Causal Perturbation Theory - Comments

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  • #26
vanhees71
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I don't know, what you mean by "real". It's a very confusing word, because it's loaden with unsharply defined philosophical meanings. What's "real" at the lab are detector responses to what we call "particles" and "fields". Relativistic QFT is a mathematical framework to predict the corresponding transition probabilities from an initial state (usually two colliding protons, leptons, or heavy ions) to a final state (depending on what you like to meausre, e.g., the Higgs production). These transition probabilities are measured as scattering cross sections. In the theory they are given by the S-matrix elements. In this sense the S-matrix elements are "real", because they can be checked by observations.

The bare electron is not observable by definition, because it doesn't interact with anything, and thus also not with the detectors used to measure cross sections. What's "real" are in some sense the asymptotic free electrons of perturbation theory, and they are quite complicated objects. In a very handwaving way you can say they are "bare electrons with their electromagnetic field around them", i.e., a "bare electron" together with a "coherent photon state".

For details, see the above cited paper by Kulish and Faddeev. For more details, see the series of papers by Kibble. A more traditional treatment can be found in Weinberg, Quantum Theory of Fields, vol. I.
 
  • #27
A. Neumaier
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So, if I understand correctly, you are saying that the (bare) electron itself is a meaningless concept. According to the theory, the EM field is what is "real", and the electron is just a localized description of the field. Is that accurate?
You don't understand correctly. Both the electromagnetic field and the electron currents are real (measurable), photons are elementary excitations of the electromagnetic field, and electrons are elementary excitation of the electron current field, hence are as real. But they are not bare - the bare photons and electrons are meaningless auxiliary constructs that do not survive the renormalization limit.

I don't know, what you mean by "real". It's a very confusing word
Real in theoretical physics is what is gauge invariant, has a well-defined dynamics in time (since reality happens in time), approximates a real world situation (it is always to some extent an idealization).

Real things include renormalized gauge invariant operator products of fields, density matrices, and what is derived from it (e.g., S-matrix elements).

Nonreal things include all bare stuff, virtual particles, and wave functions. The latter since they are not invariant under global phase shifts; the corresponding rank 1 density matrices are invariant and hence are real (though often highly idealized) according to this definition.
 
  • #28
``Together with a particular form assumed for the interaction (which has the same form as the traditional nonquadratic term in the action, but a different meaning).''

Can you, please, expand on this a little bit? How is e.g. the T1 operator-valued distribution in causal perturbation theory constructed for e.g. φ3 theory? I think I understand the inductive step from Tn-1 to Tn, but I do not understand the start of induction, unless some renormalization is hidden here to define the normal ordered power of the free fields?
 
  • #29
A. Neumaier
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I do not understand the start of induction, unless some renormalization is hidden here to define the normal ordered power of the free fields
The whole construction happens in the asymptotic space, which even for an interacting theory is a Fock space, since asymptotic particles are free by definition. In Fock space, normal ordering is all that is required to render a polynomial, local operator meaningful. Thus you can start the induction with an arbitrary local polynomial in c/a operators.

Two restrictions come in to get the most desirable properties:
1. one wants the interaction to be covariant; then the interaction must be Lorentz invariant.
2. One wants to have only finitely many renormalization conditions. This requires that the degree of the interaction is small enough to the usual renormalizable terms.
In particular, for a scalar field theory you can take the interaction to be a linear combination of the normally ordered ##\Phi^3## and ##\Phi^4## term. If you want to preserve the discrete symmetry of the free theory, only the ##\Phi^4## term qualifies.

Both conditions are met in causal perturbation theory.

However, the whole procedure makes perturbative sense also without these requirements.
In particular, for quantum field theory in curved space-time one sacrifices condition 1, with success; see work by Stefan Hollands.
For quantum gravity one sacrifices condition 2, also with success; see, e.g.,, the living Review article
Cliff P. Burgesshttp://relativity.livingreviews.org/Articles/lrr-2004-5/ [Broken]
 
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  • #30
A. Neumaier
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In the light of the recent discussion starting here, I updated this Insight article, adding in particular detail to the section ''Axioms for causal quantum field theory''.
 
