# What is the velocity of a relativistic electron?

by Juan R. Gonzlez-lvarez
Tags: electron, relativistic, velocity
 P: n/a I find different answers: ((Classical electrodynamics)) v < c ((Relativistic Quantum Mechanics)) v = c using Dirac equation for |PSI>. ((Space-time approach to QED)) v = c because the spacetime kernel K_+ is derived from the Dirac equation for |PSI>. Feynman even writes in his well-known textbook on QED [1] that {BLOCKQUOTE This result is sometimes made pausible by the argument that a precise determination of velocity implies precise determinations of position at two times. Then by the uncertainty principle, the momentum is completely uncertain and all values are equally likely, this is seen to imply that velocities near the speed of light are more probable, so that in the limit the expected value of the velocity is the speed of light. } Then in a footnote, Feynman recognizes that argument is invalid since v commutes with p. Thus, an incorrect value is justified using an incorrect argument! ((Field approach to QED)) I have computed the value of v using also the field approach to QED. I.e the value for the v corresponding to a Dirac field PHI(x,t) is again c. Conventional thinking says that position is not an observable in field QED (just a parameter) and thus v cannot be indentified with the velocity of a particle. This argument is difficult to accept because is if x and v are not observables at some fundamental level of theory, then, i) why there exist observables x and v at level of non-relativistic quantum mechanics? ii) How does one obtain classical observables x and v? Already Dirac warned about QED but his quote is almost ignored (only his related quote on renormalization is usually cited). The quote on incompatibility of QED with previous theories is {BLOCKQUOTE Most physicists are very satisfied with this situation. They argue that if one has rules for doing calculations and the results agree with observation, that is all that one requires. But it is not all that one requires. One requires a single comprehensive theory applying to all physical phenomena. Not one theory for dealing with non relativistic effects and a separate disjoint theory for dealing with certain relativistic effects. [...] For these reasons I find the present quantum electrodynamics quite unsatisfactory. }
 P: n/a On 2008-02-13, Juan R. Gonzlez-lvarez wrote: > I find different answers: > > ((Classical electrodynamics)) > > v < c > > ((Relativistic Quantum Mechanics)) > > v = c using Dirac equation for |PSI>. Eh, no. You are confused by an archaic paradox. The paradox consists of identifying the operator c*alpha (in the usual 3+1 Dirac equation notation) with the physical velocity, which quite obviously doesn't work. For a modern discussion of the Dirac electron's velocity operator, see [1] where the author discusses the Foldy-Wouthuysen transformation and how to use it to get the correct operator obeservable. See also [2] for some more discussion and references. The velocity operator thus obtained, satisfies the Heisenberg equations of motion v = dx/dt and the well known relation v = p/m in the non-relativistic limit. Note, here x is not the spatial argument of the usual Dirac wave function. It is also obtained through the Foldy-Wouthuysen transformation, or from the formula given by Newton and Wigner (again, see [2]). Once a particular time axis has been chosen, this operator x is unique (up to translations of the origin of coordinates). [...] > i) why there exist observables x and v at level of non-relativistic > quantum mechanics? They exist alright, as evidenced by any elementary treatment of QM. What you are perhaps wondering is how are the x and v observables generalized to relativistic single particle QM and to QFT. The answer to the first part of this question is contained in the references I gave above. The generalization from relativistic single-particle QM to QFT follows standard methods of second quantization [3]. > ii) How does one obtain classical observables x and v? Given that x and v exist and QM operators, the corresponding classical observables are obtained as expectation values of these operators with respect to coherent sates, which embed the classical phase space in the QM Hilbert space of states. It is important to note that the cascade QFT -> rel QM -> non-rel QM -> CM is only a limiting procedure. There are quantum effects that become non-negligible when a particle is no longer described by a coherent state. There are relativistic effects that become non-negligible when the particles speed aproaches the speed of light. There are multi-particle effects that become non-negligible when the single-particle sector cannot be sufficiently decoupled from the rest of the QFT Fock space. > Already Dirac warned about QED but his quote is almost ignored (only his > related quote on renormalization is usually cited). The quote on > incompatibility of QED with previous theories is > > {BLOCKQUOTE > Most physicists are very satisfied with this situation. They argue that if > one has rules for doing calculations and the results agree with > observation, that is all that one requires. But it is not all that one > requires. One requires a single comprehensive theory applying to all > physical phenomena. Not one theory for dealing with non relativistic > effects and a separate disjoint theory for dealing with certain > relativistic effects. [...] For these reasons I find the present quantum > electrodynamics quite unsatisfactory. } I think Dirac would have been happy today, since his dream has been achieved. [1] Paul Strange, _Relativistic Quantum Mechanics_, Cambridge (1998). Chapter 7. [2] J.P. Costellay and B.H.J. McKellarz The Foldy-Wouthuysen transformation arXiv:hep-ph/9503416 [3] Paul A. M. Dirac, _Principles of Quantum Mechanics_. Igor
