He was born inwards Johannesburg, South Africa. His woman nurture has lived at that topographic point from the wealthy years of gold, his manlike someone nurture Boris moved at that topographic point from Latvia. Throughout his childhood, they lived inwards a small-scale town but moved dorsum to Johannesburg. He was forced to earn a practical – chemic – score which he has never used inwards his life. But he returned to theoretical physics in addition to got his degrees inwards 1954 (B.A.) in addition to 1956 (PhD) inwards Cambridge in addition to Birmingham earlier he joined the Berkeley faculty inwards 1963.
Mandelstam was i of the telephone commutation forefathers of string theory in addition to he has also gotten real far inwards some of the most up-to-date fancy projects inwards string theory.
Every particle physicist knows him for the Mandelstam invariants of the \(2\to 2\) scattering,\[
second = (p_1+p_2)^2,\,\,
t = (p_1-p_3)^2,\,\,
u = (p_1-p_4)^2.
\] assuming \(\sum_{j=1}^4 p_j^\mu = 0\) in addition to the "mostly minus" metric convention. They obey\[
s+t+u = \sum_{j=1}^4 m_i^2
\] in addition to are extremely useful to delineate all the Lorentz-invariant information most the scattering energies in addition to momenta (in Mandelstam's master copy language, they are variables to delineate the "double dispersion relations"). Mandelstam introduced these symbols inwards 1958 in addition to they're of course of report an of import all the same picayune thing. The volume of his operate was much less trivial.
Mandelstam was interested inwards Tullio Regge), the ancestor of the firstly explicit stringy formulae. So he did pass a lot of fourth dimension amongst the scattering of hadrons (and also the high-energy, special-angle behaviour of the scattering amplitudes) inwards the 1960s.
H5N1 newspaper on vortices in addition to confinement has surpassed 1,000 citations. Some other papers on solitons were approximately this threshold. Independently of Sidney Coleman (building on ideas of Tony Skyrme), Mandelstam showed the kink-fermion equivalence of the sine-Gordon in addition to Thirring models inwards 2D (a co-father of bosonization/fermionization).
In the mid 1970s, he was i inwards the extremely small-scale grouping of physicists who convey worked on string theory (almost) inwards the same feel that nosotros know it today. So he reviewed the "dual resonance models", every bit string theory was called at the beginning, wrote downward the "three-string vertex" (which was of import for me when I was checking that the string plain theory interactions follow from matrix string theory, in addition to inwards our BMN interaction research), in addition to gave the correct stringy "geometric" interpretation to the previously "just algebraic" Virasoro symmetry. Mandelstam clarified the origin of factorization in addition to helped to plough the path integrals into routine operate of a string theorist. He summarized his contributions to the plain inwards the 1970s inwards these 2008 memoirs.
He worked amongst the (world sheet) supersymmetric versions of string theory every bit shortly every bit supersymmetry was discovered. So for example, Mandelstam was the firstly physicist who wrote the NSR generalizations of the Veneziano amplitude. Mandelstam was the primary guy who began to role the conformal symmetry (that he extracted from the Virasoro algebra) to calculate the scattering aAmplitude integrand on the globe canvass inwards many domains. He loved supersymmetry in addition to did lots for Her, too. In 1983 or 1987, he proved the finiteness of the \(\NNN=4\) guess theory in addition to its scale invariance to all orders of perturbation theory. He was also deeply interested inwards the Seiberg-Witten theory.
Stanley Mandelstam was also the human who constructed the firstly proof of the ultraviolet finiteness of the \(n\)-loop perturbative string theory amplitudes – he proved the finiteness of perturbative string theory (PDF). The primary obstruction he has overcome was the proof that all "corners" of the moduli infinite of the Riemann surfaces that convey the potential to create divergences may last interpreted every bit regions corresponding to infrared divergences every bit interpreted inwards a low-energy plain theory limit. Almost all afterward proofs of finiteness of string theory are only Mandelstam's cornerstone delineate a fast i on enriched yesteryear some to a greater extent than or less tardily technicalities that depend on the precise formalism how the fermions are treated etc.
Mandelstam was a dandy teacher. He has patiently taught lots of undergraduate courses in addition to was a telephone commutation someone who has built our electrical flow noesis base, every bit a grateful pupil seat it. The listing of Mandelstam's Berkeley students includes some highly familiar names such every bit Joseph Polchinski, Michio Kaku, Charles Thorn, in addition to (my in i trial co-author) Nathan Berkovits.
RIP, Prof Mandelstam.