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

[Micha]

Your approach is quite different to how they look at causality in http://aesop.phys.utk.edu/qft/2004-5/ .

It is interesting though, that the negative energy solutions (or suppressing them) play an important role in your approach as well.

 

[Micha]

Actually, it's because the photon is massless. Looking at the equation I mentioned
in Scharf, i.e., eq(2.3.36), all the terms which are non-zero for spacelike separations
are multiplied by the mass. Hence they vanish for a massless particle.

I am sure, you have your math correct.

I was trying to make sense of the following statement:

"Again, causality is due to non-trivial interference between positive-energy
modes (particles) propagating in one direction (x -> y) and negative-energy
modes (anti-particles) propagating in the opposite direction (y -> x)."

Looking at this again I must say I am confused. While photons are their own antiparticles, neutral scalar particles are as well.

 

[strangerep]

Micha,

OK, the UTK web document you quoted is essentially a course version
that combines material covered in Peskin & Schroeder, ch2, with some
extra stuff.

In particular, UTK's eq(1.4.6) is the equal-time (ie special) case of D(x-y):

<br /> D(x-y) = \frac{m}{4\pi^2\sqrt{-(x-y)^2}} ~ K_1(m \sqrt{-(x-y)^2})<br />

This equal-time expression, involving the K_1 Bessel
function, is a special case of the more general expression I mentioned
in Scharf. But even in UTK's expression, you can see immediately that
it is 0 if m=0.

Regarding the specific quote you mentioned:

Again, causality is due to non-trivial interference between
positive-energy modes (particles) propagating in one direction (x -> y)
and negative-energy modes (anti-particles) propagating in the opposite
direction (y -> x).

The discussion surrounding this in the UTK document is a bit brief.
P&S give more (on pp 28-29).

My personal opinion is that the word "due" in the quote "causality is
due to non-trivial interference..." should be regarded as an
interpretation. In their approach, I would have said "causality is
recovered by appealing to non-trivial interference...".

To understand this, I'll summarize the standard (canonical)
approach to QFT (which is basically what UTK and P&S follow):

1) Start with a classical Lagrangian function over phase space.
Ensure it is a relativistic scalar. That is, ensure the Lagrangian
is compatible with (classical) special relativity.

2) "Quantize" it, which means "construct a mapping from functions
over phase space to operators on a Hilbert space". This is
non-trivial, but most textbooks do it very quickly by saying
"promote the classical field and its conjugate momentum to
operators, and impose canonical commutation relations between them".
Then check that all the relativity transformations that were
applicable on the classical phase space are correctly represented
by operators on the Hilbert space, satisfying the Poincare commutation
relations.

Following this path, one then discovers the puzzle of the Feynman
propagator being non-zero outside the lightcone, in general. However,
this embarassment can be interpreted away by appealing to real-world
measurements. Eg, P&S say on p28: "To really discuss causality,
however, we should ask not whether particles can propagate over
spacelike intervals, but whether a meaurement performed at one point
can affect a measurement at another point whose separation from the
first is spacelike." Then they go on to show that such a relationship
between two measurements doesn't occur. However, they have to broaden
the context of their discussion to complex Klein-Gordon fields and
talk about particles and anti-particles.

My take on all this is that it's no surprise that they can derive
the result of no-effect between measurements at spacelike intervals,
because that's just basic special relativity, which was a crucial
input to the whole theory right from the start.

The difference between the above, and what I described in my
earlier post, is that the above tries to quantize the whole classical
phase space, whereas I restricted it to a mass hyperboloid first.

In one case, we find puzzling issues about the Feynman propagator
being non-zero outside the light cone. In the other, we find really
weird expressions for position states. IMHO, neither of these
approaches is entirely satisfactory (I think it's because of the
way Fourier transforms are used with gay abandon). Hence my earlier
post about the Heisenberg-Poincare group, though the latter is
still a rather speculative research topic.

