Exploring the Connection Between Quantum Mechanics and Quantum Field Theory

[A. Neumaier]

The QFT textbook by Weinberg takes a different approach

After treating the quantum mechanics of a single particle in Chapter 2, he develops in Chapter 3 the asymptotic theory of multiparticle quantum mechanics to get the properties of the S-matrix (mediating between infinite negative and infinite positive time), and relates it in Section 3.4 to measurable transition rates and cross sections for asymptotic multiparticle states in a collision. He derives the interpretation of the squared S-matrix elements as transition rates (3.4.11):

This is the master formula which is used to interpret calculations of S-matrix elements in terms of predictions for actual experiments.

This is the only formal contact between QFT and experiment in his book.

In Chapter 4 Weinberg discusses the reasons for treating the relativistic case instead as a field theory - since this (and in his opinion only this) guarantees a Lorentz invariant S-matrix. In Chapter 5 he introduces the asymptotic fields needed to describe the in- and out-states. Starting with Chapter 6 he deals with interacting QFT proper and develops the machinery needed to compute S-matrix elements from QFT. Nowhere the slightest word about what happens at finite times - one can find a discussion of this part only in books on nonequilibrium QFT such as Calzetta & Hu.

 

[Demystifier]

Nowhere the slightest word about what happens at finite times

If one removes IR divergences from QFT (e.g. by putting the whole system into a finite volume), there is no problem in principle to calculate what happens at finite time. Of course, analytical calculations are much simpler with infinite time, and this is the main reason why (even in non-relativistic QM) scattering calculations are usually performed with infinite time.

 

[A. Neumaier]

With that I strongly disagree. Yes, observers and measurements in standard formulation of quantum theory are introduced in a rather ad hoc way, but that refers equally to non-relativistic QM, non-relativistic QFT and relativistic QFT.

In the axioms for relativistic QFT (see post #30) there are field expectations and correlations functions; nothing ad hoc at all. In particular, since no reference is made to observers and measurement, their properties (and in particular Born's rule) must either be derived from the axioms or introduced by hand. There is also no reference made to particles. However, the asymptotic free fields that can be interpreted as quantum particles are derived as asymptotic concepts (at infinite time) through Haag-Ruelle theory. See. e.g., Chapter IV.3 in http://unith.desy.de/sites/site_unith/content/e20/e72/e180/e61334/e78030/QFT09-10.pdf on quantum field theory.

I don't think that this is really essential for QFT.

Well, it is not needed for pure few-particle scattering theory at zero temperature. But it is needed once you want to apply QFT at finite times and/or finite temperature; e.g., to get hydromechanics from QFT. Otherwise you have uncontrollable problems at the boundary of your volume.

 

[A. Neumaier]

If one removes IR divergences from QFT (e.g. by putting the whole system into a finite volume), there is no problem in principle to calculate what happens at finite time. Of course, analytical calculations are much simpler with infinite time, and this is the main reason why (even in non-relativistic QM) scattering calculations are usually performed with infinite time.

Standard textbooks on relativistic QFT are exclusively concerned with scattering, and there is no scattering in a box of finite volume since everything is quasiperiodic! Similarly textbooks on nonrelativistic QFT always consider the thermodynamic limit since otherwise everything becomes intractable (e.g., no continuous spectrum, no Fermi surface, no dissipation). The finite volume approximation is only the first step - the physical results appear only in the limit.

 

[Demystifier]

@A. Neumaier your way of reasoning sounds to me similar to that of mathematical statistical physicists (MSP), who rigorously prove the fact that phase transitions are only possible in infinite volume. Yet, experiments prove them wrong. Water freezes in a finite bucket. MSP then reply that what is observed in a bucket is not a true phase transition, but practical physicists then object that it is only so because MSP have chosen a bad definition of a "true" phase transition, a definition which practical physicists never approved in the first place.

The moral is that one should distinguish mathematical physics from theoretical physics. You are talking from the former point of view, while I am talking from the latter point of view. That's the main source of our disagreement.

