How can the Cauchy integral and Fourier integral produce the same result?

[OOO]

I think, the homogeneous solutions in Fourier space are just a product of delta functions for the components of the momentum vector, where you should choose only momentum vectors on mass shell: p^2-m^2 = 0

That sounds resonable because then the two Fourier profiles differ only at the location of the singularities (which are on mass shell). That would explain why we think that the propagator in momentum space is unique. It's because we tend to think only of its values off mass shell.

Edit: I have got some objections against the term product of delta functions, not sure what you mean by that. This would give discrete singularities whereas they actually form a manifold.

 

[Micha]

I worked out the propagator in one spatial dimension and this is, what I get:

For epsilon > 0:

exp(-m*abs(x))/(-2m)For epsilon < 0:

exp(m*abs(x))/(2m)It seems you get the propagator for epsilon < 0 by putting in a negative rest mass
in the propagator for epsilon >0. Mathematically, it is not surprising, that this is
possible, because only m^2 is appearing in KG.

Maybe the reason for the differences in the solutions of Hans and Zee lie in different choices of epsilon. Zee explicitly works out the properties of the +i*epsilon propagator.
Maybe Hans works implicitly or explicitly with a sum of the plus and minus epsilon solution, and so the contributions out of the light cone cancel exactly to zero. But I don't know.
Edit: Maybe this is, what Hans was trying to tell us with post #126.

 

[Micha]

Yes, partly. But as Zee remarks, in the limit epsilon to zero this amounts to $\omega_k\to\omega_k-i\epsilon$ (expanding a square root) and this is just one of those cases that you are trying to exclude.

I admit that I have always wondered why it is done this way for the KG propagator as opposed to the massless propagator. There seems to be no good reason for it, besides being slightly easier to write down (only one epsilon term, no binomial decomposition).

I don't understand.

Zee can take the epsilon out of the square root, because it is small. It is just a linear approximation in epsilon.

 

[OOO]

I don't understand.

Zee can take the epsilon out of the square root, because it is small. It is just a linear approximation in epsilon.

Yes, exactly. There is no problem with this argument at all. But by joining the two epsilons into one (or vice versa, depending where you start from) one restricts generality more than one needs to.

 

[OOO]

Maybe the reason for the differences in the solutions of Hans and Zee lie in different choices of epsilon. Zee explicitly works out the properties of the +i*epsilon propagator.
Maybe Hans works implicitly or explicitly with a sum of the plus and minus epsilon solution, and so the contributions out of the light cone cancel exactly to zero. But I don't know.

As I have said above, I am being sceptical about the ability to put all boundary conditions into moving the singularities. But if you modify your statement to "Maybe the reason for the differences in the solutions of Hans and Zee lie in different boundary conditions" I'd definitely agree.

Edit: Maybe this is, what Hans was trying to tell us with post #126.

I have just done Hans' calculation he explained in

http://chip-architect.com/physics/Higher_dimensional_EM_radiation.pdf

for the massless propagator (I think generalization to the massive one is straightforward) in 1+1 dimension. He uses lightcone coordinates $\xi=\omega-k$ and $\zeta=\omega+k$ for doing the double Fourier integral. With this trick the integral decomposes into two independent single integrals and no epsilon-tricks have to be applied. Although I don't quite understand how he used the convolution theorem in this case, I got the same result by a different method (change of variables).

I think the crucial point is that this method is not unique as well, like any other method to compute the propagator. If you do it that way, you get two "integration constants" c1(t-x) for the special solution Theta(t+x) and c2(t+x) for the special solution Theta(t-x), as far as I can see. But, in my opinion, Hans correctly assumes that the causal propagator should be zero for t<0 which rules out both "constants" and you get

$$\phi(t,x) = \Theta(t-x)\Theta(t+x) = \Theta(t)\Theta(t^2-x^2)$$

As Hans has stressed in one of his earlier posts, the textbook authors don't seem to be concerned about the past, which is why they neglect the Heaviside factor. And probably because they are historically blind the don't notice that values outside the lightcone must have their origin somewhere in the past.

 

[Micha]

Edit: I have got some objections against the term product of delta functions, not sure what you mean by that. This would give discrete singularities whereas they actually form a manifold.

Can you explain. I think, a product of delta functions for different coordinates
under a multidimensional integral is just fine. Only if you multiply delta functions for
the same integration variable, you get problems.