You also said:

It is interesting though, that the negative energy solutions
(or suppressing them) play an important role in your approach as well.

In both approaches, this is built-in from the start as axioms, in
that both approaches assume positive-energy - which is a
phenomenological expression of the fact that we don't experience
any form of backward time-travel. In my post, I used the
\Theta(E) to express this. In the canonical approach, this
is assumed implicitly in the way the Feynman propagator is chosen
(choosing which way to deform the energy integration contour).

You also asked about my "|X&gt;" states:

This means we are simply not able to produce a fully localized
single particle state in QFT. Honestly, I do not have an idea yet, how
these |X> states look like.

The |X> states look quite horrible, and I don't think they're even
well-defined. For example, the Fourier transform of the
nastily-discontinuous \Theta(E) function is something like:

<br /> -\frac{1}{it\sqrt{2\pi}} + \sqrt{\frac{\pi}{2}}~\delta(t)<br />

and that's just the start of the nightmares in trying to find an
explicit expression for a general |X&gt;.

That's why you hardly ever hear anything about such states in basic
textbooks. They're of no practical help when trying to derive
experimental consequences of QFT such as scattering cross-sections.
Unfortunately, that also encourages people to think that the ordinary
(x,t) of Minkowski are somehow physically meaningful in QFT, and then
they derive various embarrassing theorems (e.g: EPR, Reeh-Schlieder, etc)
which show that something is seriously wrong somewhere. In these situations,
it helps to think about the physically-more-relevant |X&gt; states.

 

[Micha]

strangerep,
I agree with your summary of what the link (obviously based on Peskin & Schroeder) has to say about this. I don't possesses P&S, so I would love to hear any further details from anybody. With the following text of your post, especially at the end, I have the feeling that you are slightly leaving the ground of standard QFT. The EPR effect is an established piece of standard QM, not even QFT, right? The Reeh-Schlieder effect (at least what I saw in google) seems to be a weird, but to be mathematically well established theorem of standard QFT.

I read, that in QFT the field is not to be mixed up with the wavefunction, which is a fact, that maybe has not yet been well enough appreciated in this discussion. I also think, from physical grounds, that our discussion about causality should consider, that a totally sharp localized particle in real space is not possible in QFT, because the energy needed to measure with higher and higher precision would lead to the creation of particle/antiparticle pairs, so you wouldn't know, which is the particle, you want to localize. This is why I had some sympathy for your |X> states. But I really would like to keep this discussion within standard QFT.

My take on all this is that it's no surprise that they can derive
the result of no-effect between measurements at spacelike intervals,
because that's just basic special relativity, which was a crucial
input to the whole theory right from the start.

A theory, which is consistent with special relativity right from the beginning, is all, we are asking for, I think.Back to the explanation of P&S:
Their point is, that to check, whether two measurements at different spacetime points can influence each other, you need to go the commutator of the according operators.
The vacuum amplitude for the commutator of two field operators then is nothing else as the difference of the propagators ( D(x-y) - D(y-x) ).
I am almost convinced. But I would like to understand it a little better.
Would sending a signal from A to B always involve a measurement in A? (I can understand, it involves a measurement at B). Also I would like to understand the relation to antiparticles better. Are antipartcles needed to ensure, that the propagator has the following property for spacelike intervals:
D(x-y) = D(y-x) ?

 

[OOO]

I read, that in QFT the field is not to be mixed up with the wavefunction, which is a fact, that maybe has not yet been well enough appreciated in this discussion.

This is the reason why I don't like canonical quantization at all. In the path-integral approach it is apparent from the start (at least if you do not extend it to infinite times), that the path-integral is in fact a wavefunctional which depends on final field configurations, much like the wavefunction in nonrelativistic QM depends on final particle positions.

