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

[meopemuk]

The important thing is the basic (linear) Lie algebra of quantities corresponding to observable
properties of physical systems.

I don't think we should require that all operators of observables must belong to the Lie algebra. Some of them can be expressed as Hermitian functions of Lie algebra elements. The simplest example is the operator of rest mass

M = \frac{1}{c^2}\sqrt{H^2 - c^2 \mathbf{P}^2}

Other examples are operators of velocity, spin, and position.

Eugene

 

[strangerep]

I don't think we should require that all operators of observables must belong to the Lie algebra. Some of them can be expressed as Hermitian functions of Lie algebra elements. [...]

Oops. When I said "the important thing is the basic (linear) Lie algebra of quantities ...",
I meant the things (e.g., unitary irreps) that one uses as basic building blocks to construct
one's QFT. If one thus starts from the unitary irreps of the Poincare group, it's no surprise
that things outside the whole Poincare group (like R itself) turn out to be ill-behaved.

That is, the important thing when constructing a field theory is the basic Lie algebra of observables,
not merely the group of inertial transformations. That's why I tried to use only the phrase
"H-P algebra", not "H-P group". But I suspect we're returning to subjective assessments of
what "important" means, so I'll stop here.

 

[Hans de Vries]

First, I think, the term "past" is defined by the backward lightcone and not just by some event that has t<0 in some frame. I guess the region outside the lightcone is commonly called present (even if t<0 in some frame)

The pole prescripted propagator which causes all the problems does
propagate backwards in time and does so not only for anti-particles
but for any all day live particle except for the infinite plane wave.

It's the result of requiring that the propagator instead of the wave
function contains only positive energy frequencies, For me this
is a sign of confusion. One can find the source of such a confusion
in the older texts discussing Green's function relation.

-(\Box+m^2)D(x-y)\ =\ \delta(x-y)

One sees that \delta(x-y) is interpreted as the point-particle and
D(x-y) is regarded as the subsequent wave function. Now, this is all
without consequences if one just sets i\epsilon=0 in practical calculations
and this is just what's done most of the time. It would be interesting
to look at the arguments used in the exception cases though.Regards, Hans

 

[OOO]

The pole prescripted propagator which causes all the problems does
propagate backwards in time and does so not only for anti-particles
but for any all day live particle except for the infinite plane wave.

Hans, as you say, the epsilon-propagator D is non-zero outside the lightcone and non-zero in the backward lightcone. I think we agree on it being non-zero outside the lightcone seems to cause some trouble with causality at first glance, but this is resolved by the way D is used in QFT, right ?

Now I propose that the non-vanishing amplitude of the propagator in the backward lightcone is a totally different story. In my opinion this does neither present any problems for causality (there is nothing peculiar about signals coming from the past), nor is it connected to the amplitude outside the lightcone.

The reasoning for the latter is simple: if I am able to find a propagator that vanishes for the past and out of the lightcone (which is what you have done with the Heaviside function), I am also able to find another propagator that vanishes for the future and out of the lightcone (it's the advanced propagator). I just have to flip the "sign of time" for that. But then I can simply average the retarded and advanced propagators to get another one that vanishes outside the lightcone but neither for the future nor the past. Adding further homogeneous solutions gives me the most general propagator which does not vanish anywhere.

So what I have just shown is that the amplitude outside the lightcone has nothing to do with the amplitude in the backward lightcone.

 

[Hans de Vries]

Hans, as you say, the epsilon-propagator D is non-zero outside the lightcone and non-zero in the backward lightcone.

It also is non-zero at t<0 outside the lightcone, so the propagation would
not only be instantaneous (not zero at t=0 outside the light cone). It also
would propagate to the past outside the lightcone.

Filtering out the negative energy frequencies involves a convolution in
time with f(t)=1/t. Meaning that the propagator is smeared out over the
t-axis.

The inside of the future light-cone is smeared out in the -t and +t
directions. In the vertical direction in the figure below:

\mbox{light cone:}\quad \begin{array}{c}\bigtriangledown \\ \bigtriangleup \end{array}

Going down it gets outside the lightcone and further down into negative t.Regards, Hans

PS. Filtering out negative frequencies = Multiplication with the Heaviside
step function over the E-axis in momentum space = Convolution with the
Fourier transform of the step function over the t-axis in position space.

PPS. For the transform of step-function, see entry 310 in the table here:
http://en.wikipedia.org/wiki/Fourier_transform#Distributions

 

[OOO]

It also is non-zero at t<0 outside the lightcone, so the propagation would
not only be instantaneous (not zero at t=0 outside the light cone). It also
would propagate to the past outside the lightcone.

I wouldn't call the region outside the lightcone with t<0 the "past" since this would not be a covariant definition. Just change the frame and what was the past would become the future and vice-versa. That doesn't make sense, at least semantically (whereas, of course, I have no doubt about the appropriateness of Minkowski space).