 

[A. Neumaier]

@A. Neumaier your way of reasoning sounds to me similar to that of mathematical statistical physicists (MSP), who rigorously prove the fact that phase transitions are only possible in infinite volume. Yet, experiments prove them wrong. Water freezes in a finite bucket. MSP then reply that what is observed in a bucket is not a true phase transition, but practical physicists then object that it is only so because MSP have chosen a bad definition of a "true" phase transition, a definition which practical physicists never approved in the first place.

The moral is that one should distinguish mathematical physics from theoretical physics. You are talking from the former point of view, while I am talking from the latter point of view. That's the main source of our disagreement.

The disagreement is deeper.

In relativistic QFT, the infinite volume limit, respective the infinite time limit in case of few particles at zero temperature, is essential to get Lorentz invariance, which is at the very heart of the theory. Weinberg does theoretical physics only, no mathematical physics!

Similarly, all books on nonrelativistic statistical mechanics - not only the mathematical physics books - take the infinite volume limit to produce results and phase transitions, although they remark that in some approximate sense the result is valid to good accuracy also in a finite bucket.

Theory is always an idealization compared to reality. But it is not the mathematical physicist but the theoretical physicist who makes the idealization and uses infinite times and infinite volumes. The mathematical physicist only provides additional error estimates that makes things fully rigorous.

 

[A. Neumaier]

Please tell a bit more about the Born rule of QM,about the sequence of ideal measurements...

https://en.wikipedia.org/wiki/Born_rule
It applies to ideal (von Neumann -) measurements only. More general measurements are handled by quantum operations and POVMs. https://en.wikipedia.org/wiki/Quantum_operation , https://en.wikipedia.org/wiki/POVM

 

[fxdung]

So can we combine QM and QFT?If it is able,what is the larger theory?Or do we not need to combine?

 

[bhobba]

Or do we not need to combine?

See Chapter 3 - Zee - Quantum Field Theory In A Nutshell. From page 18 - (0+1) dimensional quantum field theory is just quantum mechanics.

Thanks
Bill

 

[meopemuk]

Here is my take on this question.
There are three theories: QT = "quantum theory", QM = "quantum mechanics", and QFT = "quantum field theory".

QT says that states of a given system can be represented as unit vectors in a Hilbert space H; observables are represented by Hermitian operators in the same space H; inertial transformations form a unitary representation of the Poincare group in H; in particular, time translations are generated by the Hamilton operator; positions and momenta of particles satisfy Heisenberg commutators, etc. etc. Multiparticle states can be described by wave functions in the momentum or position representations. Squares of wave functions are interpreted as probability densities. These are fundamental rules of QT that remain valid in both QM and QFT.

QFT is a particular (most advanced) version of QT. This version recognizes that particle interactions can lead to particle creation and annihilation. The simplest example is when hydrogen atom (a 2-particle state) can emit or absorb a photon. Other examples are multiple particle creation processes in high energy collisions. Since number of particles can change, this theory is formulated in a Hilbert space that is built as a direct sum of subspaces corresponding to 0-particle, 1-particle, 2-particle, etc. sectors. The Hilbert space of QFT is called the Fock space. So, basically, QFT is QT of systems with variable numbers of particles. It can be formulated in the language of particle creation and annihilation operators, without ever mentioning "quantum fields". Quantum fields are just convenient linear combinations of creation-annihilation operators. They are useful for building Poincare invariant interaction operators in the Hamiltonian. See Weinberg's vol. 1 about that.

QM usually stands for an approximate QT, where the possibility of particle creation and destruction is ignored. (This is a reasonable approximation at low energies.) This theory can be formulated in a Hilbert space with a fixed number of particles. E.g. N=2, when we consider the hydrogen atom. This truncated Hilbert space is just one sector of the full Fock space of QFT.

So, QM and QFT are just two versions of QT having slightly different interaction Hamiltonians. The Hamiltonian of QFT contains interactions that change numbers of particles. There are no such interaction terms in the QM Hamiltonian.

