Action-based quantum field theory and causal perturbation theory

Question

What is the difference between action-based quantum field theory and causal perturbation theory?

The former can be found in any textbook on quantum field theory, the latter is exposed for QED in the book G. Scharf, Finite quantum electrodynamics, Springer, Berlin 1989. (My references are for this edition. There is also a second edition from 1995.)

Other quantum field theories including gauge theories are treated in the book G. Scharf, Quantum gauge theories: a true ghost story, Wiley, New York 2001.

This is the first of a number of planned contributions to PO where I answer questions discussed in my Theoretical Physics FAQ in an updated and polished form.

Comments

Is it a question or are you going yourself explain the difference?

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It is a question. But my last sentence announced that I'd answer this, which I did.

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Thanks for answering. You refer to his book, but we need more details on "causality".

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The 1995 edition of Scharf's Finite quantum electrodynamics is available as a PDF if your institution has an appropriate subscription to Springer: http://link.springer.com/book/10.1007%2F978-3-642-57750-5, or (with the same institutional subscription) for USD25 as a print-on-demand softcover. But I also see that Dover has taken up the book (as of now, USD22.10 from http://www.amazon.com/Finite-Quantum-Electrodynamics-Approach-Edition/dp/0486492737).

Answer 1

The main difference is that causal perturbation theory, while producing the same renormalized perturbation series as the traditional action-based approach, is from the start free of divergences, since it only works with free fields (that serve to define irreducible representations of the Poincare group with physically correct mass and spin), and nowhere introduces nonphysical terminology (such as bare coupling constants, bare or virtual particles). Thus it 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.

In contrast, canonical quantization works with physical, distribution-valued fields satisfying ill-defined nonlinear field equations, and path integral quantization uses path integrals, whose definition cannot be made rigorous at present. This lack of mathematical rigor shows in the occurrence of logical difficulties in the derivation of the formulas, although these ultimately lead to good, renormalized formulas whose predictions agree with experiment.

Scharf's book is mathematically rigorous throughout. He nowhere uses mathematically ill-defined formulas, but works throughout with mathematically well-defined distributions using the microlocal conditions appropriate to the behavior of the Green's functions. These enable him to solve recursively mathematically well-defined equations for the S-matrix by a formal power series ansatz, which is sufficient to obtain the traditional results.

Since Scharf has no action, all physics is introduced axiomatically. With a few later additions, Scharf poses the problem in Section 3.1: To find a formal series $S(g)$ of the form (3.1.1) such that the key properties
  (3.1.13) = unitarity (in the projected form (3.11.25), because of gauge invariance),
  (3.1.16) = translation invariance,
  (3.1.19) = Lorentz invariance, 
  (3.1.23) = causality,
  (3.3.2), (3.6.31) = stable vacuum state, and
  (3.6.26), (3.7.36) = stable single particle states 

hold. (Unfortunately, Scharf clouds the issue a little by not introducing all axioms in one place but postponing the last two to the place where they are needed - to fix the splitting ambiguities. But this is a matter of didactical exposition, not one of mathematical sloppiness.)

These properties axiomatically characterize a successful relativistic quantum field theory, apart from the fact that for a fully rigorous solution, $S(g)$ should be an operator-valued functional of $g$ rather than only a formal series. To single out a particular field theory on needs, in addition to the general axioms, an interaction - (3.3.1) for QED.

The causality requirement (3.1.23) is a relaxed form of the exponential law. it says that the composition of two causally unrelated unitary transformations by $S(g_1)$ and $S(g_2)$ adds canonically, thus expressing that effects in causally unrelated regions are independent.

On a heuristic level, $S(g)$ is the mathematically rigorous version of the time-ordered exponential $S(g)=Texp(\int dx g(x)T_1(x))$, where $T_1(x)$ is the physical interaction (for QED given in (3.1.1)). It is not too difficult to show that the locality properties of the quantum fields discussed in Weinberg's QFT book imply (3.1.23) at the level of rigor of Weinberg's book. This is the ultimate reason why both approaches give the same final results, though through very different routes. Only the causal perturbation theory route can claim logical coherence, due to its mathematical rigor.

On p.139, Scharf discusses perturbation theory in ordinary quantum mechanics, where working with naive time-ordering is admissible and gives finite results, and explains (in subjunctive language) why the same (i.e., the traditional approach bare theory plus perturbation theory) becomes mathematically meaningless in quantum field theory since it performs operations that are not well-defined. Then he spends Section 3.2 in showing the mathematically correct procedure, and applies it in Section 3.3 to QED. (In his other book about the true ghost story, he successfully extends the approach to other quantum field theories.)

