The formalism of non-commutative quantum mechanics and its extension to many-particle systems

[摘要] ENGLISH ABSTRACT: Non-commutative quantum mechanics is a generalisation of quantum mechanics which incorporatesthe notion of a fundamental shortest length scale by introducing non-commutingposition coordinates. Various theories of quantum gravity indicate the existence of sucha shortest length scale in nature. It has furthermore been realised that certain condensedmatter systems allow effective descriptions in terms of non-commuting coordinates. As aresult, non-commutative quantum mechanics has received increasing attention recently.A consistent formulation and interpretation of non-commutative quantum mechanics,which unambiguously defines position measurement within the existing framework of quantummechanics, was recently presented by Scholtz et al. This thesis builds on the latterformalism, extends it to many-particle systems and links it up with non-commutativequantum field theory via second quantisation. It is shown that interactions of particles,among themselves and with external potentials, are altered as a result of the fuzzinessinduced by non-commutativity. For potential scattering, generic increases are found forthe differential and total scattering cross sections. Furthermore, the recovery of a scatteringpotential from scattering data is shown to involve a suppression of high energycontributions, disallowing divergent interaction forces. Likewise, the effective statisticalinteraction among fermions and bosons is modified, leading to an apparent violation ofPauli's exclusion principle and foretelling implications for thermodynamics at high densities.

[发布机构] Stellenbosch University