  • #31
strangerep
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In the light of the recent discussion starting here, I updated this Insight article, adding in particular detail to the section ''Axioms for causal quantum field theory''.
Is there a typo or omission in the last sentence of this paragraph:
Unfortunately, models proving that QED (or other interacting local quantum field theories) exists have not yet been constructed. On the other hand, there are also no arguments proving rigorously that such models exist. For a fully rigorous solution – a problem which for interacting 4-dimensional relativistic quantum field theories is open.
?
 
  • #32
A. Neumaier
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Is there a typo or omission in the last sentence of this paragraph:
?
Thanks for pointing it out. Indeed, the paragraph was garbled. I corrected it and added some more information. Now the paragraph reads
Unfortunately, models proving that QED (or another interacting local quantum field theory in 4 spacetime dimensions) exists have not yet been constructed. On the other hand, constructions are available in 2 and 3 spacetime dimensions, and no arguments are known proving rigorously that such models cannot exist in 4 dimensions. Finding a fully rigorous construction for an interacting 4-dimensional local quantum field theory or proving that it cannot exist is therefore a widely open problem. My bet is that a rigorous construction of QED will be found one day.
 
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  • #33
atyy
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However, the whole procedure makes perturbative sense also without these requirements.
In particular, for quantum field theory in curved space-time one sacrifices condition 1, with success; see work by Stefan Hollands.
Which papers of Stefan Hollands? I took a quick look at https://arxiv.org/abs/1105.3375 which introduces an ultraviolet cutoff, then takes it to infinity, so it seems a bit different from causal perturbation theory which "nowhere introduces nonphysical entities (such as cutoffs, bare coupling constants, bare particles or virtual particles)".
 
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  • #34
A. Neumaier
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Which papers of Stefan Hollands? I took a quick look at https://arxiv.org/abs/1105.3375 which introduces an ultraviolet cutoff, then takes it to infinity, so it seems a bit different from causal perturbation theory which "nowhere introduces nonphysical entities (such as cutoffs, bare coupling constants, bare particles or virtual particles)".
Yes, it is different. Hollands is not doing causal perturbation theory since, as I said, in his work condition 1 (i.e., covariance) is sacrificed, by using a cutoff.

Note that post #29 was more generally about starting with asymptotic Fock space, not about causal perturbation theory itself, where preserving conditions 1 and 2 throughout the construction is essential. I added a clarifying sentence.
 
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  • #35
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A couple of general comments on this "Causal perturbative theory", the first is rather cosmetic. Being as discussed in the Lattice QED thread not an exactly or completely causal(in the sense of microcausal) theory due to its perturbative limitations, isn't referring to it as a "Causal theory" a bit of a misnomer? I know there are historical reasons starting with the work of Bogoliubov and I guess that it refers to the global causality of asymptotic states rather than to the "in principle" exact (micro)causality usually explained in regular QFT textbooks when explaining the locality axiom but still, maybe that's why the title of Scharf's book uses first the less confusing phrase "Finite QED".

The other comment refers to the insistence on underlining the absence of cutoffs or series truncations(as a rigorous renormalization BPHZ schema) as some constructive property of the theory, given that as also commented in the other thread, in a perturbative setting, i.e. renormalized order by order, validity only term by term is quite a "truncation" of the theory and therefore we are indeed dealing at best with effective field theories, perhaps better defined mathematically but not constructing any theory that gets us closer to the nonperturbative local QFT. Perhaps an appropriate mathematical analogy is with numerical brute force searches of Riemann hypothesis zeros outside the critical line, that always remain equally infinitely far from confirming the hypothesis.

In this sense if one assumes from the start the existence of the non-perturbative local QFT(this is what Dyson and Feynman did), this seems to me like an empty exercise in rigor and old perturbation theory was fine, and if one doesn't assume it it is mostly useless.
 
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  • #36
A. Neumaier
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isn't referring to it as a "Causal theory" a bit of a misnomer?
It is generally used; I am not responsible for the name. I think it is called causal since causality dictates the axioms and the derived conditions for the distribution splitting.
validity only term by term is quite a "truncation" of the theory
Of course; I never claimed anything else. By making approximations, any computational scheme necessarily truncates the theory and hence violates locality. This even holds in 2D and 3D, where local QFTs have been constructed rigorously but computations still need to employ approximations.

