Weyl–Heisenberg and Clifford groups and quantum physics

V. V. Kornyak

Joint Institute for Nuclear Research, Laboratory of Information Technologies

Abstract: The standard formulation of quantum mechanics uses continuously infinite sets, such as the continuous unitary group. However, the use of non-constructive infinities can generate inconsistencies and artifacts when describing physical reality. In fact, to describe quantum behavior it is sufficient to use finite subgroups of the general unitary group, namely, the Weyl-Heisenberg group and its extension, the Clifford group. We explore a version of quantum theory based on these groups that completely excludes the use of the continuous unitary group. This approach has empirically significant consequences. For example, the absence of quantum entanglement and interference between elementary particles of different types in nature receives a natural explanation. The rejection of the use of continuously infinite sets requires a revision of the concept of quantum states, namely, the replacement of the continuous projective Hilbert space of quantum states by some combinatorial set. We propose a possible approach to building constructive quantum states based on a certain set of natural criteria.

Key words and phrases: finite cyclic group, Weyl–Heisenberg group, Clifford group, quantum evolution, constructive quantum states.

UDC: 512.547.2:530.145.1