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

[nrqed]

I have some extremely basic questions in QFT.


First, P&S discuss causality in QFT in the first chapter of the book and, after showing that $<0| \phi(x)\phi(y)|0>$ does not vanish for spacelike intervals, they say "to really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals but whether a measurement performed at one point..."

What do they mean by this? I have my own interpretation but it's nontrivial and it may be wrong (they mention that so casually that it seems obvious to them). Of course, classically, it would make no sense to say that a a particle could propagate over spacelike intervals. In the quantum (and relativistic) world, the only way for me to make sense of what they are saying is that measuring the position of a particle precisely involves energies that necessarily lead to the creation of more particles than the ones already present in the system. Therefore, the very idea of checking whether a single particle that was at a point "x" at t=0 is now located at a point "y" at a time t is impossible in QFT.

Second, how is a measurement actually defined in QFT?
In NRQM, it's pretty clear. For a given hermitian operator, one finds it's eigenvalues and eigenstates, and so on. How is this defined in QFT?
For example, consider the field $\phi(x)$ itself. Now, I always thought that this is not in itself an observable, so it does not make sense to talk about *measuring* phi. And yet, P&S talk about measurements of phi in the first chapter. Is that an abuse of language?

One problem I find with QFT books is that there is no effort devoted to making the connection with NRQM. This is strange, it's the equivalent of teaching GR and never talking about how one may recover Newton's gravitation. One should be able to treat a simple sysytem (let's say the infinite square well!) and show in what way one may recover the NRQM result, within some limit! Does anyone know of a book that does that type of connection?

Another question is about the classical limit of QED. People mention that a coherent state of photons correspond to the classical limit of classical E&M. But how does that work, exactly? I know that one can then replace the creation and annihilation photon operators by their expectation values, but the field obtained is still imaginary so it's not the classical field. There has to be much more work (with many subtleties involved, no doubt) before connecting with the classical treatment of E&M.

 

[Demystifier]

One problem I find with QFT books is that there is no effort devoted to making the connection with NRQM. This is strange, it's the equivalent of teaching GR and never talking about how one may recover Newton's gravitation. One should be able to treat a simple sysytem (let's say the infinite square well!) and show in what way one may recover the NRQM result, within some limit! Does anyone know of a book that does that type of connection?

Excellent point!
I discuss this problem in a somewhat different context in Sec. VIII of
http://arxiv.org/abs/quant-ph/0609163
Sec. IX is also relevant for that issue.
I also propose a solution of this problem in
http://arxiv.org/abs/quant-ph/0406173

 

[jostpuur]

I started studying QFT from P&S too, and got stuck with nearly the same things. It's starting to look that there aren't very good answers to these questions, although I'm waiting replies to this thread still hopefully.

Instead of answering, I'll throw another question about the basics of QFT (or about relativistic theory in general). Peskin & Shroeder explain in the beginning of their book, that a propagator

$$K(x,y,T)=\int\frac{d^3p}{(2\pi)^3}e^{-i(E_{\boldsymbol{p}}T-\boldsymbol{p}\cdot(\boldsymbol{x}-\boldsymbol{y}))}$$

cannot be used, because it violates causality. Really so? If I assume a wavefunction (like in nonrelativistic theory) $$\phi(t,x)$$ to have a time evolution defined with

$$\phi(t+T,x)=\int d^3y\; K(x,y,T)\phi(t,y)$$

a brief calculation then shows that the wavefunction satisfies the KG-equation, which is Lorentz invariant. Doesn't this mean, that this propagator gives Lorentz invariant time evolution? And how could Lorentz invariant time evolution violate causality?

Besides, the integral in the propagator does not converge, but it merely behaves as a distribution when used correctly. Hence it doesn't look very smart to simply estimate if it looks zero or nonzero outside the lightcone.

My point is, that not only is the explanation on how the causality is conserved, with the measurement interpretation, confusing, but so is also the explanation on why another propagator would instead violate causality.

 

[Careful]

I started studying QFT from P&S too, and got stuck with nearly the same things. It's starting to look that there aren't very good answers to these questions, although I'm waiting replies to this thread still hopefully.