Eugene.

 

[Demystifier]

In relativistic QFT, the infinite volume limit, respective the infinite time limit in case of few particles at zero temperature, is essential to get Lorentz invariance, which is at the very heart of the theory.

Lorentz invariance is important not due to some theoretical consistency requirements, but only due to fact that this symmetry is observed in nature. This means that it is OK to break Lorentz invariance at the extent at which it does not conflict with observations. Moreover, to avoid IR and UV divergences of QFT, it is almost unavoidable to break Lorentz invariance in one way or another.

 

[Demystifier]

Theory is always an idealization compared to reality. But it is not the mathematical physicist but the theoretical physicist who makes the idealization and uses infinite times and infinite volumes. The mathematical physicist only provides additional error estimates that makes things fully rigorous.

Yes, but if such an idealization leads to a physically unacceptable result, it is theoretical physicists who will first give up of such an idealization. Usually this further complicates the theory for both theoretical and mathematical physicists. However, while a theoretical physicist will find a useful heuristic approximation to deal with the additional complication (typical example: renormalization), a mathematical physicist may give up completely and continue to deal with the idealization which he understands well (typical example: Haag theorem).

But let us not turn it into a war between theoretical and mathematical physicists. Let us be constructive instead. So let me ask you. If non-relativistic QM cannot be (rigorously) derived from relativistic QFT, is it justified to claim that relativistic QFT is more fundamental than non-relativistic QM? If yes, then how would you justify it?

 

[fxdung]

So String Theory is a typical QT and a modified of QFT?

 

[vanhees71]

If one removes IR divergences from QFT (e.g. by putting the whole system into a finite volume), there is no problem in principle to calculate what happens at finite time. Of course, analytical calculations are much simpler with infinite time, and this is the main reason why (even in non-relativistic QM) scattering calculations are usually performed with infinite time.

You can calculate a lot at finite time; the problem is the proper interpretation of the results. All this gets indeed worst when massless particles are involved.

 

[fxdung]

I have heard that it can not have position presentation for photon.But photon is experimental point particle,why there is not a probability notion for photon(there is not wave function for photon)?

 

[vanhees71]

Only in a very vague sense.

Nonrelativistic QFT is usually taken to be the statistical mechanics of gases, liquids, and solids made of nuclei and electrons, with electromagnetic interaction modeled as external field only. (To handle photons needs at least a partially relativistic setting.) As such it shares the abstract features of relativistic QFT, except that it takes the limit $c\to\infty$ to simplify the dynamics. However, the fields appearing in nonrelativistic QFT (the electron field and one interacting spacetime field for every nuclide appearing in the model - or an external periodic potential if none is modeled) are very different from those appearing in relativistic QFT (one space-time field for every elementary particle). I haven't seen any derivation of the former from the latter. There is a chain of reasoning going from quarks to hadrons to nuclides, considered as asymptotic free fields, but as far as I have seen none that would allow me to say that interacting nonrelativistic QFT is derivable from the relativistic version. It is regarded as an effective theory for the latter, but notebecause of a derivation but based on plausibility reasoning only.

No; I completely disagree!

It only works in the opposite direction, presented in all textbooks on statistical mechanics, by a two-step process of generalization and abstraction during which some features of QM are lost. First one generalizes the setting of QM by turning the number of particles - which in QM is a parameter only - into an operator acting on Fock space whose spectrum are the nonnegative numbers. Then one gets rid of all measurement issues by replacing the Born rule by the definition of ensemble expectations via $\langle X\rangle:=\mbox{tr} ~\rho X$, which no longer refers to observation and measurement. This allows one to consider arbitrarily large systems - which constitutes the second generalization - and the thermodynamic limit of infinite volume (which is needed to make it a QFT proper). Then one is at the level of field expectations and field correlations, which are the subject of QFT. Note that the notions of observation and measurement - the most controversial features of QM - are lost during this abstraction process.

