Quantum mechanics and permutation invariants of finite groups

6th International Workshop DICE2012 Spacetime – Matter – Quantum Mechanics: From the Planck Scale to Emergent Phenomena

物理学;力学

Kornyak, V.V.^1

Laboratory of Information Technologies, Joint Institute for Nuclear Research, 141980 Dubna, Russia^1

关键词: Invariants of finite groups;    Linear combinations;    Mathematical descriptions;    Permutation representation;    Quantum interference;    Quantum mechanical;    Quantum observables;    Technical difficulties;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012050/pdf
DOI  :  10.1088/1742-6596/442/1/012050

学科分类：力学，机械学

来源: IOP

【 摘 要 】

We study quantum behavior from a constructive "finite" point of view, since the introduction of continuum or other actual infinities into physics poses serious conceptual and technical difficulties without any need for these concepts in physics as an empirical science. Taking this approach, we can show that the quantum-mechanical problems can be formulated in the invariant subspaces of permutation representations of finite groups, while the quantum interferences occur as phenomena that are observable in these subspaces. The scalar products in the invariant subspaces (which are needed for formulating the Born rule-the main postulate of quantum mechanics that links mathematical description with experiment) are linear combinations of independent bilinear invariant forms of the permutation representation. A complete set of such forms for any permutation group can be easily calculated by a simple algorithm. Slightly more sophisticated algorithms are required for expressing quantum observables in terms of these forms.