 

[Micha]

As I have said above, I am being sceptical about the ability to put all boundary conditions into moving the singularities. But if you modify your statement to "Maybe the reason for the differences in the solutions of Hans and Zee lie in different boundary conditions" I'd definitely agree.

Yes, I can subscribe to this statement. But this raises a question.
If you can not get all boundary conditions by moving the singularities around,
which ones are you implicitly applying by using the epsilon prescription at all?

 

[Micha]

But, in my opinion, Hans correctly assumes that the causal propagator should be zero for t<0 which rules out both "constants" and you get

$$\phi(t,x) = \Theta(t-x)\Theta(t+x) = \Theta(t)\Theta(t^2-x^2)$$

As Hans has stressed in one of his earlier posts, the textbook authors don't seem to be concerned about the past, which is why they neglect the Heaviside factor. And probably because they are historically blind the don't notice that values outside the lightcone must have their origin somewhere in the past.

Very interesting. I need some time to look into this.

Edit: Maybe we should write a paper about this, once we understand it all. :-)

 

[OOO]

Can you explain. I think, a product of delta functions for different coordinates
under a multidimensional integral is just fine. Only if you multiply delta functions for
the same integration variable, you get problems.

Maybe I just didn't understand what expression you had in mind.

 

[OOO]

Very interesting. I need some time to look into this.

Edit: Maybe we should write a paper about this, once we understand it all. :-)

That would probably be a bit too little for a paper...

What still perplexes me is, that all the QFT textbooks I know do it the other way. Hans seems to say that for QFT it doesn't matter which propagator you take, the differences cancel anyway (and it's hard to believe that anyone has done successful calculations if it did matter). But then why not take the advanced propagator ? Strange...

 

[Micha]

Maybe I just didn't understand what expression you had in mind.

In 1+1 dimensions it would be:

harm(k0,k1) = delta(k0-p0)*delta(p0^2-*k1^2-m^2)

for some arbitrary p0

 

[OOO]

Yes, I can subscribe to this statement. But this raises a question.
If you can not get all boundary conditions by moving the singularities around,
which ones are you implicitly applying by using the epsilon prescription at all?

Very good question. I have asked that myself since the electrodynamics lecture I took years ago.

These epsilon-prescriptions seem like writing the software for an Automated Teller Machine without being able to say how much money the customer will get if he requests $100. If he gets at least some money, it has to be okay for him...

 

[Micha]

What still perplexes me is, that all the QFT textbooks I know do it the other way. Hans seems to say that for QFT it doesn't matter which propagator you take, the differences cancel anyway (and it's hard to believe that anyone has done successful calculations if it did matter). But then why not take the advanced propagator ? Strange...

Don't forget, that this Kleiss in the Field theory lecture at Cern in the link, I posted, is selling epsilon as the decay rate of the particle. So it seems, there is not only mathematics, but also Physics in it.

Edit: Hans is suspicously calm. He is figuring this all out quietly it seems. Or our discussion is just too trivial for him.

 

[OOO]

In 1+1 dimensions it would be:

harm(k0,k1) = delta(k0-p0)*delta(p0^2-*k1^2-m^2)

for some arbitrary p0

Seems to work...

 

[OOO]

Don't forget, that this Kleiss in the Field theory lecture at Cern in the link, I posted, is selling epsilon as the decay rate of the particle. So it seems, there is not only mathematics, but also Physics in it.

I'm not sure. An electron (which is, of course, no KG particle) has no decay rate. But probably unstable particles can be dealt with this way (without sending epsilon to zero). But then they don't obey Klein-Gordon but some dissipative equation.

Edit: Hans is suspicously calm. He is figuring this all out quietly it seems. Or our discussion is just too trivial for him.

As a wise man he doesn't work on weekends...

 

[Micha]

I'm not sure. An electron (which is, of course, no KG particle) has no decay rate. But probably unstable particles can be dealt with this way (without sending epsilon to zero). But then they don't obey Klein-Gordon but some dissipative equation.

The fact, that we haven't seen an electron decaying experimentally just means, that the decay time of the electron is bigger than some limit of 10^35 years or so. Remember the experiments about proton decay. It would still be clear, which sign of epsilon would be the "right" one. Kleiss goes so far to connect the sign of epsilon to the arrow in time in QFT.
Don't worry about Dirac vs. KG by the way. The denominator of the propagator is just the same.

 

[Haelfix]

Exactly what do you want the electron to decay into?