Instead of answering, I'll throw another question about the basics of QFT (or about relativistic theory in general). Peskin & Shroeder explain in the beginning of their book, that a propagator

$$K(x,y,T)=\int\frac{d^3p}{(2\pi)^3}e^{-i(E_{\boldsymbol{p}}T-\boldsymbol{p}\cdot(\boldsymbol{x}-\boldsymbol{y}))}$$

cannot be used, because it violates causality. Really so? If I assume a wavefunction (like in nonrelativistic theory) $$\phi(t,x)$$ to have a time evolution defined with

$$\phi(t+T,x)=\int d^3y\; K(x,y,T)\phi(t,y)$$

a brief calculation then shows that the wavefunction satisfies the KG-equation, which is Lorentz invariant. Doesn't this mean, that this propagator gives Lorentz invariant time evolution? And how could Lorentz invariant time evolution violate causality?

Lorentz invariance does not distinguish the past from the future, for instance in Maxwell theory you have to select the retarded propagator. The problem with locality for scalar fields stems from insisting that your solution contains positive energies only (the propagator is purely imaginary outside the lightcone). Allowing for negative energies can restore locality (but one believes such theories to be unstable once interactions are turned on).

 

[jostpuur]

I just couple of minutes ago happened to hit the url http://www.physics.ucsb.edu/~mark/qft.html on these forums, started reading it, and noticed that Srednicki explains neccecity of commutator vanishing outside the lightcone quite differently. I haven't understood it myself yet, but it certainly looks worth cheking out. On the page 46 of the pdf.

 

[Hans de Vries]

I have some extremely basic questions in QFT.

First, P&S discuss causality in QFT in the first chapter of the book and, after showing that $<0| \phi(x)\phi(y)|0>$ does not vanish for spacelike intervals, they say "to really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals but whether a measurement performed at one point..."

It's basic yes, but apparently no so simple...

Different QFT books treat this subject quite differently (Zee, Feynman,
P&S) and with different results.

A thorough treatment should handle this entirely analytically instead of
using approximations as done in most textbook. checked with numerical
simulations.

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.

The concise mathematical expression for the Green's function in 3+1
dimensions, for forward propagation is:

$$\Theta(t) \left(\ \frac{1}{2\pi}\delta(s^2)\ + \frac{m}{4\pi s} \Theta(s^2)\ \mbox{\huge J}_1(ms)\ \right), \qquad \mbox{with:}\ \ \ s^2=t^2-x^2$$

Where Theta is the Heaviside step function and J1 is the Bessel J function
of the first order. The Theta at the left selects the forward propagating
half while the other cuts off any propagation outside the light cone.

Analytically, the Bessel function goes imaginary outside the lightcone
and this is generally what becomes the part "outside the light cone" in
the form of the Bessel I and Bessel K functions which become

$$\mbox{\huge I}_1(mx)\ \rightarrow\ \frac{1}{\sqrt{2\pi mx}}\ \ e^{mx}, \ \ \ \ \mbox{\huge K}_1(mx)\ \rightarrow\ \sqrt{\frac{\pi}{2 mx}}\ \ e^{-mx}$$

for large x, typically K1 becomes exp(-mx) which is then usually given as
the part outside the light cone. However, the concise derivation of the
Green's function does produce the Heaviside step function which eliminates
the propagation outside the lightcone. (Note that in the limit of m=0 the
propagation outside the lightcone would become infinite!)

One can find many variations of the analytical expression of the Klein
Gordon propagator given above, sometimes with a negative sign for
the Dirac delta function, which is wrong since this part becomes the
photon propagator in the limit case where where m goes to zero, and
should be positive. Sometimes one sees a different normalization factor.
Also the Bessel function changes from text to text.

Feynman, in 1948, with paper and pencil as the only tools to calculate
(!) plus mathematical table books came to:

$$-\frac{1}{4\pi}\delta(s^2)\ + \frac{m}{8\pi s} \mbox{\huge H}_1^{(2)}(ms)\ \right), \qquad \mbox{with:}\ \ \mbox{\huge H}_1^{(2)}(ms)= \mbox{\huge J}_1(ms)-i\mbox{\huge Y}_1(ms)$$

There's the sign, a factor 1/2 and the Hankel function obtained from
the tables which is the Bessel equivalent of exp(ix)=cos(x)+isin(x).
This is where the "propagation outside the lightcone" started historically:

http://chip-architect.com/physics/KG_propagator_Feynman.jpgAnother very popular (modern) textbook (Zee) handles it in I.3
formula (23). This is a also a hand waving approximation leading to
the exp(-mr) light cone leaking.

