This chapter discusses the properties of matrix product state (MPS). It starts with a simple proof that the wave function generated by DMRG is an MPS. Then three different but equivalent canonical forms or representations of MPS are introduced. An MPS generally has redundant gauge degrees of freedom on each bond linking two neighboring local tensors. One can convert it into a canonical form by taking a canonical transformation to remove the gauge redundancy in the local tensors. Finally, the implementation of symmetries, including both the U(1) and SU(2) symmetries, is discussed.

This chapter discusses the properties of infinite MPS and their associated transfer matrices. The formulas for determining the expectation values of physical observables are derived and expressed using the leading eigenvectors of the transfer matrix. The concept of the string order parameter is introduced and exemplified with the AKLT state, followed by a statement on the condition for the existence of string order. Furthermore, the procedure of canonicalizing an infinite MPS with one or more than one site in a unit cell is discussed.

In this chapterwe present thefirst results on global attraction tosolitons forthe scalar wave field and the Maxwell field coupled to the charged relativistic particlewith density of charge satisfying the Wiener condition. In particular, the radiation damping in classical electrodynamics is rigorously proved for the first time (in this chapter, for the translation-invariant case). The proofs rely on calculation of energy radiation to infinity and canonical transformation to the comoving frame and use energy bound from below and the Wiener Tauberian theorem.