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

[Fra]

Here are some attempts to explain more... but keep in mind that I'm still working on it..so the comments serve to propagate ideas only.

Initial conditions of what? Wave functions? Particle positions? Something else?

Eventually this will be given a exact definition, but I mean Initial conditions of our information - prior estimates. The same set of data can in the general case be given several interpretations. We can invent concepts, like space and particles, charge... but that's just "lables".. what is a particle? I'd that that any support for that concept is in the data we have. Ie. I may be exposed to a stream of photons, propagating as nerve signals to my brain... now in principle, it's clear that the "picture" I created of the outside world is not ambigously derived from input, however in the context of live a fitness etc, one can probably consider some interpretations more or less useful, or to get back on track, to have various estimated probabilities to be successful. Against this is a subjective estimate.

The initial conditions is all our knowledge, and record of history - but constrained to the fact that our memories are limited, ultimately relating to our limited size and mass and so on. We would probably grow be be black holes if we could store every piece of information without reduction. This leads to the concept of data compression adn storage effiency, and here hte interpretation comes in. Which in turn is related to our interaction properties.

In my thinking, space, particles and other "abstractions" are emergent structures in my view. They are _selected_ as the (in context) expectedly most constructive/fit interpretations. In this sense interpretation can be thought of loosely as as choice of data reduction. The task is to reduce data storage, but loose a minimum amount of _significant_ information.

This can be linked to several interesting interpretations, black body radiation beeing a way to dispose of energy, and the distribution of the energ is what the emitter considers (subjectively) to be the least useful. But not necessarily lacking information comlpetely, this is not the same thing.

In my Bohmian proposal, probabilities emerge from our ignorance of initial particle positions, which are the quantities that are actually measured. As the probabilities are not fundamental in the Bohmian approach, the fact that an a priori relativistic probability density of particle positions is not well defined is not really a problem.

You think differently, so it's hard to see exactly, but in a certain sense perhaps your missing information of bohmian particle may be given some interpretation in my view. But I'm not sure I like the word particle though, or to assume a "shape" of what's missing. Anyway, I aim to start off at a lower level... I'm first of all trying to operatively define space in terms of correlations in a random walk... and the result of the dimensionality is nothing that can predicted from the formalism, it must come from real data... but the formalism should define the relation betwene input and prediction - the best induction. But at each stage there is uncertainty. And the result is also observer dependent. For example. I am not so sure that the simplest possible elementa we know, can GRASP the full dimensionality. How does an electron really percept reality? Of course we could never know, but that thinking is interesting can I think even in lack of a perfect answer, provide us to insight.

Anway, I hope to get back to comparing the QM equations once I've found out how to treat space and time better. My previous attempts did resemeble the bohmian formalism, but not the bohmian interpretation (of particles). I tried to consider relations connecting different probability spaces... and the phase seems to receive a special interpretation, as a way to bundle the ignorance, but this was too shaky and I stepped back again to revise the notion of space and time. Because I was uncomfortable talking about functions of space and time, before the whole issue of space and time is clearly defined from my first principles.

/Fredrik

 

[Fra]

Initial conditions of what? Wave functions? Particle positions? Something else?

Another clarification: In my (personal) rethinking here, there is at fundamental level, nothing I call wave functions. I start with "labels", and the concept of distinguishability. From there on I consider that an observer in order "to make comparasions" and perceive change, must at minimum, at least transiently somehow be able to store and compare the present with the nearest past. This gives rise to the notion of "change". Again, correlations in the observers memory (particle or system state if you like) can define things like distinguishable structure, but the distinguishability soon ends up beeing fuzzy... so you run into the concept of relative frequencies and estimated probabilites. So far there is nothing I call wave function. That is further up in the abstractions, or I think it's probably just an alternative mechanism... further on "changes" of patterns is observed, which indirectly makes dynamics observable. But not observable in the same meaning as in QM.

Eventually the system of changing but somewhat stable patterns, can be given names. And in this way I hope to understand the information theoretic connection in the hierarchy of things... from the most basic boolean observation, through emergent structures of space and dimensionality, and further structures that are really relations between the more primary patterns.

This way I hope to see the choherent line of reasoning I want. But it's a long way to go.

