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

[Hans de Vries]

To recapitulate why there is nothing wrong with the normal
relativistic position and velocity operators: If we have the
Dirac equation:

\gamma^0\ \partial_t\ \psi\ \ =\ \ -\left[\ c\ \gamma^i\ \partial_{x^i} + i\frac{mc^2}{\hbar}\ \right]\psi \qquad\qquad\qquad\qquad\qquad (1)

and we define:

\gamma^0\ H\ \ =\ \ -\gamma^0\ i\hbar\ \partial_t\ \ =\ \ i\hbar\ \left[\ c\ \gamma^i\ \partial_{x^i} + i\frac{mc^2}{\hbar}\ \right]\psi \qquad\qquad (2)

Then we find the usual result for the velocity:

v\ =\ \frac{i}{\hbar}\ [\ H,x^i\ ]\ = \ c\ \gamma^i\ /\ \gamma^0\ \ =\ \ c\ \alpha^i \qquad\qquad\qquad\qquad\qquad (3)

Then, instead of concluding that something must be terrible wrong
with the relativistic position and velocity operators we simple ask
ourself what we have to do to go from the H as defined in (2) to
get {\cal E}, the energy-density of the wave-function, and we find:

{\cal E}\ \ =\ \ \frac{1}{2E}\left(\ \bar{\psi}\ \gamma^0\ H\ \psi\ \right)

Thus to get the velocity-density we need to evaluate

v^i\ \ =\ \ \frac{1}{2E}\left(\ \bar{\psi}\ \gamma^i\ \psi\ \right) \ \ = \ \ \frac{cp^i}{E}\ \ =\ \ v^i

So, there is nothing wrong after all. Instead we could interpret
(3) as giving us some interesting information about the internal
components of the spinor wave function.Regards, Hans

 

[meopemuk]

Interesting, I always felt that the x,p relations were somehow
missing from the commutation prescription.

The Newton-Wigner position operator x is a function of Poincare generators x(P, J) and it has correct (i.e., physically acceptable) commutators with all these generators, including usual Heisenberg commutators with momentum p. So, I don't see any reason to invent an artificial substitute for x.

Eugene.

 

[meopemuk]

Hi Hans,

QM has a strictly defined formalism in which observables are represented by Hermitian operators and states are represented by vectors in the Hilbert space. The energy-densities and velocity-densities you are talking about are rather unusual notions for quantum mechanics. Are they measurable? Shall we also consider the position-density?
Why not?

Eugene.

 

[Hans de Vries]

The Newton-Wigner position operator x is a function of Poincare generators x(P, J) and it has correct (i.e., physically acceptable) commutators with all these generators, including usual Heisenberg commutators with momentum p. So, I don't see any reason to invent an artificial substitute for x.

Eugene.

Because, as you claim a few post back, the NW-position operator violates
Special Relativity and allows for faster than light signal communication, and,
yes, not really everybody finds that "physically acceptable" ...

Therefor I would prefer to use the standard operators used in classical
physics and QM also in relativistic quantum mechanics.Regards, Hans

 

[strangerep]

The X generators have perfectly reasonable physical foundations.

What are these foundations?

Er,... observations of position? ;-)

I.e., the inclusion of position observables X in the H-P algebra
just represents an attempt to have a single Lie algebra of all
the observables which are obviously relevant in a relativistic
quantum theory.

Like Hans, I too often wondered why the x,p relations are somehow
missing from the usual commutation prescription. In fact, the idea of
the H-P algebra goes right back to Heisenberg himself. But I guess it
wasn't until much later, after (axiomatic/algebraic) QFT based on
operator fields over Minkowski space was looking decidedly sick, and
alternatives like Newton-Wigner fields found to be problematic, that
people started revisiting the H-P idea.

Other generators are related to well-known inertial transformations of
reference frames (space and time translations, rotations and boosts). What finite
transformations are generated by X?

The H-P algebra should be viewed as a Lie algebra of observables.
X corresponds to the position observable. Exp(X) transformations correspond
to translations in momentum space, and this does indeed have certain
difficulties of its own (so I read).

The Newton-Wigner position operator x is a function of Poincare
generators x(P, J) and it has correct (i.e., physically acceptable)
commutators with all these generators, including usual Heisenberg
commutators with momentum p. So, I don't see any reason to invent an
artificial substitute for x.

Hmm. I certainly would never have described the X's in the H-P algebra
as artificial. Quite the opposite.

The N-W operator seems far more artificial to me (IMHO). The explicit
expression for the N-W position operator "R" involves dividing by the mass
and hamiltonian operators. That means it's not in the Lie algebra, nor even
in the (polynomial) enveloping algebra. One must extend the algebra to
some kind of division algebra, and show that it is (mathematically and
physically) sensibly well-defined on all of one's Hilbert space before such
a construction can be considered satisfactory. That's exactly where the
N-W operator runs into trouble.

