Rodolfo Figari, "SHORT Course: Mathematical Construction of Interacting Bosonic Theories"

Monday 19 Oct 2026, 09:00 → Friday 6 Nov 2026, 16:00 Europe/Rome

GSSI

Lecturer
Rodolfo Figari rodolfo.figari@gssi.it

Course Description
The course discusses mathematical structures arising in interacting scalar bosonic theories, both relativistic and non-relativistic. Starting from the harmonic oscillator and many-body quantum mechanics, we introduce Fock space, second quantization and quartic interactions. We then turn to Euclidean methods, semigroups and probabilistic representations, emphasizing their relation with statistical mechanics and constructive quantum field theory. Particular attention is devoted to the role of locality in relativistic theories and to the emergence of short-range and contact interactions in non-relativistic limits. The distinction between asymptotically shrinking interactions and genuine point interactions is stressed throughout. In the final part, zero-range many-boson Hamiltonians in three dimensions are discussed, together with their ultraviolet instabilities and the need for genuinely many-body corrections.

Course content (tentative)
Lecture 1 — The harmonic oscillator as a zero-dimensional quantum field
Lecture 2 — Many-body quantum systems I: symmetry and statistics
Lecture 3 — Many-body quantum systems II: Fock space and Bose–Hubbard
Lecture 4 — Probability and semigroups I: Brownian motion and Feynman–Kac
Lecture 5 — Probability and semigroups II: Q-space and Markov fields
Lecture 6 — From relativistic scalar fields to Euclidean QFT
Lecture 7 — Quartic bosons, asymptotic limits, and contact
Lecture 8 — Zero-range interactions I: point potentials and STM
Lecture 9 — Zero-range interactions II: instability and stability

Calendar:
The course will be held at the MLH. A premilinary calendar is as follows:

Mon, Oct. 19:   14.00 - 15.30 
Tue,   Oct. 20:     9.00 - 10.30 
Wed,  Oct. 21:     9.00 - 10.30
Thu,   Oct. 22:     9.00 - 10.30
Fri,     Oct. 23:     9.00 - 10.30

Mon,  Oct. 26:   10.45 - 12.15
Tue,   Oct. 27:     9.00 - 10.30 
Thu,   Oct. 29:    9.00 - 10.30
Fri,     Oct. 30:    9.00 - 10.30

Course requirement
Basic elements of the theory of linear operators in Hilbert spaces. 
The knowledge of the formalism Quantum Mechanics may be helpful, but it is not required. 

Connections with other courses
Introduction to the non linear Schrödinger Equation
Dynamics of Open Quantum Systems
Gross-Pitaevskii equation from Quantum Dynamics