 

[Hans de Vries]

The |X> states look quite horrible, and I don't think they're even
well-defined. For example, the Fourier transform of the
nastily-discontinuous \Theta(E) function is something like:

<br /> -\frac{1}{it\sqrt{2\pi}} + \sqrt{\frac{\pi}{2}}~\delta(t)<br />

Found a simple way to show that the seemingly innocent operation
of "filtering out the negative frequencies" is something which
devastatingly violates Lorentz invariance and leads to those crazy
situations which Eugene is talking about.

"Filtering out the negative frequencies" comes down to:

\Theta(E)\ f(E,p)\ \ \Rightarrow \ \ \frac{1}{2}~\left( \delta(t) - \frac{i}{\pi t} \right) ~* ~ f(t,x)

The latter is a convolution over the vertical t-line (think Minkovski).
Do this on a point particle at rest, which is a vertical line, and the
result is again a point particle at rest. Do this on a moving particle
on the tilted t' line and you get a smeared out line which belongs to
a particle which is not local anymore! The general formula gives:

\frac{1}{2}~\left( \delta(t) - \frac{i}{\pi t} \right)~ * ~ \delta(t-vx)<br /> \ \ =\ \ \frac{1}{2}~\left( \delta(t-vx) - \frac{i\beta}{\pi(t-vx)} \right)

So the smearing out is proportional to the speed. Clearly, what's
below the E=0 line is not necessary below the E=0 line in other
frames. The frequencies which go from - to + all belong to off-
shell propagation corresponding to imaginary mass. Regards, Hans

 

[strangerep]

Found a simple way to show that the seemingly innocent operation
of "filtering out the negative frequencies" [...] leads to those crazy situations [...]

Thanks. Doing it for a simple case makes it easier to see that something is seriously weird.

It's not surprising, because \Theta(E) is not a tempered distribution. It causes
trouble near E=0 (infrared divergences) and also at high energy where it stays stubbornly
constant to \infty (ultraviolet divergences). Scharf makes a big deal out of this.
His version of QED (based on Epstein-Glaser) relies in large part on smoothing out the
undesirable discontinuities and divergences (order by order).

Cheers.

 

[strangerep]

With the following text of your post, especially at the end, I have the feeling that you are slightly leaving the ground of standard QFT. The EPR effect is an established piece of standard QM, not even QFT, right? The Reeh-Schlieder effect (at least what I saw in google) seems to be a weird, but to be mathematically well established theorem of standard QFT.

I was just highlighting the problems., certainly not proposing an alternate theory.

The theorems I mentioned (Reeh-Schlieder in particular) fall under the auspices of
standard axiomatic QFT. This starts by postulating a set of fields over Minkowski space,
and then demands that they carry a causal representation of the Poincare group.
The Reeh-Schlieder theorem says (roughly) that if you have 2 regions of Minkowski spacetime
O1 and O2 which are spacelike separated from each other, it is nevertheless possible to reconstruct
the fields on O1 arbitrarily accurately by cyclic operations of fields in O2 upon the vacuum.
That's a serious embarrassment. In recent years, there have been papers with titles like
Newton-Wigner vs Reeh-Schlieder (or something like that - I forget).

Anyway, I wasn't really departing from standard QFT, but just showing how/where/why
some of the well-known problems occur.

I read, that in QFT the field is not to be mixed up with the wavefunction, [...]

Right, which is why I kept my posting in terms of more abstract Hilbert space states.

I also think, from physical grounds, that our discussion about causality should
consider, that a totally sharp localized particle in real space is not possible in QFT,
because the energy needed to measure with higher and higher precision would lead to the
creation of particle/antiparticle pairs, so you wouldn't know, which is the particle, you
want to localize.

That's a rationalization/interpretation. That sort of thing ought to fall out of the math
rather than being put in by hand mid-calculation. In other words, all such "physical
grounds" ought to be encoded into theory's axioms at the start, after which we just
crank the mathematical handle. If we have to inject "physical grounds" again later,
it just means our initial axioms were inappropriate and should be revised.

This is why I had some sympathy for your |X> states. But I really would like to
keep this discussion within standard QFT.