P&S then use a particular argument with particles and anti particles
which would cancel out each others propagation outside the lightcone.
to restore causality. (In chapter 2.4) This after they get the exp(-mr)
term from a similar approximation.The simplest way in which you can convince yourself that there is no
propagation outside the lightcone is by expanding like this:

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

The Fourier transform of this series leads to a series representing
the Bessel J function. The first term is the massless propagator which
is strictly on the light cone only. It's Fourier transform is the Dirac
function in the space-time version of the propagator.

The second term represent a massless propagator acting on a
massless propagator, thus the propagation on the light cone
becomes a source itself which is again propagated on the lightcone,
and so on.

Thus: None of the terms in the series has any propagation outside
the light cone, and neither does the sum of the geometric series,
The Klein Gordon propagator.Regards, Hans.
PS: related stuff:
http://functions.wolfram.com/PDF/BesselJ.pdf (also has the KG propagator)
http://en.wikipedia.org/wiki/Bessel_function
http://www.chip-architect.com/physics/Higher_dimensional_EM_radiation.pdf

With the latter paper and the series expansion you can derive the KG
propagator in any dimension.

 

[Careful]

I just couple of minutes ago happened to hit the url http://www.physics.ucsb.edu/~mark/qft.html on these forums, started reading it, and noticed that Srednicki explains neccecity of commutator vanishing outside the lightcone quite differently. I haven't understood it myself yet, but it certainly looks worth cheking out. On the page 46 of the pdf.

This argument is equivalent to demanding that spacelike separated measurements do not influence each other *statistically* (on the level of single events, this is not true - see the EPR paradox). Of course, it is perfectly legitimate to object that statistical assertions for one instant of time seem to contradict the very definition of statistics : quantum physicists interpret this again in terms of unrealized potentialities. In other words a phantom world which we will never observe.

 

[sheaf]

Second, how is a measurement actually defined in QFT?
In NRQM, it's pretty clear. For a given hermitian operator, one finds it's eigenvalues and eigenstates, and so on. How is this defined in QFT?
For example, consider the field $\phi(x)$ itself. Now, I always thought that this is not in itself an observable, so it does not make sense to talk about *measuring* phi. And yet, P&S talk about measurements of phi in the first chapter. Is that an abuse of language?

Looking at this part of the question, I was very confused by Peskin and Schroder too. I found a brief discussion in Bjorken and Drell volume 2 section 12.3 entitled "Measurability of the Field and Microscopic Causality". The last paragraph states

"In order to associate any physical content with this mathematical result , we must assume that it makes sense to attach physical meaning to the measurement of a field strength at at point, a concept already criticized in earlier paragraphs".

So I think we are in good company when we are confused ! (B and J then quote a paper by Bohr and Rosenfeld which I don't have access too, but it sounds as if it might be quite useful - Phys Rev 78 p794 (1950) )

 

[Careful]

Looking at this part of the question, I was very confused by Peskin and Schroder too. I found a brief discussion in Bjorken and Drell volume 2 section 12.3 entitled "Measurability of the Field and Microscopic Causality". The last paragraph states

"In order to associate any physical content with this mathematical result , we must assume that it makes sense to attach physical meaning to the measurement of a field strength at at point, a concept already criticized in earlier paragraphs".

So I think we are in good company when we are confused ! (B and J then quote a paper by Bohr and Rosenfeld which I don't have access too, but it sounds as if it might be quite useful - Phys Rev 78 p794 (1950) )

IMO, it is ok to use a UV cutoff (which renders the field operator at a point well defined). At extremely high energies, we need new physics anyway.