Keep in mind i epsilon really has to do with Wick rotation, with epsilon --> zero. Its justified mathematically in that sense ultimately. Typically we denote something that looks like i sigma for the decay rate in cases where the resonance is so thin that its hard to make sense off. Sigma is not a limit though, but a small and positive number.

Depending on definitions they can be used interchangeably, but ultimately one is utilized as a mnemonic for a trick, and the other is an actual physical thing.

 

[Micha]

I think, I understand now, in which direction to search for the source of our confusion.
We should remind ourselves, that, while we are calculation in natural units, where hbar = 1, there is an 1/hbar^2 sitting in front of the rest mast term.
This means, that the KG equation with finite rest mass does know about quantum mechanics, whereas the massless equation does not!
So for the massless equation we will of course rediscover the classical behaviour of the photon traveling along the light cone.
If we now look at the leaking out of the lightcone for the massive case, it goes with
exp(-m/hbar*abs(x)). From there we see, that it is clearly a quantum mechanical effect! If hbar goes to zero, there is no leaking out of the light cone! We know from things as the tunnel effect, that a quantum mechanics state can leak into classical forbidden regions with an exponentially decaying amplitude.
We can see the same thing also in a different way.
A strictly causal propagator would be completely concentrated in the point x=0 for t=0. On the other hand, the propagtor is described by 1/(p^2-m^2), so its momentum is also completely sharp.
We know from the uncertainty principle, that such states do not exist in quantum mechnacis. I think, Hans solution for the massive case should be looked at with this in mind.
By the way, in the classical electrodynmacis book of Jelitto, it is said, that the epsilon prescription is used exactly because it automatically ensures causality. I didn't follow the argument so far. Maybe this statement can be generalized to say, that the epsilon prescription is used in QFT, because it respects causality as much as possible under the laws of quantum mechancis.

 

[Micha]

Exactly what do you want the electron to decay into?

Keep in mind i epsilon really has to do with Wick rotation, with epsilon --> zero. Its justified mathematically in that sense ultimately. Typically we denote something that looks like i sigma for the decay rate in cases where the resonance is so thin that its hard to make sense off. Sigma is not a limit though, but a small and positive number.

Depending on definitions they can be used interchangeably, but ultimately one is utilized as a mnemonic for a trick, and the other is an actual physical thing.

Maybe my argument is far fetched. But I don't see how the epsilon in
1/(p^2-m^2+i*epsilon) is coming from a wick rotation.

 

[OOO]

A strictly causal propagator would be completely concentrated in the point x=0 for t=0. On the other hand, the propagtor is described by 1/(p^2-m^2), so its momentum is also completely sharp.

Momentum isn't sharp (why should it be ). I think we agree that the propagator is not a plane wave.

And remember, my numerics (as well as Hans') show no acausality if the boundary condition is Phi(t<0)=0. Sometimes it seems to me that hbar has been invented just for the purpose of having an excuse whenever one doesn't know exactly what is going on...

 

[Micha]

And remember, my numerics (as well as Hans') show no acausality if the boundary condition is Phi(t<0)=0.

Weren't you calculating the massless case only?

 

[Micha]

Momentum isn't sharp (why should it be ). I think we agree that the propagator is not a plane wave.

True. Only the modulus of the momentum is sharp.

 

[Haelfix]

Hi Micha, its a bit of a long story to see where i epsilon ultimately comes from. On one hand it seems like a bit of a hack in Feynman's prescription, and on the other it sort of appears naturally in canonical quantization (See A. Zee for this point).

Either way I think its clear by now that you guys can see that i.epsilon is related to causal structure in a certain sense (it is).

From there, enter mathematical rigor with things like the Osterwalder Schrader theorem, and its a hop leap and a jump to Wick rotation. Sorry to be vague, but its just totally nontrivial to expose this in a way that makes good sense (I only have it from theory discussion notes).

 

[OOO]

Weren't you calculating the massless case only?

You seem to refer to my analytical confirmation of Hans' result, that's true (although I have little doubt that the calculation yields a similar result for the massive case). But I was referring to my simulations (and I guess Hans has done something similar). The finite difference approximation of the massive KG equation does give a propagator that's exactly zero outside the lightcone (with the appropriate initial conditions).

True. Only the modulus of the momentum is sharp.

Neither. 1/(p^2-m^2) is nonzero for all p, not just the ones on mass shell.

 

[OOO]

Either way I think its clear by now that you guys can see that i.epsilon is related to causal structure in a certain sense (it is).

Haelfix, what (I think) we are discussing about is, whether there is one propagator that is exactly zero outside the light cone and why the textbooks don't just take this one to prove causality.