Those |X> states just express an alternate basis of the physical Hilbert space. Remember
that (in momentum space) we're restricted to a 3D mass hyperboloid, spanned by the
usual |\underline p> states. The |X> states are just an alternate basis for exactly the
same Hilbert space, but don't try and use them for practical calculations. :-)

Back to the explanation of P&S:
Their point is, that to check, whether two measurements at different spacetime points can influence each other, you need to go the commutator of the according operators.
The vacuum amplitude for the commutator of two field operators then is nothing else as the difference of the propagators ( D(x-y) - D(y-x) ).
I am almost convinced. But I would like to understand it a little better.
Would sending a signal from A to B always involve a measurement in A? (I can understand, it involves a measurement at B). Also I would like to understand the relation to antiparticles better. Are antipartcles needed to ensure, that the propagator has the following property for spacelike intervals: D(x-y) = D(y-x) ?

The antiparticle thing is a bit misleading. To follow it closely, one must go back to the
fields that make up the Lagrangian, and carry through the full quantization to multiparticle
Fock space.

But to understand the point above, it's sufficient to know two things: 1) D(x-y) is a Lorentz
invariant; 2) if x,y are spacelike-separated, there exists a continuous Lorentz transformation
that transforms (x-y) into (y-x). (P&S illustrate this with their fig 2.4, but I don't know
how to reproduce that diagram here.)

So, (1) and (2) together imply that D(x-y) = D(y-x) for spacelike-separated x,y. You don't
need to fuss around with measurements, signals or antiparticles. It's simply a consequence
of the relativity maths that D(x-y) = D(y-x), and hence [\phi(x), \phi(y)] = 0.
for spacelike x-y

(If you need more than that, with vacuum states and everything, a little more detail and
sophistication are necessary.)

 

[Avodyne]

A technical point:
\lim_{m\to 0}mK_1(mr)={1\over r}
not zero.

 

[Micha]

The theorems I mentioned (Reeh-Schlieder in particular) fall under the auspices of
standard axiomatic QFT. This starts by postulating a set of fields over Minkowski space,
and then demands that they carry a causal representation of the Poincare group.
The Reeh-Schlieder theorem says (roughly) that if you have 2 regions of Minkowski spacetime
O1 and O2 which are spacelike separated from each other, it is nevertheless possible to reconstruct
the fields on O1 arbitrarily accurately by cyclic operations of fields in O2 upon the vacuum.
That's a serious embarrassment. In recent years, there have been papers with titles like
Newton-Wigner vs Reeh-Schlieder (or something like that - I forget).

Anyway, I wasn't really departing from standard QFT, but just showing how/where/why
some of the well-known problems occur.

Ok, thanks. I am seeing a direct connection to our topic now. How can Reeh-Schlieder proove such a theorem, while Peskin & Schroeder is providing a proof for causality? But maybe to start a discussion about this interesting theorem would be too much for this already long thread, at least for now.

That's a rationalization/interpretation. That sort of thing ought to fall out of the math
rather than being put in by hand mid-calculation. In other words, all such "physical
grounds" ought to be encoded into theory's axioms at the start, after which we just
crank the mathematical handle. If we have to inject "physical grounds" again later,
it just means our initial axioms were inappropriate and should be revised.

I perfectly agree with what you say. It does not mean however, that intution is useless. It is useful as a guide to the right formal theory, if you haven't found it yet, or while you are learning it and even if you know it, intution is helpful as a shortcut to find the right answer without long calculation. Of course always with the danger of being wrong depending on how complicated the topic and how good your intution is.
And not only that, finally you have to connect your math to the real world by comparing it to measurement. So you need to know, what your math means.

The antiparticle thing is a bit misleading. To follow it closely, one must go back to the
fields that make up the Lagrangian, and carry through the full quantization to multiparticle
Fock space.