 

[Son Goku]

I was confused about the same thing after reading Peskin and Schroeder. I found a good explanation (at least I found it helpful) on the bottom Page 198 of Weinberg's "The Quantum Theory of Fields, Volume 1". Although you need to following the arguements Weinberg had been making in the first four chapters, he basically says it is best to think of the causality condition as something which is needed for the S-matrix to be Lorentz invariant, rather than thinking of it terms of measurements of field values at different points.

 

[nrqed]

I just couple of minutes ago happened to hit the url http://www.physics.ucsb.edu/~mark/qft.html on these forums, started reading it, and noticed that Srednicki explains neccecity of commutator vanishing outside the lightcone quite differently. I haven't understood it myself yet, but it certainly looks worth cheking out. On the page 46 of the pdf.

Thanks for the link! This is a fantastic book!
It is refreshing to see a QFT book that does not feel like a simple repeat of the same presentation again and again. It's clear that the author spent time thinking about presenting things from scratch and in a logical way.
I especially dislike the conventional presentation which starts with the non sequitur that one must quantize classical fields (even if the fields on starts with have no classical correspondence at all, like the Dirac field!).
It's only when I read Weinberg that I found that finally there was a textbook presenting QFT in a logical way, with the starting point that one must allow the number of particles to vary so one builds a Fock space and then it is the requirement of Lorentz invariance that leads to the need of introducing fields!
This book is closer in spirit to Weinberg but more transparent (who is very good but, let's admit it, quite heavy to follow sometimes)

 

[strangerep]

I just couple of minutes ago happened to hit the url http://www.physics.ucsb.edu/~mark/qft.html [...]

I downloaded the pdf, but the book appears to have no index!?

Can anyone tell me whether the published version has a decent index?
(Amazon doesn't have any "look inside" images, so I can't find out
that way.)

 

[meopemuk]

Great thread! It is refreshing to see so many people having the same questions that I had. It is sad that QFT textbooks did such a poor job in addressing these questions.

Of course, classically, it would make no sense to say that a a particle could propagate over spacelike intervals.

This is true.

In the quantum (and relativistic) world, the only way for me to make sense of what they are saying is that measuring the position of a particle precisely involves energies that necessarily lead to the creation of more particles than the ones already present in the system. Therefore, the very idea of checking whether a single particle that was at a point "x" at t=0 is now located at a point "y" at a time t is impossible in QFT.

This statement "localize one particle -> create more particles" is often repeated in QFT textbooks. However, I don't find it very convincing. It is true that by an accurate determination of position we increase the uncertainty of the momentum and energy of the particle. However, this does not mean that we increase the uncertainty of the *number of particles*. A relativistic (Newton-Wigner) position operator can be defined in QFT, and this operator commutes with the particle number operator. So, one can have a well-defined position of a single particle.

Then what about superluminal propagation? It seems to be well-established that a wave function of a localized particle spreads out faster than the speed of light. And the usual wisdom says that this contradicts causality, because from the point of view of a moving observer the events of particle creation and absorption would change their time order.

G. C. Hegerfeldt, "Instantaneous spreading and Einstein causality in quantum theory", Ann. Phys. (Leipzig), 7, (1998) 716.


Below I will try to explain that the conclusion about violation of causality could be premature. Instantaneous spreading and causality do not necessarily contradict each other. There are two key points in my analysis. First is that "instantaneous spreading" refers to the wavefunction, and wavefunctions must be interpreted probabilistically. The second point is that particle localization is relative. If observer at rest sees the particle as localized, then the moving observer sees that particle's wave function is spread out over entire space.

F. Strocchi, "Relativistic quantum mechanics and field theory", http://www.arxiv.org/hep-th/0401143

First take the point of view of an observer at rest O. This observer prepares particle localized at point A at t=0. At time instant t>0 he sees that the wave function has spread out superluminally. This means that the probability of finding the particle at a point B, whose distance from A is greater than ct, is non-zero. So far there is no contradiction.

Now take the point of view of observer O', which moves with a high speed relative to O. As I said above, at time t=0 (by his own clock) observer O would see particle's wave function as spread out in space. There will be a maximum at point A. But there will be also a non-zero probability of finding the particle at point B. At later times the wave function will spread out even more. However, it is important that observer O' cannot definitely say that he sees the particle as propagating from B to A. He is seeing some diffuse probability distributions at all times, from which it is impossible to say exactly what is the speed and direction of particle's propagation.

