Causal Perturbation Theory - Comments

[A. Neumaier]

A. Neumaier submitted a new PF Insights post

Causal Perturbation Theory

Continue reading the Original PF Insights Post.

 

[atyy]

When you write "As a consequence, causal perturbation theory is mathematically well-defined, and falls short of a rigorous construction of quantum field theories only in that the perturbative series obtained is asymptotic only, and that the infrared limit is not well understood.", do you mean that this method constructs relativistic QED without a UV cut-off in finite volume?

 

[samalkhaiat]

1) Regarding (P) Poincare’ covariance, I suppose you meant to write $$U(L) \ S(g) \ U^{\dagger} (L) = S(Lg) ,$$ where $L$ (an element of Poincare’ group) need not be unitary. $g(x)$ is an ordinary function and transforms according to $$g(x) \to L g(x) = g(L^{-1}x) .$$
2) Why do you attribute this framework to Epstein and Glaser? I far as I know, this formalism was first introduced by Bogoliubov and Shirkov in their 1955 paper and then included it in their classic text on QFT in the 1957 edition (translated to English in 1959). In my 1976 edition of the book, the formalism is explained in Ch.III, Sec.17&18.

 

[A. Neumaier]

@atyy: There is no UV cutoff; all UV problems are handled by mathematically safe distribution splitting. That g(x) must have compact support restricts the construction to finite volume. Letting g(x) approach 1 is the infinite volume limit, whose existence or properties are not analyzed in causal perturbation theory. The latter has for gauge theories the usual IR problems that must be handled by coherent state techniques.
QED is not constructed in finite volume as S(g) is found only as a formal series and not as an operator.

@samalkhaiat: ad 1) - indeed, my shorthand notation was supposed to mean this.
ad 2) Epstein and Glaser built on earlier work by Bogoliubov and Shirkov 1955, which is not fully rigorous as it depends on a mathematically ill-defined notion of time ordering. The contribution of Epstein and Glaser consisted in making time ordering rigorous by distribution splitting techniques borrowed from microlocal analysis.

Thanks to both of you for your comments. I updated my text to better reflect all this.

 

[vanhees71]

Great article!

Aren't the infrared problems at least FAPP solved by using the correct asymptotic states, which are rather coherent states than plane waves and finally equivalent to the usual soft-photon resummation techniques of the traditional approach (Bloch, Nordsieck, et al)? Shouldn't smoothed field operators cure both the UV and IR problems within the perturbative approach? I've to read Scharf's books in more detail, but it's of course simpler to just ask here in the forum :-).

 

[A. Neumaier]

@vanhees71: Scharf's books contain very little (QED book, Section 3.11: Adiabatic limit) about how to handle the IR problem. He does not use coherent states, although the latter is the right way to proceed. But at present, coherent state arguments in QED, while sufficient FAPP, are not mathematically rigorous. There is some rigorous IR material in the literature (under the heading ''infraparticle''), but it hasn't been incorporated so far into the causal approach.

Smoothing can be done in different ways, and traditional cutoffs are one way of doing it - the question is always how to undo the smoothing at the end to recover covariant results. The causal approach is throughout manifestly covariant; it smoothes the IR in the most general (and hence covariant) way by introducing g(x), but handles the UV part by microlocal analysis. The latter was developed to rigorously analyze classical PDEs, and it is mathematically natural to expect that one would need these techniques also for a rigorous quantum version.

 

[Greg Bernhardt]

Nice first entry @A. Neumaier!

 

[vanhees71]

I see. What I had in mind are the (of course no mathematically rigorous) papers

Kulish, P.P., Faddeev, L.D.: Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4, 745, 1970
http://dx.doi.org/10.1007/BF01066485

Kibble, T. W. B.: Coherent Soft‐Photon States and Infrared Divergences. I. Classical Currents, Jour. Math. Phys. 9, 315, 1968
http://dx.doi.org/10.1063/1.1664582

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. II. Mass-Shell Singularities of Green's Functions, Phys. Rev. 173, 1527–1535, 1968
http://dx.doi.org/10.1103/PhysRev.173.1527

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. III. Asymptotic States and Reduction Formulas, Phys. Rev. 174, 1882–1901, 1968
http://dx.doi.org/10.1103/PhysRev.174.1882

Kibble, T. W. B.: Coherent Soft-Photon States and Infrared Divergences. IV. The Scattering Operator, Phys. Rev. 175, 1624, 1968
http://dx.doi.org/10.1103/PhysRev.175.1624

 

[A. Neumaier]

Yes. The Kulish-Faddeev paper you cited counts as the definite paper on the subject. It settles the QED infrared problems on the nonrigorous level. Partial rigorous results along these lines are in

D. Zwanziger, Physical states in quantum electrodynamics, Phys. Rev. D 14 (1976), 2570-2589.

and in papers by Bagan et al. and http://http://arxiv.org/abs/hep-th/0411095 , but no synthesis with the causal approach has been tried, as far as I know.

