Problems and aspects of energy-driven wave-function collapse models

Abstract

Four problematic circumstances are considered, involving models which describe dynamical wavefunction collapse toward energy eigenstates, for which it is shown that wavefunction collapse of macroscopic objects does not work properly. In one case, a common particle position measuring situation, the apparatus evolves to a superposition of macroscopically distinguishable states (does not collapse to one of them as it should) because each such particle/apparatus/environment state has precisely the same energy spectrum. Second, assuming an experiment takes place involving collapse to one of two possible outcomes which is permanently recorded, it is shown in general that this can only happen in the unlikely case that the two apparatus states corresponding to the two outcomes have disjoint energy spectra. Next, the progressive narrowing of the energy spectrum due to the collapse mechanism is considered. This has the effect of broadening the time evolution of objects as the universe evolves. Two examples, one involving a precessing spin, the other involving creation of an excited state followed by its decay, are presented in the form of paradoxes. In both examples, the microscopic behavior predicted by standard quantum theory is significantly altered under energy-driven collapse, but this alteration is not observed by an apparatus when it is included in the quantum description. The resolution involves recognition that the statevector describing the apparatus does not collapse, but evolves to a superposition of macroscopically different states. Bz

Possibility of observing energy decoherence due to quantum gravity

“…This point was made previously in a slightly different language, but also in the time domain, by Finkelstein [8]. The present point of view is also a good way of understanding the results of the gedanken experiments discussed very recently by Pearle [9] in the context of energy-driven collapse models.…”

Quantum noise and stochastic reduction

“…The martingale relation (24) arising in the case of the energy-based stochastic Schrödinger equation can be viewed as a refinement of this principle. The supermartingale property (25) satisfied by the variance process is the essence of what is meant by a reduction process. In the case of the Schrödinger equation with a time-independent Hamiltonian, the variance of the energy is a constant of the motion.…”

“…In fact, so is the state vector |φ t , but |Ψ t and |φ t have different norms. The significance of the process {|Ψ t } is that this process is identical (modulo straightforward minor changes in notation) to the nonunitary evolution introduced and used by Pearle [24,25] for the formulation and analysis of collapse models. In particular, equation (2.1) of [25] is identical to our equation (128) above.…”

A discrete model of energy-conserved wave function collapse

“…For instance, in the CSL (Continuous Spontaneous Localization) model, the collapse due to an external noise field narrows the wave function in position space, thereby producing an increase of energy. Although it is expected that the conservation laws may be satisfied when the contributions of the noise field to the conserved quantities are taken into account (Pearle 2004;Bassi, Ippoliti and Vacchini 2005), a complete solution has not yet been found, and it is still unknown whether such a solution indeed exists.…”