So, the situation in quantum mechanics is quite different from the classical mechanics, where propagation direction and speed always have a well-defined meaning, and causality is not compatible with superluminal propagation.

 

[Demystifier]

A relativistic (Newton-Wigner) position operator can be defined in QFT, and this operator commutes with the particle number operator. So, one can have a well-defined position of a single particle.

The problem with this position operator is that it is not really relativistic covariant, but requires a preferred Lorentz frame.

 

[meopemuk]

The problem with this position operator is that it is not really relativistic covariant, but requires a preferred Lorentz frame.

Demystifier: I often hear this opinion about Newton-Wigner position operator. However, I don't know where it comes from. Could you please explain? It seems to me that Newton and Wigner introduced this operator as an explicitly relativistic thing

T. D. Newton and E. P. Wigner, "Localized states for elementary systems",
Rev. Mod. Phys., 21, (1949) 400.

 

[Hans de Vries]

It seems to be well-established that a wave function of a localized particle spreads out faster than the speed of light.

No such superluminal propagation of the wave function shows up neither
with a concise analytical treatment giving exact solutions in configuration
space, nor with extensive numerical simulations. See my old post above.

The simplest way to convince oneself may be this series development
of the Klein Gordon propagator:

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

Which becomes the following operator in configuration space:

$$\Box^{-1}\ \ -\ \ m^2\Box^{-2}\ \ +\ \ m^4\Box^{-3}\ \ -\ \ m^6\Box^{-4}\ \ +\ \ ...$$

Where $$\Box^{-1}$$ is the inverse d'Alembertian, which spreads the wave function
out on the lightcone as if it was a massless field. The second term then
retransmits it, opposing the original effect, again purely on the light cone.
The third term is the second retransmission, et-cetera, ad-infinitum.

All propagators in this series are on the lightcone. The wave function does
spread within the light cone because of the retransmission, but it does
never spread outside the light cone, with superluminal speed.Regards, Hans.

 

[meopemuk]

No such superluminal propagation of the wave function shows up neither
with a concise analytical treatment giving exact solutions in configuration
space, nor with extensive numerical simulations. See my old post above.

The simplest way to convince oneself may be this series development
of the Klein Gordon propagator:

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

Which becomes the following operator in configuration space:

$$\Box^{-1}\ \ -\ \ m^2\Box^{-2}\ \ +\ \ m^4\Box^{-3}\ \ -\ \ m^6\Box^{-4}\ \ +\ \ ...$$

Where $$\Box^{-1}$$ is the inverse d'Alembertian, which spreads the wave function
out on the lightcone as if it was a massless field. The second term then
retransmits it, opposing the original effect, again purely on the light cone.
The third term is the second retransmission, et-cetera, ad-infinitum.

All propagators in this series are on the lightcone. The wave function does
spread within the light cone because of the retransmission, but it does
never spread outside the light cone, with superluminal speed.


Regards, Hans.

Your conclusion directly contradicts results of Hegerfeldt and many other authors. Did you try to figure out what is wrong with their arguments?

 

[Hans de Vries]

Your conclusion directly contradicts results of Hegerfeldt and many other authors. Did you try to figure out what is wrong with their arguments?

Sure, see for instance this post here:

https://www.physicsforums.com/showpost.php?p=1278078&postcount=6

A concise mathematical treatment cancels everything outside the lightcone
with a Heaviside step function. Many textbook use approximations, often
quite different, missing out on this.

I derived the propagators for the Klein Gordon equation in any dimension
in configuration space analytically, and also did extensive numerical
simulations to check them. You may find some of this work here:

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

Although it mainly describes massless propagation in any dimension, (which
is interesting enough) It's still on my "to do" list to include more on the
massive propagators in this paper. Regards, Hans.

