$\zeta$-regularized lattice field theory
A talk in the Mathematical Physics series by
Tobias Hartung from University of Bath
| Abstract: | Lattice field theory is a very powerful tool to study Feynman's path integral non-perturbatively. However, it usually requires Euclidean background metrics to be well-defined. As such, there are many interesting physical situations that are not accessible using state-of-the-art lattice methods. These include matter anti-matter asymmetry, topological theories, the discrepancy of the observed and theoretically predicted amount of charge-parity (CP) violation, non-equilibrium physics, and strongly curved background geometries. The fundamental obstacle in all of these applications is the fact that either the Euclidean time lattice regularized path integral fails to produce a numerically well-defined theory, or plainly does not exist as is the case for strongly curved background geometries. On the other hand, a recently developed regularization scheme based on Fourier integral operator $\zeta$-functions can treat Feynman's path integral non-pertubatively in Lorentzian background metrics. Using the fact that the (not necessarily Euclidean) lattice regularized path integral is well-defined in the distributional sense and can be interpreted as a nuclear trace of a (usually non-traceclass) Fourier integral operator, the Fourier integral operator $\zeta$-function scheme becomes applicable to lattice field theories with Lorentzian backgrounds as well. In this talk, we will review Feynman's path integral construction and Fourier integral operator $\zeta$-functions can be used to define the path integral in cases where Feynman's original construction failed to produce well-defined continuum limits. We will then formally apply the $\zeta$-reglarizaƟon scheme to laƫce path integrals with Lorentzian backgrounds and outline the connection to the established Fourier integral operator $\zeta$-function regularization. Time permitting, we may also consider classical limits of $\zeta$-reglarized lattice field theories and the harmonic oscillator in Minkowski background explicitly. Within the CRC this talk is associated to the project(s): C6 |