# S-matrix analyticity and causality

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It is commonly stated that the analyticity of the S-matrix is the reflection of spacetime microcausality, specifically that of field observables (their commutativity if spacelike separated) if the S-matrix happens to come from a local field theory.

This is stated very explicitly but without any argument in Gribov 69, section 1.1.2. An often repeated but simplistic argument is in Eden-Landshoff-Olive-Polkinghorne 66, (1.1.1)-(1.1.5) . The only substantial argument that I am presently aware of is offered in Weinberg 95, vol 1, section 10.8..

Is there any account that does detailed justice to the claim that the analyticity of the S-matrix is the proper reflection of spacetime microcausality?

[From the discussion below:] What I'd like to see is precisely this: A derivation from the axioms of causal perturbation theory of the properties considered by Chew-Mandelstam.

edited Dec 21, 2017

I edited your question since in the discussion below it became apparent that you asked for something quite different from what your original wording suggested.

I wonder if "Causality and dispersion relations and the role of the S-matrix in the ongoing research ( pdf )"  helps. At least, it may explain the above citations ( historical sureness that analytic properties of scattering amplitudes ... arise as consequences of causal propagation properties ). In many old publications spacetime causality is often referred as the light cone theory. What is the strict analyticity definition ? does it introduce the relation by construction ?

From what I understood from Weinberg, he seems to be saying

$$\text{Microcausality}\implies \text{A very specific analytic structure}$$

instead of saying analyticity in general comes from microcausality.  This squares with Arnold's answer below.

@JuaYiyang: The same holds for the analyticity properties (e.g., dispersion relations) mentioned (by reference though not explicitly) in the OP.

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No. Analyticity and the associated dispersion relations have nothing to do with microcausality.

Analyticity of the S-matrix is basic even to nonrelativistic quantum mechanics. This is thoroughly discussed in the book ''Scattering theory of waves and particles" by R.G. Newton. Only ordinary causality (i.e., causes are always prior to effects) is involved in the derivation. In particular, the dispersion relations are derived in Sections 10.3.4; see also the references on p.296. The derivation applies to any Hamiltonian dynamical description, and in particular to the Hamiltonian formulation of QFT discussed by Weinberg.

What is specific to relativistic quantum field theory is only the crossing symmetry, whose derivation depends on microcausality in the form of spacelike commutativity. See Weinberg I, Figure 6.4 (p.269) for the simplest case, and p.467 and p.554 for other occurrences.

The book by Eden et al. tries to argue that analyticity and crossing symmetry together might be enough to fix the S-matrix. A modern, more powerful endeavor with a similar goal is causal perturbation theory. There, in order to get stronger results, stronger analyticity ($S(g)$ analytic in $g$ in the linked article) and crossing assumptions (condition (C) in the linked article) are made (but analyticity is then weakened again by working only on the formal power series level). Dispersion relations are discussed in the causal framework by Scharf in his book of finite QED (in Section 2.8, starting just before (2.8.37)).

answered Dec 20, 2017 by (15,488 points)
edited Dec 21, 2017

I wonder whether taking soft radiation into account ruins/modifies arguments about analyticity of S-matrix?

@lVladimirKalitvianski: My statements are valid for massive field theories, where there is a mass gap, so that the Haag-Ruelle scattering theory applies.

In theories containing massless fields, such as in QED, almost nothing is known rigorously, and people generally work on ad hoc assumptions. However, perturbatively, the infrared finite S-matrix for QED are rigorously constructed via causal perturbation theory in the thesis by Paweł Duch, ”Massless fields and adiabatic limit in quantum field theory”. (See also here.) The S-matrix is here an asymptotic series, of course, so that nothing is said about analyticity.

@ArnoldNeumaier: Thank you, Arnold, for your answer. From my (physicist) point of view, the masslessness of photons in QED means that we always have photons in the initial and the final states, so the appropriate mathematical description is actually via the corresponding density matrix with its machinery and analytical properties for all involved variables.

@lVladimirKalitvianski: No. The S-matrix always transforms pure in-states into pure out-states, even in the infrared-finite constructions.

The dressing of the electron states consists not of producing a mixed state but of (roughly) replacing each renormalized but infrared unphysical electron state by a tensor product with a coherent photon state, which is a superposition of infinitely many soft virtual photons of arbitrarily small energy, such that the total energy remains finite. The result is still a pure state.

No real photons are involved, but an electromagnetic Coulomb-like field assciated with the coherent state (and only figuratively composed of all soft virtual photons together).

Thanks for the pointer to Newton's book. The pointer to Weinberg is to the section I had mentioned in the question. Your remark about "stronger analycity conditions" in causal perturbation theory is vague and mysterious. What I'd like to see is precisely this: A derivation from the axioms of causal perturbation theory of the properties considered by Chew-Mandelstam.