 

[atyy]

If the causal perturbation construction is only a formal power series, is there any way to transition from that to something with physical meaning?

In the informal viewpoint, we accept that QED is only an effective theory and has a cutoff, so our starting theory is, say, lattice QED. Then we argue that if we take a Wilsonian viewpoint and run the renormalization flow down, we recover the traditionally constructed power series plus corrections suppressed in powers of the cutoff. Here the starting theory rigourously exists, since it is quantum mechanics. It has a UV cutoff and exists only in finite volume, but physically that should be ok since our experiments are low energy and always in finite volume. The part of this framework that is not rigourous is the renormalization flow to low energies. The informal thinking is that since the Wilsonian framework makes physical sense, it should be possible for it to be made rigourous one day.

Can causal perturbation theory fit within the Wilsonian framework of QED as an effective field theory?

Or does causal perturbation theory need to come from a truly Lorentz invariant theory without a UV cutoff?

 

[A. Neumaier]

Causal perturbation theory reproduces the perturbative results of standard renormalized perturbation theory, which also produces only a series expansion of the S-matrix elements.

Unfortunately, in 4D we have no method at all to construct more than a power series (or another uncontrolled approximation like a lattice simulation). One gets approximate numbers with physical meaning by the standard means - neglecting higher order terms, or numerical versions of resummation techniques such as Pade or Borel summation, hoping that these will result in good approximations.

Since causal perturbation theory is manifestly covariant, it cannot be viewed in terms of a cutoff, and hence not in terms of Wilson's renormalization (semi)group framework. However, as described here, there is still a renormalization group (a true group, unlike Wilson's). It describes (not the effect of changing the nonexistent cutoff) but the effect of a change of parameters used in the specification of the renormalization conditions. This is just a group of exact reparameterizations of the same family of QFTs.

 

[atyy]

There are rigourously constructed QFTs in 2 and 3 spacetime dimensions. Does causal perturbation theory construct a theory, or become a mathematically meaningful approximation when applied to such theories?

 

[A. Neumaier]

In 2D and 3D, causal perturbation theory only constructs the series expansion of the then rigorously existing operator S(g), but asserts nothing about the latter's existence. As such it may serve as an alternative covariant way to approximately compute things that are known to exist by the traditional (noncovariant, lattice-based) existence proofs together with Haag-Ruelle scattering theory.

 

[atyy]

In the asymptotic series I am familiar with, say, the ones that come up in Stirling's approximation, the error is on the order of the first term omitted.

Let's suppose we have rigourously constructed a relativistic QFT (say in 2D or 3D), then, if I understand correctly, the causal perturbation theory is a correct way to construct the power series. Is the error also on the order of the first term omitted?

 

[A. Neumaier]

Generically yes. In truncating any asymptotic series $\sum_k a_k \phi_k(x)$ in $x$ with nicely behaving \$phi_k(x)$ and coefficients of order 1, the error is of the order of $the first neglected $\phi_k(x)$. But the error can be much bigger than the first neglected term itself, if the coefficient $a_k$ happens to be tiny. So that the error is of the order of the first term omitted is just a rule of thumb.

With this understanding it is valid for the series calculated by causal perturbation theory (which is identical with the series calculated by other good renormalization methods).

 

[atyy]

Going back to QED in 3+1D, the match of the traditional renormalization to experimental data suggests that there is a good mathematical theory in which the outcome of traditional renormalization makes sense. It seems there are two heuristics on how to proceed.

1) the Wilsonian view, in which we can use lattice theory at some high energy, and the Lorentz invariance is not necessarily exact, but only a very good low energy approximation.

2) the causal perturbation view - we still have no rigourous theory, but a rigourous formal series that is (like traditional renormalization) nonetheless indicated by experiment to be mathematically completable in a Lorentz invariant way. At present Yang-Mills is the best candidate for constructing a rigourous 3+1D relativistic QFT, but since the experiemental match in QED is so good, what are the potential relativistic UV completions of QED? Would causal perturbation theory be consistent with either (A) asymptotic safety or with (B) the introduction of new degrees of freedom to complete the theory?

 

[A. Neumaier]

I believe that QED exists as a rigorous and mathematically complete theory, and that causal perturbation theory provides its correct perturbative expansion around the free theory. There is no proof that would establish the contrary. The arguments that would suggest its incompleteness are based on logically unjustified inference from uncontrolled approximations without any force.

But the perturbative expansion is not the whole story since there must be modifications in the IR, due to the fact that the physical electron is an infraparticle only.

 

[atyy]

Ha, ha, brave statement. So that corresponds to the informal notion of asymptotic safety, and from what you have explained, I do see that causal perturbation theory strongly suggests that to be the case.

But just to explore possibilities - if there were a rigourous 4D QFT whose fundamental degrees of freedom are different from QED's - but from which QED emerged at some lower energy. Would it be from the rigourous point of view justified to apply causal perturbation theory to such a theory? Or is it only strictly applicable in the case of asymptotic safety?

