S-Matrix Singularity Structure in the Physical Region. I. Properties of Multiple Integrals

Abstract

We obtain the real singularities and corresponding discontinuities of a class of multiple integrals over real contours. Our aim is to give a unified treatment, obtaining, by elementary mathematical methods. both previously known results and some new generalizations. In a subsequent paper the results are applied to unitarity integrals.

David Olive: his life and work

“…(a) (b) [7] Related terms are generated by other terms in the unitarity equation and together they imply a cut in (s, t)-plane with discontinuity given by [8] just as in perturbative quantum field theory (QFT). In a series of three papers (15)(16)(17) with his student, Michael Bloxham, and John Polkinghorne, David was able to extend this approach to show that for physical values of the momenta, unitarity requires scattering amplitudes to be singular on the arcs of curves where Landau (1959) had determined that singularities occur for Feynman diagrams in QFT, and that the discontinuities associated with these singularities are given by the rules obtained for perturbative QFT by Cutkosky (1960).…”

“…by an adjoint representation Higgs field, to a subgroup H = K × U(1)/Z, where Z = K ∩ U(1) is discrete; K can be identified with a 'colour' gauge group and U(1) with electromagnetism. They showed that, for any smooth solution, the magnetic charge g associated with the U(1) factor satisfies the generalized Dirac condition, exp(igQ) ∈ K , [15] where Q is the electric charge operator, which generates the U(1). If k, the element of K defined by [15], equals 1, then this condition is equivalent to the original Dirac condition [12] and this is the case for G = SU(2), where K is trivial.…”

“…They showed that, for any smooth solution, the magnetic charge g associated with the U(1) factor satisfies the generalized Dirac condition, exp(igQ) ∈ K , [15] where Q is the electric charge operator, which generates the U(1). If k, the element of K defined by [15], equals 1, then this condition is equivalent to the original Dirac condition [12] and this is the case for G = SU(2), where K is trivial. For G = SU(3), in case (ii) k = 1, but in case (i) k = 1, which is why the condition [12] can be violated in this case.…”

“…For G = SU(3), in case (ii) k = 1, but in case (i) k = 1, which is why the condition [12] can be violated in this case. Corrigan and Olive noted that, if k = 1, the generalized condition [15] implies that [12] holds for the electric charges of colour singlet particles, establishing a link between colour and fractional electric charges.…”

The Guiding Influence of Stanley Mandelstam, from S-Matrix Theory to String Theory

“…With colleagues in Cambridge, Olive developed his methods further to describe in some generality the singularities of the S-matrix at real points of the physical region of the Mandelstam variables and the corresponding discontinuities [18]. However, little progress has been made in describing singularities, real or complex, outside the physical region, which would be necessary in order to discuss the validity of the Mandelstam representation.…”