The mathematical correct procedure is determined by microlocal theory, a mathematically well-known technique for the analysis of linear partial differential equations. Microlocal theory tells when the product of two distributions is well-defined. If one understands these conditions (which in terms of physics is roughly what comes under the heading of dispersion relations, but expressed in precise mathematical terms) then one can tell precisely which splittings are mathematically valid. On case of QED, this gives a 2-dimensional space of solutions, whence there are 2 parameters that are fixed not by the requirement of correct splitting but by the requirement of stability of the vacuum and the single particle states.

The important point is that everything can be uniquely determined from the axioms and the interaction. There is neither an ambiguity nor a contradiction - everything is determined by the rules of logic in the same way as for any mathematical construction of unique objects defined by axioms (such as the real numbers).

Scharf's interaction $ej\cdot A$ in (3.3.1) looks the same as in the traditional formalism, but its interpretation is quite different as (unlike in the action-based approach, where bare quantities figure in all formulas) $j$ and $A$ are the physical (renormalized) current and vector potential to order zero, and $e$ is the physical charge. Scharf explicitly remarks that a bare charge appears nowhere. 

Similarly, the other parameter in the theory, the electron mass $m$ (which enters through the free field content of QED), is the physical mass (a zero of the self-energy, Scharf p.176).

Bogolubov & Shirkov show in their book (cited by Scharf on p. 130) how one can construct local operators from S(g). Scharf does not do this, but gives in (2.10.2) a particular example of the B&S recipe - the construction of the local current for a charged particle in an external field. More generally, constructing S(g) for a theory including sources in fact constructs the complete content of QFT. (The second edition of Scharf's book discusses this in Section 4.9.)

Scharf's construction of QED (as far as it goes) is mathematically impeccable. Indeed, it can be understood as a noncommutative analogue of the construction of the exponential function as a formal power series. The only failure of the analogy is that in the latter case, convergence can be proved, while in the former case, the series can be asymptotic only (by an argument of Dyson), and it is unknown how to modify the construction to obtain an operator-valued functional $S(g)$.

To summarize: 

Scharf solves the mathematical problem of finding S(g) satisfying the axioms stated above for the given interaction, and solves it (in perturation theory) uniquely in a mathematically consistent way. He also computes the first order results explicitly and shows that they recover the results verified by experiments. This shows that these axioms and causal perturbation theory as a method for solving them for a given interaction constitutes a good quantum field theory explaining all experiments of QED (and for other interactions most experiemnts in elementary particle physics).

Comments

Arnold, what prevents Scharf from divergent expressions, "causality"? If so, what does exactly it consist of (the physical meaning.)? Because, if he does not use equations as acausal, what is his causality? In other words, due to what doess his product of distributions with equal arguments become zero? Is this trick equivalent (in result) to the usual subtraction?

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Scharf avoids divergent expressions by never multiplying two distributions whose product is not defined. This is unrelated to causality; it is just a prerequisite for mathematical consistence, that (as you rightly criticize) is frequently ignored in the literature - but not by Scharf.

Causality is needed in order to get the right physical properties. (Note that Scharf has no action, and all physics is introduced axiomatically.)  As I said, causality is expressed by the requirement (3.1.23), which is a relaxed form of the exponential law. it says that the composition of two causally unrelated unitary transformations by $S(g_1)$ and $S(g_2)$ adds canonically, thus expressing that effects in causally unrelated regions are independent. See also the discussion on p.99-100 and in the book mentioned there.

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He never uses  product of distributions with equal arguments - these are not ``his'' as you claim!

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But he uses the same products as in the standard QED. What does make the arguments in his construction different?

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Please give a reference for ``his product of distributions with equal arguments''. I cannot find them in his text.

He uses the same letters for field operators, but he uses them very differently from how they are used in the textbooks. You should read what he is actually doing rather than projecting your prejudice on his work. 

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This passage shows that he makes the same subtractions, so his "causality" condition imposes another value to the automatically obtained one. Nothing new.

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Ha, this seems an interesting way to look at QFT I have not yet heard of before. Can things, like the standard model for example, be described in this way too ?

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Yes, see the ghost story book! There is even work on gravity, but this is incomplete. More work on on causal perturbation theory can be found at http://scholar.google.com

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This is not the product of distributions at the same point that you had claimed. (Also, you seem to use a different edition of the book than I. I don't find an equation (3.2.56) in my copy.)

Finally if two sources make the same operation somewhere, it doesn't mean that they have the same faults. What is wrong in making a well-defined subtraction in the course of Scharf's argument? He get the correct results without any logical blunder or approximation or reinterpretation of terms. 