My emphasis was on that all truncations are covariant and hence relativistic in the standard sense of the word. Only locality is slightly violated.
not constructing any theory that gets us closer to the nonperturbative local QFT.
The complete, infinite order construction satisfies all axioms, and hence the local commutation rules, in the sense of formal power series. In this sense it is closer to nonperturbative local QFT.
if one assumes from the start the existence of the non-perturbative local QFT(this is what Dyson and Feynman did), this seems to me like an empty exercise in rigor and old perturbation theory was fine
Old perturbation theory only defines the perturbative S-matrix, but not the operators, and hence not the finite time dynamics. Thus it lacks much of what makes a quantum theory well-defined.
Perhaps an appropriate mathematical analogy is with numerical brute force searches of Riemann hypothesis zeros outside the critical line, that always remain equally infinitely far from confirming the hypothesis.
This is a valid comparison. Indeed, the Riemann hypohesis, global existence of solutions of the the Navier-Stokes equations, and the construction of an interacting local QFT in 4D are three of the 6 open MIllnium problems. They share the fact that numerically, everything of interest is established without doubt but the mathematical techniques to produce rigorous arguments are not sufficiently developed to lead to success on this level.
 
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  • #37
vanhees71
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Then I have to ask again: What else do you gain in (vacuum) QFT than the S-matrix?

In the standard approach, the S-matrix is what describes the observable effects like cross sections and decay rates. It deals with transition-probability rates between asymptotic free initial states (where you have a physical definition of the states as "particles") and asymptotic free final states.

Is there an idea, that there are physically observable interpretations of states defined by the "transient" field operators and if so what is it and how can it be measured?
 
  • #38
A. Neumaier
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What else do you gain in (vacuum) QFT than the S-matrix?
I had already answered this:

BPHZ perturbation theory, say, only defines the perturbative S-matrix, but neither operators nor Wightman N-point functions, and hence no finite time dynamics. Moreover nothing for states different from the ground states. Thus it lacks much of what makes a quantum theory well-defined. Quite independent of what can be measured.

For comparison, if all that ordinary quantum mechanics could compute for a few particle system were its ground state and the S-matrix, we would have the status quo of 1928, very far from the current state of the art in few particle quantum mechanics.
 
  • #39
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not an exactly or completely causal (in the sense of microcausal) theory due to its perturbative limitations, isn't referring to it as a "Causal theory" a bit of a misnomer? I know there are historical reasons starting with the work of Bogoliubov and I guess that it refers to the global causality of asymptotic states rather than to the "in principle" exact (micro)causality usually explained in regular QFT textbooks when explaining the locality axiom
I think the name ''microcausality'' for the spacelike commutation rule, though quite common, is the real misnomer. The commutation rule is rather characteristic of locality (''experiments at the same time but different places can be independently prepared'') , as indicated by the title of Haag's book. Locality is intrinsically based on spacelike commutation and cannot be discussed without it or directly equivalent properties.

On the other hand, the relation between causality and spacelike commutation is indirect, restricted to relativistic QFT. Moreover, the relation works only in one direction since spacelike commutation requires a notion of causality for its definition, while causality can be discussed easily without spacelike commutation.

Indeed, causality (''the future does not affect the past'') is conceptually most related to dispersion relations (where causal arguments enter in an essential way throughout quantum mechanics, even in the nonrelativistic case) and to Lorentz invariance (already in classical mechanics where ''microcausality'' is trivially valid). Both figure very prominently in causal perturbation theory.
maybe that's why the title of Scharf's book uses first the less confusing phrase "Finite QED".
Scharf's QED is finite order by order, just as causal perturbation theory is causal order by order. Since Scharf does not produce finite results in the limit of infinite order you should complain against the appropriateness of the label ''finite'' with the same force as you complain against the label ''causal'' in causal perturbation theory.
 