Because of this loss, one can go only part of the way back if one tries to reverse the direction, going from nonrelativistic QFT to QM. One can consider a fixed number of particles and restrict to the eigenspace of the number operator with fixed eigenvalue $N$. This produces (restricting for simplicity to a single scalar field) the Hilbert space of totally symmetrized wave functions in $N$ 3-dimensional position coordinates $x_i$. On this Hilbert space, only those operators (constructed from the field operators in Fock space) have a meaning that commute with the number operator. This is not enough to construct position and momentum operators for the individual particles but only for their center of mass. One sees already here that one needs to make additional assumptions to recover traditional quantum mechanics.

Worse, since in the QFT description both observers and measurements are absent, one has to introduce observers and measurements and their properties by hand! In particular, the Born rule of QM, that tells what happens in a sequence of ideal measurements, must be postulated in addition to what was inherited from QFT! Unless the concept of observers and measurement are fully defined in quantum mechanical terms so that one could deduce their properties. While this seems not impossible, it certainly hasn't been done so far!

In non-relativistic QFT you often have the special case that a particle number is conserved. Then, in the microcanonical ensemble, you can entirely work in the eigenspace of the total-number operator, and then everything can be formulated in the "first-quantization formulation" aka. wave mechanics. In this special case QM and nrel. QFT are fully equivalent. However, even in the non-relativistic many-body theory the QFT formulation is more flexible and you can work with quasiparticles (phonons, plasmons and various other, partially pretty exotic, quasiparticles come to mind) whose number is not conserved, and this is the usual way condensed-matter problems are (approximately) solved with great success.

I'm, however, not aware of any systematic treatment of the non-relativistic limit of relativistic many-body QFT.

 

[meopemuk]

You can calculate a lot at finite time; the problem is the proper interpretation of the results. All this gets indeed worst when massless particles are involved.

Could you please mention some examples of finite-time calculations in a relativistic renormalized QFT, such as QED? Are they comparable with experiments?

 

[Demystifier]

I have heard that it can not have position presentation for photon.But photon is experimental point particle,why there is not a probability notion for photon(there is not wave function for photon)?

Even without a position operator one can introduce a POVM for a photon-position measurement.

 

[A. Neumaier]

to avoid IR and UV divergences of QFT, it is almost unavoidable to break Lorentz invariance in one way or another.

Only during the intermediate calculations. In the final renormalized result expressed in terms of cohrent states, where the continuum limit (for UV) and the infinite volume limit (for IR) are taken, Lorentz invariance of the S-matrix is restored exactly.

if such an idealization leads to a physically unacceptable result, it is theoretical physicists who will first give up of such an idealization.

But the idealization of Lorentz symmetry didn't lead to physically unacceptable results. On the contrary, it is verified to extreme accuracy and assumed to be valid by almost all physicists working on the smallest and the largest scales. Enen when giving up continuous spacetime one doesn't give up Lorentz symmetry! If we did, we wouldn't have any guidance left for restricting the possibilities...

is it justified to claim that relativistic QFT is more fundamental than non-relativistic QM? If yes, then how would you justify it?

it is more fundamental since at the level of unitary evolution (i.e., for isolated systems where external measurements are impossible due to lack of interaction), quantum mechanics is clearly visible to be a low energy approximation of QFT.

The differences only show in the treatment of measurement. QFT is silent about measurement and only talks about mean fields and correlations, whereas QM makes additional assumptions that allow the analysis of particles with macroscopic devices at finite times. These additional assumptions are in conflict with unitary evolution, which is considered acceptable because of the unavoidable interaction wih the measurement device. Again QFT is more fundamental since it is conceptually more parsimonious and does not require (and strictly speaking not even allow) an exernal classical world.

