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Stanley Mandelstam: 1928-2016

A string theorist whose surname is known to every particle physicist, Stanley Mandelstam, Berkeley's emeritus professor, died on the Brexit referendum Thursday at age of 87.

He was born in Johannesburg, South Africa. His mother has lived there from the wealthy years of gold, his father Boris moved there from Latvia. Throughout his childhood, they lived in a small town but moved back to Johannesburg. He was forced to earn a practical – chemical – degree which he has never used in his life. But he returned to theoretical physics and got his degrees in 1954 (B.A.) and 1956 (PhD) in Cambridge and Birmingham before he joined the Berkeley faculty in 1963.

Mandelstam was one of the key forefathers of string theory and he has also gotten very far in some of the most up-to-date fancy projects in string theory.

Every particle physicist knows him for the Mandelstam invariants of the \(2\to 2\) scattering,\[

s = (p_1+p_2)^2,\,\,
t = (p_1-p_3)^2,\,\,
u = (p_1-p_4)^2.

\] assuming \(\sum_{j=1}^4 p_j^\mu = 0\) and the "mostly minus" metric convention. They obey\[

s+t+u = \sum_{j=1}^4 m_i^2

\] and are extremely useful to describe all the Lorentz-invariant data about the scattering energies and momenta (in Mandelstam's original language, they are variables to describe the "double dispersion relations"). Mandelstam introduced these symbols in 1958 and they're of course an important yet trivial thing. The bulk of his work was much less trivial.

Mandelstam was interested in quantization of gravity at least since 1962 and he was one of the fathers of the bootstrap program, a moral foundation for string theory that was born later, and Regge theory (with Tullio Regge), the ancestor of the first explicit stringy formulae. So he did spend a lot of time with the scattering of hadrons (and also the high-energy, special-angle behavior of the scattering amplitudes) in the 1960s.

A paper on vortices and confinement has surpassed 1,000 citations. Some other papers on solitons were close to this threshold. Independently of Sidney Coleman (building on ideas of Tony Skyrme), Mandelstam showed the kink-fermion equivalence of the sine-Gordon and Thirring models in 2D (a co-father of bosonization/fermionization).

In the mid 1970s, he was one in the extremely small group of physicists who have worked on string theory (almost) in the same sense that we know it today. So he reviewed the "dual resonance models", as string theory was called at the beginning, wrote down the "three-string vertex" (which was important for me when I was checking that the string field theory interactions follow from matrix string theory, and in our BMN interaction research), and gave the right stringy "geometric" interpretation to the previously "just algebraic" Virasoro symmetry. Mandelstam clarified the origin of factorization and helped to turn the path integrals into routine work of a string theorist. He summarized his contributions to the field in the 1970s in these 2008 memoirs.

He worked with the (world sheet) supersymmetric versions of string theory as soon as supersymmetry was discovered. So for example, Mandelstam was the first physicist who wrote the NSR generalizations of the Veneziano amplitude. Mandelstam was the main guy who began to use the conformal symmetry (that he extracted from the Virasoro algebra) to calculate the scattering amplitude integrand on the world sheet in many domains. He loved supersymmetry and did lots for Her, too. In 1983 or 1987, he proved the finiteness of the \(\NNN=4\) gauge theory and its scale invariance to all orders of perturbation theory. He was also deeply interested in the Seiberg-Witten theory.

Stanley Mandelstam was also the man who constructed the first proof of the ultraviolet finiteness of the \(n\)-loop perturbative string theory amplitudes – he proved the finiteness of perturbative string theory (PDF). The main obstacle he has overcome was the proof that all "corners" of the moduli space of the Riemann surfaces that have the potential to produce divergences may be interpreted as regions corresponding to infrared divergences as interpreted in a low-energy field theory limit. Almost all later proofs of finiteness of string theory are simply Mandelstam's cornerstone trick enriched by some more or less tedious technicalities that depend on the precise formalism how the fermions are treated etc.

Mandelstam was a keen teacher. He has patiently taught lots of undergraduate courses and was a key person who has built our current knowledge base, as a grateful student put it. The list of Mandelstam's Berkeley students includes some highly familiar names such as Joseph Polchinski, Michio Kaku, Charles Thorn, and (my once co-author) Nathan Berkovits.

RIP, Prof Mandelstam.

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