The only covariant definition of the term "past" is that it is represented by the backward lightcone.

 

[Hans de Vries]

I wouldn't call the region outside the lightcone with t<0 the "past"
The only covariant definition of the term "past" is that it is represented by the backward lightcone.

Granted.

Regards, Hans

PS. However, a succession of two such propagation steps can get
you in the past light cone, propagating information from the current
to the past, if normal propagation to the past isn't bad enough...

 

[Hans de Vries]

The reasoning for the latter is simple: if I am able to find a propagator that vanishes for the past and out of the lightcone (which is what you have done with the Heaviside function), I am also able to find another propagator that vanishes for the future and out of the lightcone (it's the advanced propagator). I just have to flip the "sign of time" for that. But then I can simply average the retarded and advanced propagators to get another one that vanishes outside the lightcone but neither for the future nor the past. Adding further homogeneous solutions gives me the most general propagator which does not vanish anywhere.

So what I have just shown is that the amplitude outside the lightcone has nothing to do with the amplitude in the backward lightcone.

By choosing to integrate in the positive time direction one does
indeed "sneak in" causality. Physically integrating backwards in
time by itself already violates time ordering causality. So, I don't
believe in propagators which are non-zero for t<0.

Zee doesn't either on page 109, in II.2 where he calls "an electron
going backward in time": Poetic but confusing metaphorical language..


Regards, Hans.

 

[Micha]

Now, this is all
without consequences if one just sets i\epsilon=0 in practical calculations
and this is just what's done most of the time. It would be interesting
to look at the arguments used in the exception cases though.Regards, Hans

I think, this is exactly the right question to ask. Does it matter somewhere, when calculating Feynman diagrams (which is always done for practical (experimental) purposes, as far as I know, in momentum space) if I take the +i*epsilon or -i*epsilon or maybe even a ++ or -- epsilon prescription? If not, doing the Fourier transform to real space, should be regarded just as a mathematical exercise, and textbooks should put in a warning to not take this result seriously because of its ambiguity.

 

[Avodyne]

I think, this is exactly the right question to ask. Does it matter somewhere, when calculating Feynman diagrams (which is always done for practical (experimental) purposes, as far as I know, in momentum space) if I take the +i*epsilon or -i*epsilon or maybe even a ++ or -- epsilon prescription?

Yes, it matters. Changing the sign of epsilon changes how one does the Wick rotation to euclidean space, and this in turn changes the sign of every one-loop diagram. Among other things, this would mean that in quantum electrodynamics we would have charge antiscreening rather than screening, and hence a negative beta function. Cool, but wrong.

 

[Micha]

Yes, it matters. Changing the sign of epsilon changes how one does the Wick rotation to euclidean space, and this in turn changes the sign of every one-loop diagram. Among other things, this would mean that in quantum electrodynamics we would have charge antiscreening rather than screening, and hence a negative beta function. Cool, but wrong.

Ok, sorry. I should have taken a more careful look at Zees book. Here the +i*epsilon comes from the fact, that you want a factor -epsilon*phi^2 in the exponent of the path integral to let it go to zero for large phi.

 

[OOO]

Granted.

Regards, Hans

PS. However, a succession of two such propagation steps can get
you in the past light cone, propagating information from the current
to the past, if normal propagation to the past isn't bad enough...

Yes, I agree with this completely. But vice-versa, you won't be able to combine two propagation steps inside the lightcone to get out of the lightcone. That's what I wanted to emphasize. So [nonzero; out of the lightcone] means trouble with causality, but [nonzero; inside the past lightcone] does not.

 

[OOO]

By choosing to integrate in the positive time direction one does
indeed "sneak in" causality. Physically integrating backwards in
time by itself already violates time ordering causality. So, I don't
believe in propagators which are non-zero for t<0.

I don't understand your reasoning. Isn't it quite natural to think of a present event (at 0,0) as being caused by something in the infinite past and causing something else in the infinite future. In my opinion that's what a time-symmetric propagator could be trying to tell us.

Zee doesn't either on page 109, in II.2 where he calls "an electron
going backward in time": Poetic but confusing metaphorical language.

This reference to authority stands a bit isolated among your criticism of Zee and other textbook authors...

 

[Hans de Vries]

I don't understand your reasoning. Isn't it quite natural to think of a present event (at 0,0) as being caused by something in the infinite past and causing something else in the infinite future. In my opinion that's what a time-symmetric propagator could be trying to tell us.

But the Green's function is defined as the response of a field on a
perturbation at (0,0). Of course, a point in the past would contribute
the same to (0,0) as (0,0) would contribute to that point mirrored
into the future light cone, but the Green's function is defined with
the cause at (0,0), while the inverse Green's function is used to track
back the source of the field.

