Discussion About Non-commutative Phase-space And Time-space

In this thesis we try to construct a representation for noncommutative space or time-space based on some kind of structure similar to the Bopp's shift which avoids any con?ictsbetween the representation and the symmetry or requirement of inversive representation. Sym-metry plays critical role in the discussion. Meanwhile, we apply the"commutative representation"of 3-dimensional NC phase-space to the harmonic oscillator system, which also further simpli?esthis representation. Then, we try to build this"commutative representation"of 4-dimensionaltime-space based on the symmetry under Lorentz transformations.In chapter 1 we mainly introduced the history of NC space and space-time, the process ofprofound, subsequent developing, and some characteristic of this NC space-time.In Chapter 2 we mainly introduce the recent study about NC space/time-space, includingsome new form in modern physics. We give an emphasized introduction about an important mathtool–Moyal-Weyl product, and it's deformation, kind of useful skill–Bopp's shift. Many physicistincluding my tutors Kang Li and Sayip did some exploration about the possible form of Bopp'sshift, which initiated this thesis.In the third Chapter, we try to construct the"commutative representation"of 3-dimensionalNC phase-space which maintains it's inversive form, based on 3-dimensional rotation symmetry.Then we apply this representation to harmonic oscillator system, simplify the representation byusing some characteristic of harmonic oscillator. Then we make a little further discussion on it.In 4th Chapter we do some exploratory job about representation of NC time-space based onLorentz transformation symmetry.Finally we summarize the research about those NC space/time-space representations in thisthesis, and ponder it's physical meaning.