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Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics

Response to Dr. Pashby: Time operators and POVM observables in quantum mechanics

Gordon N. Fleming

Published Sep 22, 2014• 2 citations
Publisher
DOI
10.1016/j.shpsb.2014.08.010

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H R W R y y( ) ( )= (11.15) This system is the general case of a stationary state. This equation is also called the eigenvalue equation in which W is the eigenvalue of the system (constant) and y is the eigenfunction (wavefunction). If a possible value of observable A is defined as · Ò = =ÚA R t A R t dR R t A R tY Y Y Y* *( , ) ( , ) ( , ) ( , ) (11.16)substituting Eq. 11.16 in Eq. 11.15, we get A R t A R e iWt Y System ( , ) ( )= · Ò -y (11.17) Next, if the Hamiltonian operator is separated into unperturbed and perturbation parts as H H H = +0 ¢ (11.18) we will finally have a solution to the energy W and the wavefunction y as W W m H m k H m = + + - + π ¢ ¢ 2 (11.19) m k k H m W W = + - + π  ¢ (11.20) 11.3 Semi-Classical Quantum Mechanical TreatmentHere we will determine the dielectric response of matter under the light frequency by solving a time-dependent perturbation theory (Fig. 11.2).

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No Third Slot: A Local Entropy–Phase Closure of "Quantum Gravity" Within Standard Quantum Mechanics

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Within orthodox quantum mechanics, the only empirical content of any physical claim is a change in the outcome statistics of a declared measurement. In the modern formulation of measurement theory this declaration is an instrument (a family of completely positive maps) or equivalently a POVM together with an implementation. In curved spacetimes, localized measurement theory supplies the appropriate notion of a “local interface”: a finite laboratory worldtube with finite switching, whose induced observables lie in the causal hull and whose instruments compose causally. This paper proves a closure result: for any declared localized measurement, all geometric dependence (metric, connection, curvature, horizons) can affect outcome statistics only through exactly two operational apertures: • Phase / support aperture: the implemented localized instrument or channel on the accessible algebra, including any differential proper‑time or propagation‑dependent phase and any geometry‑dependent support of the induced observable. • Entropy aperture: the accessibility restriction itself (restriction of the global state to the accessible algebra), equivalently a trace over inaccessible degrees of freedom, with the corresponding mixedness/entropy as the interface invariant. There is no third independent slot in the quantum‑mechanical probability rule into which “quantum gravity” can be inserted. Common‑mode phases are operationally null. Sub‑resolution “micro‑selection” is inadmissible without a refined POVM/instrument. And “mediator” talk without a mediator POVM is non‑identifiable, because Stinespring dilations are never unique. The closure is robust: within finite‑response classes, small perturbations of couplings or geometry produce proportionally small changes in outcome statistics. The result does not deny the possibility of new physics beyond orthodox QM. It eliminates a specific move that has repeatedly appeared in quantum‑gravity discourse: treating curvature as an additional, independently specifiable ingredient of orthodox quantum‑mechanical prediction at a local interface.

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Physics Letters A•Aug 1, 1994•106 citations
The problem of quantum time and evolution of quantum systems, where time is not a parameter, is considered. In our model, following some earlier works, time is represented by a quantum operator. In this paper, similarly to the position operators in the Schrödinger representation of quantum mechanics, this operator is a multiplication-type operator. It can be also represented by an appropriate positive operator-valued measure (POVM) which together with the 3D position operators/measures provide a quantum observable giving a position in the quantum spacetime. The quantum evolution itself is a stochastic process based on Lüder’s projection postulate. In fact, it is a generalization of the unitary evolution. This allows to treat time and generally the spacetime position as a quantum observable, in a consistent and observer-independent way.
H R W R y y( ) ( )= (11.15) This system is the general case of a stationary state. This equation is also called the eigenvalue equation in which W is the eigenvalue of the system (constant) and y is the eigenfunction (wavefunction). If a possible value of observable A is defined as · Ò = =ÚA R t A R t dR R t A R tY Y Y Y* *( , ) ( , ) ( , ) ( , ) (11.16)substituting Eq. 11.16 in Eq. 11.15, we get A R t A R e iWt Y System ( , ) ( )= · Ò -y (11.17) Next, if the Hamiltonian operator is separated into unperturbed and perturbation parts as H H H = +0 ¢ (11.18) we will finally have a solution to the energy W and the wavefunction y as W W m H m k H m = + + - + π ¢ ¢ 2 (11.19) m k k H m W W = + - + π  ¢ (11.20) 11.3 Semi-Classical Quantum Mechanical TreatmentHere we will determine the dielectric response of matter under the light frequency by solving a time-dependent perturbation theory (Fig. 11.2).
Within orthodox quantum mechanics, the only empirical content of any physical claim is a change in the outcome statistics of a declared measurement. In the modern formulation of measurement theory this declaration is an instrument (a family of completely positive maps) or equivalently a POVM together with an implementation. In curved spacetimes, localized measurement theory supplies the appropriate notion of a “local interface”: a finite laboratory worldtube with finite switching, whose induced observables lie in the causal hull and whose instruments compose causally. This paper proves a closure result: for any declared localized measurement, all geometric dependence (metric, connection, curvature, horizons) can affect outcome statistics only through exactly two operational apertures: • Phase / support aperture: the implemented localized instrument or channel on the accessible algebra, including any differential proper‑time or propagation‑dependent phase and any geometry‑dependent support of the induced observable. • Entropy aperture: the accessibility restriction itself (restriction of the global state to the accessible algebra), equivalently a trace over inaccessible degrees of freedom, with the corresponding mixedness/entropy as the interface invariant. There is no third independent slot in the quantum‑mechanical probability rule into which “quantum gravity” can be inserted. Common‑mode phases are operationally null. Sub‑resolution “micro‑selection” is inadmissible without a refined POVM/instrument. And “mediator” talk without a mediator POVM is non‑identifiable, because Stinespring dilations are never unique. The closure is robust: within finite‑response classes, small perturbations of couplings or geometry produce proportionally small changes in outcome statistics. The result does not deny the possibility of new physics beyond orthodox QM. It eliminates a specific move that has repeatedly appeared in quantum‑gravity discourse: treating curvature as an additional, independently specifiable ingredient of orthodox quantum‑mechanical prediction at a local interface.