1943–1959: S-matrix theory
String theory is an outgrowth of
S-matrix theory,^{
[1]} a research program begun by
Werner Heisenberg in 1943^{
[2]} (following
John Archibald Wheeler's 1937 introduction of the S-matrix),^{
[3]} picked up and advocated by many prominent theorists starting in the late 1950s and throughout the 1960s, which was discarded and marginalized in the mid 1970s^{
[4]} to disappear by the 1980s. It was forgotten because some of its mathematical methods were alien, and because
quantum chromodynamics supplanted it as an experimentally better qualified approach to the
strong interactions.^{
[5]}
The theory was a radical rethinking of the foundation of physical law. By the 1940s it was clear that the
proton and the
neutron were not pointlike particles like the electron. Their
magnetic moment differed greatly from that of a pointlike
spin-½ charged particle, too much to attribute the difference to a small perturbation. Their interactions were so strong that they scattered like a small sphere, not like a point. Heisenberg proposed that the strongly interacting particles were in fact extended objects, and because there are difficulties of principle with extended relativistic particles, he proposed that the notion of a space-time point broke down at nuclear scales.
Without space and time, it is difficult to formulate a physical theory. Heisenberg believed that the solution to this problem is to focus on the observable quantities—those things measurable by experiments. An experiment only sees a microscopic quantity if it can be transferred by a series of events to the classical devices that surround the experimental chamber. The objects that fly to infinity are stable particles, in quantum superpositions of different momentum states.
Heisenberg proposed that even when space and time are unreliable, the notion of momentum state, which is defined far away from the experimental chamber, still works. The physical quantity he proposed as fundamental is the
quantum mechanical
amplitude for a group of incoming particles to turn into a group of outgoing particles, and he did not admit that there were any steps in between.
The
S-matrix is the quantity that describes how a collection of incoming particles turn into outgoing ones. Heisenberg proposed to study the S-matrix directly, without any assumptions about space-time structure. But when transitions from the far-past to the far-future occur in one step with no intermediate steps, it is difficult to calculate anything. In
quantum field theory, the intermediate steps are the fluctuations of fields or equivalently the fluctuations of virtual particles. In this proposed S-matrix theory, there are no local quantities at all.
Heisenberg proposed to use
unitarity to determine the S-matrix. In all conceivable situations, the sum of the squares of the amplitudes must be equal to 1. This property can determine the amplitude in a quantum field theory order by order in a perturbation series once the basic interactions are given, and in many quantum field theories the amplitudes grow too fast at high energies to make a unitary S-matrix. But without extra assumptions on the high-energy behavior, unitarity is not enough to determine the scattering, and the proposal was ignored for many years.
Heisenberg's proposal was reinvigorated in the 1956 when
Murray Gell-Mann recognized that
dispersion relations—like those discovered by
Hendrik Kramers and
Ralph Kronig in the 1920s (see
Kramers–Kronig relations)—allow a notion of causality to be formulated, a notion that events in the future would not influence events in the past, even when the microscopic notion of past and future are not clearly defined. He also recognized that these relations might be useful in computing observables for the case of strong interaction physics.^{
[6]} The dispersion relations were
analytic properties of the S-matrix,^{
[7]} and they imposed more stringent conditions than those that follow from unitarity alone. This development in S-matrix theory was based on Murray Gell-Mann and
Marvin Leonard Goldberger's (1954) discovery of
crossing symmetry, another condition that the S-matrix had to fulfil.^{
[8]}^{
[7]}
Prominent advocates of the new "dispersion relations" approach were
Stanley Mandelstam^{
[9]} and
Geoffrey Chew,^{
[10]} both at
UC Berkeley at the time. Mandelstam discovered the
double dispersion relations, a new and powerful analytic form, in 1958,^{
[9]} and believed that it would be the key to progress in the intractable strong interactions.