 P: n/a Igor Khavkine wrote on Fri, 15 Feb 2008 20:47:23 +0000: > On 2008-02-13, Juan R. Gonzlez-lvarez wrote: >> I find different answers: >> >> ((Classical electrodynamics)) >> >> v < c >> >> ((Relativistic Quantum Mechanics)) >> >> v = c using Dirac equation for |PSI>. > > Eh, no. > > You are confused by an archaic paradox. The paradox consists of > identifying the operator c*alpha (in the usual 3+1 Dirac equation > notation) with the physical velocity, which quite obviously doesn't > work. For a modern discussion of the Dirac electron's velocity operator, > see [1] where the author discusses the Foldy-Wouthuysen transformation > and how to use it to get the correct operator obeservable. See also [2] > for some more discussion and references. > > The velocity operator thus obtained, satisfies the Heisenberg equations > of motion v = dx/dt and the well known relation v = p/m in the > non-relativistic limit. Note, here x is not the spatial argument of the > usual Dirac wave function. It is also obtained through the > Foldy-Wouthuysen transformation, or from the formula given by Newton and > Wigner (again, see [2]). Once a particular time axis has been chosen, > this operator x is unique (up to translations of the origin of > coordinates). Actually the author of reference [1] does not maintain your own distinction between "correct" and "incorrect" velocity operators. He is saying something *different* and, in fact, he is supporting my point. By velocity operator i did mean its standard definition, obviously. >From [1] one reads: {BLOCKQUOTE [emphasis mine] We have seen that the *standard* velocity operator gives the electron speed as equal to the velocity of the light. } Next he discusses a *different* velocity operator (his eq. 7.41) and adds {BLOCKQUOTE [emphasis mine] These two can be *reconciled* if we identify (7.41) with the *average* velocity of the particle. } Then the *full* motion of the electron can be divided into two parts {BLOCKQUOTE [emphasis mine] Firstly there is the *average* velocity (7.41) and secondly there is very rapid oscillatory motion which ensures that if an instantaneous measurement of the *velocity* of the electron could be done it would give c. } It seems that the author of [1] think one cannot measure velocities when says "if an instantaneous measurement [...] could be done". This argument usually goes over discussion of sequential measurements, thus position at two times and then taking the limit when time interval goes to zero. This is also the point on [1]. This argument however is not acceptable, which is also the remark done by Feynman in the footnote on the page i cited (see sci.physics.foundations, where my post is not truncated, for the reference). Feynman (and myself) point is that v commutes with p, therefore, instantaneous measurement of v are possible (at least in theory). Now we would analyze the author [1] is really showing. He splits the "velocity of the electron" into two parts, say v = + Dv he identifies , the *average* velocity, with {BLOCKQUOTE the average motion of the particle, i.e. that given by classical relativistic formulae } Thus does not describe the full velocity of the *quantum* particle. And the author gives a value c for the latter. A simple transformation cannot change that fact. >> i) why there exist observables x and v at level of non-relativistic >> quantum mechanics? > > They exist alright, as evidenced by any elementary treatment of QM. What > you are perhaps wondering is how are the x and v observables generalized > to relativistic single particle QM and to QFT. > > The answer to the first part of this question is contained in the > references I gave above. The generalization from relativistic > single-particle QM to QFT follows standard methods of second > quantization [3]. I was just asking an inverse question. We know x is a classical parameter in relativistic quantum field theory. Thus the (unsolved) question is: How a non-observable transform into an observable in the non- relativistic limit? This is one of inconsistencies troubled Dirac. >> ii) How does one obtain classical observables x and v? > > Given that x and v exist and QM operators, the corresponding classical > observables are obtained as expectation values of these operators with > respect to coherent sates, which embed the classical phase space in the > QM Hilbert space of states. > > It is important to note that the cascade QFT -> rel QM -> non-rel QM -> > CM is only a limiting procedure. There are quantum effects that become > non-negligible when a particle is no longer described by a coherent > state. There are relativistic effects that become non-negligible when > the particles speed aproaches the speed of light. There are > multi-particle effects that become non-negligible when the > single-particle sector cannot be sufficiently decoupled from the rest of > the QFT Fock space. You missed my point, again. If relativistic quantum field theory says that position is not an observable. How it is supposed that something cannot be observed at some supposed fundamental level turns into a observable in the classical limit? >> Already Dirac warned about QED but his quote is almost ignored (only >> his related quote on renormalization is usually cited). The quote on >> incompatibility of QED with previous theories is >> >> {BLOCKQUOTE >> Most physicists are very satisfied with this situation. They argue that >> if one has rules for doing calculations and the results agree with >> observation, that is all that one requires. But it is not all that one >> requires. One requires a single comprehensive theory applying to all >> physical phenomena. Not one theory for dealing with non relativistic >> effects and a separate disjoint theory for dealing with certain >> relativistic effects. [...] For these reasons I find the present >> quantum electrodynamics quite unsatisfactory. } > > I think Dirac would have been happy today, since his dream has been > achieved. Difficult to believe since Dirac issues remain unsolved at present date. In [hep-th0501222], It is stated that Dirac maintained his criticism on QED in a talk on 1983. Moreover the quote is rather large and finalizes with Dirac hope for the development of a full and consistent relativistic quantum mechanics. Several groups in all the world are now searching one. > [1] Paul Strange, _Relativistic Quantum Mechanics_, Cambridge (1998). > Chapter 7. > > [2] J.P. Costellay and B.H.J. McKellarz > The Foldy-Wouthuysen transformation > arXiv:hep-ph/9503416 > > [3] Paul A. M. Dirac, _Principles of Quantum Mechanics_. > > Igor -- I follow http://canonicalscience.org/en/misce...guidelines.txt
P: n/a