I have now browsed through Roger G. Newton's book "Scattering theory of waves and particles" that you pointed to. I see discussion of scattering in classical field theory and in quantum mechanics. Maybe I am missing it: Where does the author discuss the S-matrix (and its analyticity) in (perturbative) quantum field theory?

@ArnoldNeumaier: As there is no physical mechanism of preventing the real photons from existing (and you do not mention it), then they exist in the initial and final states. They are not only those emitted with charges in question, but also background photons of the experimental setup. This is the reality to describe. Generally, it is not a coherent field. Now, we can cool our experimental setup and keep it cold, so no black-body radiation exists in the background. Also, we somehow get rid of the accelerator fields and have ideally only a filed of a constantly moving charge with the velocity $v$. Naturally, it is a Lorentz-transformed Coulomb field. But we are speaking of S-matrix, i.e., of interactions of charges, and this interaction makes radiation different from Lorentz-transformed Coulomb fields. This radiated field has its own variables and parameters entering the scattering matrix. Experiments and calculations show that the soft spectrum of this radiation is always present because it is easy to excite (probability is close to unity). Also, we include such processes in our experiments all together, i.e.,we deal with inclusive picture. In my opinion, an adequate description of such situations is in terms of density matrix, especially keeping in mind that we never get rid of radiation appearing in the experimental setup.

@lVladimirKalitvianski: Real photons also exist asymptotically, and are represented in the S-matrix context as pure photon states. The total in-state is a superposition of a tensor product of dressed electron states and photon states. Just as for in-states in in nonrelativistic QM, except that in the latter case we have ordinary bound states in place of dressed electron states and photon states. Nowhere is there a need for density matrices in the theoretical part.

Of course, in a real experiment, it is impossible to produce pure states, so one has to work with density matrices. But this is true already in nonrelativistic problems, so there is no need to mention this in the theoretical analysis (which is always idealized).

@UrsSchreiber: Newton discusses QM only, but this applies also to the canonical formalism of QFT. I added details in my answer. If this is not enough, please edit your original question and state there which properties you want to see derived. Your question was misleading since it sounded as if you looked for some sort of equivalence or characterization rather than just a proof of the dispersion relations.

Real photons also exist asymptotically, and are represented in the S-matrix context as pure photon states. The total in-state is a superposition of a tensor product of dressed electron states and photon states.

It is a statement to prove. Otherwise they are in mixed states. As you do not provide anything, I think you are wrong here. Note, the density matrix formalism includes pure state exotics, but certainly the former is a more general formalism.

@VladimirKalitvianski: No. The S-matrix is defined as linear operator, and hence acts on pure states! It is never defined in the context of mixed states. The latter is handled by classical averaging.

In any case, the thesis by Pawel Duch constructs perturbatively an S-matrix of the kind I described.

@ArnoldNeumaier: Your objections ("No, etc.") imply that I say something wrong, whereas what I say is correct. Again, take an S-matrix element and then calculate a measurable cross section. It is an inclusive cross section! It contains at least a new variable, say, the energy "resolution" $E$. This thing corresponds to a mixed state transition probability. (Why do you insert after @ something like l or | in front of my name?)

@VladimirKalitvianski: The | was a typo. Concerning inclusinve cross sections of the most general kind, we have the formula $Pr(\rho_{out}|\rho_{in})=Tr\rho_{out} S^*\rho_{in}S$. This reduces in the special case of pure states to the traditional textbook formula $Pr(\psi_{out}|\psi_{in})=|\langle \psi_{out}|S|\psi_{in}\rangle|^2$.

In both cases, $S$ is the same S-matrix, which (as any unitary operator) maps pure states into pure states.

Thus nothing is needed beyond a proper construction of the S-matrix. Perturbatively, with the thesis by Duch, everything is rigorously in place. The only problem is that nobody yet knows how to sum the resulting formal power series in the correct way, and thus give it an unambiguous numerical meaning.

@ArnoldNeumaier: Yes, there is something that goes beyond the perturbative S-matrix. That is why summation of the soft photon contributions is so necessary. As I mentioned many times, following textbooks, the numerical value of $\alpha$ itself is not sufficient to say it is a small parameter. $\alpha$ goes always in combination (product) of other problem variables, and for soft photons the corresponding combination is big. For that reason we have IR divergence when we deal exclusively with S-matrix elements. Summation of the soft photon contributions is an essential step in calculations meaningful functions. The easiest way to see it is to look at the solution for the photon emission by a classical current $J_{\rm{cl.}}$. The solution - coherent states - is not an analytical function of $\alpha$ and cannot be expanded in the series without IR divergences; this is what we encounter in S-matrix. Thus the density matrix formalism is a must in theories with massless excitations.

