nLab John Elias Roberts

Not to be confused with John A. G. Roberts? working in dynamical systems.

John Elias Roberts was a mathematical physicist working on the mathematical foundations of quantum mechanics and quantum field theory in terms of AQFT.

He was born in England, but his father came from the Llŷn Peninsula. He worked in Rome at Tor Vergata for a long time, before living his last few years in Goettingen.

Selected writings

John Roberts wrote his PhD thesis on the notion of rigged Hilbert spaces, a way of making Dirac‘s description of quantum mechanics precise. After that he followed the Haag-Kastler approach for axiomatizing quantum theory and became one of its central proponents.

Early on he suggested in

• John E. Roberts, Mathematical Aspects of Local Cohomology, in: Algèbres d’opérateurs et leurs applications en physique mathématique, Colloques Internationaux du Centre National de la Recherche Scientifique (C.N.R.S) 274, Paris (1979) 321–332 [ISBN:2-222-02441-2, pdf, pdf]

that local nets of quantum observables (as formulated in algebraic quantum field theory) should carry a notion of cohomology – or rather of nonabelian cohomology – with coefficients in an ∞-category.

Motivated by this he was one of the first to consider a notion of strict ∞-categories. He conjectured that these are characterized by their ∞-nerves being complicial sets. This led Ross Street to develop the notion of orientals and to formulate a precise version of this conjecture, which was proven by Dominic Verity. An account of this development is in:

• Ross Street, pages 9-10 of: An Australian conspectus of higher category theory, talk at Institute for Mathematics and its Applications Summer Program: $n$-Categories: Foundations and Applications at the University of Minnesota (Minneapolis, 7–18 June 2004), in: Towards Higher Categories, The IMA Volumes in Mathematics and its Applications 152, Springer (2010) 237-264 [pdf, pdf, doi:10.1007/978-1-4419-1524-5]

Later Roberts proved together with Doplicher what is now one of the central results in AQFT, the Doplicher-Roberts reconstruction theorem – a version of Tannaka duality – which in the context of AQFT serves to intrinsically characterize the superselection sectors of a QFT. See also DHR superselection theory.

On $C^\ast$-categories and introducing the special case of $W^\ast$-categories:

• P. Ghez, Ricardo Lima, John E. Roberts, $W^\ast$-categories, Pacific J. Math. 120 1 (1985) 79-109 [euclid:pjm/1102703884]

category: people