Narrow Results

We continue with our study of vague groups. We also examine the notion of a vague field. For a pair (X, μ), where μ is a vague binary operation on X, we define in the language of fuzzy equalities, the notion of an identity and of an inverse of an element in X. We show that these definitions are equivalent to the original definitions of an identity and inverse of an element in a vague group. We further show how a vague group can be constructed using an ascending chain of normal subgroups. This construction of vague groups is transitive of first and second order [3]. We show that a preimage of a vague group under a homomorphism f is a ker f-vague group. Construction of a vague field using a descending chain of subfields is introduced.

In this paper, we study the symmetries of (a particular kind of) coherent states defined in the framework of the Galois quantum theory introduced in a previous publication. The configuration and the momentum spaces of this theory are given by finite and discrete abelian groups, namely the Galois group G = Gal(L : K) of a Galois field extension (L : K) and its unitary dual Ĝ ≐ Hom_gr (G, U(1)). The main interest of this quantum theory is that it is possible to define coherent states with indeterminacies in the position q ∊ G and the momentum χ ∊ Ĝ encoded by subgroups of G and Ĝ respectively. First, we show that the group of automorphisms of a coherent state with indeterminacy H ⊆ G in the position is H × H^⊥, where H^⊥ is the annihilator of H in Ĝ and encodes the corresponding indeterminacy in the momentum. Second, we show that the quantum numbers that completely define such coherent states fix an irreducible unitary representation of H × H^⊥. These results generalize the group-theoretical interpretation of the limit cases of Heisenberg indeterminacy principle proposed in previous publications to states with non-zero indeterminacies in both q and p. According to this interpretation, a quantum coherent state describes a structure-endowed system characterized by a group of automorphisms acting in an irreducible unitary representation fixed by the quantum numbers that define the state.