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I will read more carefully the book at home to show where he arrives at the same products.

As to "what is wrong", I think the physical meaning of $j\cdot A$ is an interaction that inevitably includes the undesirable self-induction. There may be many ways to get rid of its contribution: introducing bare parameters to cancel it, introducing conterterms to subtract it, even putting this contribution to zero as a way of imposing "physical condition" to our results, etc. All this is nothing but discarding this effect. "Causality" in this book is also boils down to obtaining the desirable result instead of undesirable one. And the problem here is not only in ill-defined products of distributions, no - with cut-offs one can make them finite and even small. The problem is that they are not necessary at all, they modify the phenomenological coefficients of equations and make the results disagree with simple experiments. So the physical effect of self-induction should be removed completely. That what is done by "renormalization".

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'' that inevitably includes the undesirable self-induction.'' It includes a finite self-energy computed by Scharf to first nontrivial order in Section 3.7; cf. the discussion p.176. This contribution is consistent with experimental results. Why should this be undesirable?

Nowhere is any attempt to get rid of this (or any other) contribution, and there is no need (or use) at all of an UV-cutoff. The consequences of the axioms are developped with full rigor, and the results happen to agree with experiment, proving the physical consistency of the approach. The logical consistency is already guaranteed by only using valid mathematical operations.

You want to impose your own preferences that are irrelevant to theory and interpretation, this is why we never reach an agreement.

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Scharf writes (in the first edition on page 176) "Since $\hat{\Sigma}(p = m) = 0$ according to (3.7.43), the physical mass $m$ remains unchanged." It is possible, Arnold, when the etire correction to $m$ is discarded (his trick). The remainder (although called a finite self-energy correction) is not a self-induction, but the desirable "radiation reaction" term. In QM in the ground bound state it gives a real energy shift; in CED the energy shift exists too, but it is "imaginary" (a line width) because no "ground state" is possible. In QM the excited states have complex shifts in analogy with CED.

Then he continues (following the mainstream):

"Our point of view will always be that the undetermined parameters in the theory should be fixed in such a way that the quantities appearing in the lowest orders agree with the physical quantities. Then only one physical mass $m$ and charge $e$ appear in the theory. This is in fact the only possible point of view which is physically correct because the asymptotically incoming and outgoing particles have this physical mass and charge, too."

This confirms my words about constants being physical in the regions out of external fields: I disagree with your reasoning about bare mass in absence of external fields. Also, he admits undetermined parameters in solutions which is strange because he should have none. Anyway, fixing "undetermined parameters" in solutions is worse than fixing the initial equations right.

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The crucial part of the development seems to me to be Section 3.2, "Splitting of Causal Distributions", which you gloss in half a sentence, rather than the way in which causality is enforced, Eq. (3.1.23), using test functions that have disjoint supports in time, which you devote most energy to in your Answer. AFAICT, splitting implements the Epstein-Glaser subtractions, without which there would be divergences ["the correct distribution splitting with the right singular order \(\omega\) is terribly important. Incorrect distribution splitting leads to ultraviolet divergences", p181, 1995 edition of the Scharf]; it seems a reasonably well-organized way to implement the Epstein-Glaser procedure, but I don't see that it gives a clearer motivation for them, at least not as Scharf presents the mathematics.

The problem I have is that I can't see how to motivate a correct distribution splitting instead of an incorrect distribution splitting, and the only help you give in your Answer is that Scharf presents "the mathematically correct procedure". My somewhat vague impression is that Scharf works with a multiplication of generalized functions that satisfies axioms such as those of Colombeau, and that something of that sort is sufficiently consistent and unique for the purposes of QFT, up to the usual worries due to Dyson.

Scharf and Lagrangian QFT seem not much different in terms of these considerations.

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" I can't see how to motivate a correct distribution splitting instead of an incorrect distribution splitting":

Microlocal theory tells when the product of two distributions is well-defined. If one understands these conditions (which in terms of physics is roughly what comes under the heading of dispersion relations, but expressed in precise mathematical terms) then one can tell precisely which splittings are mathematically valid. On case of QED, this gives a 2-dimensional space of solutions, whence there are 2 parameters that are fixed not by the requirement of correct splitting but by the requirement of stability of the vacuum and the single particle states.  

Vladimir objects to these undetermined parameters, although Scharf determines them. He is like someone complaining about a solution of 5 linear equations in 5 variables by Gaussian elimination because there is an intermediate stage of the determination of the solution where two variables are still undetermined (and the others depend on their values). 

