Tensor network representation for numerical calculations and its application for lattice QFT.

In this talk, I try to introduce the tensor representation and its application relating to my recent works. The tensor network representation is often used to calculate the system on the discrete spacetime without a sign problem. The tensor renormalization group (TRG) is an example. TRG introduces the tensor representation of the physical quantities based on the path integral formalism and efficiently contracts it by singularvalue decomposition (SVD). I show numerical calculations for the lattice quantum field theory (QFT), including our analysis of the CP(1) model, with the recent progress of the TRG. I show our triad TRG method as an example of progress. I also mention the relation between TRG and PEPS which is another tensor representation method.
As a future application, I show our study of the Casimir effect on the lattice for the realistic Dirac/Weyl semimetal by using our simple definition of the Casimir energy for free lattice fermions. I also briefly mention our work on lattice QCD as an ultimate target of TRG.