## What is the velocity of a relativistic electron?

On 2008-02-16, Juan R. <juanrgonzaleza@canonicalscience.com> wrote:
> Igor Khavkine wrote on Fri, 15 Feb 2008 20:47:23 +0000:
>
>> On 2008-02-13, Juan R. Gonzlez-lvarez <Juan@canonicalscience.com> wrote:
>>> I find different answers:
>>>
>>> ((Classical electrodynamics))
>>>
>>> v < c
>>>
>>> ((Relativistic Quantum Mechanics))
>>>
>>> v = c using Dirac equation for |PSI>.

>>
>> Eh, no.
>>
>> You are confused by an archaic paradox. The paradox consists of
>> identifying the operator c*alpha (in the usual 3+1 Dirac equation
>> notation) with the physical velocity, which quite obviously doesn't
>> work. For a modern discussion of the Dirac electron's velocity operator,
>> see [1] where the author discusses the Foldy-Wouthuysen transformation
>> and how to use it to get the correct operator obeservable. See also [2]
>> for some more discussion and references.
>>
>> The velocity operator thus obtained, satisfies the Heisenberg equations
>> of motion v = dx/dt and the well known relation v = p/m in the
>> non-relativistic limit. Note, here x is not the spatial argument of the
>> usual Dirac wave function. It is also obtained through the
>> Foldy-Wouthuysen transformation, or from the formula given by Newton and
>> Wigner (again, see [2]). Once a particular time axis has been chosen,
>> this operator x is unique (up to translations of the origin of
>> coordinates).