 

[Feeble Wonk]

But the perturbative expansion is not the whole story since there must be modifications in the IR, due to the fact that the physical electron is an infraparticle only.

I've been trying to follow your discussion, with something (significantly) less than complete understanding. But I'd appreciate it if you could expand on this comment. Are you saying that the physical electron itself does not actually exist other than as a focal point for it's cloud of virtual photons?

 

[A. Neumaier]

Causal perturbation theory can be applied to any quantum field theory for which you can write down a free field theory and a first order term for the S-matrix consistent with causality. Everything else is then determined. This is completely independent of asymptotic safety. The standard model is a theory of which QED is the low energy limit, and yes, you can apply causal perturbation theory. Any other fundamental theory if it satisfies the two criteria just mentioned works as well.

 

[A. Neumaier]

The physical electron in QED exists, but it is (like an electron in experiments, but unlike a - nonexistent - bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood. In a nonperturbative construction of QED, the infraparticle structure must be explicitly represented. This means that one would have to do perturbation theory starting in place of the Hilbert space of a free field with a Hilbert space featuring infraparticles instead. People have been trying to build such a Hilbert space but nobody so far has married it with a perturbative construction in the spirit of causal perturbation theory - except in case of an electron in an external electromagnetic field, which is already quite technical.

(Described in terms of bare stuff - which is frequently done though it is meaningless imagery - the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)

 

[vanhees71]

Just to add another point of view of (non-rigorous) renormalization. A cutoff is not renormalization but regularization, and you have to take the regularization to the "physical point" in order to achieve a Lorentz covariant S-matrix. In the usual framework, cutoff-regularization is not a wise choice, because it complicates the calculations. What's common to all regularization schemes is that you somehow have to introduce an energy-momentum scale.

The breakthrough in some sense was the discovery of dimensional regularization by 't Hooft during his PhD work (adviced by Veltman). There the physics of the scale is a bit obscured, because it comes in very formally by assuming that the coupling parameters in the Lagrangian keep their energy-momentum dimension at arbitrary space-time dimensions. E.g., to keep the em. coupling in QED dimensionless you have to introduce a scale factor $\mu$ in the loop-integration measure. The great thing is that you have very easily defined "mass-independent" renormalization schemes (minimal subtraction or modified minimal subtraction, which is the standard use in QCD, leading to the definition of the physical scale parameter $\Lambda_{\text{QCD}}$ (see the Review of Particle Physics).

Another point of view is not to regularize at all but to do the subtractions directly at the level of the loop-integral's integrands (BPHZ renormalization). There the introduction of a scale, particularly in the case of theories where massless degrees of freedom are involved, becomes very natural: You cannot subtract at 0 external momenta of the Feynman diagrams, because you hit the cuts of the proper vertex functions in the complex energy plane. So you have to subtract at some space-like momentum, which introduces the energy-momentum scale. Also you should not choose an on-shell renormalization scheme, because this introduces artificial IR problems if massless degrees of freedom are present.

The RG can be formulated (particularly simple for mass-independent renormalization schemes) from the invariance of the physical properties (i.e., S-matrix elements) from the choice of the energy-momentum scale (of course only approximately in the perturbative sense). For an RG treatment of the simple $\phi^4$ model in the BPHZ scheme see

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[A. Neumaier]

Indeed, causal perturbation theory is just the rigorous version of BPHZ renormalization.

 

[Feeble Wonk]

The physical electron in QED exists, but it is (like an electron in experiments, but unlike a - nonexistent - bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood. In a nonperturbative construction of QED, the infraparticle structure must be explicitly represented. This means that one would have to do perturbation theory starting in place of the Hilbert space of a free field with a Hilbert space featuring infraparticles instead. People have been trying to build such a Hilbert space but nobody so far has married it with a perturbative construction in the spirit of causal perturbation theory - except in case of an electron in an external electromagnetic field, which is already quite technical.

(Described in terms of bare stuff - which is frequently done though it is meaningless imagery - the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)

 

[Feeble Wonk]

The physical electron in QED exists, but it is (like an electron in experiments, but unlike a - nonexistent - bare electron) inseparable from the electromagnetic field carried by its charge. This implies that the asymptotic states (which in perturbation theory are treated incorrectly as free Dirac states without a field) are in fact more complicated objects, called infraparticles. The latter exist in a far more real sense than the Dirac electrons. But their behavior is mathematically not fully understood.
>>>>>>
(Described in terms of bare stuff - which is frequently done though it is meaningless imagery - the infraparticle consists of a virtual bare electron plus a cloud of infinitely many virtual soft photons; this is manifested in traditional S-matrix calculations by summing over corresponding outgoing states to remove the IR divergences.)

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?

 

[vanhees71]

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.

 

[A. Neumaier]

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.

 

[philosophus]

``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 T_n-1 to T_n, 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?

 

[A. Neumaier]

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 into 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
http://relativity.livingreviews.org/About/authors.html#burgess.cliffhttp://relativity.livingreviews.org/Articles/lrr-2004-5/

 

[A. Neumaier]

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''.