/Fredrik

 

[Anonym]

Both probabilistic and charge-current density behavior are extensively proven(and both must be explained by any theory describing the underlaying physics).

Please look at A.Tonomura et al, “Double –biprism electron interferometry”, Applied Physics Letters, 84(17), 3229 (2004); Fig. 3(b),(c),(d) and (a).

I see E. Schrödinger, Zs. Phys., 14,664 (1926) coherent wave packet. Absence of the relevant set-up parameters prevent the geometrical optics calculations to be sure. Please, provide your comment/explanations.

If I am right, it is impossible with M.Born statistical approach.

Regards, Dany.

 

[meopemuk]

That is exactly my point too. Both are proven experimentally. And both should be explained by a single coherent self-consistent theory. The problem is that we do not seem to have such a theory, or at least not a widely accepted one. Instead, we have TWO widely accepted theories (nonrelativistic QM and QFT) that we frequently mix in an incoherent manner.

What about Weinberg's approach in his volume 1? I think it is an ideal way to formulate QFT. The limit to non-relativistic QM and the probabilistic interpretation of wavefunctions are readily available. Quantum fields and wavefunctions are well-separated.

Eugene.

 

[Micha]

I

I did extensive numerical simulations of Klein Gordon propagation
(in many different spatial dimensions) and one never sees any
propagation outside the light cone. Also analytically one doesn't see
anything outside the light cone.

Hi Hans,
I also checked this and I think, Feynman and the textbooks are right.
The propagator 1/(p^2-m^2 + i*epsilon) (written in Fourier space)
is not strictly zero outside the lightcone, when written
in real space, although dropping of fast.

I checked this in 1+1 dimensional spacetime.
First I did the integral over energy by integration along contours.
Then I put t=0 (i.e. +0, because of the teta function in the result you have to decide).
Now I am left with a one dimensional integral and I can easily check its
value as a function of x numerically.

Of course, my analysis is very primitive, but therefore I can not see, where the error
should lie.

For example, I think, you can do better and solve the second integral analytically by integrating along the branch cut. This way you get a factor exp(-kx) in the remaining integral pointing to an exponential drop off as well.

I can only guess, but maybe the problem with your analysis is, that you work with the massless propagator, and this
gets singular.

 

[Hans de Vries]

Hi Hans,
I also checked this and I think, Feynman and the textbooks are right.
The propagator 1/(p^2-m^2 + i*epsilon) (written in Fourier space)
is not strictly zero outside the lightcone, when written
in real space, although dropping of fast.

There is a lot of confusion and disagreement between different physicist and
different textbook. Pauli's paper "Spin and Statistics" disagrees with Feynman
in the analytical result. Pauli then uses the commutation argument as well to
claim there is no propagation outside the lightcone. That is he literally, quote:
"postulates" this to be the case without giving the math...


Regards, Hans

 

[Micha]

There is a lot of confusion and disagreement between different physicist and
different textbook. Pauli's paper "Spin and Statistics" disagrees with Feynman
in the analytical result. Pauli then uses the commutation argument as well to
claim there is no propagation outside the lightcone. That is he literally, quote:
"postulates" this to be the case without giving the math...Regards, Hans

Solving the integral is a well defined and rather simple mathematical question.
We should be able to settle on the right answer without referring to
either history or commutation relations. With this for the moment I just mean the question,
whether the propagator is zero outside the light cone or not.
I claim, this question can be settled in 1+1 dimensions.
Of course a full analytical answer in 4d is even better, if it is correct.

 

[Hans de Vries]

I can only guess, but maybe the problem with your analysis is, that you work with the massless propagator, and this
gets singular.

Both the massive and massless case have poles in momentum space and they need
careful attention. In position space you do not get infinities for any t smaller than
infinity. Why? The reason for the infinities is the plane wave representation which
stretches from x = +infinity to x = -infinity. Contributions from farther and farther
away regions keep coming in and, at the pole frequency, they all add up. The result
is infinite at t=infinte.

Any physical process doesn't continue until t=infinite nor is infinite in size.

One can obtain the analytical Green's function with the series development:

\frac{1}{p^2-m^2}\ =\ \frac{1}{p^2}+\frac{m^2}{p^4}+\frac{m^4}{p^6}+\frac{m^6}{p^8}+...