But I guess it's a subjective judgement whether something is/isn't "artificial".

 

[meopemuk]

X corresponds to the position observable. Exp(X) transformations correspond
to translations in momentum space, and this does indeed have certain
difficulties of its own (so I read).

That's what I am having problems with. For Poincare group generators H, P, J, K, the corresponding finite transformations Exp(H), Exp(P), Exp(J), Exp(K) have a well-defined and easily observable physical meaning as transformations of inertial reference frames. On the other hand, momentum-space translations Exp(X) are rather abstract things. That's why I don't feel like treating X on the same footing as other generators.

But I guess it's a subjective judgement whether something is/isn't "artificial".

Here I should agree with you. Relativistic quantum theory is so weird. It seems that no matter which approach we take, we face artificial or counterintuitive things of one kind or another. Most often, these things cannot be directly measured. So, which of them are less "artificial" remains largely a matter of taste. Apparently, we have different tastes for these things. I don't think it's bad.

Eugene.

 

[OOO]

This leakage can be used to transmit information superluminally. However, I am not convinced that these facts can be used to build a machine that sends signals back to the past. If the possibility of such a machine can be proved, then we are in a deep trouble.

Hi meopemuk,

I haven't followed the thread for the last few days. But what struck me was your statement above about signals being sent back to the past. I feel I have to object to this.

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).

Second, even if it was possible to "send signals back to the past" (the backward lightcone), this would not be interpreted by the observer as such. Think of the advanced propagator in classical ED: what we would see in such a case was a spherical wave that runs towards a specific spacetime point, and when it reaches it, it vanishes and creates a delta-source (the reverse of what the retarded propagator does: create a delta source in connection with an outgoing spherical wave). So I guess the term "sending signals back to the past" is a bit misleading because it implies that the past can be changed, which it can't. Rather signals seemingly sent to the past correspond to very unlikely course of events (in some sense similar to the broken roof tile that heals magically and flies back to the roof...).

Do you agree with that ?

 

[meopemuk]

So I guess the term "sending signals back to the past" is a bit misleading because it implies that the past can be changed, which it can't.

I agree with you here. I think it is impossible to send signals back to the past even if it is quite possible to have particle wavefunctions that propagate superluminally or action-at-a-distance interactions. You can find in the literature designs of "time machines" that are based on superluminal signals. These "time machines" should supposedly allow you to make terrible things, like killing your grandfather before you were born, etc.
I think that these designs involve subtle flaws that wouldn't allow them to work as intended.

Eugene.

 

[strangerep]

X corresponds to the position observable. Exp(X) transformations correspond
to translations in momentum space, and this does indeed have certain
difficulties of its own (so I read).

That's what I am having problems with. For Poincare group generators H, P, J, K, the
corresponding finite transformations Exp(H), Exp(P), Exp(J), Exp(K) have a well-defined
and easily observable physical meaning as transformations of inertial reference frames.
On the other hand, momentum-space translations Exp(X) are rather abstract things.
That's why I don't feel like treating X on the same footing as other generators.

If we write R (the N-W position operator) as (for example):

R = -c^2H^{-1}K - \frac{i \hbar c^2 P}{2H^2} - \frac{cP \times W}{MH(Mc^2 + H)}<br />

then one could also ask what exp(R) means. Since it satisfies (at least formally)
the Heisenberg CRs, I think you'd reach an identical conclusion that R generates
translations in momentum space.

The important thing is the basic (linear) Lie algebra of quantities corresponding to observable
properties of physical systems. Then the task is to go further and construct a 4D
infinite-dimensional interacting quantum theory rigorously from the algebra, which of
course no one has done.

 

[meopemuk]

If we write R (the N-W position operator) as (for example):

R = -c^2H^{-1}K - \frac{i \hbar c^2 P}{2H^2} - \frac{cP \times W}{MH(Mc^2 + H)}<br />

then one could also ask what exp(R) means. Since it satisfies (at least formally)
the Heisenberg CRs, I think you'd reach an identical conclusion that R generates
translations in momentum space.

Yes, it is true that \exp (R) performs translations in momentum space. But it is important to realize that there is no inertial transformations of reference frames, which correspond to such a translation. (Boost transformations come close, but their action on definite momentum states is more complicated than simple translation) So, \exp (R) is not a representative of any Poincare group element. This is the reason why R does not belong to the Poincare Lie algebra and should be constructed as a (rather complicated) function of proper Lie algebra elements that you correctly reproduced.

Eugene.

 

[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.