$\zeta$-regularized vacuum expectation values

Abstract:
Computing vacuum expectation values is paramount in studying Quantum
Field Theories (QFTs) since they provide relevant information for
comparing the underlying theory with experimental results. However,
unless the ground state of the system is explicitly known, such
computations are very difficult and Monte Carlo simulations generally
run months to years on state-of-the-art high performance computers.
Additionally, there are various physically interesting situations, in
which most numerical methods currently in use are not applicable at all
(e.g., the early universe or settings requiring Lorentzian backgrounds).
Thus, new algorithms are required to address such problems in QFT.

In recent joint work with K. Jansen (NIC, DESY Zeuthen), I have shown
that $\zeta$-functions of Fourier integral operators can be applied to
regularize vacuum expectation values with Euclidean and Lorentzian
backgrounds and that these $\zeta$-regularized vacuum expectation values
are in fact physically meaningful. In order to prove physicality, we
introduced a discretization scheme which is accessible on a quantum
computer. Using this discretization scheme, we can efficiently
approximate ground states on a quantum device and henceforth compute
vacuum expectation values. Furthermore, the Fourier integral operator
$\zeta$-function approach is applicable to Lattice formulations in
Lorentzian background.