The analytical results then coincide with the numerical simulations and there is
no propagation outside the light cone. There is no need to use commutation
arguments to preserve causality. Regards, Hans

 

[Hans de Vries]

One can obtain the analytical Green's function with the series development:

\frac{1}{p^2-m^2}\ =\ \frac{1}{p^2}+\frac{m^2}{p^4}+\frac{m^4}{p^6}+\frac{m^6}{p^8}+...

The analytical results then coincide with the numerical simulations

The procedure to solve this analytically is as follows:
1) Solve the 1+1d case for 1/p²

2) Extend this to the full series given above for the 1+1d case. The series
becomes a Bessel J function of order zero.

3) Extend this to any dimensional space using the "inter-dimensional operator"
The Bessel J function of order zero becomes one of first order in 3+1d space
and the total result is:\Theta(t) \left(\ \frac{1}{2\pi}\delta(s^2)\ + \frac{m}{4\pi s} \Theta(s^2)\ \mbox{\huge J}_1(ms)\ \right), \qquad \mbox{with:}\ \ \ s^2=t^2-x^2
Steps 1) and 3) can be found in my paper here:
http://chip-architect.com/physics/Higher_dimensional_EM_radiation.pdf

Step 1: See page 3: "IV Derivation of the propagators" ({\cal H}\equiv\Theta is Heaviside step-function)
Step 3: The ïnter dimensional operator is proved at page 5: section V.Now for step 2 I'll write up another post.Regards, Hans

 

[Hans de Vries]

Now for step 2 I'll write up another post.

First we need the series expansion for the Bessel function which you can find here:
http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/02/

In our case we need:

J_0(s)\ =\ \sum_{k=0}^\infty\ \frac{(-1)^k}{(k!)^2} \left( \frac{s}{2} \right)^{2k}

Written out:

J_0(s)\ =\ 1\ -\ \left( \frac{s}{2} \right)^2\ +\ \frac{1}{4}\left( \frac{s}{2} \right)^4\ -\ \frac{1}{36}\left( \frac{s}{2} \right)^6 ...Now we need to go back to page 3 of my paper "IV derivation of the propagators"
to get the higher order terms of the series.

The series in position space is:

\Box^{-1}\ \ -\ \ m^2\Box^{-2}\ \ +\ \<br /> m^4\Box^{-3}\ \ -\ \ m^6\Box^{-4}\ \ +\ \ ...

For each extra term the procedure is:

a) Integrate over the t=+r line.
b) Integrate over the t=-r line.
c) Multiply by -m²

We start with the first term which we know and which is a simple Heaviside step
function. The paper uses {\cal H} instead of \Theta for the Heaviside step function.
Now we get the series:

\Theta(s^2)J_0(ms)\ \ =\ \ \Theta(s^2) \left(\ 1\ -\ \left( \frac{ms}{2} \right)^2\ +\ \frac{1}{4}\left( \frac{ms}{2} \right)^4\ -\ \frac{1}{36}\left( \frac{ms}{2} \right)^6 ...\right)
Regards, Hans

 

[Hans de Vries]

For each extra term the procedure is:

a) Integrate over the t=+r line.
b) Integrate over the t=-r line.
c) Multiply by -m²

Use the fact that:

s^2\ =\ (t+r)(t-r)\ =\ u v

Keep the terms expressed in the u and v and:

a) Integrate over u.
b) Integrate over v.
c) Multiply by -m²

Where u and v are independent (orthogonal) coordinates.Regards, Hans

 

[Micha]

Your machinery is quite impressive.
Yet, I think, if it predicts no propagation outside the light cone,
there must be an error somewhere.

Let's focus just on the 1+1 dimensional case for that
to keep things simple, ok?

Forget my remark about the massless propagator being singular.
It is just that I focused on the case with finite rest mass so far,
although the massless propagator should be also finite and probably even
simpler to calculate.

Your geometrical series for the propagator as such is obviously right.
I didn't follow your calculation from there so far.
But did you ever do the calculation in the way suggested by Zee in his
Quantum field theory in a nutshell?
You mentioned this book yourself in your earlier post.
The integration along contours for the integral over energy
is rather straightforward and the result can be found in the book.
The calculation is done for the 3+1d case, but you can copy this
step for the 1+1d case.

