MIT Infinite Dimensional Algebra Seminar (Fall 2024)

In the geometric approach to quantum theory that I suggested several years ago we take as a starting point the set of states. This viewpoint is more general than the standard approach where the states are considered as density matrices and the algebraic approach where the states are identified with positive functionals on an associative algebra with involution. A large class of examples including conventional quantum mechanics can be constructed from classical theory where our devices can measure only a part of observables.

In a geometric approach one can derive the formulas for probabilities analyzing interaction with a random environment. If the theory is translation-invariant we can define a notion of particle and scattering of particles.

The quantum difference equations are K-theoretic analogs of Dubrovin connection in quantum cohomology. In my talk I show that over $p$-adic fields the quantum difference equations of Nakajima quiver varieties are equipped with the Frobenius automorphism $z\to z^p$.

I show that the corresponding Frobenius intertwiner is a partition function of quasimaps with special boundary conditions. I describe an explicit formula for the degree zero term of the intertwiner and explain the connection with works of Dwork, Sperber and Kedlaya.