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Quantum Jumps in Indefinite-Metric Spaces: Extending the Continuous Coherent-Flight Interpretation to Krein-Space Quantization and the CPT Inner Product

P. P. Afxenti, A. Andreou

Published Aug 5, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.11019.45609

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Schroumloumloumldinger equation for joint bidirectional motion in time
The conventional time-dependent Schrödinger equation describes only unidirectional time evolution of the state of a physical system, i.e. forward or, less commonly, backward. This paper proposes a generalized quantum dynamics for the description of joint, and interactive, forward and backward time evolution within a physical system. The principal mathematical assumption for bidirectional evolution in general is that the space of states should be taken to be not merely a Hilbert space, but a more restricted entity known as a Kreĭn space, which is a complex Hilbert space with a Hermitian operator that has eigenvalues +1 and −1 only, and that therefore gives rise to an indefinite metric. The vector subspaces of states with positive or negative norm with respect to the indefinite metric will—for open channels—be construed to be states in forward or, respectively, backward evolution along the time axis. The quantum dynamics is generated by a pseudo-Hermitian Hamiltonian operator and conserves inner products with respect to the indefinite metric. Input and output states are defined in physically plausible ways such that the output comprises both reflected and transmitted states from a zone of interaction in time; a unitary transformation between input and output states is obtained from the pseudounitary transformation between the initial and final states. Three applications are studied: (1) a formal theory of collisions in terms of perturbation theory; (2) a relativistically invariant quantum field theory for a system that kinematically comprises the direct sum of two quantized real scalar fields, such that one subfield evolves forward and the other backward in time, and such that there is dynamical coupling between the subfields; (3) an argument that in the latter field theory, the dynamics predicts that in a range of values of the coupling constants, the expectation value of the vacuum energy of the universe is forced to be zero to high accuracy.