And then there is only a single integral left, which you can
evaluate for t=0 and x finite numerically.

 

[Hans de Vries]

Your geometrical series for the propagator as such is obviously right.
I didn't follow your calculation from there so far.

\Box^{-1}\ - \ m^2\Box^{-2}\ +\ <br /> m^4\Box^{-3}\ - \ m^6\Box^{-4}\ + \ ...

Using the series is sufficient to guarantee causality and SR because
none of the terms of this series has propagation outside the light cone.
\Box^{-1} is the photon propagator on the light cone. \Box^{-2} is a re-emission,
which is again on the light-cone, the third time is the second re-emission
and so on, all on the light-cone...


Regards, Hans

 

[Hans de Vries]

But did you ever do the calculation in the way suggested by Zee in his
Quantum field theory in a nutshell?
You mentioned this book yourself in your earlier post.
The integration along contours for the integral over energy
is rather straightforward and the result can be found in the book.
The calculation is done for the 3+1d case, but you can copy this
step for the 1+1d case.

You can find the source of all this in chapter 17 and 18 of Feynman's:

"The theory of fundamental processes"

It's is a non-physical artifact of perturbation theory where there are
pairs of diagrams which have a physical meaning. Independently
they have no physical meaning because they can be converted into
each other via a Lorentz transform, and only together they respect
causality and special relativity.

Now the term "non-physical" here is mine because Feynman himself,
In the spirit of his mentor John Archibald Wheeler (The good man is still
with us) had no problem with particles going not only forward but also
backward in time, or going at any speed faster than c.

Feynman cuts the propagator in two with his pole prescription which
leads to these pair of diagram's which have to be considered together
to get both poles back and special relativity restored.


Regards, Hans

 

[OOO]

It is frightening (or should I say enlightening) to see what a considerable amount of dissent can be generated by seemingly "very simple QFT questions". I didn't understand neither P&S' argument nor any other I have found in textbooks. It's also quite remarkable how much all the reputable authors avoid making definite observable statements for which they may be held responsible.

I have also done numerical simulations of KG and it seems quite obvious to me that there is no propagation outside the light cone. If one discretizes the d'Alembertian by finite differences of second order (for stability analysis see e.g. Numerical Recipes), one gets something like (in 1+1 dimension)

\Psi(t+1,x)+\Psi(t-1,x)-2\Psi(t,x) - (\Psi(t,x+1)+\Psi(t,x-1)-2\Psi(t,x)) = -m^2 \Psi(t,x)

If one solves for the next timestep one gets

\Psi(t+1,x) = -\Psi(t-1,x)+ \Psi(t,x+1)+\Psi(t,x-1) -m^2 \Psi(t,x)

The maximum speed with which information can propagate through this lattice is 1 (the speed of light in lattice units). You see this immediately from the above equation: the next time step is only influenced by the two adjacent lattice sites (and the one in the middle of course).

Of course it is not a strict proof but I think very convincing and very graphic (if you've seen the sim).

 

[Micha]

It's is a non-physical artifact of perturbation theory where there are
pairs of diagrams which have a physical meaning. Independently
they have no physical meaning because they can be converted into
each other via a Lorentz transform, and only together they respect
causality and special relativity.
Regards, Hans

The term "propagation outside the lightcone" seems to be loaded with mystery.
So let's avoid it and talk about the Fourier transform of the function
1/(p^2-m^2)

We are just talking about the Green's function of the classical Klein-Gordon
equation. Why would we need pertubation theory for this?

Making a distinction of cases for t>0 and t<0 in solving the integral over energy
is a completely valid mathematical step. If you see any problems, please
indicate them on mathematical grounds.

@OOO
Interesting argument.
Do you think, it will hold for an infinite Dirac delta peek?

 

[Micha]

If 1+1 dimensions is too hard to do, why not going to 1 dimension for a while.
I claim the propagator here is
1/(2m)*exp(-m*abs(x)).

I think this also indicates, that the massless propagator really gets singular here.

I should mention, that I talk about the Fourier transform of the function 1/(p^2+m^2) here. This
means, I talk about one spacelike dimension here. You can go to the timelike case
by substituting m=i*m. This means, you get oscillatory behaviour in a timelike dimension,
and exponential decay in a spacelike dimension.