 

[jostpuur]

I did not fully understand Hans de Vries's post, but here's one point: Does values of function $$f(x)$$ at $$x\neq 0$$ have any effect on the quantity

$$\lim_{L\to\infty}\int_{-\infty}^{\infty}\frac{1}{\pi x}\sin(Lx) f(x)dx ?$$

One might think that they do have some effect, because

$$\lim_{L\to\infty} \frac{1}{\pi x}\sin(Lx) \neq 0$$

for all x and particularly for $$x\neq 0$$, but such conclusion is wrong, because still

$$\lim_{L\to\infty}\frac{1}{\pi x}\sin(Lx) = \delta(x)$$

Now what Peskin & Shroeder do in their book is, that they study the propagator outside the light cone with some approximations, note that it is not zero, and conclude that causality is violated. I dare to say that that is a false conclusion. You cannot find out where contribution comes in certain integrals by merely checking if something is zero or not.

Your conclusion directly contradicts results of Hegerfeldt and many other authors. Did you try to figure out what is wrong with their arguments?

I'm not sure what Hegerfeldt is saying, but if P&S belongs to that "other authors", then this was my response to this matter.

EDIT:
I got a feeling that this has more to do with the propagator in which $$1/(2E_p)$$ factor is missing.

MORE EDIT:
Anyway, I feel this point I made here is important, but since I am a bit lost with these propagators, I started another thread with a topic "what is propagator", where I took the role of the one who is asking questions. If you think I'm not doing things right with this $$1/(2E_p)$$ factor, feel free to explain me there.

 

[Hans de Vries]

Here is a confirmation on the Wolfram function side which has a page
giving the Green function of the Klein Gordon equation in 3+1 D:

http://functions.wolfram.com/BesselAiryStruveFunctions/BesselJ/31/02/

Now look at the last term, containing the Bessel function and see how it's
canceled outside the light cone by the Heaviside step function $$\Theta(t-|x|)$$

Regards, Hans.

 

[Hans de Vries]

EDIT:
I got a feeling that this has more to do with the propagator in which $$1/(2E_p)$$ factor is missing.

If one does a Fourier transform of the propagator $$1/(p^2-m^2)$$ via an
intermediate step: First only the 1D Fourier transform energy-to-time,
then one gets an expression as at the start of P&S (2.54) on page 29.

Now compare (2.54) with (2.50), the latter is the one which leads them to
the claim of violation of causality. You see that one of the two terms in
(2.54) is missing in (2.50)

They proceed by stating that causality is conserved by including both
particles and anti-particles in the theory, where the anti particles now
give the second (missing) term in (2.50) to arrive at (2.54). This then
would restore causality back again.

They give a rather complicated interpretation: The antiparticle going from
B to A should cancel the effects from the particle going from A to B.

Of course, In the path integral approach the Klein Gordon propagator is used
as is, and is in no way split in two parts as is done in P&S in the context of
canonical quantization of the real and complex Klein Gordon fields.You can find P&S (2.54) back in Zee, chapter I3 equation (23) on page 23.
Here however, this formula, by using an approximation, is used to claim
propagation outside the lightcone, which thereby would contradict P&S...Regards, Hans.

 

[meopemuk]

Hans, jostpuur,

I am not sure that QFT propagators give correct description of particle probability amplitudes evolving in the position space. P&S presume that propagators can do that, but they never prove it. Propagators are good for calculations of S-matrix elements in perturbation theory, but in these calculations their dependence on x gets integrated out.

I have an (old-fashioned) idea that in order to describe the position-space wave function one first needs to define the position operator. In relativistic quantum theory this is the Newton-Wigner position operator. Superluminal spreading of Newton-Wigner eigenfunctions is discussed in section 10.1 of http://www.arxiv.org/physics/0504062

Hegerfeldt's discussion is even more general as he doesn't specify the position operator explicitly. You can find his article on the web http://www.arxiv.org/quant-ph/9809030

 

[Demystifier]

Demystifier: I often hear this opinion about Newton-Wigner position operator. However, I don't know where it comes from. Could you please explain? It seems to me that Newton and Wigner introduced this operator as an explicitly relativistic thing

T. D. Newton and E. P. Wigner, "Localized states for elementary systems",
Rev. Mod. Phys., 21, (1949) 400.

For an explicit and detailed critique of the Newton-Wigner theory, see
T. O. Philips, Phys. Rev. 136 (1964) B893.
Or even better, read what Newton and Wigner have written by themselves in their paper. After Eq. (12), in the paragraph that begins with "It may be well to remember ...", they write:
"Hence our operators q^k have no simple covariant meaning under relativistic transformations."
Actually, the whole paragraph is devoted to that point, but in this sentence it is the most explicit.

