to Mandl's book, the first just authored by Mandl, and the second with a coauthor whose name is Shaw, or something similar to that name. The 2nd one is much better I am told.

It's interesting that in the link you have, the author says that the free propagator is a function of (x'-x), but that this is not valid for an interacting propagator. Both Peskin and Schroeder and Srednicki say otherwise: the interacting propagator is still a function of (x'-x), although more complicated.

 

[malawi_glenn]

Ok, do you have pages in Peskin which I can compare with?

 

[RedX]

the paragraph above eqn. 7.14 in PS say that the coordinate dependence is an integral over p of the form f(p)*e^ip(x-y). also eqns. 7.5 and 7.6 say it, but for a spin-0 field. srednicki has it on eqns. 62.26 for a spin 1/2 field, and 13.6 for a spin-0 field.

 

[Deathless]

i couldn't derive anything in your link. relativistic quantum mechanics can be really hard.

in quantum field theory, it's easy to see that a antiparticle is a negative energy particle going backwards in time. you do two things: in your feynman diagrams draw all the arrows backwards, and have the 4-momenta of every external particle be it's negative. That Feynman diagrams have all the arrows going backwards: this means that the final state becomes the initial state, and the initial state becomes the final state - this is in keeping with going backwards in time. Also mutliplying the 4-momentum by negative one changes the negative energy solution (corresponding to antiparticles) to positive energy: this is in keeping that a negative energy antiparticle is like a positive energy particle when it goes backwards in time. Finally, when multiplying the 4-momentum by negative one, the 3-momentum also reverses direction, and this is in keeping with the notion that 3-momentum reverses when going backwards in time.

Amazing! A Feymann diagram with arrows drawn backwards?!
My knowledge here is more basic than yours - having just drawn Feymann diagrams and never made more than a single calculation from it - so hope you will humour my lack of knowledge.
But - a question here

Can you simply draw the arrows backwards just because you want to?
What justification is there for this and how would you apply these conditions when considering Feymann diagrams where particles and antiparticles are present - which is the majority

Well the propagators here are derived in the same way as one does in e.g. Mandl's textbook. Also we had the same propagators in my quantum field theory in statistical mechanics class.

But maybe you are correct that one will see this clearer when doing quantum field theory, on a more modern approach.

But at the moment, I am satisfied, it was just that stress that the author seemed to have on that minus sign which really confused me.

Wow! more insights!
Mandl's textbook - could you tell me the full name of the book please

 

[malawi_glenn]

Deathless: You can start looking at the page where I was referring to and then you can pick up "Quantum Field Theory" by Franz Mandl, wiley.

 

[Deathless]

Thanks. Will do.

EDIT: - yes seen it - last 2 equations. t-> - infinity!

will study it in more detail when I have the time