 

[Micha]

@OOO
Another thought:
You might as well solve your equation for
Psi(x+1) and then you get a spacelike correlation instead
of a timelike.

 

[OOO]

Interesting argument.
Do you think, it will hold for an infinite Dirac delta peek?

You can't do an infinite delta peak in lattice numerics. But as an approximation you can use boundary conditions where Psi does not vanish at exactly one site (with finite but arbitrarily large value however; this doesn't really matter since the KG equation is linear).

What you get then is two peaks running left and right (1+1 dimension) or a circular wavefront (1+2 dimension). If there's dispersion (m != 0) then there are oscillations lagging behind the wavefront(s), but not before it (which would be outside the light cone). In the case of the massless equation m=0 (e.g. photons, apart from polarization) then there are no such oscillations (within numerical accuracy), one gets a sharp "retarded propagator".

So yes, this also holds for an approximate delta peak, and that's really the test case, which is why you mentioned it.

 

[OOO]

@OOO
Another thought:
You might as well solve your equation for
Psi(x+1) and then you get a spacelike correlation instead
of a timelike.

What for ?

 

[Micha]

What for ?

You are solving for Psi(t+1), because you know, that t is the time variable.
On mathematical grounds, x and t are just symbols, so why not solving
for Psi(x+1) and see a propagation in space?

I think, that you get only timelike propagation and not spacelike, because
you are already putting this in as an assumption..

 

[OOO]

You are solving for Psi(t+1), because you know, that t is the time variable.
On mathematical grounds, x and t are just symbols, so why not solving
for Psi(x+1) and see a propagation in space?

I think, that you get only timelike propagation and not spacelike, because
you are already putting this in as an assumption..

I think I don't know what you mean. The difference between space and time is not just one between mathematical symbols. It's a difference in sign ! You may, of course, try to solve the equations that way. But, to make a long story short, this scheme becomes unstable. You actually said it in one of your posts above: exchanging t<->x amounts to m<->i*m and this exchanges oscillatory solutions with exponentially damped and (more importantly !) excited ones. You'd have to select your boundary conditions very carefully in order to have only damped solutions.

Anyway, this does only make sense from a mathematical POV.

 

[reilly]

Hans -- your discussion in post 5 is one of the best I've ever read.

In my usual simple minded way, I'll suggest that the standard retarded Green's Function, like in E&M, prohibits propagation outside the light cone almost by definition -- it all has to do with those tricky i epsilons and contours of integration -- beloved by Electrical Engineers.

A personal note: a few years ago I became interested in the spreading of wave packets for the Dirac Equation. Initially I thought that the edges of the packet spread more quickly than Special Relativity allowed. But after extensive calculations, I rediscovered the retarded propagator and decided that my initial hunch was incorrect, which it is.
Regards, Reilly Atkinson

 

[Micha]

I think I don't know what you mean. The difference between space and time is not just one between mathematical symbols. It's a difference in sign ! You may, of course, try to solve the equations that way. But, to make a long story short, this scheme becomes unstable. You actually said it in one of your posts above: exchanging t<->x amounts to m<->i*m and this exchanges oscillatory solutions with exponentially damped and (more importantly !) excited ones. You'd have to select your initial conditions very carefully in order to have only damped solutions.

Anyway, this does only make sense from a mathematical POV.

Yes, I know about the sign difference and I know there is no propagation in space, but
only exponential damping. My whole point is, there is correlation along the spacelike dimension which accounts for the exponential decay. Remember, the discussion is,
whether we have exponential decay in the spacelike dimension or if the propagtor is exactly zero for spacelike dimensions.

 

[OOO]

Yes, I know about the sign difference and I know there is no propagation in space, but
only exponential damping. My whole point is, there is correlation along the spacelike dimension which accounts for the exponential decay. Remember, the discussion is,
whether we have exponential decay in the spacelike dimension or if the propagtor is exactly zero for spacelike dimensions.

Hmm, still not sure what you mean. If there was correlation along spacelike separation then you couldn't specify initial conditions. But you can. Choose any Psi(t,x) and Psi(t-1,x) for all x, that you like, and you will get perfectly definite time evolution.

 

[Micha]

Hmm, still not sure what you mean.

