Indefinite metric vectors are absolutely required as the physical states in Minkowski space because that is indefinite metric space and the physical space-time. For example, Maxwell equations are wave equations in Minkowski space. However, traditional Quantum theory ordinarily has been studied only in definite metric space, i.e., Hilbert space. There are no clear expression for indefinite metric vectors. Here we show a wave function example using Dirac's delta function for indefinite metric vectors in Minkowski space. In addition, we show the vectors can interfere with itself. This example also suggests indefinite metric will be absolutely required.
AbstractThe microscopic reversibility of physical dynamics presents a fundamental challenge tothe macroscopic emergence of thermodynamic irreversibility. In this work, we formalize thealgebraic structure governing finite-resolution observation on a closed, measure-preservingHamiltonian flow. Differentiating the coarse-grained phase-space density under resolutionevolution σ(t) on a dense Schwartz domain, we derive the dissipative master equation∂tρσ = (Lσ + Dσ ∆z )ρσ directly from scale flow. To embed this dissipative dynamics within aglobally measure-preserving Hamiltonian framework (|J| = 1), we construct a pseudo-unitaryembedding on a Krein space equipped with an indefinite metric η, generalizing Bateman’sdual Hamiltonian framework to arbitrary nonlinear potentials V (q). We prove via continuousLyapunov operator analysis that the metric is uniquely off-diagonal, η =(︃ 0 QQ 0)︃ , strictlyeliminating block-diagonal metrics due to the Hurwitz spectrum of the dissipative generator.As a direct structural consequence of this canonical completion, the accessible causal channelcapacity for an observer subject to thermodynamic dissipation is algebraically constrainedto strictly Tr(Πcausal)/ dim Hflow = 1/2, laying the groundwork for a possible connection toholographic projection bounds.