It is very likely that some time in the future people will be able to show in which way the additional assumptions of QM can be fully justified from QFT, by modeling the system of few particles + detector as a pure QFT system. The current trend (decoherence theory) treats it instead as a pure quantum mechanical system, with the unavoidable result that it can only shift the Heisenberg cut between system and observer to a different location. Since the measurement postulate is built in directly into the foundations, it is impossible to resolve the measurement riddle within QM! This is the deepest root of the interpretation problem in quantum mechanics. It cannot go away unless QM is understood as an approximation to a theory whose axioms are independent of measurement. QFT (with the Wightman axioms) is such a theory, and I predict that some time in the future, it will solve the measurement problem in a satisfactory way.

 

[Demystifier]

it is more fundamental since at the level of unitary evolution (i.e., for isolated systems where external measurements are impossible due to lack of interaction), quantum mechanics is clearly visible to be a low energy approximation of QFT.

OK, we can agree on that.

The differences only show in the treatment of measurement. QFT is silent about measurement and only talks about mean fields and correlations, whereas QM makes additional assumptions that allow the analysis of particles with macroscopic devices at finite times. These additional assumptions are in conflict with unitary evolution, which is considered acceptable because of the unavoidable interaction wih the measurement device. Again QFT is more fundamental since it is conceptually more parsimonious and does not require (and strictly speaking not even allow) an exernal classical world.

It is very likely that some time in the future people will be able to show in which way the additional assumptions of QM can be fully justified from QFT, by modeling the system of few particles + detector as a pure QFT system. The current trend (decoherence theory) treats it instead as a pure quantum mechanical system, with the unavoidable result that it can only shift the Heisenberg cut between system and observer to a different location. Since the measurement postulate is built in directly into the foundations, it is impossible to resolve the measurement riddle within QM! This is the deepest root of the interpretation problem in quantum mechanics. It cannot go away unless QM is understood as an approximation to a theory whose axioms are independent of measurement. QFT (with the Wightman axioms) is such a theory, and I predict that some time in the future, it will solve the measurement problem in a satisfactory way.

But we cannot agree on that. It is true that QFT books usually don't talk about measurement axioms, but I think it's only because the writers of these books don't want to repeat what has already been said in books on QM. If QFT could offer some new insight on the measurement problem, writers of QFT books would not miss the opportunity to say something about it.

 

[fxdung]

So what we must account for statistical ensemble for QFT?We only make into acount on the particles(quanta) or also on the quantum states of every particles?I think quantum statistical mechanics account both.

 

[A. Neumaier]

So what we must account for statistical ensemble for QFT? We only make into acount on the particles(quanta) or also on the quantum states of every particles?I think quantum statistical mechanics account both.

In relativistic QFT one talks about the state of a system extending over all spacetime. The (canonical, grand canonical, etc.) ensemble is the label attached in statistical mechanics to particular macroscopic states of the form $e^{-S/\hbar}$ with a nice expression for $S$. Thus in QFT, the notion of an ensemble can be taken as a synonym for the state of the macroscopic system. Nowhere in statistical mechanics (and hence in QFT) is made use of the assumption that an ensemble is interpreted in the sense of a collection of many identically prepared macroscopic objects - it is only an ensemble of many microscopic particles! Therefore the predictions of QFT apply to each single macroscopic object. Already Gibbs, who introduced the notion of an ensemble towards the end of the 19th century, noted (in his 1901 book - still quite readable!) that one must consider a statistical mechanics ensemble as a fictitious collection of copies of which only one is realized!

 

[A. Neumaier]

It is true that QFT books usually don't talk about measurement axioms, but I think it's only because the writers of these books don't want to repeat what has already been said in books on QM. If QFT could offer some new insight on the measurement problem, writers of QFT books would not miss the opportunity to say something about it.

Indeed they say something new [compared to QM foundations, but very old in terms of the physics] on measurement, and they don't miss the opportunity to say it!

