Irreducible representations of the Heisenberg algebra in Krein spaces

Abstract

The representations of the Heisenberg algebra in Krein spaces, more generally in weakly complete inner product spaces, are classified under general regularity and irreducibility conditions. Besides the Fock representation, two other types appear; one with negative, the other with a two-sided discrete spectrum of the number operator.

Infinite infrared regularization and a state space for the Heisenberg algebra

“…by Mnatsakanova, Morchio, Strocchi, and Vernov in [8]. There, it was pointed out that this issue is somewhat more difficult to handle than in the positive definite case, which is covered by the Stone-von Neumann uniqueness theorem, see [12,Chapter IV].…”

“…then implies statement iv). Assertion v) follows from iv) and the fact that the sum in (8) converges to an element of N τ .…”

“…The somewhat exotic representation of the Heisenberg algebra above does not fit into the classification of [8], see also [10]. Rather it corresponds to the 'counterexample' in the appendix of [8].…”

“…Rather it corresponds to the 'counterexample' in the appendix of [8]. As explained there, L 2 0 (R) naturally decomposes into two irreducible subspaces of 'left-' and 'right-movers', i.e., states with support on the negative, respectively positive half-axis in momentum space, by closure of the domains V ± = {f ∈ V | f (p) = 0 for p ≶ 0} in the Krein topology.…”

“…. Thus using (8), equation (7) extends to a definition of a quadratic normed topology on the τ -complete space V τ , i.e., a Hilbert majorant topology on that space:…”