>
> Actually the author of reference [1] does not maintain your own
> distinction between "correct" and "incorrect" velocity operators. He
> is saying something *different* and, in fact, he is supporting my
> point.
>
> By velocity operator i did mean its standard definition, obviously.
>>From [1] one reads:

>
> {BLOCKQUOTE [emphasis mine]
> We have seen that the *standard* velocity operator gives the electron
> speed as equal to the velocity of the light.
> }

Velocity is defined by experiment. If the "standard" velocity operator
does not reproduce experimental measurements, then it is still not the
*correct* one. It is only unfortunate that it has somehow become the
"standard" one.

In any case, reference [1] describes how to obtain an operator that does
reproduce experimental observations of electron velocities. If you were
wondering if such an operator exists, then that's your answer. If you
were wondering why this is not the "standard" velocity operator, then
the answer has more to do with history than physics.

[...]
> Now we would analyze the author [1] is really showing. He splits the
> "velocity of the electron" into two parts, say
>
> v = <v> + Dv
>
> he identifies <v>, the *average* velocity, with
>
> {BLOCKQUOTE
> the average motion of the particle, i.e. that given by classical
> relativistic formulae
> }
>
> Thus <v> does not describe the full velocity of the *quantum*
> particle.
>
> And the author gives a value c for the latter. A simple transformation
> cannot change that fact.

I am eagerly awaiting your publications on the measurement of the "full
velocity of the quantum particle". Until then, we only have the usual
velocity measurement experiments, which are already well reproduced by
the velocity operator obtained with the help of the Foldy-Wouthuysen
transformation.

>>> i) why there exist observables x and v at level of non-relativistic
>>> quantum mechanics?

>>
>> They exist alright, as evidenced by any elementary treatment of QM. What
>> you are perhaps wondering is how are the x and v observables generalized
>> to relativistic single particle QM and to QFT.
>>
>> The answer to the first part of this question is contained in the
>> references I gave above. The generalization from relativistic
>> single-particle QM to QFT follows standard methods of second
>> quantization [3].

>
> I was just asking an inverse question. We know x is a classical
> parameter in relativistic quantum field theory. Thus the (unsolved)
> question is:
>
> How a non-observable transform into an observable in the non-
> relativistic limit?

Let x - position coordinates (parameter)
X - position observable (operator)
|x> - an eigenstate of the X operator
chi(x) - position-space wave function
Psi(x) - field operator
|chi> - single-particle state
|0> - Fock vacuum

1. Project onto position eigenstates: chi(x) = <x|chi>.
2. Embed state in Fock space: |chi> = int dx chi(x) Psi(x)^* |0>.
3. Promote X to operator on Fock space: X = int dx x Psi(x)^* Psi(x).

On the LHS of step 3, we have an operator X (an observable), while on
the RHS of the same equality we have integration over a coordinate x (a
field theoretic parameter). Step 3 is the answer to your question.

This is a generic method for promoting single-particle operators to Fock
operators. In any particular situation, there will be differences.
Again, the Foldy-Wouthuysen transformation helps you pick out the right
position operator. See sections 59-65 of [3], for the general
prescription.

>> I think Dirac would have been happy today, since his dream has been
>> achieved.

>
> Difficult to believe since Dirac issues remain unsolved at present
> date.
>
> In [hep-th0501222], It is stated that Dirac maintained his
> criticism on QED in a talk on 1983.

Yes, it is unfortunate that in 1983 theoretical physics lost one of its
great thinkers. It is also unfortunate that Dirac can no longer join us
in a discussion of how his dream of a consistent treatment of
relativistic quantum mechanics has been achieved. I guess all we can do
now is rely on existing evidence and our own judgement, rather than the
opinions of someone whose mind can no longer be altered.