In books on nonequilibrium statistical mechanics it is very obvious that whatever they compare with experiment has nothing at all to do with the kind of idealized measurements considered in QM. They talk about field expectations (such as the energy density and mass density) and certain coefficients in the formula for the state of the macroscopic system (such as local temperature and local chemical potential), and relate them to thermodynamic observables, which are measured in the ordinary engineering way. If mentioned at all, Born's rule with its assumption of external measurement is eliminated in the very first few pages of the books in favor of the formula $\langle X\rangle = \mbox{tr}~\rho X$. This formula is a much more general and much more useful axiom for quantum physics! It doesn't have the problematic baggage that the traditional, ill-defined foundations of QM have.

 

[Demystifier]

Indeed they say something new [compared to QM foundations, but very old in terms of the physics] on measurement, and they don't miss the opportunity to say it!

In books on nonequilibrium statistical mechanics it is very obvious that whatever they compare with experiment has nothing at all to do with the kind of idealized measurements considered in QM. They talk about field expectations (such as the energy density and mass density) and certain coefficients in the formula for the state of the macroscopic system (such as local temperature and local chemical potential), and relate them to thermodynamic observables, which are measured in the ordinary engineering way. If mentioned at all, Born's rule with its assumption of external measurement is eliminated in the very first few pages of the books in favor of the formula $\langle X\rangle = \mbox{tr}~\rho X$. This formula is a much more general and much more useful axiom for quantum physics! It doesn't have the problematic baggage that the traditional, ill-defined foundations of QM have.

It seems to imply that quantum statistical mechanics is based on QFT, not on QM.

But one can certainly study quantum statistical mechanics based on QM, without using QFT. (If not for photons, one can certainly do that for non-relativistic electrons.) So can quantum statistical mechanics based on QM tell us something about measurement which "pure" QM can't?

 

[A. Neumaier]

It seems to imply that quantum statistical mechanics is based on QFT, not on QM.

But one can certainly study quantum statistical mechanics based on QM, without using QFT. (If not for photons, one can certainly do that for non-relativistic electrons.) So can quantum statistical mechanics based on QM tell us something about measurement which "pure" QM can't?

Well, quantum statistical mechanics for macroscopic objects is always based on expectations and correlations only, which is the QFT setting. Even though one starts with the QM1 setting - since this is already known, by the way physics education happens everywhere -, one drops the connection to the QM foundations once the QFT fondations are established. Thus the former serve only as a motivation.

However there is a mix of quantum mechanical and field theoretic reasoning in some treatments of the measurement process. They treat the environment as a macroscopic system - typically heavily idealized as an infinite size heat bath -, and then treat system + detector + environment by statistical mechanics. See, e.g., the Lectures on dynamical models for quantum measurements by Nieuwenhuizen, Perarnau-Llobet, and Balian. It is this line of reasoning that ultimately should solve the measurement problem.

 

[Demystifier]

Well, quantum statistical mechanics for macroscopic objects is always based on expectations and correlations only, which is the QFT setting.

It is not only a QFT setting. Even in QM you have correlation functions such as $\langle x(t_1) x(t_2)\rangle$.

 

[fxdung]

What is the difference between the predictions of quantum statistical mechanics and of classical physics?Is classical physics is the result of classical statistical mechanics?

 

[A. Neumaier]

It is not only a QFT setting. Even in QM you have correlation functions such as $\langle x(t_1) x(t_2)\rangle$.

In the Schroedinger picture, which is the basis of the usual axiomatization of QM, this object doesn't exist.

Time correlations only exists after casting QM in the form of a 1+0-dimensional QFT (the Heisenberg picture), where state vectors do not evolve in time and therefore the Born rule no longer applies.

 

[A. Neumaier]

What is the difference between the predictions of quantum statistical mechanics and of classical physics?Is classical physics is the result of classical statistical mechanics?

The quantum predictions difeerr form the classical predictions by corrections of order $\hbar$. Since this is a very small quantity in macroscopic units, the corrections are negligible in macroscopic cases that can both be described classically and quantum mechanically.
In particula, one gets classical hydrodynamics and elasticity theory as macroscopic limits of classical or quantum statistical mechanics applied to the appropriate conditions, without or with the quantum corrections, respectively.

