Very simple QFT questions

Micha

I see. I think, it is ok to be confused.

Let me ask another question. Do you think, the leaking of the lightcone is physical? Can it be measured?

OOO

As this thread has shown, many people seem to say, that it can't, because causality has to be preserved if special relativity is to make any sense. On the other hand there are advocats of superluminal propagation. I'm not in the position to question either side, because I'm still trying to learn that stuff too.

My personal suspicion is, that the idea of superluminal propagation could well be grounded in a psychological motivation to fuel esotericism and/or science fiction movies.

Avodyne

1) The Feynman propagator does not vanish outside the lightcone. Explicit expressions (in four spacetime dimensions) are given in Appendix C of Relativistic Quantum Fields by Bjoken and Drell.

2) The i-epsilon prescription that leads to the Feynman propagator corresponds to taking the vacuum expection value of the time-ordered product of two free fields. Time-ordered products of fields are relevant because they are related (by the LSZ reduction formula) to scattering amplitudes.

3) Causality is related to the commutator of two fields; this should vanish outside the lightcone, so that a measurement of the field at one point does not affect the measurement at a spacelike separated point.

jostpuur

I must say I'm surprised by this debate about the propagators. It seems to be always going on in some thread.

If physicists used more rigor mathematics to justify their conclusions about this propagator problem, we probably wouldn't have this debate. The physicists always have the policy, that they don't need to understand the math, as long as their calculations work. Now, as a consequence, there is no agreement about the behaviour of the relativistic propagator.

OOO

Edit: this post is obsolete. Avodyne's statement is compatible with what Peskin & Schroeder say. Sorry.

Welcome to this delicate discussion Avodyne. What you say is interesting because it adds a little to my confusion. In Peskin & Schroeder, eqs. 2.51 and 2.52, the authors calculate the quantity

[tex]<0|\phi(x)\phi(y)|0> =: D(x-y)[/tex]

and afterwards they explicitely say: "So again we find that outside the light-cone, the propagation amplitude is exponentially vanishing but nonzero.". I've hacked their intermediate result into maple and it seems to me they are right.

Because the above D(x-y) reduces to the time-ordered product (i.e. the Feynman propagator) in the special case [itex]x^0>y^0[/itex], it seems that the Feynman propagator is nonzero outside the lightcone too. At least if one believes in what they have done with D(x-y).

Anyway I'll have a look at Bjorken-Drell. Meanwhile I'm at a point where nothing comes as a surprise...

Micha

Why is Avodyne's remark confusing you? What you say is in agreement with what he said, isn't it?

Micha

Could somebody say then in one sentence, what is the exact physical meaning of the Feynman propagator?

Edit: I'd say, it is the amplitude to find a particle at spacetime point y, when you have found one at earlier spacetime point x.

OOO

Thanks for pointing that out, Micha. My brain has become a knot. I'm not even able to read properly.

jostpuur

I tried to ask about this here http://physicsforums.com/showthread.php?t=176563

I don't feel like I got satisfying answer. The answer seems to be, that the propagator doesn't necessarily mean anything. It just works when used correctly to compute scattering amplitudes.

Hans de Vries

At the end all the confusion is unlikely to have any influence on real calculations since these are all done in momentum space, not in position space, and non physical results are massaged away with other prescriptions.

Leaking outside the light cone with exp-m at t=0 would mean instantaneous propagation at infinite speed over micron size distances in the case of neutrinos. The size of living species.


Regards, Hans

Avodyne

The Feynman propagator has no direct physical meaning. It simply appears as a component in the calculation of infinite-time scattering amplitudes.

I don't think this is correct; at least, I've never seen a calculation that shows it to be correct. (One has to be careful about the meaning of position in quantum field theory, so there are some subtleties. But it doesn't even have the right dimensions.)

meopemuk

Hi Avodyne,

I agree with you completely. Propagators are formal quantities used in calculations of the S-matrix amplitudes. Position-space propagators cannot be interpreted as propagation amplitudes (from point to point) or time-dependent wave functions. Such interpretation can be found in some QFT textbooks, but it 1) has zero experimental support; 2) leads to numerous theoretical contradictions.

Eugene.

Hans de Vries

For those interested: from Pauli's famous 1940 paper, Spin and Statistics:


Pauli's Spin and Statistics

To be compared with Feynman's:

Feynman's propagator in position space.

Although Pauli's propagators are worse. (zero'th order Bessels rather than first order). Pauli, quote, "expressively postulates" commutation outside the light cone to overrule the Green's function.

Peshkin & Schroeder's remarks about anti-particles canceling the non-causality stem from the second link. Chapter 18 of "Fundamental processes": Taking only one pole violates relativity, any physical process has diagrams with the other pole as well (anti-particle) to restore Lorentz invariance.


Regards, Hans

Haelfix

Hi Hans, what paper or book is that Feynman link too? Hellishly hard to find some of his old papers nowdays.

Anyway that should settle the confusion as expected.

Hans de Vries

It's in this nice book from his 1959-60 Caltech lectures:

The Theory of Fundamental Processes


Regards, Hans

PS: more copies here: amazon.com

Micha

Notice, that in the link Feyman does take serious the leaking out of the lightcone of the propagator named after him as a physical effect.

If the modern view is apperently different, ok.

EDIT: What I ask myself, is, how to we design an experiment to check this?

meopemuk

That's exactly the point. How can you measure propagators?

Eugene.