Discretization of Bosonic System

Abstract

Chapter 4 addressed the Lie groups $\mathop {\mathrm{U}}\nolimits (r)$ and $\mathop {\mathrm{SU}}\nolimits (r)$ composed of unitary matrices on the space $\mathbb {C}^r$. Dependently of the physical situation, all of unitary matrices do not necessarily have the equal possibility to physically realize. When we realize them artificially, it is natural that several specific transformations can be easily realized. Due to such a situation, it is important to address the symmetry of a specific subgroup as well as that of the Lie groups $\mathop {\mathrm{U}}\nolimits (r)$ and $\mathop {\mathrm{SU}}\nolimits (r)$. On the other hand, the Heisenberg representation discussed in Chap. 7 is a representation on an infinite-dimensional space, but the Lie group of interest has the finite dimension. Besides, the representation is irreducible. Hence, we might construct a small subgroup of $\mathop {\mathrm{U}}\nolimits (r)$ or $\mathop {\mathrm{SU}}\nolimits (r)$ such that it plays a similar role to $\mathop {\mathrm{U}}\nolimits (r)$ or $\mathop {\mathrm{SU}}\nolimits (r)$ if we employ a similar structure as the Heisenberg representation. For this purpose, we address the discretizations of the Heisenberg representation and the representation of $\mathop {\mathrm{Sp}}\nolimits (2r,\mathbb {R})$ that corresponds to the squeezing operation.