This reference to authority stands a bit isolated among your criticism of Zee and other textbook authors...

OK But I only pointed to Zee's skeptical remarks about Feynman's
original ideas of electrons going back in time, to show that I'm not
alone in my reservations. The "reference to authority" wasn't intended
towards you, since you were not claiming that anyway if I understood
you correctly.

I don't think I'm criticizing Zee, I'm only discussing a piece of sideline math
which comes to us from the early days of QED copied from one textbook
to another.Regards, Hans.

 

[OOO]

But the Green's function is defined as the response of a field on a
perturbation at (0,0). Of course, a point in the past would contribute
the same to (0,0) as (0,0) would contribute to that point mirrored
into the future light cone, but the Green's function is defined with
the cause at (0,0), while the inverse Green's function is used to track
back the source of the field. .

Yes, if that's the definition. I admit that I often mentally switch back to classical electrodynamics where the propagators are obviously defined that way. What we are doing in ED is one of two cases:

1) Move the charge on a predefined trajectory (which amounts to imposing a constraint on it) and calculate the fields that are generated from this movement. The retarded Lienard-Wiechert potentials give us in some sense the "minimal" fields (neither initial nor past fields are non-zero). Of course this must be artificial because there must have been some fields which caused the charge to move like it did in the first place.

2) Apply some fields and calculate the movement of the charge due to Lorentz force.

The combination of both is not possible in classical ED because the energy momentum conservation proves to be incomplete, ie. there is no energy-momentum tensor for the matter field.

Now this is different in QED, where the matter field is included in the Lagrangian and thus in the energy-momentum tensor. So what I was thinking of is that singling out a retarded propagator out of the infinitely many ones is not necessary any more since we have left the realm of moving charges by constraints.

Therefore I conjecture that our insisting on the retarded propagator as the "real" propagator is a reverb of this "moving charges by hand" business. In this sense I also conjecture that a time symmetric propagator is an expression of the fact that an incoming spherical wave causes charge movement and this again causes an outgoing wave, so in sum we are describing a scattering process. On the other hand if we used a retarded propagator only, then causality is violated because the cause of the electron (or charged pion in case of KG) movement is missing from the description.

OK But I only pointed to Zee's skeptical remarks about Feynman's
original ideas of electrons going back in time, to show that I'm not
alone in my reservations. The "reference to authority" wasn't intended
towards you, since you were not claiming that anyway if I understood
you correctly.

You did. I also felt a lot better if the books could clearly explain why they do things the way they do. But there seem to be slightly too many excuses around there, or probably things are just too complicated to be explained to such drooling idiots like us. And so we have to keep on thinking and sometimes change our minds about things. I still can't say I'm sure about causality in QFT...

 

[Micha]

Let me sum up the state of our discussion:

1. The Feynman propagator does leak out of the light cone.
2. The Feynman propagator is not just another of an infinite number of Greensfunctions. It IS the amplitude for a particle moving from one space-time point to another.
In Zee, chapter I.8. (14), the Feynman propagator is derived from canonical field theory as an integral over space. You can make this a 4 dimensional integral and get in a unique way the +i*epsilon prescription.

This leaves the question open, how the propagator goes together with causality. I think, we have in this thread rediscovered, that we need antiparticles to restore causality, because clearly, with particles only we are stuck at this point.
Indeed I found a statement in this link:
http://aesop.phys.utk.edu/qft/2004-5/2-5.pdf

"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)."

I didn't follow the math so far, but I tend to believe, it is true.
Two observations fit nicely:

1. The leaking out of the lightcone get bigger for lighter particles. Cleary for lighter particles, it is easier to create particle-antiparticle pairs
2. The photon propagator does not show any leaking out of the lightcone. This must be, because the photon is its own antiparticle.

 

[Micha]

I further read in http://aesop.phys.utk.edu/qft/2004-5/ , and I am still convinced, that it can give us the answer to our question about causality.
Notice, that the Feynman propagator is defined as the time ordered product of the two field operators at the two space-time points. Thus we cut the propagator into two pieces, the positive time propagator gives the propagation of the particle only, whereas the negative time propagator gives the propagation of the antiparticle. Clearly this is, what you need in Feynman diagrams, because you work with particle/antiparticle eigenstates.
You can also see this from the fact, that eg. for t>0 only the positive energy pole is contributing.
But for the propagation of a real particle, we have to consider both particle and antiparticle propagation.

 

[jostpuur]

Let me sum up the state of our discussion:

Good idea. I haven't been following the discussion anymore, but I'm interested in any conclusions you may end with.

Have you yet come to agreement about how precisely is the propagation amplitude related to the spatial probability densities? As I noted in my question, it at least is not related by the same equation

<br /> \Psi(t,y) = \int d^3x\; K(t-t_0, y,x) \Psi(t_0,x)<br />

(where K is the propagation amplitude) as it is in the non-relativistic QM.