But to understand the point above, it's sufficient to know two things: 1) D(x-y) is a Lorentz
invariant; 2) if x,y are spacelike-separated, there exists a continuous Lorentz transformation
that transforms (x-y) into (y-x). (P&S illustrate this with their fig 2.4, but I don't know
how to reproduce that diagram here.)

So, (1) and (2) together imply that D(x-y) = D(y-x) for spacelike-separated x,y. You don't
need to fuss around with measurements, signals or antiparticles. It's simply a consequence
of the relativity maths that D(x-y) = D(y-x), and hence [\phi(x), \phi(y)] = 0.
for spacelike x-y

(If you need more than that, with vacuum states and everything, a little more detail and
sophistication are necessary.)

I agree, to show, that [\phi(x), \phi(y)] = 0. is true, is only mathematics.
You don't need to talk about antiparticles. (Although it would be nice to understand, why other
people see a connection here.)
But in order to say, that [\phi(x), \phi(y)] = 0. means, that causality is preserved, you need to have a concept of causality first, to which you can connect. And the best definition of causality in physics I know, is, that you can not send a signal from A to B with a speed greater than the speed of light.

 

[Micha]

Small addition: The concept of causality behind [\phi(x), \phi(y)] = 0 obviously is: Two measurements at spacelike distances can not influence each other. This is for me a respectable concept of causality, too. I am only not sure, whether the two definitions of causality, we then have, are necessarily equivalent.

 

[strangerep]

A technical point:
\lim_{m\to 0}mK_1(mr)={1\over r}
not zero.

Oops! OK, thanks.

This means the over-simplified UTK formula is not adequate to illustrate
that massless photons propagate on the lightcone. One must use the more general
formula from Scharf, which has a \Theta(-x^2) multiplying that term,
not to mention a \delta(x^2) elsewhere. So it all gets horribly messy
near the lightcone.

 

[strangerep]

How can Reeh-Schlieder proove such a theorem, while Peskin
& Schroeder is providing a proof for causality?

I'm not familiar with the details of the Reeh-Schlieder proof, but others have
suggested the roots lie in the non-locality of the vacuum state itself.

But in order to say, that [\phi(x), \phi(y)] = 0. means, that causality is
preserved, you need to have a concept of causality first, to which you can connect.
And the best definition of causality in physics I know, is, that you can not send a signal
from A to B with a speed greater than the speed of light.

Small addition: The concept of causality behind [\phi(x), \phi(y)] = 0. obviously is:
Two measurements at spacelike distances can not influence each other. This is for me a
respectable concept of causality, too. I am only not sure, whether the two definitions of
causality, we then have, are necessarily equivalent.

They're equivalent - because tachyons have never been physically observed.

[and yes, this thread is getting too long]

 

[Hans de Vries]

Micha,

OK, the UTK web document you quoted is essentially a course version
that combines material covered in Peskin & Schroeder, ch2, with some
extra stuff.

In particular, UTK's eq(1.4.6) is the equal-time (ie special) case of D(x-y):

<br /> D(x-y) = \frac{m}{4\pi^2\sqrt{-(x-y)^2}} ~ K_1(m \sqrt{-(x-y)^2})<br />

This equal-time expression, involving the K_1 Bessel
function, is a special case of the more general expression I mentioned
in Scharf.

This expression is also used in Weinberg's treatment of the subject:
Volume one, chapter 5.2: Causal scalar fields.

Weinberg refrains from an explicit picture of outside the lightcone
propagating particles and counter propagating anti particles which
cancel each other in his treatment.


Regards, Hans

 

[Micha]

They're equivalent - because tachyons have never been physically observed.

I am too stupid to clearly see this equivalence.

Imagine I had a source of radioactive particles and the source was shielded by some lead. Now in order to send a signal I would remove the lead. Is removing the lead a measurement? Mhm, probably yes. But is it a measurement of the radioactive particles I am sending? Is it nonsense to think about this altogether?

 

[geoffc]

For me, it made matters much worse when I had to "unlearn" everything about the measurement process and the notion of "particles" when looking at quantum field theory in curved spacetime.