> Moreover the quote is rather large and finalizes with Dirac hope for
> the development of a full and consistent relativistic quantum
> mechanics.
>
> Several groups in all the world are now searching one.

I wish them the best of luck.

>> [1] Paul Strange, _Relativistic Quantum Mechanics_, Cambridge (1998).
>> Chapter 7.
>>
>> [2] J.P. Costellay and B.H.J. McKellarz
>> The Foldy-Wouthuysen transformation
>> arXiv:hep-ph/9503416
>>
>> [3] Paul A. M. Dirac, _Principles of Quantum Mechanics_.

Igor

 P: n/a Juan R. wrote; > (QED continues to > work only for infinite time scattering processes between free fields > over a classical space-time). This is not true. One can compute - nonrigorously, in renormalized perturbation theory - many time-dependent things, namely via the Schwinger-Keldysh (or closed time path = CPT) formalism; see, e.g., http://theory.gsi.de/~vanhees/publ/green.pdf See also the entry S9c. What about relativistic QFT at finite times? in my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physics-faq.txt Arnold Neumaier
 P: n/a Arnold Neumaier wrote on Wed, 27 Feb 2008 12:33:20 +0000: > Juan R. wrote; > >> (QED continues to >> work only for infinite time scattering processes between free fields >> over a classical space-time). > > This is not true. One can compute - nonrigorously, in renormalized > perturbation theory - many time-dependent things, namely via the > Schwinger-Keldysh (or closed time path = CPT) formalism; see, e.g., > http://theory.gsi.de/~vanhees/publ/green.pdf {BLOCKQUOTE [1] The closed time path formalism proposed by Schwinger and refined by many others, notable Keldysh, has been frequently used to study nonequilibrium situations. But a close inspection of the formalism reveals that it is more of a steady state theory than a truly time-dependent theory. } "steady state theory" = "time-independent theory" See also the prose before and after (eq 13) and after (eq 36) in the pfd you are citing. Moreover, the SK formalism is based in inconsistent mixture of elements from contradictory theories, e.g. Feynman diagrams for S-matrix (only defined for infinite times) are mixed with a LvN equation for finite times. See also for extra criticism on SK [1]. > > See also the entry > S9c. What about relativistic QFT at finite times? > in my theoretical physics FAQ at > http://www.mat.univie.ac.at/~neum/physics-faq.txt I agree when says: {BLOCKQUOTE Current 4D QFT is based on perturbation theory for free (i.e., asymptotic in- and out-) states; therefore it gives only predictions that relate the in- and out-states. [...] In nonrelativistic QM, one has a well-defined dynamics at finite times, given by the Schroedinger equation. This dynamics can be recast in terms of Feynman path integrals. Unfortunately, this does not extend to the relativistic case. [...] But I haven't seen a single article that gives meaning (i.e., infrared and ultraviolet finite, renormalization scheme independent properties) to, say, quantum electrodynamics states at finite t and their propagation in time. People don't even know what an initial state should be in a relativistic QFT (i.e., from which space to take the states at finite t); so how can they know how to propagate it... Thus the standard theory gives an S-matrix (or rather an asymptotic series for it) but not a dynamics at finite times. } and disagree with other claims. I can see you do not cite any of the classical results over which relativistic QFT was developed. For instance, the Peirls-Landau inequality is the reason S = U(-infinity, +infinity) and that QFT lacks dynamical description at finite times. It is interesting you cite the recent work of my colleague Eugene V. Stefanovich. He agrees with me on the non-existence of finite time relativistic QFT. E.g. see section 9.1.2 on [2]. You also write in the FAQ {BLOCKQUOTE The missing consistent dynamical theory in 4D relativistic QFT may also have consequences for the foundations of quantum mechanics. Clearly, measurements happen in finite time, hence cannot be described at present in a fundamental way (i.e., beyond the nonrelativistic QM approximation). Thus foundational studies based on nonrelativistic QM are naturally incomplete. This implies that it is quite possible that a solution of the unresolved issues in relativistic QFT are related to the unresolved issues in quantum measurement theory. } One of main points of Eugene (and others, including me) affirms there is *not* quantum dynamical relativistic theory based in Minkowski space-time unification: "The Einstein-Minkowski 4-dimensional spacetime is an approximate concept as well." Do not forget that relativistic QFT breaks with quantum mechanics and treats spacetime as a *classical* object. See references and quotations in my previous message. [1] A unified formalism of thermal quantum field theory. 1994. Int. J. of Mod. Phys A 9(14), 2363. Chu, H; Umezawa, H. [2] http://arxiv.org/abs/physics/0504062 -- I apply http://canonicalscience.org/en/misce...guidelines.txt
