Narrow Results

This paper addresses the dilation problem on (dual) frames for Krein spaces. We characterize Riesz bases for Krein spaces and equivalence ($(J_1,J_2)$-unitary equivalence) between frames for Krein spaces; prove that every frame (dual frame pair) for a Krein space can be dilated to a Riesz basis (dual Riesz basis pair) for a larger Krein space, and that the corresponding $J$-orthogonal complementary frame ($J$-joint complementary frame) is unique up to equivalence ($(J_1,J_2)$-joint equivalence). Also we illustrate that two equivalent Parseval frames for Krein spaces need not be $(J_1,J_2)$-unitarily equivalent and that not every Parseval frame can be dilated to a $J$-orthonormal basis for a larger Krein space, and derive a result on matrices of finite size as application.

We make an explicit calculation of Casimir effect through an alternative approach of field quantization for a spherical shell in which only the radial part of fluctuation has been taken into account. In this method, negative norm states have been utilized to obtain a natural renormalized theory with the condition that they cannot be affected by the physical boundary conditions. It is shown that by using this method, the exact result is obtained without any appearance of infinity.

The Euler–Heisenberg effective action at the one-loop for a constant electromagnetic field is derived in Krein space quantization with Ford's idea of fluctuated light-cone. In this work, we present a perturbative but convergent solution of the effective action. Without using any renormalization procedure, the result coincides with the famous renormalized Euler–Heisenberg action.

Haag's theorem is extended to the case of regular representations of the canonical commutation relations in a nondegenerate indefinite inner product space.

In this paper, the existence of the Weyl representation for the canonical commutation relations algebras was proved in a Krein space.