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  • #40
vanhees71
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I had already answered this:

BPHZ perturbation theory, say, only defines the perturbative S-matrix, but neither operators nor Wightman N-point functions, and hence no finite time dynamics. Moreover nothing for states different from the ground states. Thus it lacks much of what makes a quantum theory well-defined. Quite independent of what can be measured.

For comparison, if all that ordinary quantum mechanics could compute for a few particle system were its ground state and the S-matrix, we would have the status quo of 1928, very far from the current state of the art in few particle quantum mechanics.
Yes, that's formally clear. What I don't see is what you gain physics wise. What is the physical observable related to finite-time dynamics. I also don't understand, why you say that in standard PT you don't get Wightman functions. Of course you can calculate them perturbatively within the usual formalism.
 
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  • #41
A. Neumaier
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Yes, that's formally clear. What I don't see is what you gain physics wise. What is the physical observable related to finite-time dynamics.
What do you man? The Hamiltonian together with the Schrödinger equation tell how a state changes in a finite time. But in the textbook formalism, the Hamiltonian comes out infinite and cannot be used.
I also don't understand, why you say that in standard PT you don't get Wightman functions. Of course you can calculate them perturbatively within the usual formalism.
The textbook formalism only gives the time-ordered N-point functions. How do you time-unorder them?
 
  • #42
vanhees71
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The question is, what I gain from finite-time states in QFT. The problem is first, how to interpret them. All you need are transition probabilities between observable states, which have a physical meaning. The closest to something finite in time are, e.g., long-base-line experiments for neutrinos, which you handle with the usual S-matrix theory using wave packets as asymptotic initial an final states or coincidence measurements of multi-photon states. Also here you need the corresponding correlation functions, which can be evaluated perturbatively, and afaik that's all that's needed in quantum optics to describe the observables.

The "fixed-ordered Wightman functions" should be calculable by deriving the corresponding Feynman rules for them, but again my question: to calculate which observable predictions do you need them for?
 
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  • #43
A. Neumaier
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The question is what I gain from finite-time states in QFT.
If you are only interested in the interpretation of collision experiments, nothing. But if you want to do simulations in time of what happens, you need it. You are doing such simulations, so it is strange why you ask.

As you well know, this goes far beyond BPHZ, which is the textbook material that was under discussion above. Simulations in time are usually done with the CTP formalism, which can produce all required information in a nonperturbative way, given some pertubatively computed input.

The latter must be renormalized, which is done in an a nonrigorous hoc way. Presumably it can be placed on a more rigorous basis by using the causal techniques.

This may even resolve some of the causality issues reported in the CTP literature. (This was maybe around 10 years ago; I haven't followed it up, are these problems satisfactorily resolved by now?)
 
  • #44
A. Neumaier
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The "fixed-ordered Wightman functions" should be calculable by deriving the corresponding Feynman rules for them
But there is no Dyson series for them, so Feynman rules cannot be derived in the textbook way.

And CTP only produces 2-point Wightman functions, but not the ##N##-point functions for ##N>2##.
 
  • #45
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I think the name ''microcausality'' for the spacelike commutation rule, though quite common, is the real misnomer. The commutation rule is rather characteristic of locality (''experiments at the same time but different places can be independently prepared'') , as indicated by the title of Haag's book. Locality is intrinsically based on spacelike commutation and cannot be discussed without it or directly equivalent properties.

On the other hand, the relation between causality and spacelike commutation is indirect, restricted to relativistic QFT. Moreover, the relation works only in one direction since spacelike commutation requires a notion fo causality for its definition, while causality can be discussed easily without spacelike commutation.

Indeed, causality (''the future does not affect the past'') is conceptually most related to dispersion relations (where causal arguments enter in an essential way throughout quantum mechanics, even in the nonrelativistic case) and to Lorentz invariance (already in classical mechanics where ''microcausality'' is trivially valid). Both figure very prominently in causal perturbation theory.
I basically agree with the above. There is a degree of mixing between the different terms in certain contexts though. In the free theory like in the classical you mention there is a causality and microcausality trivial overlapping, that relativistically is reinforced by the symmetries that include time reversals and spacetime translations, the latter is also in the putative non-perturbative local QFT. There you have the triad of Poincaré covariance, locality and unitarity inextricably united through the analytic dispersion relations extended from the Kramers relations to the whole complex plane.