 

[Demystifier]

In the Schroedinger picture, which is the basis of the usual axiomatization of QM, this object doesn't exist.

Time correlations only exists after casting QM in the form of a 1+0-dimensional QFT (the Heisenberg picture), where state vectors do not evolve in time and therefore the Born rule no longer applies.

Then we only disagree on terminology. If by "QFT" you really mean Heisenberg picture and by "QM" you really mean Schrodinger picture, then I can agree with you. But I would prefer to use the standard terminology. Besides, did you know that QFT can be formulated in the Schrodinger picture?

 

[fxdung]

It may be that QM and classical physics applied to macro objects have the same results,but classical and quantum statistical mechanics give different results because the systems is agregate of quantum and classical particles(where two different physics applied to micro particles).So I do not know classical physics is results of what.

 

[A. Neumaier]

[

Then we only disagree on terminology. If by "QFT" you really mean Heisenberg picture and by "QM" you really mean Schrodinger picture, then I can agree with you. But I would prefer to use the standard terminology.

For the purposes of foundations, I call QFT that part of quantum theory where only expectations and correlation functions are asserted to have meaning related to experiment, and QM that part of quantum theory where the Schroedinger equation is used and Born's rule relates it to experiments. This naturally divides quantum physics in two nearly disjoint parts with completely different ontologies.

Besides, did you know that QFT can be formulated in the Schrodinger picture?

Well, there is the so-called functional Schroedinger equation, which is occasionally useful. But it is an amputation rather than a formulation of QFT since one loses in the process not only manifest covariance but also all time-correlation information. But covariance (in the relativistic case) and correlation functions (in general) are the bread and butter of most QFT applications.

 

[Demystifier]

For the purposes of foundations, I call QFT that part of quantum theory where only expectations and correlation functions are asserted to have meaning related to experiment, and QM that part of quantum theory where the Schroedinger equation is used and Born's rule relates it to experiments.

Then you should have said that at the beginning, to avoid all the misunderstandings that this non-standard terminology caused.

 

[vanhees71]

Could you please mention some examples of finite-time calculations in a relativistic renormalized QFT, such as QED? Are they comparable with experiments?

For a pretty academic example, see

http://arxiv.org/abs/1208.6565

There we were modest and came to the conclusion that one has to define quantities carefully using the idea of asymptotic states.

 

[vanhees71]

[
For the purposes of foundations, I call QFT that part of quantum theory where only expectations and correlation functions are asserted to have meaning related to experiment, and QM that part of quantum theory where the Schroedinger equation is used and Born's rule relates it to experiments. This naturally divides quantum physics in two nearly disjoint parts with completely different ontologies.

Well, there is the so-called functional Schroedinger equation, which is occasionally useful. But it is an amputation rather than a formulation of QFT since one loses in the process not only manifest covariance but also all time-correlation information. But covariance (in the relativistic case) and correlation functions (in general) are the bread and butter of most QFT applications.

That's a bit strange a view. Usually you distinguish nonrelativistic QT in the "first-quantization" and the "second-quantization formalism". The former describes systems of a fixed number of particles and can be formulated as wave mechanics, realizing the Hilbert space as $L^2(\mathbb{R}^{3N},\mathbb{C}^{2s+1})$ for $N$ particles of spin $s$ and the latter describes any many-body system of particles and/or quasiparticles be their number conserved or not. The 2nd-quantization formalism is fully equivalent for the 1st-quantization formalism if particle number is conserved and you deal with states of a fixed particle number.

Also there is no difference between the Schrödinger and the Heisenberg picture (at least not as far as I'm aware of, because I've not heard about problems like with the interaction picture in the case of relativistic QFT, where the latter strictly speaking doesn't exist due to Haag's theorem). It's just two equivalent mathematical descriptions of the same theory. They are just related by a unitary time-dependent transformation, and observables (including correlation functions of gauge invariant observables) thus do not depend on which picture you use to evaluate them.

In all cases the Born rule is used to associate formal quantities with real-world observables.