 P: n/a On 2008-03-07, Juan R. González-Álvarez wrote: > Arnold Neumaier wrote on Wed, 27 Feb 2008 12:33:20 +0000: >> Juan R. wrote; >> >>> (QED continues to >>> work only for infinite time scattering processes between free fields >>> over a classical space-time). >> >> This is not true. One can compute - nonrigorously, in renormalized >> perturbation theory - many time-dependent things, namely via the >> Schwinger-Keldysh (or closed time path = CPT) formalism; see, e.g., >> http://theory.gsi.de/~vanhees/publ/green.pdf > > {BLOCKQUOTE [1] > The closed time path formalism proposed by Schwinger and refined by many > others, notable Keldysh, has been frequently used to study nonequilibrium > situations. But a close inspection of the formalism reveals that it is > more of a steady state theory than a truly time-dependent theory. > } > > "steady state theory" = "time-independent theory" > [1] A unified formalism of thermal quantum field theory. 1994. Int. J. > of Mod. Phys A 9(14), 2363. Chu, H; Umezawa, H. This reference is from the authors of a formalism called "thermo field dynamics" (many papers and at least one textbook has been written on the subject). From your quote it appears that the authors perceive the Schwinger-Keldysh formalism as a direct competitor and alternative, distinct from theirs. The point of both, BTW, is to capture time dependent phenomena in field theory. Unfortunately, the authors of [1] are mistaken. A very clear side by side comparison of "thermo field dynamics" and the Schwinger-Keldysh formalism reveals that the latter completely reproduce and is more general than the former. Lawrie, I. D. On the relationship between thermo field dynamics and quantum statistical mechanics J. Phys. A: Math. Gen. 27 (1994) 1435-1452 > See also the prose before and after (eq 13) and after (eq 36) in the pfd > you are citing. > > Moreover, the SK formalism is based in inconsistent mixture of elements > from contradictory theories, e.g. Feynman diagrams for S-matrix (only > defined for infinite times) are mixed with a LvN equation for finite > times. See also for extra criticism on SK [1]. Perhaps you'd like to share some specific criticism of (eq 13) and (eq 36) from van Hees's notes and why the Schwinger-Keldysh formalism appear inconsistent to you. >> See also the entry >> S9c. What about relativistic QFT at finite times? >> in my theoretical physics FAQ at >> http://www.mat.univie.ac.at/~neum/physics-faq.txt > I can see you do not cite any of the classical results over which > relativistic QFT was developed. For instance, the Peirls-Landau > inequality is the reason S = U(-infinity, +infinity) and that QFT lacks > dynamical description at finite times. It's unclear what you mean here. Peierls and Landau only wrote two papers together, neither of which dealt with this topic. > It is interesting you cite the recent work of my colleague Eugene V. > Stefanovich. He agrees with me on the non-existence of finite time > relativistic QFT. E.g. see section 9.1.2 on [2]. Eugene's misconceptions regarding the content of QED have been pointed out numerous times in this group. Igor
 P: n/a Juan R. González-Álvarez wrote (in the thread: What is the velocity of a relativistic electron?): > Arnold Neumaier wrote on Wed, 27 Feb 2008 12:33:20 +0000: >> Juan R. wrote; >> >>> (QED continues to >>> work only for infinite time scattering processes between free fields >>> over a classical space-time). >> This is not true. One can compute - nonrigorously, in renormalized >> perturbation theory - many time-dependent things, namely via the >> Schwinger-Keldysh (or closed time path = CPT) formalism; see, e.g., >> http://theory.gsi.de/~vanhees/publ/green.pdf > > {BLOCKQUOTE [1] > The closed time path formalism proposed by Schwinger and refined by many > others, notable Keldysh, has