Concerning the summation result, it is effectively reduced to another choice of the initial approximation containing already $\alpha$ in a non-trivial way. The remaining series has (hopefully) much better numerical behaviour than an asymptotic series with the zero convergence radius. I guess, it has a finite radius of convergence and is an analytical function of $\alpha$.

@VladimirKalitvianski: You should make yourself acquainted with these new results (from 2017!)

The perturbative S-matrix constructed by Duch has no infrared divergences anymore - it is different from the textbook S-matrix because it uses dressed asymptotic electron states rather than the customary Dirac electron states. Thus a summation over soft photon states (leading to density matrices) is no longer necessary, and your critique (valid for the usual textbook treatment) does not apply anymore. In a sense, with the dressing, Duch accomplishes what you wanted to achieve with your proposals.

@ArnoldNeumaier: It would be a great relief to me, if he did it. I have already browsed his thesis and I have not seen anything interesting. I will read his thesis more carefully now.

CONTINUATION: I browsed it again and found the following:

1) The Title "Massless fields and adiabatic limit in quantum field theory" sounds intriguing.

2) In the Abstract the phrase: "In this approach each interaction term is multiplied by a switching function which vanishes rapidly at infinity, and thus plays the role of an infrared regularization" sounds familiar.

3) Statements on page 114-115: "The Wightman and Green functions characterize the local properties of the theory and are well-defined even in models with long-range interactions such as the quantum electrodynamics and the Yang-Mills theories. These theories suffer from the infrared problem which emerges when one investigates the scattering of particles. For example, in the case of the quantum electrodynamics the strong adiabatic limit does not exist even for the first order correction to the scattering matrix. Another manifestation of
the infrared problem is a non-standard form of the mass-shell singularities of the Green
functions [Kib68] which makes impossible the application of the LSZ reduction formula
114 in the computation of the S-matrix elements. For this reason, in theories with long-range
interactions one usually restricts attention to the so-called inclusive cross sections which,
however, characterize the scattering event only partially. The correct definition of the
scattering matrix in models with long-range interactions is still an unsolved problem.
" are disappointing. The author achieved nothing and it is so because he obviously does not feel the physics. All his constructions, theorems, lemmas, etc. resulted in nothing useful. And it is certainly not what I conceived in my pet (electronium) theory.

Nevertheless I found some originality in his thesis, namely, his notation made me wonder whether I fell behind the modern science forever or not.

@VladimirKalitvianski: You achieved nothing with your electronium model, not even Lorentz invariance!

On the other hand, Duch achieves something highly nontrivial, namely the (covariant) weak adiabatic limit. Its relevance is explained at the beginning of Chapter 4. It gives a full perturbative and IR-finite construction of the Wightman and Green's functions, i.e., the field correlators of all orders and the time-ordered correlation functions of all orders. The latter give the ingredients inherent in the perturbative expression for the S-matrix.

The strong limit, if it exists, must equal the weak limit, and then gives the same expression for the perturbative series for the S-matrix. Thus from a computational point of view, everything has been achieved. What is missing on the perturbative level is a purely technical mathematical step, namely showing that the strong limit of the time-ordered correlation functions exists. On p.57 this is promised to second order in work to be published elsewhere.

(Of course, all this does not even touch the nonperturbative issues in constructing QED. Thus there still remain important unsolved mathematical problems in the foundations of QED.)

@ArnoldNeumaier: As a physicist, I see nothing new in his work. Anyway, the shameful adiabatic limit has been used well before him. Inclusive cross sections mean precisely this: the charge interaction with soft modes is taken into account exactly since:

1) it exists always and

2) it is very necessary.

Dr. Duch does not understand this.

@ArnoldNeumaier: It was you who started first.

@UrsSchreiber: The review by D. Iagolnitzer,

"The Analyticity Program in Axiomatic Quantum Field Theory." Rigorous Quantum Field Theory (2007): 141-159.

gives an overview over various aspects of analyticity in quantum field theory, and its relation to causality. Iagolnitzer also wrote a book about scattering (Scattering in quantum field theories: the axiomatic and constructive approaches, 2014), which might interest you.

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The relation between analyticity and causality is some completely general phenomenon, well-known e.g., in signal processing. Its simplest incarnation is the fact that if a transfer function is supported on the positive real axis of time, then it's Fourier transform can be analytically continued to the upper half-plane. (But I recognise that this is probably not enough to fully answer the question).

For relativistic theories, causality is essentially equivalent to locality (if you can go outside the light cone, then by an appropriate Lorentz transformation you can go backward in time). But of course, it is not the case for non-relativistic theories, which can be both causal and non-local (as mentioned in Arnold Neumaier's answer).

answered Dec 20, 2017 by (5,140 points)

I know the claims and the simpler incarnations. What I am after are actual derivations not restricted to toy models.

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