The important point is that everything can be uniquely determined from the axioms and the interaction. There is neither an ambiguity nor a contradiction - everything is determined by the rules of logic in the same way as for any mathematical construction of unique objects defined by axioms (such as the real numbers).

Microlocal theory is quite different from Colombeau theory (though there are relations). In Colombeau theory each distribution is realized by a continuum of generalized functions whose product always exists. The ambiguitiy recurs there since many different generalized function may be obtained by squaring a generalized function equivalent as a distribution to the delta distribution.

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@ArnoldNeumaier: Arnold, you simplify my objections. Even with divergent $\delta m$ and $\delta e$ people (physicists and mathematicians) "determine" these ill-defined (or undetermined, as you like) corrections as zero from physical requirements imposed to solutions. In my opinion, the physical requirements must be imposed to the equations. Then there is no need in (re)defining solutions.

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Scharf solves the mathematical problem of finding S(g) satisfying the axioms stated above for the given interaction, and solves it (in perturation theory) uniquely in a mathematically consistent way. This is indisputable.

He also computes the first order results explicitly and shows that they recover the results verified by experiments. This is also indisputable.

As a consequence he is entitled to claim that these axioms and causal perturbation theory as a method for solving them for a given interaction gives a good theory explaining all experiments of QED (and for other interactions most experiemnts in elementary particle physics).

The fact that you have objections about the philosophy expressed in some of the formulations doesn't change a thing in this logic. Your objections (regarding infinities, deletions, undetermined or ill-defined parameters) are completely irreleant. As the past has shown you won't change your position, but it is so far from reasonable that I won't discuss it further.

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OK, don't discuss it. But then, don't write about me either, please. I prefer to discuss physics and mathematics rather than my personality.

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@ArnoldNeumaier: Please, don't take me wrong, I am writing it here just to express my disappointment. You wrote: "He also computes the first order results explicitly and shows that they recover the results verified by experiments. This is also indisputable."

Yes, he derives a Klein-Nishina formula, for example. He borrowed everything from the standard QED, he knows what he wants to obtain and he obtains it. In particular, the first order Klein-Nishina formula. He forgot simply to multiply it by zero. The final QED result is (among others) that there is no elastic process of scattering, i.e., without radiations. To obtain a non zero result, one has to sum up soft diagrams to all orders, otherwise the result is zero.

Somewhere else you wrote that nobody, including me, has a clue how to make the QED series convergent in the absolute sense. They are asymptotic, by Dyson's argument. But Dyson forgot that summing up soft corrections to all orders is equivalent to taking some part (a soft one, which is the strongest) into account exactly. This exact formula contains the charge or $\alpha$ in a non-trivial way and leaves behind another series whose convergence was not studied. There is no need to expand this exact formula in series. The perturbative remainder gives then another series, probably as convergent as any regular Taylor series. So, if we manage to take some (the strongest) part of interaction into account exactly, we may get a better convergent series. You are right, G. Scharf does not discuss this possibility, but I am aware of this possibility.

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Dyson's argument for the non-convergence of asymptotic series in QFT also holds for QED with massive photons, where there are no infrared problems, in particular no soft photons. Thus accounting correctly for soft photons (which is done not by Scharf but by Kulish-Faddeev and by Zwanziger) alone cannot account for the convergence problems.

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@ArnoldNeumaier: Your argument is not convincing since I speak of taking into account some part of interaction exactly in the initial approximation and of not developing it in series. In case of massive photons the Dyson argument still goes about everything developped in series.

For example, the function $f(x)=e^{-x}-\frac{x}{1+x}$ has a finite radius of convergence, if expanded in Taylor series. If we manage to sum up effectively a part of this series into $h_1(x)=-\frac{x}{1+x}$ (by a direct summation or by choosing another initial approximation for $f(x)=h_1(x)+h_2(x)$), then the remaining series of $h_2(x)$ will be as convergent as the expansion of $e^{-x}$.

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Rather than polluting other discussions with your half-baked ideas, bake them first: Do - for QED proper (i.e., with gauge invariance and Poincare invariance)  - the stuff you claim can be done, and we can discuss the results.

You always choose toy examples in which essential parts of the difficulties are missing, so they have no convincing power. In your current example, both sequences converge for small $x$, hence this says nothing about Dyson's argument. 

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@ArnoldNeumaier: As you know, I am not in the right position to carry out this research, I am not emplyed in academia and I do not have any freedom.

As to my toy function, it shows the very essential thing - choosing a better initial approximation makes the residual series better convergent. In paticular, you may apply my function to a strong coupling case $x \approx 1$ to see the benefits of my idea.