It is so easy to say "amplitude to propagate", but it doesn't mean anything without clear meaning in terms of spatial probability density.

 

[strangerep]

1. The Feynman propagator does leak out of the light cone.

Depends what you precisely mean by "leak". The standard Feynman propagator
is indeed non-zero outside the light-cone. That's a mathematical fact, as derived
(for example) in Scharf's "Finite Quantum Electrodynamics" pp64-69. See in
particular eq(2.3.36) and the discussion on the following page 69.

The photon propagator does not show any leaking out of the lightcone.
This must be, because the photon is its own antiparticle.

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.

This leaves the question open, how the propagator goes together with causality.

As I tried to explain before, the problem is that a naive Hilbert space whose basis
vectors correspond to ordinary 4D Minkowski space is not a physically-meaningful
Hilbert space. I'll run through the construction again...

Start with a 4-dimensional vector space, denoting an arbitrary vector is denoted as |k>.
That is, it's a 4-momentum vector space, but it is not yet a Hilbert space, nor does it
correspond to a relativistic particle type. It doesn't even have an inner product yet, so
expressions like <k|k'> do not yet have any meaning.

To turn this k-space into a Hilbert space for a relativistic particle of mass m, we restrict
a subspace of those vectors which satisfy k^2 = m^2, and also satisfy E &gt; 0,
where E := \sqrt{m^2 + {\underline{p}}^2}, and \underline{ p} denotes 3-momentum.
That is, we restrict to only those |k> vectors on the mass hyperboloid corresponding to mass=m.

Any vector in the restricted space (the mass hyperboloid) can thus be written
<br /> |\underline{p}&gt; ~=~ \Theta(E) ~\delta^{(4)}(m^2 - E^2 + {\underline{p}}^2) |k&gt;<br />
where \Theta(E) is a step function restricting to +ve energy.

With these restrictions, the subspace consisting only of these |\underline{p}&gt;
vectors can be made into a Hilbert space by defining an inner product of the form:
<br /> &lt;\underline{p} | \underline{p&#039;}&gt; ~=~ \delta^{(3)}(\underline{p} - \underline{p&#039;})<br />
(Depending one's conventions, there might also be a factor involving E on
the RHS, but that's not important here.) Note also that these |\underline{p}&gt;
vectors do not span the original |k> vector space in any sense.

Now let's think about trying to change to a position basis. That's easy for the original
|k> space:
<br /> |x&gt; ~:= \int d^4k ~ e^{ikx} ~ |k&gt;<br />
This gives 4D Minkowski vector space. Unfortunately, it's useless as a Hilbert space,
because it's not the same space as our physical Hilbert space above consisting of |\underline{p}&gt; vectors.
To get position-like vectors in the physical Hilbert space, we must do something
like the following instead:

<br /> |X&gt; ~:= \int d^4k ~ e^{ikX} ~ \Theta(E) ~\delta^{(4)}(m^2 - E^2 + {\underline{p}}^2) ~ |k&gt;<br />
where here I've used capital "X" so we can remember that it's different from the
previous unphysical |x&gt; vectors. The above is equivalent to:

<br /> |X&gt; ~\sim~ F_x[\Theta(E)] ~*~ F_x[\delta^{(4)}(m^2 - E^2 + {\underline{p}}^2)] ~*~ |x&gt;<br />

where F_x[f(k)] denotes the (inverse) 4D Fourier transform of f(k),
and "*" denotes (4D) convolution in x-space.

Summary: the physically-meaningful position basis vectors are the |X>, and not the |x>.
Each |X> is a complicated convolution of the |x> vectors with all the forward lightcones
in x-space. From an x-space viewpoint, the |X>'s do indeed seem non-local, but that doesn't
matter because "non-locality in x space" means "non-locality in physically irrelevant x-space".
Only the |X> vectors have physical meaning. Indeed, Hilbert space inner products are only
defined between |X>-type vectors.

That's why it also doesn't matter that the Feynman propagator is non-zero outside the
lightcones in x-space. Our Hilbert space is restricted to relativistically correct states
on the mass hyperboloid, and it doesn't matter how things look in x-space. Only X-space
matters, but we almost never use the latter in calculations. Rather we mostly use the
|\underline{p}&gt; 3-momentum Hilbert space (hyperboloid for mass=m).[Hmm... Maybe the Newton-Wigner construction has some merit after all. :-)]

 

[Micha]

Good post, strangerep.

To repeat this in my own words, in ordinary QM, if we find a particle at x0, we would assign to it a wavefunction |x0> = delta(x-x0). Now, in QFT, delta(x-x0) is not a vector in Hilbert space and we are forced to choose |X0>. (Actually choosing a delta function is already an idealization in ordinary QM and we would choose a very narrow wavepackage.)
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.

 

[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