A great explanation of the difficult is described in the introduction of:
http://arxiv.org/abs/gr-qc/9707062

An article that discusses of the notion of "particle" and "measurement" as it pertains to Unruh radiation is here:
http://xxx.lanl.gov/abs/0710.5373

 

[strangerep]

I am too stupid [...]

Trust me... everyone feels stupid now and then. Only the totally ignorant and braindead
never feel stupid. Don't worry about it.

Imagine I had a source of radioactive particles and the source was shielded by some lead. Now in order to send a signal I would remove the lead. Is removing the lead a measurement? Mhm, probably yes. But is it a measurement of the radioactive particles I am sending? Is it nonsense to think about this altogether?

You're over-complicating it. The crux is that any type of event at x cannot affect any
type of event at y, if x-y is spacelike. That is, information cannot be propagated between
x and y.

(A measurement at x is just a type of event at x.)

 

[Micha]

Trust me... everyone feels stupid now and then. Only the totally ignorant and braindead
never feel stupid. Don't worry about it.You're over-complicating it. The crux is that any type of event at x cannot affect any
type of event at y, if x-y is spacelike. That is, information cannot be propagated between
x and y.

(A measurement at x is just a type of event at x.)

Now you replaced the word measurement by the word event. While this might give us a better feeling, because the word event seems innocent and is surely less mysterious then the word measurement in the context of quantum mechanics, I don't see, what this really explains. Especially I am not able to make a clear connection between events and creation and annihilation operators.

 

[strangerep]

Now you replaced the word measurement by the word event. While this might give us a better feeling, because the word event seems innocent and is surely less mysterious then the word measurement in the context of quantum mechanics, I don't see, what this really explains.

I was just using "event at x" as a generic term for anything that happens at x,
ie: any change of configuration of the quantum field at x. To perform a measurement
within a region A of spacetime involves the creation and annihilation of initial/final
states within A. (Think of preparation and detection.) The point is that any side effects
of these operations within A cannot affect the result of some other measurement off in
another spacetime region B, if A and B are spacelike-separated.

Especially I am not able to make a clear connection between events and creation and annihilation operators.

Think of each operator as a mapping from one state to another. Ie: a change in the
configuration of the quantum field.

 

[Micha]

Thanks, strangerep,
I think I just forgot the basic starting point of quantum mechanics: Every observable is associated with an operator.

But I want to say another thing:
Look what wikipedia has now on the propagator:
http://en.wikipedia.org/wiki/Propagator
It looks like a very nice summary of our thread with some details added.

 

[Haelfix]

Since this thread has resurfaced, i'd like to point out a neat little paper I read on this subject (with regards to curved spacetimes)

arXiv:0709.1483. In it they try to show that causality of the classical theory implies microcausality of the quantum picture.

Now, there is something a little fishy going on in at least part of their derivation, but i'll leave it to the reader to see if they can spot my objections. I do agree with the premise though, at least in part.

 

[jostpuur]

Look what wikipedia has now on the propagator:
http://en.wikipedia.org/wiki/Propagator
It looks like a very nice summary of our thread with some details added.

What does \delta(t,t&#039;) mean in the first equation?

 

[strangerep]

What does \delta(t,t&#039;) mean in the first equation?

It's just a Dirac delta function. (Look at the next line, where it says that \delta(x,x&#039;)
is a Dirac delta fn, even though they've used "\delta(x-x&#039;)" just before that.)
Their notation is a bit inconsistent.

 

[tiny-tim]

thread approaching mass of black hole … HELP!

I started to read this thread - but it's RIDICULOUSLY LONG!

Themes and sub-themes have come and gone, and there could be almost anything in the middle pages.

I can't plough through all that just in case.

Shouldn't it be someone's job to split it up every so often?

Or perhaps once a thread gets a certain size, there should be an inhibitor encouraging new contributors to start new threads?