Of course in the perturbative theory this has to be spoiled a little, and (always assuming the existence of the non-perturbative theory it approximmates which is what allows the necessary deformation of the free fields) locality splits from the other two.

Scharf's QED is finite order by order, just as causal perturbation theory is causal order by order. Since Scharf does not produce finite results in the limit of infinite order you should complain against the appropriateness of the label ''finite'' with the same force as you complain against the label ''causal'' in causal perturbation theory.
Well, I think I complained about the concepts behind it(which is what I care about) enough in other threads to get my point across. My comment as I said was purely about the word "causal" and its connotations and conflation with locality and (micro)causality that are known to create endless confusions in Bell-like discussions.
 
  • #46
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Wightman propagators obviously belong to the axiomatic QFT proposal, I guess by what you get with "the standard perturbative theory" vanhees71 refers to the plain Feynman propagator.
 
  • #47
A. Neumaier
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Wightman propagators obviously belong to the axiomatic QFT proposal, I guess by what you get with "the standard perturbative theory" vanhees71 refers to the plain Feynman propagator.
Yes, but he had asked what can be computed from approximate field operators that are not accessible by standard perturbative theory. I gave as examples the unordered N-point correlation functions. These are defined independent of the Wightman axioms; the latter just specify their desired covariance and laocality conditions.
 
  • #48
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Since CPT is UV-complete, does it suggest a physically motivated regularization scheme for old-fashioned QFT?
 
  • #49
A. Neumaier
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Since CPT is UV-complete, does it suggest a physically motivated regularization scheme for old-fashioned QFT?
It is a physically motivated renormarization scheme replacing old-fashioned QFT, making regularization unnecessary. Why should one still want to regularize?
 
  • #50
vanhees71
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If you are only interested in the interpretation of collision experiments, nothing. But if you want to do simulations in time of what happens, you need it. You are doing such simulations, so it is strange why you ask.

As you well know, this goes far beyond BPHZ, which is the textbook material that was under discussion above. Simulations in time are usually done with the CTP formalism, which can produce all required information in a nonperturbative way, given some pertubatively computed input.

The latter must be renormalized, which is done in an a nonrigorous hoc way. Presumably it can be placed on a more rigorous basis by using the causal techniques.

This may even resolve some of the causality issues reported in the CTP literature. (This was maybe around 10 years ago; I haven't followed it up, are these problems satisfactorily resolved by now?)
Yes, I'm doing simulations what's going on in heavy-ion collisions, but it's far from the claim that you have a physical interpretation of "transient states". Take, e.g., the calculations I've done for dilepton production, i.e., the production of ##\text{e}^+ \text{e}^-##- and ##\mu^+ \mu^-##-pairs from a hot and dense strongly interacting medium.

The medium itself is described by either a fireball ("blastwave") parametrization of a hydrodynamic medium or in a coarse-graining approach with a relativistic transport-model simulation. In any case one maps the many-body system's evolution to a local-thermal-equilibrium situation.

For the dilepton-production rates we use spectral functions from an equilibrium-qft calculation. The QFT observable here is the thermal electromagnetic-current autocorrelation function, i.e., the "retarded" expectation value of ##\hat{j}^{\mu}(x) \hat{j}^{\nu}(y)## wrt. the grand-canonical ensemble statistical operator. This can be evaluated either in terms of the Matsubara formalism and then analytically continued to the corresponding retarded two-point function or you directly use the Schwinger-Keldysh real-time formalism to directly evaluate this retarded two-point function. What enters the dilepton-production-rate formula is the imaginary part of the retarded two-point function, the famous McLerran formula.

So everything from QFT is within the standard formalism of thermal correlation functions. There's no need to physically interpret transient states all "particles" observed are calculated in the sense of the usual concept of asymptotic free states.

The description of the bulk medium is in terms of semi-classical transport theories, behind which when looked at them from the point of view of many-body QFT, also is the interpretation of "particles" in terms of asymptotic free states.
 

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