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This is a bad excuse. You carry around these ideas for many years, and I don't believe that you have less spare time than me. (I have a full-time job in math and do all physics in my spare time.) If you had spent the time defending your ideas in the internet with applying it to QED proper, you'd have either solved it successfully (and then proper ground for discussion) or (far more likely) detected the obstacles that make it unlikely that your ideas will work.

In particular, the difficulty is not in generating an idea but to show that your idea is compatible with gauge invariance and Poincare invariance. Nothing in your contributions over the years suggests that one can do this.

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I am guilty in nothing, Arnold. I do my best in my situation.

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You are guilty of calling well-established work wrong, without sufficient justification except that it violates your personal philosophy. This is what earns you all the downvotes and a growing resentment of anyone who discusses with you in vain.

You are not entitled to such judgments unless you can do better. It is not enough to have half-baked ideas for re-deriving things in your own philosophy that in anyone else's judgment don't lead anywhere.

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I refer to a Feynman lecture to show that I am not alone to worry about these issues. We can remember the other fathers of QED. Nothing has changed since that time. I also clearly discuss the moments when we modify things because these things are wrong. I cannot present more justifications than that. I am afraid you just do not read it. I notice that "renormalizators" won with their number, that is why there are so many downvotes. As I said, despite their voting down and chasing me, I always speak of physics and mathematics to show where it gets wrong. So I do my best in tough conditions. My toy examples are purposed to show how and when we make mistakes. You never discussed my models concretely. You reject them without discussing. Meanwhile, you yourself use toy models in your explanations and nobody blames you for that. You want me to bake up a whole thing on my own whereas it took generations of well-paid physicists to develop the standard QED. Are you serious, Arnold? Maybe it is already enough of chasing me? If you have some concrete objections to my physical and mathematical arguments, present them without discussing my personality. Otherwise it is personal attacks.

Answer 2

The principal aspect of Scharf's construction is that it only constructs nonlocal S-matrix observables, endeavoring to make no hypotheses about local structure, except that the term \(T_1(x)\) is presented in terms of local fields, for example in Section 3.4. Although QFT (as it is presented in Itzykson&Zuber, for example) does not calculate local observables such as smeared Wightman functions for interacting fields, nonetheless local observables are in principle part of the theory.

This particularly shows up in the different definition of locality, in terms of the S-matrix, Equation 3.1.23 of Scharf 1995, \(S(g_1+g_2)=S(g_1)S(g_2)\), whenever \(g_1\) and \(g_2\)are temporally separated, in contrast to the conventional requirement in terms of the local field, as microcausality.

As my comment on Arnold's self-Answer suggests, it seems that the construction of S-matrix elements is quite similar to the Epstein-Glaser procedure, and Feynman diagrams play an almost identical role in the two approaches, precisely because of my note above, that the term \(T_1(x)\) is presented in terms of local fields (and takes the same form as the interaction Lagrangian density).

Comments

Hi Arnold, I admit that I by no means correctly understand causal perturbation theory. But how is physics valid at different scales linked in this approach. In the Wilsonion EFT framework there are no infinities either, at the scales where infinities used to be because people tried to apply a certain EFT beyond its domain of validity, just different operators become relevant whereas others can be neglected if I understand this correctly. Is this Wilsonian picture kept in causal perturbation theory, or do things work different? Sorry if I am making a mess of things, this is why I don't ask a "regular" question about it ...

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Classical electrodynamics is not the topic here, and I am not interested in it since,  for almost 70 years, it is superseded by QED.

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To get physics at different scales you need to use a renormalization scheme with an intrinsic physical scale (i.e., one independent of the cutoff), which is most accurate at energies close to the value of the scale. This is absent in Scharf's causal perturbation theory treatment of QED; there the scale corresponds to the infrared fixed point. This is the appropriate scale for electromagnetic effects, which are visible at very low (e.g., everyday) energies. (See, however, the remark on other renormalization conditions in this answer.)

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It would be better if you turned it into a proper question and link to it from here, as the discussion will then be easier found by later users.

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At the level where classical ED is a valid approximation to QED the finite terms are negligibly small. Thus there is a problem only if one extrapolates CED outside its range of validity.

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I agree. I am just saying that people made such test (elementary, to be exact) charges act on themselves with the full self-field rather than with the radiated field only, so they obtained the self-induction effect bringing a correction to the phenomenological mass (which was already good). For a finite radius the $m_{em}$ is finite, but not necessary. It is discarded. Or the self-interaction $j\cdot A$ is furnished with counter-terms subtracting the extra mass corrections. It means the original idea of self-action is not completely perfect.
 

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It is not I who extrapolates CED, but "fundamental" physicists.