From there, enter mathematical rigor with things like the Osterwalder Schrader theorem, and its a hop leap and a jump to Wick rotation. Sorry to be vague, but its just totally nontrivial to expose this in a way that makes good sense (I only have it from theory discussion notes).

I have looked for the Osterwalder Schrader theorem before (in a different context) but I haven't found something useful.

Could you please tell me the reference to a publication or textbook that deals with the OS theorem.

 

[Haelfix]

I like Hans argument for the vanishing of the propagator offshell, but its not necessary really.

The commutator argument works fine either as a consequence of the Smatrix satisfying certain properties, or even just defining it as such (alla Weinberg who just imposes microcausality).

Constructive field theory makes all of this painfully rigorous as they go to great lengths to spell out the requirements and axioms necessary for a field to be causal. I don't have a good introductory link, but Streeter-Wightman probably can get you started.

This whole business is really a very old debate, that was done 40-50 years ago, i'd imagine it would be quite hard to track down the appropriate papers absent some old proffessor who remembers things.

 

[OOO]

I like Hans argument for the vanishing of the propagator offshell, but its not necessary really.

The commutator argument works fine either as a consequence of the Smatrix satisfying certain properties, or even just defining it as such (alla Weinberg who just imposes microcausality).

I think there must be something wrong with QFT (at least pedagogically) if Peskin & Schroeder are forced to introduce a concept on page 30 of their book, that, as you say, works fine, but the reason of which can't be understood without axiomatic quantum field theory.

I mean there is not even the slightest hint of a justification for why we should calculate the commutator by shifting the poles by ++ and not by +- epsilon. The first possibility yields the retarded propagator (which is zero outside the lightcone) and the second possibility yields the Feynman propagator (which is nonzero outside the cone, although being exponentially damped for larger separation). As P&S say:

"The p^0 integral of (2.58) can be evaluated according to four different contours, of which that used in (2.54) is only one."

So we are expected to learn by heart that the retarded propagator is used for calculating the commutator and the Feynman propagator is used for Feynman graphs. That's like learning zoology ! So what is all this jabbering for, that P&S involve into ?

Constructive field theory makes all of this painfully rigorous as they go to great lengths to spell out the requirements and axioms necessary for a field to be causal. I don't have a good introductory link, but Streeter-Wightman probably can get you started.

Thanks. I have looked into the Streater-Wightman book at Amazon but it doesn't seem to mention the Osterwalder-Schrader theorem in the index.

 

[Micha]

Neither. 1/(p^2-m^2) is nonzero for all p, not just the ones on mass shell.

True.

 

[Micha]

@OOO
I am confused by post 167.
I don't have the book of P&S at hand. I never saw two different propagators advocated.
From checking the integral I think, that the plus plus propagator (1/(p^2-m^2+i*epsilon)) is leaking out of the lightcone. And the reason is, it is doing this already at t=0.
This is, why there is no contraction with your numerical simulations.

 

[OOO]

@OOO
I am confused by post 167.
I don't have the book of P&S at hand. I never saw two different propagators advocated.
From checking the integral I think, that the plus plus propagator (1/(p^2-m^2+i*epsilon)) is leaking out of the lightcone. And the reason is, it is doing this already at t=0.
This is, why there is no contraction with your numerical simulations.

I have reread that part of P&S and to me it's confusing as well. P&S first calculate the vacuum expectation value of two field operators: <0|Phi(x)Phi(y)|0>. For this they show that there is the "leakage outside the lightcone", i.e. it does not vanish outside the light cone.

Then they say: this is void of meaning because it's not what we measure, so consider commutators instead. For simplicity they calculate the commutator as an expectation value again (valid because commutator is a c-number): <0| [Phi(x)Phi(y)] |0>. They show that this yields the 4D Fourier transform of what we have been discussing here all the time: 1/(p^2-m^2). Then they say that, by moving both poles into the lower half plane (so I should have rather said -- instead of ++, sorry), one gets a propagator (retarded) that vanishes outside the lightcone, which is what they seemed to expect.

Finally, they knock our socks off by saying that the (--) prescription is not the only one, but the Feynamn propagator is obtained by the (+-) prescription, or simply by p^2-m^2+iepsilon, and, again, this one does not vanish outside the lightcone.

 

[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]

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?

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]

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.

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

$$<0|\phi(x)\phi(y)|0> =: D(x-y)$$

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 $x^0>y^0$, 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...