The origins of the S-matrix. S-matrix theory (SMT)

theory (SMT). The concept of an S-matrix was introduced by John Wheeler in 1937 in the context of a theoretical description of the scattering of light nuclei. In modern notation, the S-matrix consists of elements Sαβ, where α, β= 1,2, . . . , N, that give the relative strengths of the asymptotic forms of the wave function for various reaction channels:

ψ_αβ⁺ (r_β)→ i

2k_βr_β[δ_αβe^−ikαrα−S_αβe^ikβrβ],

wheree^−ikαrα represents the incident wave, ande^ikβrβ represents the scattered wave. From calculus emerges the expression of the total cross section:

σ_t=σ_sc+σ_r = 2π

k²_α[1−ReS_αα] = 4π

k_αImf_αα,

whereσscis the elastic (scattering) cross section,σris the interaction (reaction) cross section, and f_αα is the forward scattering amplitude. This expression forms the so-calledoptical theorem, which establishes the crucial result that the total cross section is completely fixed once the imaginary part of the forward scattering amplitude is known. Therefore once the elements of the scattering matrixS_αβ are given, all the cross sections can be calculated by their means.

Apparently independently of Wheeler, Heisenberg proposed at the begin-ning of the 1940s a program whose central entity was a matrix which he termed the ‘characteristic matrix’ of the scattering problem. He conceived this pro-gram as an alternative to QFT. His explicit aim was to avoid any reference to a Hamiltonian or to an equation of motion, and instead to base his theory only upon observable quantities. Heisenberg’s programmatic papers outlined the hope for a theory capable of predicting the behavior of all observed par-ticles, together with their properties, based on symmetries and constraints on the S-matrix: unitarity and analyticity.

As appears very much in nonrelativistic quantum mechanics, the basic probability amplitudes allowed for the prediction of the outcomes of experi-ments. The output of Heisenberg’s theory consists of those probability am-plitudes which directly correspond to measurable quantities. The S-matrix elements connect the initial, asymptotically free stateψi to the final, asymp-totically free stateψ_f:

ψ_f =S_{f i}ψ_i,

whereS_{f i} represents the overlap of ψ_f with a given ψ_i asS_{f i} =< ψ_i |ψ_f >.

Thus, the chance of beginning in an asymptotic state ψ_i and ending in the asymptotic stateψf is

|< ψ_i |ψ_f >|²=|S_{f i}|² .

Since the probability of starting in ψi and ending up is some allowed stateψ must be unity, the conservation of probability requires the unitarity relation

S⁺S=I.

However, for a relativistic quantized field theory, it has never been demon-strated that any realistic interacting field system of fields has a unitary solu-tion. While solutions of formal power series have been generated, satisfying unitarity to each order of the expansion, the series itself cannot be shown to converge.

The success of the renormalized QFT did undercut Heisenberg’s original motivation for his S-matrix program – the inability of QFT to produce finite and unique results. However, in the late 1950s and early 1960s, the inability of the even renormalized QFT to provide quantitative results for the strong interactions led once more to a new S-matrix program. Jeffrey Chew, the father of the renewed S-matrix, surmised that

finally we have within our grasp all the properties of the S-matrix that can be inferred from field theory and that future development of an under-stating of strong interactions will be expedited if we eliminate from our thinking such field-theoretical notions as Lagrangians, bare masses, bare coupling constants, and even the notion of elementary particles. (Chew 1962: 2)

Apart from unitarity, other properties of the S-matrix proved to be important for the development of the program. Unitarity’s twin principle, analyticity, had been already related in QFT to the causality constraints upon the scat-tering processes. The suggestion was made that the S-matrix be considered an analytic function of its energy variable, so that scattering data could (in prin-ciple) constrain the values of bound-states energies. Thirring pointed out that the causality requirement in QFT can be implemented by demanding that the effects of the field operatorsφ(x) andφ(y) acting at spacelike separated points x and y be independent. This condition was used for the derivation within perturbation theory, of dispersion relations for forward photon-scattering to the lowest order in e². The key point here is that these relations were derived from causality constraints.

A noticeable feature of this theoretical development is the pragmatic char-acter of the S-matrix approach. Marvin Goldberger (1955) stated it quite clearly:

We made rules as we went along, proceeded on the basis of hope and conjecture, which drove purists mad. (Goldberger 1955: 155)

Quite often, key mathematical instruments were used without rigorous foun-dation. Instead, intuition and previous empirical success of similar procedures motivated numerous theoretical decisions. Thus, in his 1961 paper on the applications of single variable dispersion relations, Goldberger remarked:

It is perhaps of historical interest to relate that almost all the important philosophical applications of the nonforward dispersion relations were car-ried out before even the forward scattering relations were proved rigor-ously. (Goldberger 1961: 196)

It was only in 1957 that a rigorous proof for the forward case was provided, but no proof for all scattering angles nor for massive particles was ever given.