I don't mind short threads that wander off-topic, or long threads that stay on-topic.

But, please, no long threads that meander all over the place!

Am I the only one who gets turned off by threads like this one?

 

[Hans de Vries]

I made some nice progress here, writing the chapter on the Klein Gordon equation
of my textbook.

http://chip-architect.com/physics/Book_Chapter_Klein_Gordon.pdf" (see 8.9 and further) For instance I found out why we can get away with 'sloppy' math like using.

<br /> \frac{-1}{E^2-p^2}~~~~ \mbox{and}~~~~\frac{-1}{E^2-p^2-m^2}<br />

as the propagators for the photon and KG equation. They are symmetric in
E(nergy) and thus symmetric in time. This means that the electromagnetic
field would have advanced potentials as well as retarded potentials. Clearly
in conflict with experiment.

It turns out that the difference between the two-sided propagators (forward
and backward in time) and the fully causal, forward-in-time-only propagators
is only in the poles. For instance:

Causal, forward in time Photon propagator

<br /> D^{\triangledown}(E,p)\ =\ \frac{-1}{E^2-p^2}\ \ +\ \ \frac{\pi}{2ip}\bigg( \delta(E-p)-\delta(E+p) \bigg)<br />

The extra term only propagates on-shell. It modifies the poles. It does not influence
first order Feynman diagrams since the virtual photons of those diagrams can not be
real. (real electrons can not emit a single real photon)
The same is true for the Klein Gordon equation:

Causal, forward in time Klein Gordon propagator

<br /> D^{\triangledown}(E,p)\ =\ \frac{-1}{E^2-p^2-m^2}\ \ +\ \ \frac{\pi}{2i\sqrt{p^2+m^2}}\bigg( \delta(E-\sqrt{p^2+m^2})-\delta(E+\sqrt{p^2+m^2}) \bigg)<br />

Again, only the poles are different. First order diagrams are unaffected.
The extra deltas have an amplitude in the poles which is infinitely much
higher as the normal poles. This is proved with Rayleigh's theorem.

We can reorganize the forward propagator on a pole-by-pole base to
study the behavior at the poles, for instance for the photon

<br /> D^{\triangledown}(E,p)\ =\ <br /> -\frac{1}{2p} \left( \frac{1}{E-p}+i\pi\delta(E-p)\right)\ \ \ +\ \ \ \frac{1}{2p}\left( \frac{1}{E+p}+i\pi\delta(E+p)\right)<br />

Going slowly through a pole gives an amplitude which can be symbolically
written as:

-\infty ~~~~ \rightarrow ~~~~ i\infty^2 ~~~~ \rightarrow ~~~~ +\infty <br />

Where the middle term i\infty^2 symbolizes the contribution from the delta.

A compelling argument can be made that the addition of the delta functions
is a physical requirement. Any real life photon has a finite lifetime and is therefor
not exactly on-shell. Its frequency spectrum extends to both sides of the pole
which propagate with opposite sign, and therefor, would destructively interfere.

The destructive interference would be 100% in the exactly symmetric case.
Minimal changes in frequency would move the spectrum to either side of the
pole and the destructive interference would disappear. The on-shell propagation
would be ill-defined.

Only the addition of the delta functions at the poles makes the on-shell propagation
well-behaved since the magnitude of the deltas is much higher as that of the
reciprocal functions.Regards, Hans

 

[Micha]

Hans, maybe this sloppy mathematics works for you.
But I will certainly not use it, when the "real" mathematics is so clear and easy
as presented in wikipedia.

Note also, that the advanced propagator, which is surely the causal propagator in classical physics, is not the one to be used in quantum field theory. Instead the Feynman propagator has to be used, because it contains both the positive and negative energy solutions (particles and antiparticles). As explained in wikipedia, while it is nonzero outside the light cone, causality is ensured by the commutators of space-like separated field operators being zero.

