# 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 (13,209 points)
edited Dec 21, 2017

@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.

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.

@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: It was you who started first.

@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.

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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 (5,000 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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