Dispersion relations were simply assumed to hold in these cases as well and were successfully applied to the analysis of both electromagnetic and strong interaction scattering data.

Murray Gell-Mann (1956) put together the list of theoretical constraints on the S-matrix: Lorenz invariance, crossing, analyticity, unitarity, and asymp-totic boundary conditions. He expected them to be in principle enough to specify the scattering amplitudes of QFT without having to resort to any spe-cific Lagrangian. It is worth reemphasizing that this expectation was actually the main attraction of SMT. Certainly, Gell-Mann did not consider it to be anti-QFT, but rather an alternative to the standard QFT approach of us-ing Lagrangians. Relatively soon afterwards (1958), Chew expressed the even bolder hope that the S-matrix equations might provide a complete dynamical description for strong interaction physics.

Nonetheless, the main actors on the HEP scene put SMT and QFT in direct opposition. In 1961, Chew emphatically rejected QFT, not so much for being incorrect as for being empty or useless for strong interactions:

Whatever success theory has achieved in this area [strong interactions]

it is based on the unitarity of the analytically continued S-matrix plus symmetry principles. I do not wish to assert as does Landau [1960] that conventional field theory is necessarily wrong, but only that it is sterile with respect to strong interactions and that, like an old soldier, it is destined not to die but just to fade away. (Chew 1961: 3)

Ironically, the last phrase best describes the fate of SMT itself.

Lev Landau was notorious for being most dismissive of QFT:

The only observables that could be measured within the framework of the relativistic quantum theory are momenta and polarizations of freely

moving particles, since we have an unlimited amount of time for measur-ing them thanks to momentum conservation. Therefore the only relations which have a physical meaning in the relativistic quantum theory are, in fact, the relations between free particles, i.e. different scattering ampli-tudes of such particles. (Landau 1960: 97)

At the same time, SMT had declared adversaries. For example, the prominent field theorist Francis Low (of whom Chew had been a student!) maintained that

The S-matrix theory replaced explicit field theory calculations because nobody knew how to do calculations in a strongly coupled theory. I believe that very few people outside the Chew orbit considered S-matrix theory to be a substitute for field theory. (Low 1985; correspondence with Cushing 1990)

Most participants in the debate had a rather pragmatic and conciliatory position. Stanley Mandelstam maintained that preference for one theory over the other depends on which one is more likely to provide the best results rather than on any question of principle. Similar views were expressed by Gell-Mann and Goldberger.

The S-matrix was eventually abandoned as an independent program due to its insurmountable calculational complexity, as well as to several concep-tual difficulties – for example, the physical basis of the analyticity constraints (Chew 1961), and the anomaly that the electromagnetism constituted to SMT.

Improvements were achieved with succeeding versions of the program (the du-ality version and the topological SMT version), but only at the price of even more complex mathematics. On the other hand, neither calculational com-plexity, nor conceptual muddiness were absent from the ‘winning camp’, QFT – suffice it to remind one of the ad hoc character of some renormalization manoeuvres.

Let us draw the conclusions of this sketchy presentation. First, if QFT had not run into difficulties (divergent integrals at higher orders of the per-turbation expansion, and inapplicability to strong interactions), it is unlikely that SMT would have ever existed. But theorists were able to have QFT over-come the divergence problems through renormalization, and in the late 1970s, to formulate a local gauge principle along with a mechanism of spontaneous symmetry breaking, which allowed QFT to cope with the strong interactions.

Second, each time QFT made a comeback, the overwhelming majority of scientists re-embraced it. The emphasis on this flipping back and forth between QFT and SMT is of great significance for our argumentative strategy.

By underscoring the repeated pragmatic changes of the theoretical apparatus,

the fragility of the ontological commitment in either QFT or SMT is shown.

This in turn, invites an instrumentalist interpretation of the two theories.

Third, SMT has never been properly falsified. It was abandoned mainly because of its calculational complexity. However, QFT is also a highly complex mathematical construct. The comparatively better prospects of development of QFT turned out to be decisive, yet this kind of judgment allows for social factors to come into play. For example, as Pickering (1984) aptly observes, QFT’s progress depended upon the choice made by experimentalists to study a class of very rare scattering events, which QFT could handle, to the exclusion of a much more common class of events, which it could not. It is thus not irrelevant that the HEP community displays a pyramidal organization, on the apex of which few dominant personalities make crucial theoretical decisions, which in turn open (or close) research programs.