 

[meopemuk]

For an explicit and detailed critique of the Newton-Wigner theory, see
T. O. Philips, Phys. Rev. 136 (1964) B893.
Or even better, read what Newton and Wigner have written by themselves in their paper. After Eq. (12), in the paragraph that begins with "It may be well to remember ...", they write:
"Hence our operators q^k have no simple covariant meaning under relativistic transformations."
Actually, the whole paragraph is devoted to that point, but in this sentence it is the most explicit.

Thank you for the Philips reference. I'll pick it up on my next trip to the library.

Regarding the "non-covariance" of the Newton-Wigner operator, I don't see anything wrong with it. Apparently, particles localized in the reference frame at rest don't look like localized in the moving frame. So what? Special relativity has taught us that many things previously considered absolute are, in fact, observer-dependent. Localization is just one of these relative things. That's how I see it.

 

[Haelfix]

I like Hans's and Zee's argument. Its funny that you guys all had problems with the P&S derivation, I had the same problem some years ago, and rereading it I still don't buy it at all, and I think its quite wrong (even though the answer is correct).

Ultimately Weinberg's approach is probably the most logically sound from first principles (assuming you keep following things through several chapters), but it does of course somewhat sweep things under the rug a tiny bit, as it deals with the Smatrix perse, which has the wonderful property of blurring all these messy details out.

 

[Demystifier]

Regarding the "non-covariance" of the Newton-Wigner operator, I don't see anything wrong with it. Apparently, particles localized in the reference frame at rest don't look like localized in the moving frame. So what? Special relativity has taught us that many things previously considered absolute are, in fact, observer-dependent. Localization is just one of these relative things. That's how I see it.

Special relativity has thought us that observer dependent things cannot depend on observer in an arbitrary way, but in a specific mathematically precisely defined way. This specific law of transformation is called - relativistic covariance. If a physical quantity does transform in that way, but in some other way, then the requirement of covariance is not satisfied. In such cases, this law of transformation is not really a pure consequence of relativity.

An example of non-covariant transformation of a physical quantity is the Unruh effect. It is argued that it does not fully respect relativity because it is a quantum effect. This is just another example why relativity and quantum theory do not seem to be mutually consistent.

 

[Demystifier]

Ultimately Weinberg's approach is probably the most logically sound from first principles (assuming you keep following things through several chapters), but it does of course somewhat sweep things under the rug a tiny bit, as it deals with the Smatrix perse, which has the wonderful property of blurring all these messy details out.

S-matrix is defined in the momentum space. If you do a formal Fourier transform into the position space, it is not consistent to interpret it as a probability amplitude in the position space. At least not in the relativistic case. That is the problem with QFT.

 

[Demystifier]

Your conclusion directly contradicts results of Hegerfeldt and many other authors. Did you try to figure out what is wrong with their arguments?

As I understand it, the following is wrong with the Hegerfeldt reasoning. Consider a wavefunction that is a delta-function initially, thus vanishing everywhere except at a point. Now let the time evolution of this wave function be given by the Klein-Gordon equation. One can show that there is a solution that will be nonvanishing (almost) everywhere at an infinitesimal time after the initial one. Apparently, it looks like if something instantaneously appeared at positions at which nothing existed initially. This is the essence of the Hegerfeldt paradox, as I understand it. Now what is wrong with this reasoning? The point is that the Klein-Gordon equation is a SECOND order differential equation in time derivatives. Consequently, the initial condition is given not only by the initial wave function, but also by the initial time derivative of the wave function. In other words, it is not true that nothing existed at other positions initially; although the wave function vanished, its time derivative did not vanish. Therefore, nothing propagated instantaneously; the wave function appeared there because the time derivative of the wave function was already there.

If, instead, you start with a wave function for which both the initial wave function and the initial time derivative of the wave function are delta-functions, nothing will look as instantaneous propagation.