been frequently used to study nonequilibrium > situations. But a close inspection of the formalism reveals that it is > more of a steady state theory than a truly time-dependent theory. > } This ''close inspection'' cannot have a deep basis. How would it then be possible that the paper E. Calzetta and B. L. Hu, Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation, Phys. Rev. D 37 (1988), 2878-2900. derives finite-time Boltzmann-type kinetic equations from quantum field theory using the CTP formalism? > Moreover, the SK formalism is based in inconsistent mixture of elements > from contradictory theories, e.g. Feynman diagrams for S-matrix (only > defined for infinite times) are mixed with a LvN equation for finite > times. See also for extra criticism on SK [1]. Any current formalism for interacting quantum field theory is inconsistent and ill-defined when viewed from a logical point of view. Nevertheless, people derive from it predictions, often very accurate ones. >> See also the entry >> S9c. What about relativistic QFT at finite times? >> in my theoretical physics FAQ at >> http://www.mat.univie.ac.at/~neum/physics-faq.txt > > I agree when says: > > {BLOCKQUOTE > Current 4D QFT is based on perturbation theory for free > (i.e., asymptotic in- and out-) states; therefore it gives only > predictions that relate the in- and out-states. This referred to the standard textbook approach. I improved the FAQ by providing clarifying words. > But I haven't seen a single article that gives meaning (i.e., > infrared and ultraviolet finite, renormalization scheme independent > properties) to, say, quantum electrodynamics states at finite t and > their propagation in time. Nevertheless, there are very useful approximations towards that goal, which I had mentioned in the part that follows, but which you didn't quote. > I can see you do not cite any of the classical results over which > relativistic QFT was developed. For instance, the Peirls-Landau > inequality is the reason S = U(-infinity, +infinity) and that QFT lacks > dynamical description at finite times. There is little point in citing old results which no longer reflect the current state of the art. > It is interesting you cite the recent work of my colleague Eugene V. > Stefanovich. He agrees with me on the non-existence of finite time > relativistic QFT. E.g. see section 9.1.2 on [2]. Quoting a paper does not mean that I consent to every statement in it. A few years ago we had extensive discussions here on s.p.r. where you can find out what I think of his work. Many of his claims at that time were immature, and what you write below about his current views does not indicate that things have changed. > You also write in the FAQ > > {BLOCKQUOTE > The missing consistent dynamical theory in 4D relativistic QFT > may also have consequences for the foundations of quantum mechanics. The point here is ''consistent'', which means logically impeccable. The theory of Stefanovich has the same logical flaws as all current 4D quantum field theory - no convergence theory which would prove that the objects to which perturbative or nonperturbative approximations are computed are logically well-defined. I had read the first version of his online book (your reference [2]), and it presents a view which completely ignores much of what has been established in the field. His arguments are provably wrong in 2D quantum field theory. So there is no reason to trust them in 4D. > Clearly, measurements happen in finite time, hence cannot > be described at present in a fundamental way (i.e., beyond the > nonrelativistic QM approximation). Thus foundational > studies based on nonrelativistic QM are naturally incomplete. > This implies that it is quite possible that a solution of the > unresolved issues in relativistic QFT are related to the unresolved > issues in quantum measurement theory. > } > > One of main points of Eugene (and others, including me) affirms there is > *not* quantum dynamical relativistic theory based in Minkowski space-time > unification: "The Einstein-Minkowski 4-dimensional spacetime is an > approximate concept as well." I find this view mistaken and not supported by the existing research literature. > Do not forget that relativistic QFT breaks with quantum mechanics No. It is fully based on quantum mechanics, as my FAQ amply demonstrates. [2] http://arxiv.org/abs/physics/0504062 Arnold Neumaier