Me neither. Maybe correlation is even the wrong word. After all we know, that wave packets and information can only travel inside the light cone, right?

Remember, I was only claiming, that the Fourier transform of the function 1/(p^2-m^2) in two dimensions with Minkowski metric is not exactly zero at (x,0).
You help me please on why you think, you can proof me (and Feynman and the textbooks) wrong without doing the integral, or if I am right, what this exactly means for the time evolution of the KG equation.

 

[Anonym]

Hans -- your discussion in post 5 is one of the best I've ever read.

I am confused. Do you mean post #6 in this session by Hans de Vries?

a few years ago I became interested in the spreading of wave packets for the Dirac Equation. Initially I thought that the edges of the packet spread more quickly than Special Relativity allowed. But after extensive calculations, I rediscovered the retarded propagator and decided that my initial hunch was incorrect, which it is.

Provided you mean post #6, Hans de Vries discuss KG and not Dirac. Please reproduce the relevant summary of your calculations that allowed your decision. Notice that it means that V.A. Fock was wrong.

Regards, Dany.

 

[OOO]

Me neither. Maybe correlation is even the wrong word. After all we know, that wave packets and information can only travel inside the light cone, right?

Remember, I was only claiming, that the Fourier transform of the function 1/(p^2-m^2) in two dimensions with Minkowski metric is not exactly zero at (x,0).
You help me please on why you think, you can proof me (and Feynman and the textbooks) wrong without doing the integral, or if I am right, what this exactly means for the time evolution of the KG equation.

I didn't say anything about calculating the propagator in position representation from its momentum representation, did I ? If you feel the urge to do that, I won't stop you, but I won't try to find your respective mistakes either.

I did say that, on lattice-discretizing the KG equation, the impulse response, which is nothing but the retarded propagator of the difference equations, shows no sign of having values outside the cone.

Of course, we could go on and on and on, talking about different things, but I was thinking that you referred to what I was saying. If this is not the case then I apologize for that misconception.

 

[Micha]

I didn't say anything about calculating the propagator in position representation from its momentum representation, did I ? If you feel the urge to do that, I won't stop you, but I won't try to find your respective mistakes either.

I did say that, on lattice-discretizing the KG equation, the impulse response, which is nothing but the retarded propagator of the difference equations, shows no sign of having values outside the cone.

Of course, we could go on and on and on, talking about different things, but I was thinking that you referred to what I was saying. If this is not the case then I apologize for that misconception.

Ok, let me explain, where I am coming from.

I am trying to learn quantum field theory by the book of Zee.

Meanwhile I read post #6 from Hans, which says, that the propagator of KG in spacetime representation is exactly zero for spacelike intervals. This contracticts the book of Zee, which Hans himself mentions in his discussion.
So I tried to check the issue myself and now I am convinced, that Zee is right and Hans is wrong. If this is true, I think this might be interesting for other people as well, who are reading this thread.

Now you came in saying
"I have also done numerical simulations of KG and it seems quite obvious to me that there is no propagation outside the light cone."
So I thought you were supporting Hans result. I guess this is just a misunderstanding
about mixing up the two statements

1. "The propagator in spacetime representation is strictly zero outside the light cone."

and

2. "There is no propagation outside the lightcone."

I now think, you mean, that no wave packets can travel outside the light cone, and if so, I think, you are completely right.

To sum up, I think, that 1 is wrong and 2 is right.

 

[OOO]

I am trying to learn quantum field theory by the book of Zee.

I love that extraordinary book. Very much fun to read.

So I thought you were supporting Hans result. I guess this is just a misunderstanding
about mixing up the two statements

"The propagator in spacetime representation is zero outside the light cone."

and

"There is no propagation outside the lightcone."

I now think, you mean, that no wave packets can travel outside the light cone, and if so, I think, you are completely right.

Indeed I don't know the difference between these two statements. So this must be the problem. What I was trying to say is the first statement (derived from numerical evidence). So how can there be propagation outside the lightcone when the propagator is zero there ?

Edit: As you say, you think 1 is wrong, I say numerical simulation gives no indication for that. Thus, actually, I wanted to support what Hans said. I didn't claim anything about wave packets or group velocity, but about the maximum speed of interactions represented by the behaviour of the wavefront of the KG propagator.