 

[meopemuk]

Special relativity has thought us that observer dependent things cannot depend on observer in an arbitrary way, but in a specific mathematically precisely defined way. This specific law of transformation is called - relativistic covariance. If a physical quantity does transform in that way, but in some other way, then the requirement of covariance is not satisfied. In such cases, this law of transformation is not really a pure consequence of relativity.

Are you then saying that a particle that looks localized for observer ar rest must also look localized for any moving observer? In other words, that localization does not depend on boosts? But one can find the following contradiction in these views.

It goes without saying that space translations and rotations of observer do not affect particle localization. I think we can also agree that time translations do change localization - this is the famous wave-packet spreading effect. However, within Poincare group any time translation can be always represented as a product of space translations and boosts. This follows from commutation relations of the Poincare group, in particular

$$[\mathcal{K}_i, \mathcal{P}_j] = -\frac{1}{c^2} \mathcal{H} \delta_{ij}$$

where K, P, and H are generators of boosts, space translations, and time translations, respectively. So, if both K and P do not change localization, the same must be true for H. So, we got a contradiction.

 

[meopemuk]

S-matrix is defined in the momentum space. If you do a formal Fourier transform into the position space, it is not consistent to interpret it as a probability amplitude in the position space. At least not in the relativistic case. That is the problem with QFT.

In scattering theory, the fundamental quantity is the S-operator, which is basis-independent. Matrix elements of this operator on momentum eigenvectors form the momentum space S-matrix, but you can also choose to find matrix elements of the S-operator in any other representation, including the position representation. This is not convenient, but possible.

 

[Demystifier]

In scattering theory, the fundamental quantity is the S-operator, which is basis-independent. Matrix elements of this operator on momentum eigenvectors form the momentum space S-matrix, but you can also choose to find matrix elements of the S-operator in any other representation, including the position representation. This is not convenient, but possible.

In the relativistic case, there is no position representation because there is no position basis because there is no position operator. Fourier transform is not a transition to the position representation.

 

[meopemuk]

Now let the time evolution of this wave function be given by the Klein-Gordon equation... The point is that the Klein-Gordon equation is a SECOND order differential equation in time derivatives. Consequently, the initial condition is given not only by the initial wave function, but also by the initial time derivative of the wave function.

It is often said in textbooks that the Klein-Gordon equation is a relativistic analog of the Schroedinger equation, and therefore it can be used for the description of time evolution. Then, as you correctly pointed out, initial conditions should include the time derivative of the wave function in addition to the wave function itself. In other word, the state of the particle at time $t + \Delta t$ is determined not only by its state at time $t$, but also by the "state derivative" at time $t$.

This statement is in contradiction with the fundamental equation for time evolution in quantum mechanics

$$|\Psi(t) \rangle = \exp(\frac{i}{\hbar}Ht) |\Psi(0) \rangle$$

where H is the Hamiltonian. This equation show, in particular, that the state at time $t + \Delta t$ is determined by the state at time $t$, the Hamiltonian, and nothing else.

 

[Demystifier]

Are you then saying that a particle that looks localized for observer ar rest must also look localized for any moving observer?

No, I am saying the transformation from one observer to another must be covariant.

 

[Demystifier]

This statement is in contradiction with the fundamental equation for time evolution in quantum mechanics

$$|\Psi(t) \rangle = \exp(\frac{i}{\hbar}Ht) |\Psi(0) \rangle$$

where H is the Hamiltonian. This equation show, in particular, that the state at time $t + \Delta t$ is determined by the state at time $t$, the Hamiltonian, and nothing else.

Exactly! Relativistic QM does not satisfy this axiom. This is also closely related to the fact that relativistic QM cannot be interpreted probabilistically.

 

[meopemuk]

In the relativistic case, there is no position representation because there is no position basis because there is no position operator. Fourier transform is not a transition to the position representation.

I still cannot see what could go wrong with choosing the Newton-Wigner operator as a relativistic generalization of position? I think this operator provides a perfect description of position in relativistic quantum theory.

From another point of view, there must be *some* position operator in relativistic quantum theory. You cannot just say: there is no operator, so I am not going to consider the position representation. Position is the most basic observable in physics, and it remains measurable in both non-relativistic and relativistic physics. If position is an observable, then in quantum theory there must be an operator corresponding to it. It is just inevitable.