 

[Hans de Vries]

Hans, maybe this sloppy mathematics works for you.
But I will certainly not use it, when the "real" mathematics is so clear and easy
as presented in wikipedia.

1) The article in Wikipedia has many errors. It's not for nothing that it says in the
header that the article needs an expert.
http://en.wikipedia.org/wiki/Talk:Propagator

2) Sloppy mathematics certainly doesn't work for me. That's exactly the reason
for me to go to the bottom of things.

Note also, that the advanced propagator, which is surely the causal propagator in classical physics, is not the one to be used in quantum field theory.

The advanced potentials come from the backward propagator which goes from the future
to the past. This is the part you want to remove from the propagator and that is what I did.

Instead the Feynman propagator has to be used, because it contains both the positive and negative energy solutions (particles and antiparticles). As explained in wikipedia, while it is nonzero outside the light cone,

We have discussed the single pole Feynman propagator here before. It violates
special relativity as you say. In practice one doesn't use the pole prescription
when evaluating Feynman diagrams. The expression without the prescription,
simply: 1/(p^2-m^2) supports both positive and negative solutions as well.

causality is ensured by the commutators of space-like separated field operators being zero.

One can postulate this as Pauli did in his 1940 paper on spin-statistics, but that's
hardly satisfying. Regards, Hans.

 

[Micha]

1) The article in Wikipedia has many errors. It's not for nothing that it says in the
header that the article needs an expert.
http://en.wikipedia.org/wiki/Talk:Propagator

I know this article from before. It already had this comment on it, but since then it has been completely reworked. And now it contains nicely everything, what I learned from this thread.
I think, they just didn't remove the comment yet.

I do not want to discuss the whole thing all over again here. But since a contribution of yours already caused a lot of confusion in this thread earlier (if for nobody than certainly for me), I felt I had to make this remark. Nothing personal.

Edit: It seems I confused the advanced with the retarded propagator in my earlier post. I mean a propagator being non zero in the future of course.

 

[Hans de Vries]

I know this article from before. It already had this comment on it, but since then it has been completely reworked. And now it contains nicely everything, what I learned from this thread.
I think, they just didn't remove the comment yet.

The article has many errors

The use a Klein Gordon propagator in position space which is totally wrong,
it has the wrong Bessel function (Bessel Y1 instead of J1). The Greens function
has the wrong argument. It misses out on the delta function..What is presented as a causal propagator violates the basic laws of the Fourier
transform. A causal propagator has to respect the Kramers Kronig relation. A
real forward-only propagator has a Fourier transform which has an even real
part and an odd imaginary part, the two are related via the Hilbert transform.

I do not want to discuss the whole thing all over again here. But since a contribution of yours already caused a lot of confusion in this thread earlier (if for nobody than certainly for me), I felt I had to make this remark. Nothing personal.

This just is a confusing subject. There are so many different versions going
around of the Klein Gordon propagator in position space that textbook authors
avoid writing about it because they are not sure which one to choose.

That alone is a reason to discuss these things here.Regards, Hans

 

[Hans de Vries]

What is presented as a causal propagator violates the basic laws of the Fourier
transform. A causal propagator has to respect the Kramers Kronig relation. A
real forward-only propagator has a Fourier transform which has an even real
part and an odd imaginary part, the two are related via the Hilbert transform.

I may have been a bit too impulsive here, one would have to prove that
the causal propagator:

\frac{-1}{(E-i\epsilon)^2 -p^2-m^2}

respects the Kramers Kronig relation. It would have to be equal to my
expression for which I'm quite sure that it does:

<br /> \frac{-1}{E^2-p^2-m^2}\ \ +\ \ \frac{\pi}{2i\sqrt{p^2+m^2}}\bigg( \delta(E-\sqrt{p^2+m^2})-\delta(E+\sqrt{p^2+m^2}) \bigg)<br />

Note that both are always real except in the poles where they become
imaginary delta functions.

I would actually be quite happy if they turn out to be the same. It's
late now, tomorrow more.Regards, Hans